FUSION SYSTEMS IN ALGEBRA AND TOPOLOGY

MICHAEL ASCHBACHER, RADHA KESSAR, AND BOB OLIVER

Introduction

Let G be a finite group, p a prime, and S a Sylow p-subgroup of G. Subsets of S are said to be fused in G if they are conjugate under some element of G. The term “fusion” seems to have been introduced by Brauer in the fifties, but the general notion has been of interest for over a century. For example in his text The Theory of Groups of Finite Order [Bu] (first published in 1897), Burnside proved that if S is abelian then the normalizer in G of S controls fusion in S. (A subgroup H of G is said to control fusion in S if any pair of tuples of elements of S which are conjugate in G are also conjugate under H.)

Initially information about fusion was usually used in conjunction with transfer, as in the proof of the normal p-complement theorems of Burnside and Frobenius, which showed that, under suitable hypotheses on fusion, G possesses a normal p-complement: a normal subgroup of index |S| in G. But in the sixties and seventies more sophisticated results on fusion began to appear, such as Alperin’s Fusion Theorem [Al1], which showed that the family of normalizers of suitable subgroups of S control fusion in S, and Goldschmidt’s Fusion Theorem [Gd3], which determined the groups G possessing a nontrivial abelian subgroup A of S such that no element of A is fused into S ∖ A.

In the early nineties, Lluis Puig abstracted the properties of G-fusion in a Sylow subgroup S, in his notion of a Frobenius category on a finite pgroup S, by discarding the group G and focusing instead on isomorphisms between subgroups of S. (But even earlier in 1976 in [P1], Puig had already considered the standard example F_{S}(G) of a Frobenius category, defined below.) Puig did not publish his work until his 2006 paper [P6] and his 2009 book [P7]. In the interim his approach was taken up and extended by others, who introduced alternate terminology which by now has become standard, and which we therefore have adopted here. In particular Puig’s Frobenius categories are now referred to as “saturated fusion systems” in most of the literature.

2000 Mathematics Subject Classification. Primary 20E25. Secondary 20D20, 20C20, 55R35, 20J05.

Key words and phrases. Finite groups, fusion, modular representations, classifying space, p-completion.

M. Aschbacher was partly supported by NSF-0504852 and NSF DMS-0969009.

B. Oliver was partially supported by UMR 7539 of the CNRS; by project ANR BLAN08-2 338236, HGRT; and by the DNRF through a visiting professorship at the Centre for Symmetry and Deformation in Copenhagen.

A fusion system F on a finite p-group S is a category whose objects are the subgroups of S, with the set Hom_{F}(P, Q) of morphisms from P to Q consisting of monomorphisms from P into Q, and such that some weak axioms are satisfied (see Definition I.2.1 for the precise conditions). The standard example of a fusion system is the category F_{S}(G), where G is a finite group, S ∈ Syl_{p}(G), and the morphisms are those induced by conjugation in G. A fusion system F is saturated if it satisfies two more axioms (Definition I.2.2), which hold in F_{S}(G) as a consequence of Sylow’s Theorem.

Many classic results on fusion in a Sylow subgroup S of a finite group G can be interpreted as results about the fusion system F = F_{S}(G). Burnside’s Fusion Theorem becomes F_{S}(G) = F_{S}(N_{G}(S)) when S is abelian. Alperin’s Fusion Theorem says that F is generated by certain “local” subcategories of F (cf. Theorem I.3.5). Goldschmidt’s Fusion Theorem says that an abelian subgroup A of S is “normal” in F (cf. Definition I.4.1) when no element of A is fused into S∖A, and goes on to use this fact to determine G.

Puig created his theory of Frobenius categories largely as a tool in modular representation theory, motivated in part by work of Alperin and Broue in [AB]. Later, homotopy theorists used this theory to provide a formal setting for, and prove results about, the p-completed classifying spaces of finite groups. As part of this process, objects called p-local finite groups associated to abstract fusion systems were introduced by Broto, Levi and Oliver in [BLO2]; these also possess interesting p-completed classifying spaces. Finally, local finite group theorists became interested in fusion systems, in part because methods from local group theory proved to be effective in the study of fusion systems, but also because certain results in finite group theory seem to be easier to prove in the category of saturated fusion systems.

These three themes — the application of fusion systems in modular representation theory, homotopy theory, and finite group theory — together with work on the foundations of the theory of saturated fusion systems, remain the focus of interest in the subject. And these are the four themes to which this volume is devoted.

This book grew out of a workshop on fusion systems at the University of Birmingham in July–August of 2007, sponsored by the London Mathematical Society and organized by Chris Parker. At that workshop there were three series of talks, one each on the role of fusion systems in modular representation theory, homotopy theory, and finite group theory, given by Kessar, Oliver, and Aschbacher, respectively. It was Chris Parker’s idea to use those talks as the point of departure for this book, although he unfortunately had to pull out of the project before it was completed. We have extracted material on the foundations of the theory of fusion systems from the various series and collected them in Part I of the book, where we have also included proofs of many of the most basic results. Then the talks have been updated and incorporated in Parts II through IV of the book, which describe the state of the art of the role of fusion systems in each of the three areas.

David Craven has also written a book on fusion systems [Cr2], which also can trace its origins to the 2007 workshop in Birmingham, and which should appear at about the same time as this one. The two books are very different in style — for example, his is intended more as a textbook and ours as a survey — and also very different in the choice of topics. In this way, we expect that the two books will complement each other.

The theory of fusion systems is an emerging area of mathematics. As such, its foundations are not yet firmly established, and the frontiers of the subject are receding more rapidly than those of more established areas. With this in mind, we have two major goals for the volume: first, collect in one place the various definitions, notation, terminology, and basic results which constitute the foundation of the theory of fusion systems, but are currently spread over a number of papers in the literature. In the process we also seek to reconcile differences in notation, terminology, and even basic concepts among papers in the literature. In particular, there is a discussion of the three existing notions of a “normal subsystem” of a saturated fusion system. Second, we seek to present a snapshot of the important theorems and open problems in our four areas of emphasis at this point in time. Our hope is that the book will serve both as a basic reference on fusion systems and as an introduction to the field, particularly for students and other young mathematicians.

The book is organized as follows. Part I contains foundational material about fusion systems, including the most basic definitions, notation, concepts, and lemmas. Then Parts II, III, and IV discuss the role of fusion systems in local finite group theory, homotopy theory, and modular representation theory, respectively. Finally the book closes with an appendix which records some of the basic material on finite groups which is well known to specialists, but perhaps not to those who approach fusion systems from the point of view of representation theory or homotopy theory.

We have received help from a large number of people while working on this project. In particular, we would like to thank Kasper Andersen, David Craven, and Ellen Henke for reading large parts of the manuscript in detail and making numerous suggestions and corrections. Many of the others who have assisted us will be acknowledged in the introductions to the individual Parts.

Notation: We close this introduction with a list of some of the basic notation involving finite groups used in all four Parts of the book. Almost all of this notation is fairly standard.

For x, g ∈ G, we write ^{g}x = gxg^{−1} for the conjugate of x under g, and let c_{g}: G → G be conjugation by g, defined by c_{g}(x) = ^{g}x. Set x^{g} = g^{−1}xg and for X ⊆ G, set ^{g}X = c_{g}(X) and X^{g} = c_{g^{−1}}(X). Let N_{G}(X) = {g ∈ G |^{g}X = X} be the normalizer in G of X and C_{G}(X) = {g ∈ G | xg = gx for all g ∈ G} be the centralizer in G of X. Write ⟨X⟩ for the subgroup of G generated by X.

Similar notation will be used when conjugating by an isomorphism of (possibly distinct) groups. For example (when group homomorphisms are composed from right to left), if φ: G ⟶^{≅} H is an isomorphism of groups, and α ∈ Aut(G) and β ∈ Aut(H), we write ^{φ}α = φαφ^{−1} ∈ Aut(H) and β^{φ} = φ^{−1}βφ ∈ Aut(G).

We write H ≤ G, H < G, or H ⊴ G to indicate that H is a subgroup, proper subgroup, or normal subgroup of G, respectively. Observe, for H ≤ G, that c: g ↦ c_{g} is a homomorphism from N_{G}(H) into Aut(H) with kernel C_{G}(H); we write Aut_{G}(H) for the image c(N_{G}(H)) of H under this homomorphism. Thus Aut_{G}(H) is the automizer in G of H and Aut_{G}(H) ≅ N_{G}(H)/C_{G}(H). The inner automorphism group of H is Inn(H) = Aut_{H}(H) = c(H), and the outer automorphism group of H is Out(H) = Aut(H)/Inn(H).

We write Syl_{p}(G) for the set of Sylow p-subgroups of G. When π is a set of primes, a π-subgroup of G is a subgroup whose order is divisible only by primes in π. We write O_{π}(G) for the largest normal π-subgroup of G, and O^{π}(G) for the smallest normal subgroup H of G such that G/H is a π-group. We write p′ for the set of primes distinct from p; we will be particularly interested in the groups O_{p}(G), O_{p′}(G), O^{p}(G), and O^{p′}(G). Sometimes we write O(G) for O_{2′}(G).

As usual, when P is a p-group for some prime p, we set Ω_{1}(P) = ⟨g ∈ P | g^{p} = 1⟩.

As for specific groups, C_{n} denotes a (multiplicative) cyclic group of order n, and D_{2^{k}}, SD_{2^{k}}, Q_{2^{k}} denote dihedral, semidihedral, and quaternion groups of order 2^{k}. Also, A_{n} ≤ S_{n} denote alternating and symmetric groups on n letters.

FUSION SYSTEMS IN ALGEBRA AND TOPOLOGY 5

Throughout the book, p is always understood to be a fixed prime. All p-groups are assumed to be finite.

Contents

Part I. Introduction to fusion systems 10 1. The fusion category of a finite group 10 2. Abstract fusion systems 12 3. Alperin’s fusion theorem 16 4. Normal and central subgroups of a fusion system 22 5. Normalizer fusion systems 26 6. Normal fusion subsystems and products 30 7. Fusion subsystems of p-power index or of index prime to p 37 8. The transfer homomorphism for saturated fusion systems 43 9. Other definitions of saturation 50

Part II. The local theory of fusion systems 54 1. Notation and terminology on groups 56 2. Fusion systems 56 3. Saturated fusion systems 58 4. Models for constrained saturated fusion systems 59 5. Factor systems and surjective morphisms 61 6. Invariant subsystems of fusion systems 65 7. Normal subsystems of fusion systems 67 8. Invariant maps and normal maps 70 9. Theorems on normal subsystems 73 10. Composition series 76 11. Constrained systems 83 12. Solvable systems 85 13. Fusion systems in simple groups 88 14. Classifying simple groups and fusion systems 92 15. Systems of characteristic 2-type 97

Part III. Fusion and homotopy theory 108

1. Classifying spaces, p-completion, and the Martino-Priddy

conjecture 111 1.1. Homotopy and fundamental groups 111

FUSION SYSTEMS IN ALGEBRA AND TOPOLOGY 7

1.2. CW complexes and cellular homology 114 1.3. Classifying spaces of discrete groups 115 1.4. The p-completion functor of Bousfield and Kan 118 1.5. Equivalences between fusion systems of finite groups 121 1.6. The Martino-Priddy conjecture 122 1.7. An application: fusion in finite groups of Lie type 123

2. The geometric realization of a category 124 2.1. Simplicial sets and their realizations 125 2.2. The nerve of a category as a simplicial set 127 2.3. Classifying spaces as geometric realizations of categories 129 2.4. Fundamental groups and coverings of geometric realizations130 2.5. Spaces of maps 134

3. Linking systems and classifying spaces of finite groups 138 3.1. The linking category of a finite group 138 3.2. Fusion and linking categories of spaces 140 3.3. Linking systems and equivalences of p-completed classifying

spaces 143

4. Abstract fusion and linking systems 144

4.1. Linking systems, centric linking systems and p-local finite

groups 145

4.2. Quasicentric subgroups and quasicentric linking systems 148 4.3. Automorphisms of fusion and linking systems 157 4.4. Normal fusion and linking subsystems 160 4.5. Fundamental groups and covering spaces 163 4.6. Homotopy properties of classifying spaces 166 4.7. Classifying spectra of fusion systems 170 4.8. An infinite version: p-local compact groups 172

5. The orbit category and its applications 173 5.1. Higher limits of functors and the bar resolution 175 5.2. Constrained fusion systems 180 5.3. Existence, uniqueness, and automorphisms of linking

systems 187

5.4. Some computational techniques for higher limits over orbit categories 194

8 MICHAEL ASCHBACHER, RADHA KESSAR, AND BOB OLIVER

5.5. Homotopy colimits and homotopy decompositions 202 5.6. The subgroup decomposition of |L| 205 5.7. An outline of the proofs of Theorems 4.21 and 4.22 209 5.8. The centralizer and normalizer decompositions of |L| 212

6. Examples of exotic fusion systems 214 6.1. Reduced fusion systems and tame fusion systems 215 6.2. The Ruiz-Viruel examples 217 6.3. Saturated fusion systems over 2-groups 219 6.4. Mixing related fusion systems 220 6.5. Other examples 220

7. Open problems 221

Part IV. Fusion and Representation theory 225

1. Algebras and G-algebras 227 1.1. Ideals and Idempotents 227 1.2. G-algebras 231 1.3. Relative trace maps and Brauer homomorphisms 232

2. p-permutation algebras, Brauer pairs and fusion systems 237 2.1. p-permutation algebras and the Brauer homomorphisms 237 2.2. (A, G)-Brauer pairs and inclusion. 240 2.3. (A, b, G)-Brauer pairs and inclusion. 246 2.4. (A, b, G)-Brauer pairs and fusion systems 248

3. p-permutation algebras and saturated fusion systems 249 3.1. Saturated triples 249 3.2. Normaliser systems and saturated triples 254 3.3. Saturated triples and normal subgroups 256 3.4. Block fusion systems 258 3.5. Fusion systems of blocks of local subgroups 260

4. Background on finite group representations 263 4.1. Ordinary and modular representations 264 4.2. p-modular systems 266 4.3. Cartan and decomposition maps 267 4.4. Ordinary and Brauer characters 271

FUSION SYSTEMS IN ALGEBRA AND TOPOLOGY 9

5. Fusion and structure 275 5.1. The three main theorems of Brauer 275 5.2. Relative projectivity and representation type 279 5.3. Finiteness conjectures 280 5.4. Source algebras and Puig’s conjecture 283 5.5. Külshammer-Puig classes 285 5.6. Nilpotent blocks and extensions 290 5.7. Counting Conjectures 292

6. Block fusion systems and normal subgroups. 298 7. Open Problems 302

Appendix A. Background facts about groups 305

References 311

List of notation 319

Index 322

Part I. Introduction to fusion systems

This part is intended as a general introduction to the book, where we describe the properties of fusion systems which will be used throughout. We begin with the basic definitions of fusion systems of finite groups and abstract fusion systems, and give some versions of Alperin’s fusion theorem in this setting. Afterwards, we discuss various topics such as normal and central subgroups of fusion systems, constrained fusion systems, normal fusion subsystems, products of fusion systems, the normalizer and centralizer fusion subsystems of a subgroup, and fusion subsystems of p-power index or of index prime to p.

1. The fusion category of a finite group

For any group G and any pair of subgroups H, K ≤ G, we define

Hom_{G}(H, K) = {φ ∈ Hom(H, K) |

φ = c_{g} for some g ∈ G such that ^{g}H ≤ K}.

In other words, Hom_{G}(H, K) is the set of all (injective) homomorphisms from H to K which are induced by conjugation in G. Similarly, we write Iso_{G}(H, K) for the set of elements of Hom_{G}(H, K) which are isomorphisms of groups.

Definition 1.1. Fix a finite group G, a prime p, and a Sylow p-subgroup S ∈ Syl_{p}(G). The fusion category of G over S is the category F_{S}(G) whose objects are the subgroups of S, and which has morphism sets

Mor_{F_{S}(G)}(P, Q) = Hom_{G}(P, Q).

Many concepts and results in finite group theory can be stated in terms of this category. We list some examples of this here. In all cases, G is a finite group and S ∈ Syl_{p}(G).

• Alperin’s fusion theorem [Al1], at least in some forms, is the statement that F_{S}(G) is generated by automorphism groups of certain subgroups of S, in the sense that each morphism in F_{S}(G) is a composite of restrictions of automorphisms of those subgroups.

• Glauberman’s Z^{∗}-theorem [Gl1] says that when p = 2 and O_{2′}(G) = 1, then Z(G) = Z(F_{S}(G)), where Z(F_{S}(G)) is the “center” of the fusion category in a sense which will be made precise later (Definition 4.3).

• A subgroup H ≤ G which contains S controls fusion in S if F_{S}(H) = F_{S}(G). Thus Burnside’s fusion theorem states that when S is abelian, F_{S}(G) = F_{S}(N_{G}(S)), and that every morphism in F_{S}(G) extends to an automorphism of S.

• By a theorem of Frobenius (cf. [A4, 39.4]), G has a normal p-complement (a subgroup H ⊴ G of p-power index and order prime to p) if and only if F_{S}(G) = F_{S}(S).

• The focal subgroup theorem says that

S ∩ [G, G] = 𝔣𝔬𝔠(F_{S}(G)) =^{def} ⟨x^{−1}y | x, y ∈ S, x = ^{g}y for some g ∈ G⟩ ,

and is thus described in terms of the category F_{S}(G). By the hyperfocal subgroup theorem of Puig [P5, § 1.1], S ∩O^{p}(G) can also be described in terms of the fusion category F_{S}(G) (see Section 7).

The following lemma describes some of the properties of these fusion categories, properties which help to motivate the definition of abstract fusion systems in the next section.

Lemma 1.2. Fix a finite group G and a Sylow p-subgroup S ∈ Syl_{p}(G).

(a) For each P ≤ S, there is Q ≤ S such that Q is G-conjugate to P and N_{S}(Q) ∈ Syl_{p}(N_{G}(Q)). For any such Q, C_{S}(Q) ∈ Syl_{p}(C_{G}(Q)) and Aut_{S}(Q) ∈ Syl_{p}(Aut_{G}(Q)).

(b) Fix P, Q ≤ S and g ∈ G such that ^{g}P = Q. Assume N_{S}(Q) ∈ Syl_{p}(N_{G}(Q)). Set

N = {x ∈ N_{S}(P) |^{g}x ∈ N_{S}(Q)·C_{G}(Q)} .

Then there is h ∈ C_{G}(Q) such that ^{hg}N ≤ S.

Proof. (a) Fix T ∈ Syl_{p}(N_{G}(P)). Since T is a p-subgroup of G, there is g ∈ G such that ^{g}T ≤ S. Set Q = ^{g}P. Then Q ⊴ ^{g}T ≤ S, and gT ∈ Syl_{p}(N_{G}(Q)) since c_{g} ∈ Aut(G). Also, ^{g}T ≤ N_{S}(Q), and since ^{g}T ∈ Syl_{p}(N_{G}(Q)), ^{g}T = N_{S}(Q). The last statement now holds by Lemma A.3, upon identifying Aut_{X}(Q) with N_{X}(Q)/C_{X}(Q) for X = G, S.

(b) Since N_{S}(Q) normalizes C_{G}(Q), N_{S}(Q)·C_{G}(Q) is a subgroup of G. By assumption, ^{g}N ≤ N_{S}(Q)·C_{G}(Q). Since N_{S}(Q) ∈ Syl_{p}(N_{G}(Q)) and N_{S}(Q)·C_{G}(Q) ≤ N_{G}(Q), N_{S}(Q) is also a Sylow subgroup of N_{S}(Q)·C_{G}(Q). Hence there is h ∈ C_{G}(Q) such that^{h}(^{g}N) ≤ N_{S}(Q); i.e.,^{hg}N ≤ N_{S}(Q). □

2. Abstract fusion systems

The notion of an abstract fusion system is due to Puig. The definitions we give here are modified versions of Puig’s definitions (given in [P6]), but equivalent to them. The following is what he calls a “divisible S-category”.

Definition 2.1 ([P6], [BLO2]). A fusion system over a p-group S is a category F, where Ob(F) is the set of all subgroups of S, and which satisfies the following two properties for all P, Q ≤ S:

• Hom_{S}(P, Q) ⊆ Mor_{F}(P, Q) ⊆ Inj(P, Q); and

• each φ ∈ Mor_{F}(P, Q) is the composite of an F-isomorphism followed by an inclusion.

Composition in a fusion system F is always given by composition of homomorphisms. We usually write Hom_{F}(P, Q) = Mor_{F}(P, Q) to emphasize that the morphisms in F actually are group homomorphisms, and also set Aut_{F}(P) = Hom_{F}(P, P). Note that a fusion system over F contains all inclusions incl_{P}^{Q}, for P ≤ Q ≤ S, by the first condition (it is conjugation by 1 ∈ S). The second condition means that for each φ ∈ Hom_{F}(P, Q), φ: P ⟶ φ(P) and φ^{−1} : φ(P) ⟶ P are both morphisms in F.

Fusion systems as defined above are too general for most purposes, and additional conditions are needed for them to be very useful. This leads to the concept of what we call a “saturated fusion system”: a fusion system satisfying certain axioms which are motivated by properties of fusion in finite groups. The following version of these axioms is due to Roberts and Shpectorov [RS].

Definition 2.2. Let F be a fusion system over a p-group S.

• Two subgroups P, Q ≤ S are F-conjugate if they are isomorphic as objects of the category F. Let P^{F} denote the set of all subgroups of S which are F-conjugate to P.

• A subgroup P ≤ S is fully automized in F if Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)).

• A subgroup P ≤ S is receptive in F if it has the following property: for each Q ≤ S and each φ ∈ Iso_{F}(Q, P), if we set

N_{φ} = N_{φ}^{F} = {g ∈ N_{S}(Q) |^{φ}c_{g} ∈ Aut_{S}(P)},

then there is φ ∈ Hom_{F}(N_{φ}, S) such that φ|_{Q} = φ.

• A fusion system F over a p-group S is saturated if each subgroup of S is F-conjugate to a subgroup which is fully automized and receptive.

We also say that two elements x, y ∈ S are F-conjugate if there is an isomorphism φ ∈ Iso_{F}(⟨x⟩, ⟨y⟩) such that φ(x) = y, and let x^{F} denote the F-conjugacy class of x.

The fusion category F_{S}(G) of a finite group G clearly satisfies the conditions in Definition 2.1, and thus is a fusion system. It also satisfies the saturation conditions by Lemma 1.2.

Theorem 2.3 (Puig). If G is a finite group and S ∈ Syl_{p}(G), then F_{S}(G) is a saturated fusion system.

Proof. By Lemma 1.2(a), each subgroup P ≤ S is G-conjugate to a subgroup Q ≤ S such that N_{S}(Q) ∈ Syl_{p}(N_{G}(Q)), and each such subgroup Q is fully automized in F_{S}(G). By Lemma 1.2(b), Q is also receptive in F_{S}(G), and thus F_{S}(G) is saturated. □

A saturated fusion system F over a p-group S will be called realizable if F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G), and will be called exotic otherwise. Examples of exotic fusion systems will be described in Section III.6.

There are several, equivalent definitions of saturated fusion systems in the literature. We discuss here the definition of saturation which was given in [BLO2]. Two other definitions, the original one given by Puig and another one by Stancu, will be described and shown to be equivalent to these in Section 9 (Proposition 9.3).

In order to explain the definition in [BLO2], and compare it with the one given above, we first need to define two more concepts.

Definition 2.4. Let F be a fusion system over a p-group S.

• A subgroup P ≤ S is fully centralized in F if |C_{S}(P)| ≥ |C_{S}(Q)| for all Q ∈ P^{F}.

• A subgroup P ≤ S is fully normalized in F if |N_{S}(P)| ≥ |N_{S}(Q)| for all Q ∈ P^{F}.

For example, when F = F_{S}(G) for some finite group G and some S ∈ Syl_{p}(G), then by Lemma 1.2(a), a subgroup P ≤ S is fully normalized (centralized) in F_{S}(G) if and only if N_{S}(P) ∈ Syl_{p}(N_{G}(P)) (C_{S}(P) ∈ Syl_{p}(C_{G}(P))).

Definition 2.4 is different from the definition of “fully normalized” and “fully centralized” in [P6, 2.6], but it is equivalent to Puig’s definition when working in saturated fusion systems. This will be discussed in much more detail in Section 9.

The following equivalent condition for a fusion system to be saturated, stated in terms of fully normalized and fully centralized subgroups, was given as the definition of saturation in [BLO2, Definition 1.2].

Proposition 2.5 ([RS, Theorem 5.2]). Let F be a fusion system over a p-group S. Then F is saturated if and only if the following two conditions hold.

(I) (Sylow axiom) Each subgroup P ≤ S which is fully normalized in F is also fully centralized and fully automized in F.

(II) (Extension axiom) Each subgroup P ≤ S which is fully centralized in F is also receptive in F.

Proposition 2.5 is an immediate consequence of the following lemma.

Lemma 2.6 ([RS]). The following hold for any fusion system F over a p-group S.

(a) Every receptive subgroup of S is fully centralized.

(b) Every subgroup of S which is fully automized and receptive is fully normalized.

(c) Assume P ≤ S is fully automized and receptive. Then for each Q ∈ P^{F}, there is a morphism φ ∈ Hom_{F}(N_{S}(Q), N_{S}(P)) such that φ(Q) = P. Furthermore, Q is fully centralized if and only if it is receptive, and is fully normalized if and only if it is fully automized and receptive.

Proof. (a) Assume P ≤ S is receptive. Fix any Q ∈ P^{F} and any φ ∈ Iso_{F}(Q, P). Since P is receptive, φ extends to some φ ∈ Hom_{F}(N_{φ}, S), where N_{φ} contains C_{S}(Q) by definition. Thus φ sends C_{S}(Q) injectively into C_{S}(P), and so |C_{S}(P)| ≥ |C_{S}(Q)|. Since this holds for all Q ∈ P^{F}, P is fully centralized in F.

(b) Now assume P is fully automized and receptive, and fix Q ∈ P^{F}. Then |C_{S}(Q)| ≤ |C_{S}(P)| by (a), and |Aut_{S}(Q)| ≤ |Aut_{S}(P)| since Aut_{F}(Q) ≅ Aut_{F}(P) and Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)). Thus

|N_{S}(Q)| = |C_{S}(Q)|·|Aut_{S}(Q)| ≤ |C_{S}(P)|·|Aut_{S}(P)| = |N_{S}(P)| .

Since this holds for all Q ∈ P^{F}, P is fully normalized in F.

(c) Assume P is fully automized and receptive, and fix Q ∈ P^{F}. Choose ψ ∈ Iso_{F}(Q, P). Then ^{ψ}Aut_{S}(Q) is a p-subgroup of Aut_{F}(P), and hence is Aut_{F}(P)-conjugate to a subgroup of Aut_{S}(P) since P is fully automized. Fix α ∈ Aut_{F}(P) such that ^{αψ}Aut_{S}(Q) is contained in Aut_{S}(P). Then N_{αψ} = N_{S}(Q) (see Definition 2.2), and so αψ extends to some homomorphism φ ∈ Hom_{F}(N_{S}(Q), S). Since φ(Q) = αψ(Q) = P, Im(φ) ≤ N_{S}(P).

If Q is fully centralized, then φ(C_{S}(Q)) = C_{S}(P). Fix R ≤ S and β ∈ Iso_{F}(R, Q). For g ∈ N_{β}, ^{β}c_{g} ∈ Aut_{S}(Q) implies ^{αψβ}c_{g} ∈ Aut_{S}(P) since αψ extends to a homomorphism defined on N_{S}(Q), and thus g ∈ N_{αψβ}. Since P is receptive, αψβ extends to a homomorphism χ ∈ Hom_{F}(N_{β}, N_{S}(P)). For each g ∈ N_{β}, ^{β}c_{g} = c_{h} for some h ∈ N_{S}(Q) by definition of N_{β}, so c_{φ(h)} = ^{φ}c_{h} = c_{χ(g)}, and thus χ(g) ∈ Im(φ)·C_{S}(P) = Im(φ). Thus Im(χ) ≤ Im(φ), so χ factors through some β ∈ Hom_{F}(N_{β}, N_{S}(Q)) with β|_{R} = β, and this proves that Q is receptive.

If Q is fully normalized, then φ is an isomorphism. Hence φ sends C_{S}(Q) onto C_{S}(P), so Q is fully centralized and hence receptive. Also, Aut_{S}(Q) ≅ N_{S}(Q)/C_{S}(Q) is isomorphic to Aut_{S}(P) ≅ N_{S}(P)/C_{S}(P), and Aut_{F}(Q) ≅ Aut_{F}(P) since Q ∈ P^{F}. So Aut_{S}(Q) ∈ Syl_{p}(Aut_{F}(Q)) since Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)), and Q is fully automized. □

We end this section with an example, which describes how to list all possible saturated fusion systems over one very small 2-group.

Example 2.7. Assume S ≅ D_{8}: the dihedral group of order 8. Fix generators a, b ∈ S, where |a| = 4, |b| = 2, and |ab| = 2. Set T_{0} = ⟨a^{2}, b⟩ and T_{1} = ⟨a^{2}, ab⟩: these are the only subgroups of S isomorphic to C_{2} × C_{2}. Then the following hold for any saturated fusion system F over S.

(a) Since S is fully automized and Aut(S) is a 2-group, Aut_{F}(S) = Inn(S).

(b) If P = ⟨a⟩ and Q = P or S, then Hom_{F}(P, Q) = Hom_{S}(P, Q) = Hom(P, Q).

(c) The subgroups T_{0} and T_{1} are both fully normalized in F, and hence are fully automized and receptive. So if T_{1} ∈ T_{0}^{F}, then by Lemma 2.6(c), there is α ∈ Aut_{F}(S) such that α(T_{0}) = T_{1}. Since this contradicts (a), T_{0} and T_{1} cannot be F-conjugate.

(d) Set P = ⟨a^{2}⟩, and let Q ≤ S be any subgroup of order 2. Since P ⊴ S, P is fully normalized in F and hence fully automized and receptive. So if Q ∈ P^{F} (and Q ≠ P), then by Lemma 2.6(c) again, there is some φ ∈ Hom_{F}(T_{i}, S), where T_{i} = N_{S}(Q) (i = 0 or 1), such that φ(Q) = P. Also, φ(T_{i}) = T_{i} by (c).

(e) By (a–d), F is completely determined by Aut_{F}(T_{0}) and Aut_{F}(T_{1}). Also, for each i, Aut_{S}(T_{i}) ≤ Aut_{F}(T_{i}) ≤ Aut(T_{i}), and hence Aut_{F}(T_{i}) has order 2 or 6.

Thus there are at most four saturated fusion systems over S. Denote these fusion systems F_{ij}, where i = 0 if |Aut_{F}(T_{0})| = 2, i = 1 if |Aut_{F}(T_{0})| = 6, and similarly j = 0, 1 depending on |Aut_{F}(T_{1})|. Then F_{00} is the fusion system of S ≅ D_{8} itself, F_{01} ≅ F_{10} are isomorphic to the fusion system of S_{4}, and F_{11} is isomorphic to the fusion system of A_{6}. In particular, all four of the F_{ij} are saturated.

3. Alperin’s fusion theorem

We next prove a version of the Alperin-Goldschmidt fusion theorem for abstract fusion systems. Before doing this, we must list some important classes of subgroups in a given fusion system. All of these are modelled on analogous definitions for p-subgroups of groups.

Definition 3.1. Let F be a fusion system over a p-group S.

• For each P ≤ S, set Out_{F}(P) = Aut_{F}(P)/Inn(P) and Out_{S}(P) = Aut_{S}(P)/Inn(P). Thus Out_{S}(P) ≤ Out_{F}(P) ≤ Out(P).

• A subgroup P of S is F-centric if C_{S}(Q) = Z(Q) for all Q ∈ P^{F}. Equivalently, P is F-centric if P is fully centralized in F and C_{S}(P) = Z(P).

• A subgroup P of S is F-radical if Out_{F}(P) is p-reduced; i.e., if O_{p}(Out_{F}(P)) = 1. We say P is F-centric-radical if it is F-centric and F-radical.

• Let F^{cr} ⊆ F^{c} ⊆ F denote the full subcategories whose objects are the F-centric-radical, and F-centric, subgroups of S, respectively.

• Let F^{c} = Ob(F^{c}) and F^{cr} = Ob(F^{cr}) denote the sets of F-centric and F-centric-radical subgroups of S, respectively.

When G is a finite group, then a p-subgroup P ≤ G is called p-centric if Z(P) ∈ Syl_{p}(C_{G}(P)). If S ∈ Syl_{p}(G) and F = F_{S}(G), then P ≤ S is easily seen to be F-centric if and only if P is p-centric in G. However, P being F-radical is not the same as being a radical p-subgroup of G: P is radical in G (by definition) if O_{p}(N_{G}(P)/P) = 1, while P is F-radical if and only if O_{p}(N_{G}(P)/(P ·C_{G}(P))) = 1. For example, if S is abelian and normal in G, then every subgroup of S is F-radical, but the only radical p-subgroup of G is S itself. As a second example, if G is dihedral of order 24, S ∈ Syl_{2}(G), and P = O_{2}(G) is the normal cyclic subgroup of order 4, then P is a radical 2-subgroup of G but is not F-radical (since Out_{G}(P) has order 2).

Definition 3.2. Let F be a fusion system over a p-group S. A subgroup P of S is F-essential if P is F-centric and fully normalized in F, and Out_{F}(P) contains a strongly p-embedded subgroup (Definition A.6).

Note that Puig’s definition of “F-essential” in [P6, § 5.4] is different from this one. It is, however, equivalent (except for our requirement that the subgroup be fully normalized) when the fusion system is saturated.

The next proposition describes the key property of essential subgroups.

Proposition 3.3. Let F be a saturated fusion system over a p-group S.

(a) Each F-essential subgroup of S is F-centric-radical and fully normalized in F.

(b) Fix a proper subgroup P of S which is fully normalized, and let H_{P} ≤ Aut_{F}(P) be the subgroup generated by those α ∈ Aut_{F}(P) which extend to F-isomorphisms between strictly larger subgroups of S. Then either P is not F-essential and H_{P} = Aut_{F}(P); or P is F-essential and H_{P} /Inn(P) is strongly p-embedded in Out_{F}(P).

Proof. (a) By definition, each F-essential subgroup P ≤ S is F-centric and fully normalized. Since Out_{F}(P) contains a strongly p-embedded subgroup, O_{p}(Out_{F}(P)) = 1 by Proposition A.7(c), and hence P is also F-radical.

(b) Fix a proper subgroup P of S which is fully normalized in F. In particular, Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)). If P is not F-centric, then each α ∈ Aut_{F}(P) extends to some F-automorphism of C_{S}(P)·P > P since P is receptive, and so H_{P} = Aut_{F}(P) in this case. So we can assume P is F-centric. Also, N_{S}(P) > P since P < S, so Out_{S}(P) ≠ 1 since P is F-centric, and thus p||H_{P} /Inn(P)| since H_{P} ≥ Aut_{S}(P).

We claim that

H_{P} = ⟨α ∈ Aut_{F}(P) |^{α}Aut_{S}(P) ∩ Aut_{S}(P) > Inn(P)⟩.  (1)

To see this, fix any α ∈ Aut_{F}(P) such that ^{α}Aut_{S}(P)∩Aut_{S}(P) > Inn(P), and recall from Definition 2.2 that

N_{α} = {g ∈ N_{S}(P) |^{α}c_{g} ∈ Aut_{S}(P)}.

Then ^{α}Aut_{N_{α}}(P) = ^{α}Aut_{S}(P) ∩ Aut_{S}(P) > Inn(P), so N_{α} > P. By the extension axiom, α extends to a morphism in Hom_{F}(N_{α}, S), and thus α ∈ H_{P}. Conversely, if α ∈ Aut_{F}(P) extends to α ∈ Hom_{F}(Q, S) for some Q > P, ^{α}Aut_{Q}(P) ≤ Aut_{S}(P), and

αAut_{S}(P) ∩ Aut_{S}(P) ≥ ^{α}Aut_{Q}(P) > Inn(P).

This proves (1). Hence by Proposition A.7(b), either H_{P} = Aut_{F}(P), in which case Out_{F}(P) contains no strongly p-embedded subgroup and P is not F-essential; or H_{P} < Aut_{F}(P), in which case H_{P} /Inn(P) is strongly p-embedded in Out_{F}(P) and P is F-essential. □

There is always a “universal” fusion system U over any p-group S which contains all other fusion systems over S: for each P, Q ≤ S, Hom_{U}(P, Q) is the set of all monomorphisms from P to Q. Also, the intersection of two fusion systems over S is clearly again a fusion system over S. This allows us, in certain cases, to talk about the “smallest fusion system” over a given S which satisfies certain given conditions.

Definition 3.4. Fix a p-group S.

(a) Fix another p-group T, and an isomorphism φ: S ⟶^{≅} T. For any fusion system F over S, let ^{φ}F be the fusion system over T defined by setting

Hom_{^{φ}F}(P, Q) = {^{φ}ψ ∈ Hom(P, Q) | ψ ∈ Hom_{F}(φ^{−1}(P), φ^{−1}(Q))}

for all P, Q ≤ T.

(b) For any set 𝔛 of monomorphisms between subgroups of S and/or fusion systems over subgroups of S, the fusion system generated by 𝔛, denoted ⟨𝔛⟩_{S}, is the smallest fusion system over S (not necessarily saturated) which contains 𝔛. Thus ⟨𝔛⟩_{S} is the intersection of all fusion systems over S which contain 𝔛, and the morphisms in ⟨𝔛⟩_{S} are the composites of restrictions of homomorphisms in the set 𝔛 ∪ Inn(S) and their inverses. We write ⟨𝔛⟩ = ⟨𝔛⟩_{S} when the choice of S is clear.

In these terms, the “Alperin-Goldschmidt fusion theorem” for fusion systems says that a saturated fusion system is generated by certain automorphism groups. This result is originally due to Puig (see [P6, Corollary 5.10] for a linearlized version of the theorem). It is modeled on the original fusion theorems of Alperin [Al1] and Goldschmidt [Gd1].

Theorem 3.5. Fix a p-group S and a saturated fusion system F over S. Then

F = ⟨Aut_{F}(P) | P = S or P is F-essential⟩_{S} .

Proof. Set E = ⟨Aut_{F}(P) | P = S or P is F-essential⟩, and assume E ⫋ F. Fix a morphism φ ∈ Hom_{F}(P, Q) ∖ Hom_{E}(P, Q), chosen such that |P | is maximal subject to this constraint. Since Aut_{E}(S) = Aut_{F}(S), P < S. Since φ factors as an isomorphism in F followed by an inclusion (which lies in E), we can assume φ ∈ Iso_{F}(P, Q).

Choose R ∈ P^{F} which is fully normalized in F. By Lemma 2.6(c), there are ψ_{1} ∈ Hom_{F}(N_{S}(P), N_{S}(R)) and ψ_{2} ∈ Hom_{F}(N_{S}(Q), N_{S}(R)) such that ψ_{1}(P) = R and ψ_{2}(Q) = R. Since P, Q < S, P < N_{S}(P) and Q < N_{S}(Q) (see Lemma A.1), and so ψ_{1}, ψ_{2} ∈ Mor(E) by the maximality of |P |. Upon replacing φ by (ψ_{2}|_{Q})φ(ψ_{1}|_{P})^{−1}, we can assume φ ∈ Aut_{F}(R) where R is fully normalized. If R is F-essential, then Aut_{F}(R) = Aut_{E}(R) by definition of E. If R is not F-essential, then by Proposition 3.3(b), Aut_{F}(R) is generated by automorphisms which extend to strictly larger subgroups, and Aut_{F}(R) = Aut_{E}(R) by the maximality of |P |. So in either case, φ ∈ Mor(E), contradicting the original assumption on φ. Thus E = F. □

As an immediate consequence of Theorem 3.5, we get the Alperin-Goldschmidt fusion theorem in its more explicit form.

Theorem 3.6. Fix a p-group S and a saturated fusion system F over S. For each P, Q ≤ S and each φ ∈ Hom_{F}(P, Q), there are subgroups P = P_{0}, P_{1}, …, P_{k} = Q, subgroups R_{i} ≥ ⟨P_{i−1}, P_{i}⟩ (i = 1, …, k) which are F-essential or equal to S, and automorphisms φ_{i} ∈ Aut_{F}(R_{i}), such that φ_{i}(P_{i−1}) ≤ P_{i} for each i, and φ = (φ_{k}|_{P_{k−1}}) ∘ ⋯ ∘ (φ_{1}|_{P_{0}}).

The following is one very elementary consequence of Theorem 3.5 or 3.6.

Corollary 3.7. Let F be a saturated fusion system over the p-group S. Assume, for each P ≤ S, that Aut_{F}(P) is a p-group. Then F is the fusion system of the group S.

Proof. Since all automorphism groups are p-groups, there are no F-essential subgroups. Hence by Theorem 3.5 or 3.6, F is generated by Aut_{F}(S) = Inn(S), and thus F = F_{S}(S). □

We ended the last section by laboriously describing, almost directly from the definitions, all saturated fusion systems over a dihedral group of order 8. With the help of Theorem 3.5, we can now do the same thing much more quickly for arbitrary 2-groups which are dihedral, semidihedral, or quaternion.

Example 3.8. Assume S is a dihedral, semidihedral, or quaternion group of order 2^{n} ≥ 16. Fix generators a, b ∈ S, where |a| = 2^{n−1}, and where |b| = 2 if S is semidihedral. For each i ∈ ℤ, set T_{i} = ⟨a^{2^{n−2}}, a^{i}b⟩ ≅ C_{2}^{2} if |a^{i}b| = 2, or T_{i} = ⟨a^{2^{n−3}}, a^{i}b⟩ ≅ Q_{8} if |a^{i}b| = 4. Then the following hold for any saturated fusion system F over S.

(a) If α ∈ Aut(S) has odd order, then α|_{⟨a⟩} = Id, and hence α = Id_{S} by Lemma A.2. Thus Aut(S) is a 2-group, and hence Aut_{F}(S) = Inn(S) since S is fully automized.

(b) By similar arguments, the only subgroups of S whose automorphism groups are not 2-groups are the T_{i}. Hence these are the only subgroups which could be F-essential.

(c) For each i, Out(T_{i}) ≅ S_{3} and Out_{S}(T_{i}) ≅ C_{2}. Hence Out_{F}(T_{i}) must be one of these two groups.

(d) If i ≡ j (mod 2), then T_{i} and T_{j} are S-conjugate, and hence Out_{F}(T_{i}) ≅ Out_{F}(T_{j}).

Thus there are at most four distinct saturated fusion systems over S. We denote these F_{ij}, where i = 0 if |Out_{F}(T_{0})| = 2, i = 1 if |Out_{F}(T_{0})| = 6, and simiarly for j = 0, 1 depending on |Out_{F}(T_{1})|. All of these are fusion systems of finite groups. For example, when S is semidihedral of order 2^{n} with n ≥ 4 (hence T_{0} ≅ C_{2}^{2} and T_{1} ≅ Q_{8}), then for any odd prime power q ≡ 2^{n−2} − 1 (mod 2^{n−1}), F_{00} is the fusion system of S itself, F_{01} is the fusion system of GL_{2}(q), F_{10} is the fusion system of a certain extension of P SL_{2}(q^{2}) by C_{2} (the extension by the product of a field and a diagonal automorphism), and F_{11} is the fusion system of P SL_{3}(q).

In many situations, when we want to prove that a fusion system over S is saturated, it is very useful to know that we need prove the saturation axioms only for certain subgroups of S. Before stating a theorem, we first need some terminology for formulating this.

Definition 3.9. Let F be a fusion system over a p-group S, and let H be a set of subgroups of S.

• Let F|_{H} ⊆ F be the full subcategory with object set H.

• The fusion system F is H-generated if F = ⟨F|_{H}⟩_{S}.

• Assume H is closed under F-conjugacy. Then F is H-saturated if each F-conjugacy class in H contains a subgroup which is fully automized and receptive in F.

By Lemma 2.6, a fusion system F is H-saturated (when H is closed under F-conjugacy) if and only if the subgroups in H all satisfy the Sylow axiom and the extension axiom (see Proposition 2.5). Hence the above definition is equivalent to that in [5a1, Definition 2.1].

Theorem 3.5 (together with Proposition 3.3(a)) implies that each saturated fusion system F is H-generated, when H is the set of subgroups which are F-centric-radical and fully normalized in F. The next theorem can be thought of as a partial converse to this. It was first shown by Puig (see [P6, Theorem 3.8]) in the special case where H = F^{c}.

Theorem 3.10 ([5a1, Theorem 2.2]). Let F be a fusion system over a p-group S. Let H be a set of subgroups of S closed under F-conjugacy such that F is H-saturated and H-generated. Assume also that each F-centric subgroup of S not in H is F-conjugate to some subgroup P ≤ S such that

Out_{S}(P) ∩ O_{p}(Out_{F}(P)) ≠ 1 .  (∗)

Then F is saturated.

Theorem 3.10 can be very useful, for example, when one is working with a fusion system F over a p-group S, where morphisms are known explicitly only among subgroups in a certain family H. For example, in certain cases, morphisms are defined explicitly between subgroups in H, and then among other subgroups by taking composites of restrictions. In this situation (where F is H-generated by construction), if H contains all F-centric subgroups (or satisfies the slightly weaker hypothesis in the theorem), then to prove F is saturated, it suffices to check the saturation axioms for subgroups in H.

It is natural to ask whether condition (∗) in Theorem 3.10 can be replaced by the condition that H contain all subgroups which are F-centric-radical. An example was given in [5a1] to show that this is not the case, but Kasper Andersen recently pointed out to us that this example was in error. So we give a different example here of a fusion system which is not saturated, but which is H-generated and H-saturated for some set H of subgroups which includes all of those which are F-centric-radical.

Example 3.11. Consider the following matrices in SL_{3}(ℤ/4):

Z = ( \begin{matrix}−1 & 0 & −1 \\ 0 & −1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , X_{1} = ( \begin{matrix}−1 & 0 & 0 \\ 2 & 1 & 1 \\ 0 & 0 & −1 \end{matrix} ) , X_{2} = ( \begin{matrix}−1 & −1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix} ) ,

A_{1} = ( \begin{matrix}0 & 1 & 0 \\ −1 & −1 & 0 \\ 2 & 0 & 1 \end{matrix} ) , A_{2} = ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 2 & −1 & −1 \end{matrix} ) .

Set

T = ⟨Z, X_{1}, X_{2}⟩, T_{i} = ⟨Z, X_{i}⟩, H_{i} = ⟨T, A_{i}⟩ (i = 1, 2), G = ⟨H_{1}, H_{2}⟩.

Let pr: SL_{3}(ℤ/4) ⟶ GL_{3}(2) be the natural projection, and set

N = Ker(pr) = O_{2}(SL_{3}(ℤ/4)) ≅ C_{2}^{8}

(the group of matrices I +2M for M ∈ M_{3}(ℤ/4) of trace zero). Straightforward computations show that T ≅ D_{8}, T_{i} ≅ C_{2}^{2}, and T_{i} ⊴ H_{i} ≅ S_{4}. Thus pr sends each of these subgroups injectively into GL_{3}(ℤ/2). In contrast, one easily sees that pr(G) = GL_{3}(2), but the element (A_{2}X_{2}A_{1})^{3} is a nontrivial element of N and hence of O_{2}(G) = G∩N. Also, O_{2}(G) ⊴ ⟨G, N⟩ = SL_{3}(ℤ/4), no proper nontrivial subgroup of N is normal in SL_{3}(ℤ/4), and we conclude that O_{2}(G) = N and G = SL_{3}(ℤ/4).

Set A = C_{4}^{3}, with the canonical action of G = SL_{3}(ℤ/4), and let Γ = A ⋊ G be the semidirect product. Set S = AT, and set F = ⟨Inn(S), Aut_{AH_{1}}(AT_{1}), Aut_{AH_{2}}(AT_{2})⟩ as a fusion system over S. Set H = {S, AT_{1}, AT_{2}}. Then F is H-generated by construction. Also, F is Hsaturated: all three subgroups in H are fully automized in F, and the AT_{i} are receptive since each α ∈ N_{Aut_{F} (AT_{i})}(Aut_{S}(AT_{i})) = Aut_{S}(AT_{i}) extends to S (note that Out_{F}(AT_{i}) ≅ N_{H_{i}}(T_{i})/T_{i} ≅ S_{3}). However, F is not saturated, since A^{F} = {A} but A is not fully automized (Aut_{S}(A) is not a Sylow 2-subgroup of Aut_{F}(A) = G). Theorem 3.10 does not apply in this case since condition (∗) fails to hold for the F-centric subgroup A:

O_{2}(Aut_{F}(A)) ∩ Aut_{S}(A) = N ∩ T = 1 .

We claim that F^{cr} ⊆ H. If P ≤ S is such that P A/A is cyclic of order 2 or 4 or P A = S, then Aut_{F}(P) = Aut_{S}(P) since no element of order three in Aut_{F}(AT_{i}) (for i = 1 or 2) restricts to an automorphism of P. If P A = AT_{i} and P < AT_{i}, then Aut_{F}(P) = Aut_{AH_{i}}(P), and either P ∉ F^{c}, or 1 ≠ Out_{AT_{i}}(P) ≤ O_{2}(Out_{F}(P)) (hence P is not F-radical). If P ≤ A, then either P < A and P ∉ F^{c}, or P = A, Aut_{F}(P) = ⟨H_{1}, H_{2}⟩ = G, and P is not F-radical since O_{2}(G) = N ≠ 1. Thus F^{cr} ⊆ H (and in fact, the two sets are equal).

4. Normal and central subgroups of a fusion system

We next look at normal and central subgroups of a fusion system, as well as strongly and weakly closed subgroups.

Definition 4.1. Let F be a fusion system over a p-group S. Fix a subgroup Q of S.

• Q is central in F if Q ⊴ S, and for all P, R ≤ S and all φ ∈ Hom_{F}(P, R), φ extends to a morphism φ ∈ Hom_{F}(P Q, RQ) such that φ|_{Q} = Id_{Q}.

• Q is normal in F (denoted Q ⊴ F) if Q ⊴ S, and for all P, R ≤ S and all φ ∈ Hom_{F}(P, R), φ extends to a morphism φ ∈ Hom_{F}(P Q, RQ) such that φ(Q) = Q.

• Q is strongly closed in F if no element of Q is F-conjugate to an element of S∖Q. More generally, for any P ≤ S which contains Q, Q is strongly closed in P with respect to F if no element of Q is F-conjugate to an element of P ∖Q.

• Q is weakly closed in F if Q^{F} = {Q}. More generally, for any P ≤ S which contains Q, Q is weakly closed in P with respect to F if Q is the only subgroup of P in Q^{F}.

The following implications, for any F and any Q ≤ S, follow immediately from Definition 4.1:

Q central ⟹ Q normal ⟹ Q strongly closed

⟹ Q weakly closed ⟹ Q ⊴ S.

Note that when G is a finite group, S ∈ Syl_{p}(G), and Q ⊴ S is normal (central) in F_{S}(G), then this need not imply that Q is normal (central) in G. What it does imply is that N_{G}(Q) (C_{G}(Q)) controls p-fusion in G (i.e., has the same fusion system over S).

The following lemma gives another, simpler criterion for a subgroup to be central — at least, when the fusion system is saturated.

Lemma 4.2. Let F be a saturated fusion system over a p-group S. For any subgroup Q in S, Q is central in F if and only if x^{F} = {x} for each x ∈ Q.

Proof. Assume x^{F} = {x} for each x ∈ Q. In particular, Q ≤ Z(S). Fix P, R ≤ S and φ ∈ Hom_{F}(P, R), and set R_{0} = φ(P). Choose T ∈ P^{F} which is fully centralized in F, and fix some χ ∈ Iso_{F}(R_{0}, T). Since Q ≤ Z(S) and T is receptive in F, ψ = χφ extends to ψ ∈ Hom_{F}(P Q, S), and χ extends to χ ∈ Iso_{F}(R_{0}Q, S). For each x ∈ Q, since x^{F} = {x}, χ(x) = x = ψ(x). Hence χ(R_{0}Q) = T Q = ψ(P Q), and χ^{−1} ∘ ψ ∈ Hom_{F}(P Q, RQ) extends φ and is the identity on Q. Thus Q is central in F.

The converse is clear: if Q is central in F, then x^{F} = {x} for each x ∈ Q. □

For example, when S is a cyclic 2-group, or a quaternion 2-group of order ≥ 8, then S has a unique involution z. Hence z^{F} = {z}, and ⟨z⟩ is central in F, for any saturated fusion system F over S.

It follows immediately from the definitions of normal and central subgroups that if Q_{1}, Q_{2} ≤ S are normal (central) in F, then so is Q_{1}Q_{2}. So it makes sense to talk about the maximal normal or central subgroup in F.

Definition 4.3. For any fusion system F over a p-group S,

• O_{p}(F) ⊴ S denotes the largest subgroup of S which is normal in F; and

• Z(F) ≤ Z(S) denotes the largest subgroup of S which is central in F.

Recall that for any finite group G, Z^{∗}(G) ≤ G is the subgroup such that Z^{∗}(G)/O_{2′}(G) = Z(G/O_{2′}(G)). So Glauberman’s Z^{∗}-theorem [Gl1, Corollary 1] can be interpreted to say that for S ∈ Syl_{2}(G), Z(F_{S}(G)) = Z^{∗}(G) ∩ S.

The next proposition is very elementary.

Proposition 4.4. If Q is normal in a fusion system F over a p-group S, then each characteristic subgroup of Q is also normal in F.

Proof. Assume U is characteristic in Q, where Q ⊴ F. Then for each P, R ≤ S and each φ ∈ Hom_{F}(P, R), there is φ ∈ Hom_{F}(P Q, RQ) such that φ|_{P} = φ and φ(Q) = Q, and φ(U) = U since U char Q. Thus U ⊴ F. □

One easy consequence of Alperin’s fusion theorem is the following useful criterion for a subgroup to be normal in a fusion system. Conditions (a) and (b) in the following proposition were shown in [5a1, Proposition 1.6] to be equivalent.

Proposition 4.5. Let F be a saturated fusion system over a p-group S. Then for any Q ≤ S, the following conditions are equivalent:

(a) Q is normal in F.

(b) Q is strongly closed in F, and Q ≤ P for each P ∈ F^{cr}.

(c) If P ≤ S is F-essential or P = S, then P ≥ Q and Q is Aut_{F}(P)- invariant.

Proof. (a ⇒ b) Assume Q ⊴ F. Clearly, Q is strongly closed in F.

For each P ∈ F^{cr} and each α ∈ Aut_{F}(P), α extends to some α ∈ Aut_{F}(P Q) such that α(Q) = Q, and hence α normalizes Aut_{P Q}(P). Thus Aut_{P Q}(P) ⊴ Aut_{F}(P), and so Aut_{P Q}(P) = Inn(P) since P is F-radical. Since P is also F-centric, it follows that N_{P Q}(P) = P, and hence that P Q = P (Lemma A.1). So P ≥ Q.

(b ⇒ c) If P is F-essential or P = S, then P ∈ F^{cr} by Proposition 3.3(a). Hence P ≥ Q, and Q is Aut_{F}(P)-invariant since it is strongly closed in F.

(c ⇒ a) If (c) holds for Q, then by Theorem 3.6, every morphism in F is a composite of restrictions of automorphisms of subgroups which contain Q and which leave it invariant. Hence each φ ∈ Hom_{F}(P, R) in F extends to some φ ∈ Hom_{F}(P Q, RQ) which sends Q to itself, and thus Q ⊴ F. □

As one special case of Proposition 4.5, if F is a saturated fusion system over an abelian p-group S, then the only F-centric subgroup is S itself, and hence S ⊴ F. Translated to groups, this is just Burnside’s fusion theorem: if G has an abelian Sylow p-subgroup S, then N_{G}(S) controls p-fusion in G.

The next proposition gives a different characterization of normal subgroups in a fusion system; one which is also very useful.

Proposition 4.6. Fix a saturated fusion system F over a p-group S. Then for any subgroup Q of S, Q ⊴ F if and only if there exists a series 1 = Q_{0} ≤ Q_{1} ≤ ⋯ ≤ Q_{n} = Q of subgroups such that

(a) for each 1 ≤ i ≤ n, Q_{i} is strongly closed in F; and

(b) for each 1 ≤ i ≤ n, [Q, Q_{i}] ≤ Q_{i−1}.

Proof. Assume first there is a series of subgroups Q_{i} ≤ Q which are strongly closed in F and satisfy (b). We claim that Q ⊴ F. By Proposition 4.5, it suffices to show Q is contained in each member U of F^{cr}. Fix such a U, choose k maximal subject to Q_{k} ≤ U, and assume k < n. Set R = U ∩ Q_{k+1} < Q_{k+1}, and consider the series

1 = Q_{0} ≤ Q_{1} ≤ ⋯ ≤ Q_{k} ≤ R ≤ U .  (1)

Since the Q_{i} are strongly closed, each term in (1) is Aut_{F}(U)-invariant (i.e., Aut_{F}(U) acts on each Q_{i} and on R). Also, [N_{Q_{k+1}}(U), U] ≤ U ∩ Q_{k+1} = R, [N_{Q_{k+1}}(U), R] ≤ [Q_{k+1}, Q_{k+1}] ≤ Q_{k}, and [N_{Q_{k+1}}(U), Q_{i}] ≤ [Q, Q_{i}] ≤ Q_{i−1} for each 0 < i ≤ k. In other words, Aut_{Q_{k+1}}(U) centralizes each factor group in (1), so Aut_{Q_{k+1}}(U) ≤ O_{p}(Aut_{F}(U)) by Lemma A.2. Hence Aut_{Q_{k+1}}(U) ≤ Inn(U) since U is F-radical, so N_{Q_{k+1}}(U) ≤ UC_{S}(U), and UC_{S}(U) = U since U is F-centric. Thus N_{Q_{k+1}U}(U) = U, Q_{k+1}U = U by Lemma A.1, and this contradicts the original assumption that Q_{k+1} ≰ U. We conclude that k = n, Q ≤ U for each U ∈ F^{cr}, and hence Q ⊴ F.

Conversely, assume Q ⊴ F, and let 1 = Q_{0} ≤ ⋯ ≤ Q_{n} = Q be the ascending central series for Q. Then each Q_{i} is characteristic in Q, hence is normal in F by Proposition 4.4, and hence is strongly closed in F. The series {Q_{i}} thus satisfies (a) and (b), completing the proof. □

We now list some easy corollaries of Proposition 4.6. Recall (Definition 4.3) that we write O_{p}(F) for the largest normal subgroup in a fusion system F.

Corollary 4.7. The following hold for any saturated fusion system F over a p-group S.

(a) An abelian subgroup Q of S is normal in F if and only if Q is strongly closed in F.

(b) O_{p}(F) ≠ 1 if and only if there is a nontrivial abelian subgroup of S which is strongly closed in F.

Proof. (a) Let Q be an abelian subgroup of S. If Q ⊴ F, then Q is strongly closed by Proposition 4.5. Conversely if Q is strongly closed then Q ⊴ F by Proposition 4.6, since conditions (a) and (b) of 4.6 hold with respect to the series 1 = Q_{0} ≤ Q_{1} = Q.

(b) If there is a nontrivial strongly closed abelian subgroup Q, then Q ⊴ F by (a), so O_{p}(F) ≠ 1. Conversely, if Q = O_{p}(F) ≠ 1, then Z(Q) ≠ 1, and hence Z(Q) is a nontrivial abelian subgroup of S which is normal in F (hence strongly closed) by Proposition 4.4. □

We next look at constrained fusion systems, which will play an important role in Parts II and IV.

Definition 4.8. Fix a saturated fusion system F over a p-group S.

• F is constrained if there is a normal subgroup Q ⊴ F which is Fcentric; equivalently, if O_{p}(F) ∈ F^{c}.

• If F is constrained, then a model for F is a finite group G such that S ∈ Syl_{p}(G), F_{S}(G) = F, and C_{G}(O_{p}(G)) ≤ O_{p}(G).

• G(F) denotes the class of all finite groups which are models for F.

The definition of a constrained fusion system is motivated by the analogous terminology for groups: a finite group G is p-constrained if O_{p}(G/O_{p′}(G)) contains its centralizer in G/O_{p′}(G). We say that G is strictly p-constrained if G is p-constrained and O_{p′}(G) = 1; equivalently, C_{G}(O_{p}(G)) ≤ O_{p}(G). Thus the models for F are the finite, strictly pconstrained groups which realize F.

The next theorem says that each constrained fusion system has models, and that they are unique up to isomorphism in a strong sense. This result was originally conjectured by Puig in unpublished notes.

Theorem 4.9 (Model theorem for constrained fusion systems). Let F be a constrained saturated fusion system over a p-group S. Fix Q ⊴ S which is F-centric and normal in F. Then the following hold.

(a) There are models for F; i.e., G(F) ≠ ∅.

(b) If G_{1} and G_{2} are two models for F, then there is an isomorphism φ: G_{1} ⟶^{≅} G_{2} such that φ|_{S} = Id_{S}.

(c) For any finite group G containing S as a Sylow p-subgroup such that Q ⊴ G, C_{G}(Q) ≤ Q, and Aut_{G}(Q) = Aut_{F}(Q), there is β ∈ Aut(S) such that β|_{Q} = Id_{Q} and F_{S}(G) = ^{β}F. Thus there is a model for F which is isomorphic to G.

Proof. This is shown in [5a1, Proposition C], except for the strong uniqueness property (b), which is shown in [A5, 2.5]. A different proof (of all three parts of the theorem) is given in Part III (Theorem III.5.10). □

5. Normalizer fusion systems

Let F be a fusion system over a p-group S. For any subgroup Q ≤ S and any group of automorphisms K ≤ Aut(Q), set Aut_{F}^{K}(Q) = K ∩ Aut_{F}(Q), Aut_{S}^{K}(Q) = K ∩ Aut_{S}(Q), and

N_{S}^{K}(Q) = {x ∈ N_{S}(Q) | c_{x} ∈ K} .

Thus N_{S}^{Aut(Q)}(Q) = N_{S}(Q) is the usual normalizer, and N_{S}^{{1}}(Q) = C_{S}(Q) is the centralizer. Also, for any monomorphism φ ∈ Hom(Q, R), we write

φK = {^{φ}χ | χ ∈ K} ≤ Aut(φ(Q)) .

Definition 5.1. Let F be any fusion system over S. For any subgroup Q ≤ S and any group of automorphisms K ≤ Aut(Q), Q is fully Knormalized in F if |N_{S}^{K}(Q)| ≥ |N_{S}^{^{φ}K}(φ(Q))| for all φ ∈ Hom_{F}(Q, S).

Thus Q is fully normalized or fully centralized in F if and only if it is fully K-normalized, for K = Aut(Q) or K = 1, respectively. The following proposition, most of which was proven in [BLO2, Proposition A.2], describes the key properties which characterize fully K-normalized subgroups in a saturated fusion system. Part of the proposition has already been established in Lemma 2.6(c): the case where K = Aut(Q).

Proposition 5.2. Let F be a saturated fusion system over a p-group S. Fix a subgroup Q ≤ S and a group of automorphisms K ≤ Aut(Q). Then the following three conditions are equivalent:

(a) Q is fully K-normalized in F.

(b) Q is fully centralized in F and Aut_{S}^{K}(Q) ∈ Syl_{p}(Aut_{F}^{K}(Q)).

(c) For each P ≤ S and each φ ∈ Iso_{F}(P, Q), there are homomorphisms χ ∈ Aut_{F}^{K}(Q) and φ ∈ Hom_{F}(P ·N_{S}^{K^{φ}}(P), S) such that φ|_{P} = χ ∘ φ.

Proof. (a ⇒ b) Assume Q is fully K-normalized in F. Choose P ≤ S which is F-conjugate to Q, and fully automized and receptive in F, and fix ψ ∈ Iso_{F}(Q, P). By Lemma 2.6(a), P is fully centralized.

Fix T ∈ Syl_{p}(Aut_{F}^{^{ψ}K}(P)) such that T ≥ ^{ψ}Aut_{S}^{K}(Q). Since Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)), there is α ∈ Aut_{F}(P) such that ^{α}T ≤ Aut_{S}(P). Set φ = α ∘ ψ ∈ Iso_{F}(Q, P). Then

Aut_{S}^{^{φ}K}(P) = Aut_{S}(P) ∩ Aut_{F}^{^{φ}K}(P) ≥ ^{α}T ∈ Syl_{p}(Aut_{F}^{^{φ}K}(P)) ,

and hence Aut_{S}^{^{φ}K}(P) = ^{α}T is also a Sylow subgroup. We thus have

|C_{S}(Q)|·|Aut_{S}^{K}(Q)| = |N_{S}^{K}(Q)| ≥ |N_{S}^{^{φ}K}(P)| = |C_{S}(P)|·|Aut_{S}^{^{φ}K}(P)|;

since Q is fully K-normalized, while

|C_{S}(Q)| ≤ |C_{S}(P)|  and  |Aut_{S}^{K}(Q)| ≤ |Aut_{S}^{^{φ}K}(P)|

since P is fully centralized and Aut_{S}^{^{φ}K}(P) is a Sylow p-subgroup. So all of these inequalities are equalities, Q is fully centralized, and Aut_{S}^{K}(Q) ∈ Syl_{p}(Aut_{F}^{K}(Q)).

(b ⇒ c) Assume Aut_{S}^{K}(Q) ∈ Syl_{p}(Aut_{F}^{K}(Q)) where Q is fully centralized in F. Fix φ ∈ Iso_{F}(P, Q), and set L = K^{φ} ≤ Aut(P). Then ^{φ}Aut_{S}^{L}(P) is a p-subgroup of Aut_{F}^{K}(Q), so there is χ ∈ Aut_{F}^{K}(Q) such that ^{χφ}Aut_{S}^{L}(P) ≤ Aut_{S}^{K}(Q). Since Q is fully centralized, it is receptive, and hence χφ extends to a morphism φ ∈ Hom_{F}(P ·N_{S}^{L}(P), S). Thus Q satisfies the condition in (c).

(c ⇒ a) Assume Q satisfies the extension condition in (c). Thus for each φ ∈ Hom_{F}(Q, S), there exist

χ ∈ Aut_{F}^{K}(Q) and φ ∈ Hom_{F}(φ(Q)·N_{S}^{^{φ}K}(φ(Q)), S)

with φ|_{φ(Q)} = χ ∘ φ^{−1}. In particular, φ(N_{S}^{^{φ}K}(φ(Q))) ≤ N_{S}^{K}(Q), and so |N_{S}^{^{φ}K}(φ(Q))| ≤ |N_{S}^{K}(Q)|. Thus Q is fully K-normalized in F. □

We can now define the normalizer fusion subsystems in a fusion system.

Definition 5.3. Let F be a fusion system over a p-group S, and fix Q ≤ S and K ≤ Aut(Q). Let N_{F}^{K}(Q) ⊆ F be the fusion system over N_{S}^{K}(Q) where for P, R ≤ N_{S}^{K}(Q),

Hom_{N_{F}^{K}(Q)}(P, R) = {φ ∈ Hom_{F}(P, R) | ∃ φ ∈ Hom_{F}(P Q, RQ)

with φ|_{P} = φ, φ(Q) = Q, and φ|_{Q} ∈ K} .

As special cases, set N_{F}(Q) = N_{F}^{Aut(Q)}(Q) and C_{F}(Q) = N_{F}^{{1}}(Q): the normalizer and centralizer fusion systems, respectively, of Q.

It follows immediately from Definitions 4.1 and 5.3 and Lemma 4.2 that for any saturated fusion system F over a p-group S, and any Q ≤ S, Q ⊴ F if and only if N_{F}(Q) = F, and Q ≤ Z(F) (Q is central in F) if and only if C_{F}(Q) = F.

Definition 5.3 is motivated by the following proposition, which relates normalizers in fusion systems with those in finite groups.

Proposition 5.4. Assume F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G), and fix Q ≤ S and K ≤ Aut(Q). Then Q is fully K-normalized in F if and only if N_{S}^{K}(Q) ∈ Syl_{p}(N_{G}^{K}(Q)). If this is the case, then N_{F}^{K}(Q) = F_{N_{S}^{K}(Q)}(N_{G}^{K}(Q)).

Proof. By Proposition 5.2, Q is fully K-normalized in F if and only if Q is fully centralized and Aut_{S}^{K}(Q) ∈ Syl_{p}(Aut_{F}^{K}(Q)). Also, by Lemma 1.2, Q is fully centralized in F if and only if C_{S}(Q) ∈ Syl_{p}(C_{G}(Q)). Since we can identify Aut_{S}^{K}(Q) = N_{S}^{K}(Q)/C_{S}(Q) and Aut_{G}^{K}(Q) = N_{G}^{K}(Q)/C_{G}(Q), Lemma A.3 now implies that Q is fully K-normalized if and only if N_{S}^{K}(Q) ∈ Syl_{p}(N_{G}^{K}(Q)).

Now assume N_{S}^{K}(Q) ∈ Syl_{p}(N_{G}^{K}(Q)). For P, R ≤ N_{S}^{K}(Q), a morphism φ ∈ Hom_{G}(P, R) has the form φ = c_{g} for some g ∈ N_{G}^{K}(Q) if and only if φ extends to some φ ∈ Hom_{G}(P Q, RQ) such that φ(Q) = Q and φ|_{Q} ∈ K. Thus N_{F}^{K}(Q) is the fusion system of N_{G}^{K}(Q) over N_{S}^{K}(Q). □

Of course, for the normalizer fusion systems to be very useful, they must be saturated.

Theorem 5.5 ([P6, Proposition 2.15], [BLO2, Proposition A.6]). Fix a saturated fusion system F over a p-group S. Assume Q ≤ S and K ≤ Aut(Q) are such that Q is fully K-normalized in F. Then N_{F}^{K}(Q) is a saturated fusion system over N_{S}^{K}(Q).

Proof. Set S_{0} = N_{S}^{K}(Q) and F_{0} = N_{F}^{K}(Q) for short. For each P ≤ S_{0}, set

K_{P} = {α ∈ Aut(P Q) | α(P) = P, α(Q) = Q, α|_{Q} ∈ K} .

We will show that

(a) each subgroup of S_{0} is F_{0}-conjugate to a subgroup P such that P Q is fully K_{P} -normalized in F;

(b) if P ≤ S_{0} and P Q is fully K_{P} -normalized in F, then P is fully automized in F_{0}; and

(c) if P ≤ S_{0} and P Q is fully K_{P} -normalized in F, then P is receptive in F_{0}.

The theorem then follows immediately from Definition 2.2.

(a) Fix R ≤ S_{0}. Let φ ∈ Hom_{F}(RQ, S) be such that φ(RQ) is fully φK_{R}-normalized. Set R_{1} = φ(R) and Q_{1} = φ(Q). Since Q is fully Knormalized by assumption, by Proposition 5.2, there are χ ∈ Aut_{F}^{K}(Q) and ψ ∈ Hom_{F}(N_{S}^{^{φ}K}(Q_{1})·Q_{1}, S) such that χ ∘ ψ|_{Q_{1}} = (φ|_{Q})^{−1}. Since R ≤ S_{0} = N_{S}^{K}(Q), φ(R) ≤ N_{S}^{^{φ}K}(Q_{1}). Thus ψ∘φ ∈ Hom_{F}(RQ, S), ψ∘φ|_{Q} = χ^{−1} ∈ K, and so ψ ∘ φ|_{R} ∈ Hom_{F_{0}}(R, S_{0}). Set P = ψ(R_{1}) = ψ ∘ φ(R).

Set L = ^{φ}K_{R}, so that K_{P} = ^{ψφ}K_{R} = ^{ψ}L. Then

N_{S}^{K_{P}} (P Q) = N_{S}(P) ∩ N_{S}^{K}(Q) and N_{S}^{L}(φ(RQ)) = N_{S}(R_{1}) ∩ N_{S}^{^{φ}K}(Q_{1}).

Hence ψ(N_{S}^{L}(φ(RQ))) ≤ N_{S}^{K_{P}} (P Q); and P Q is fully K_{P} -normalized in F since φ(RQ) is fully L-normalized in F.

(b) Assume P ≤ S_{0}, and P Q is fully K_{P} -normalized in F. By Proposition 5.2, Aut_{S}^{K_{P}} (P Q) ∈ Syl_{p}(Aut_{F}^{K_{P}} (P Q)). The images of these two subgroups under restriction to P are Aut_{S_{0}}(P) and Aut_{F_{0}}(P), respectively, so Aut_{S_{0}}(P) ∈ Syl_{p}(Aut_{F_{0}}(P)), and P is fully automized in F_{0}.

(c) Assume P ≤ S_{0}, and P Q is fully K_{P} -normalized in F. Fix R ≤ S_{0} and φ ∈ Iso_{F_{0}}(R, P), and let N_{φ} = N_{φ}^{F_{0}} be as in Definition 2.2. Set

L = {α ∈ Aut(P Q) | α|_{P} ∈ Aut_{S_{0}}(P), α|_{Q} ∈ K} ≤ K_{P} .

For each g ∈ N_{S}^{K_{P}} (P Q), g ∈ S_{0} since c_{g}|_{Q} ∈ K, and thus g ∈ N_{S_{0}}(P). So Aut_{S}^{L}(P Q) = Aut_{S}^{K_{P}} (P Q). Since P Q is fully K_{P} -normalized in F, it is also fully L-normalized.

By definition of F_{0} = N_{F}^{K}(Q), φ extends to some φ ∈ Iso_{F}(RQ, P Q) such that φ|_{Q} ∈ K. By Proposition 5.2, there are χ ∈ Aut_{F}^{L}(P Q) and ψ ∈ Hom_{F}(N_{φ}·Q, N_{S}^{K}(Q)·Q) such that ψ|_{RQ} = χ ∘ φ. Since χ|_{P} ∈ Aut_{S_{0}}(P), there is g ∈ N_{S_{0}}(P) such that χ|_{P} = c_{g}|_{P}. Then c_{g}^{−1} ∘ ψ|_{N_{φ}} ∈ Hom_{F_{0}}(N_{φ}, S_{0}) and c_{g}^{−1} ∘ ψ|_{R} = φ; and this finishes the proof that P is receptive in F_{0}. □

The following is one easy application of Theorem 5.5.

Lemma 5.6. Let F be a saturated fusion system over a p-group S. Assume Q ≤ P ≤ S are such that Q ∈ F^{c}. Let φ, φ′ ∈ Hom_{F}(P, S) be such that φ|_{Q} = φ′|_{Q}. Then there is x ∈ Z(Q) such that φ′ = φ ∘ c_{x}.

Proof. This was shown in [BLO2, Proposition A.8], but we give here a different, shorter proof. Since φ ∘ c_{g} = c_{φ(g)} ∘ φ for each g ∈ Q, it suffices to show that φ′ = c_{y} ∘ φ for some y ∈ Z(φ(Q)). Upon replacing P by φ′(P), Q by φ(Q) = φ′(Q), and φ by φ ∘ (φ′)^{−1}, we can assume that φ′ = incl_{P}^{S} and φ|_{Q} = Id_{Q}. We must show that φ = c_{x} for some x ∈ Z(Q). Since Q is subnormal in P, it suffices to prove this when Q ⊴ P.

Set K = Aut_{P} (Q). Since Q is F-centric, it is fully centralized. Since Aut_{F}^{K}(Q) = Aut_{S}^{K}(Q) = K, Q is fully K-normalized by Proposition 5.2. Hence by Theorem 5.5, the normalizer subsystem N_{F}^{K}(Q) over N_{S}^{K}(Q) = P ·C_{S}(Q) = P is saturated. Also, since φ|_{Q} = Id, Aut_{φ(P)}(Q) = Aut_{P} (Q) = K. Thus φ(P) ≤ N_{S}^{K}(Q), and φ ∈ Mor(N_{F}^{K}(Q)).

It thus suffices to prove that N_{F}^{K}(Q) = F_{P} (P). To show this, by Corollary 3.7, it suffices to show that all automorphism groups in N_{F}^{K}(Q) are p-groups. Assume otherwise: then there are R ≤ P and α ∈ Aut_{F}(R) such that Q ≤ R, α(Q) = Q, α|_{Q} ∈ K, and α ≠ Id_{R} has order prime to p. Since α|_{Q} ∈ K and K is a p-group, α|_{Q} = Id_{Q}. Hence for g ∈ R, g and α(g) have the same conjugation action on Q, and g^{−1}α(g) ∈ C_{R}(Q) ≤ Q. Thus α induces the identity on R/Q, so α = Id_{R} by Lemma A.2, and this is a contradiction. □

6. Normal fusion subsystems and products

Several different definitions of normal fusion subsystems are found in the literature. That there are differences is not surprising, since the conditions one wants to impose depend to a great extent on how one needs to use them. The most restrictive definition is due to Aschbacher [A5], and is what we call here a normal fusion subsystem. The definitions of Linckelmann [Li3] and Oliver [O4] are equivalent to what we call here a weakly normal subsystem, while that of Puig [P6] is equivalent to our invariant subsystem.

Definition 6.1. Fix a prime p, and a fusion system F over a p-group S.

• A (saturated) fusion subsystem of F is a subcategory E ⊆ F which is itself a (saturated) fusion system over a subgroup T ≤ S.

• A fusion subsystem E ⊆ F over T ⊴ S is F-invariant if T is strongly

closed in F, and the two following conditions hold:

– (invariance condition) ^{α}E = E for each α ∈ Aut_{F}(T), and

– (Frattini condition) for each P ≤ T and each φ ∈ Hom_{F}(P, T), there are α ∈ Aut_{F}(T) and φ_{0} ∈ Hom_{E}(P, T) such that φ = α ∘ φ_{0}.

• A fusion subsystem E ⊆ F over T ⊴ S is weakly normal in F (E ˙⊴ F) if E and F are both saturated and E is F-invariant.

• A fusion subsystem E ⊆ F over T ⊴ S is normal (E ⊴ F) if E is weakly

normal, and

– (Extension condition) each α ∈ Aut_{E}(T) extends to some α ∈ Aut_{F}(T C_{S}(T)) such that [α, C_{S}(T)] ≤ Z(T).

• F is simple if it contains no proper nontrivial normal fusion subsystem.

Note that the fusion system of a simple group need not be simple. For example, the fusion systems of the group A_{5} (at any prime p = 2, 3, 5) are not simple.

The Frattini condition in Definition 6.1 is motivated by the Frattini argument (see Proposition A.4(a)), which says that when H ⊴ G and T ∈ Syl_{p}(H), then G = N_{G}(T)·H. This is one step in the proof of the following proposition, which says that when H ⊴ G are finite groups, the fusion system of H is normal in the fusion system of G.

Proposition 6.2. Fix a finite group G, a normal subgroup H ⊴ G, and a Sylow subgroup S ∈ Syl_{p}(G), and set T = S ∩ H ∈ Syl_{p}(H). Then F_{T} (H) ⊴ F_{S}(G).

Proof. Since H ⊴ G, no element of T is G-conjugate to any element of S∖T. Hence T is strongly closed in F_{S}(G). The invariance condition holds for F_{T} (H) ⊆ F_{S}(G) since conjugation by g ∈ N_{G}(T) (or by c_{g}) sends Hom_{H}(P, Q) to Hom_{H}(^{g}P,^{g}Q). By the Frattini argument (see Proposition A.4(a)), G = N_{G}(T)·H, so each morphism in F_{S}(G) between subgroups of T factors as a morphism in F_{T} (H) followed by one in Aut_{G}(T). This proves the Frattini condition in Definition 6.1, and finishes the proof that F_{T} (H) is weakly normal in F_{S}(G).

By the Frattini argument again, this time applied to C_{H}(T)·C_{S}(T) ⊴ N_{H}(T)·C_{S}(T) and the Sylow p-subgroup C_{S}(T),

N_{H}(T)·C_{S}(T) = C_{H}(T)·C_{S}(T)·N_{N_{H}(T)}(C_{S}(T)) .

Thus for each g ∈ N_{H}(T), g = ah for some a ∈ C_{HS}(T) and some h ∈ N_{N_{H}(T)}(C_{S}(T)), and hence c_{g} ∈ Aut_{H}(T) extends to c_{h} ∈ Aut_{G}(T ·C_{S}(T)) where [c_{h}, C_{S}(T)] ≤ H ∩ C_{S}(T) = Z(T). This proves the extension condition, and finishes the proof that F_{T} (H) ⊴ F_{S}(G). □

As another example, when F is a saturated fusion system over S and Q ⊴ S, then Q ⊴ F if and only if F_{Q}(Q) ⊴ F.

The following is one example of a pair of saturated fusion systems E ⊆ F, where E is weakly normal in F but not normal. It is the smallest member of the class of examples appearing after Proposition 6.7.

Example 6.3. Set p = 3, G_{1} = G_{2} = S_{3}, and G = G_{1} × G_{2}, where we regard G_{1} and G_{2} as subgroups of G. Fix S_{i} ∈ Syl_{3}(G_{i}) (i = 1, 2), and set S = S_{1} × S_{2} ∈ Syl_{3}(G). Let G_{0} < G be the (unique) subgroup of index two which contains neither G_{1} nor G_{2}. Set F = F_{S}(G_{0}) and E = F_{S_{1}}(G_{1}). Then E ⊆ F: it is the full subcategory whose objects are the subgroups of S_{1}. Also, E is saturated since E = F_{S_{1}}(G_{1}), and E is F-invariant since Aut_{E}(S_{1}) = Aut_{F}(S_{1}). Thus E ˙⊴ F. However, E is not normal in F, since for g ∈ G_{1} of order two, there is no extension of c_{g} ∈ Aut_{G_{1}}(S_{1}) to α ∈ Aut_{G_{0}}(S) such that [α, S] ≤ S_{1}.

The fusion systems of Example 6.3 are a special case of the more general situation which will be described in Proposition 6.7(c). A still more general theorem of David Craven, describing the difference between normal and weakly normal subsystems, will be stated in the next section (Theorem 7.8).

We next list some of the different, but equivalent, definitions of what we here call F-invariant fusion subsystems. In most cases (see [Li3, § 3], [P6, § 6.4], and [A5, § 3]), some version of the strong invariance condition (c) or (d) below was used.

When F is a fusion system over S and T ≤ S, we let F|_{≤T} denote the full subcategory of F with objects the subgroups of T. This is always a fusion system over T, but not, in general, saturated.

Proposition 6.4 ([A5, Theorem 3.3]). Let F be a saturated fusion system over a p-group S, and let E ⊆ F be a fusion subsystem (not necessarily saturated) over a subgroup T of S. Assume T is strongly closed in F. Then the following conditions are equivalent:

(a) E is F-invariant.

(b) ^{α}E = E for each α ∈ Aut_{F}(T), and F|_{≤T} = ⟨Aut_{F}(T), E⟩.

(c) ^{α}E = E for each α ∈ Aut_{F}(T), and Aut_{E}(P) ⊴ Aut_{F}(P) for each P ≤ T.

(d) (strong invariance condition) For each pair of subgroups P ≤ Q ≤ T, each φ ∈ Hom_{E}(P, Q), and each ψ ∈ Hom_{F}(Q, T), ψφ(ψ|_{P})^{−1} ∈ Hom_{E}(ψ(P), ψ(Q)).

Proof. (a ⇒ d) Fix φ and ψ as in (d). Since E is F-invariant, we can write ψ = α ∘ ψ_{0} for some ψ_{0} ∈ Hom_{E}(Q, T) and some α ∈ Aut_{F}(T). Then

φ′ =^{def} ψ_{0}φ(ψ_{0}|_{P})^{−1} ∈ Hom_{E}(ψ_{0}(P), ψ_{0}(Q)), and so ψφ(ψ|_{P})^{−1} = ^{α}φ′ ∈

Mor(^{α}E) = Mor(E).

(d ⇒ c) Clear.

(c ⇒ b) Assume E satisfies the hypothesis in (c), but ⟨E, Aut_{F}(T)⟩ ⫋ F|_{≤T}. Fix a morphism φ ∈ Hom_{F}(P, Q) (for P, Q ≤ T) which is not in ⟨E, Aut_{F}(T)⟩, chosen such that |P | is maximal subject to these conditions. By Theorem 3.6 (and since T is strongly closed), we can assume P = Q = R ∩ T for some F-essential subgroup R ≤ S, and φ = φ|_{P} for some φ ∈ Aut_{F}(R). Set G = Aut_{F}(R), and let H be the set of all α ∈ G such that α|_{P} ∈ Aut_{E}(P). Then H ⊴ G, since Aut_{E}(P) ⊴ Aut_{F}(P) by assumption. Set U = Aut_{S}(R) ∩ H ∈ Syl_{p}(H).

By the Frattini argument (see Proposition A.4(a)), G = H·N_{G}(U). Hence φ = φ_{0} ∘ β, where φ_{0} =^{def} φ_{0}|_{P} ∈ Aut_{E}(P), and β normalizes Aut_{S}(R) ∩ H. Set N = {g ∈ N_{S}(R) | c_{g} ∈ H}. Then β normalizes Aut_{N}(R), and hence β extends to an element of Aut_{F}(N) by the extension axiom (and since R is fully normalized). So β|_{P} extends to an F-automorphism of N ∩ T ≥ N_{T R}(R) ∩ T > R ∩ T = P (Lemma A.1), so β|_{N∩T} is in ⟨E, Aut_{F}(T)⟩ by the maximality assumption on P, and thus φ ∈ ⟨E, Aut_{F}(T)⟩.

(b ⇒ a) Assume ^{α}E = E for all α ∈ Aut_{F}(T). Then for any φ ∈ Hom_{E}(P, Q) and any α ∈ Aut_{F}(T), φ ∘ α|_{α^{−1}(P)} = α|_{α^{−1}(Q)} ∘ (φ^{α}) where φ^{α} = ^{α^{−1}}φ ∈ Hom_{E}(α^{−1}(P), α^{−1}(Q)).

Thus any composite of E-morphisms and restrictions of elements of Aut_{F}(T) can be rearranged so that the E-morphisms come first and are followed by the restriction of one F-automorphism of T. So the Frattini condition holds if F|_{≤T} = ⟨E, Aut_{F}(T)⟩. □

We next look at products of fusion systems. These are defined in the obvious way.

Definition 6.5. For any pair F_{1} and F_{2} of fusion systems over p-groups S_{1} and S_{2}, F_{1} × F_{2} is the fusion system over S_{1} × S_{2} generated by the set of all (φ_{1}, φ_{2}) ∈ Hom(P_{1} × P_{2}, Q_{1} × Q_{2}) for φ_{i} ∈ Hom_{F_{i}}(P_{i}, Q_{i}).

The first thing to check is that the product of two saturated fusion systems is saturated.

Theorem 6.6 ([BLO2, Lemma 1.5]). Assume F_{1} and F_{2} are fusion systems over p-groups S_{1} and S_{2}, respectively. Then for all P, Q ≤ S_{1} × S_{2}, if P_{i} and Q_{i} denote the images of P and Q under projection to S_{i},

Hom_{F_{1}×F_{2}}(P, Q) = {(φ_{1}, φ_{2})|_{P} | φ_{i} ∈ Hom_{F_{i}}(P_{i}, Q_{i}), (φ_{1}, φ_{2})(P) ≤ Q} .

(1)

If F_{1} and F_{2} are both saturated, then so is F_{1} × F_{2}.

Proof. Set S = S_{1} × S_{2} and F = F_{1} × F_{2} for short. For each subgroup P ≤ S, let P_{i} ≤ S_{i} be the image of P under projection, and set P̂ = P_{1}×P_{2}. Thus P ≤ P̂ in all cases.

Let F_{∗} be the category whose objects are the subgroups of S, and whose morphisms are those given by (1). This is a fusion system: Mor(F_{∗}) contains Inn(S) and is closed under restrictions. By definition, it contains all morphisms (φ_{1}, φ_{2}) ∈ Hom(P_{1} × P_{2}, Q_{1} × Q_{2}) for φ_{i} ∈ Hom_{F_{i}}(P_{i}, Q_{i}), and every morphism in F_{∗} is the restriction of such a morphism. Thus F_{∗} = F by definition of F = F_{1} × F_{2}.

Now assume F_{1} and F_{2} are both saturated. To prove that F is saturated, we must show that each subgroup P of S is F-conjugate to one which is fully automized and receptive in F. It suffices to do this when each P_{i} is fully automized and receptive in F_{i} (i = 1, 2).

Since Aut_{F}(P̂) = Aut_{F_{1}}(P_{1}) × Aut_{F_{2}}(P_{2}) and similarly for Aut_{S}(P̂), and since each P_{i} is fully automized in F_{i}, P̂ is fully automized in F. Each α ∈ Aut_{F}(P) extends to a unique automorphism α̂ ∈ Aut_{F}(P̂) by (1), and we use this to identify Aut_{F}(P) with a subgroup of Aut_{F}(P̂). Choose β ∈ Aut_{F}(P̂) such that Aut_{S}(P̂) contains a Sylow p-subgroup of βAut_{F}(P) = Aut_{F}(β(P)). Set P^{∗} = β(P). Thus Aut_{S}(P^{∗}) = Aut_{F}(P^{∗}) ∩ Aut_{S}(P̂) is a Sylow p-subgroup of Aut_{F}(P^{∗}), and so P^{∗} is fully automized in F.

It remains to show that P^{∗} is receptive. Fix Q ≤ S and φ ∈ Iso_{F}(Q, P^{∗}), let φ_{i} ∈ Iso_{F_{i}}(Q_{i}, P_{i}^{∗}) be such that φ = (φ_{1}, φ_{2})|_{Q}, and set φ̂ = (φ_{1}, φ_{2}) ∈ Iso_{F}(Q̂, P̂^{∗}). Let N_{φ} = N_{φ}^{F} ≤ N_{S}(Q) and Nφ̂ ≤ N_{S}(Q̂) be as in Definition 2.2. Then Nφ̂ = N_{φ_{1}}^{F_{1}} × N_{φ_{2}}^{F_{2}}, each φ_{i} extends to N_{φ_{i}}^{F_{i}} since P_{i}^{∗} is receptive in F_{i}, and thus φ̂ (and φ) extend to a morphism defined on Nφ̂. But N_{φ} ≤ Nφ̂, so φ extends to a morphism defined on N_{φ}; and this proves P^{∗} is receptive. □

The following proposition helps to illustrate the difference between normal and weakly normal subgroups. It can be thought of as a version for fusion systems (though much more elementary) of a theorem of Goldschmidt [Gd2]. Goldschmidt’s theorem says that if S = S_{1} × S_{2} ∈ Syl_{2}(G), where S_{1} and S_{2} are strongly closed in S with respect to G and O(G) = 1, then there are H_{1}, H_{2} ⊴ G such that S_{i} ∈ Syl_{2}(H_{i}) and H_{1} ∩ H_{2} = 1.

Proposition 6.7 ([AOV, Proposition 3.3]). Let F be a saturated fusion system over a p-group S = S_{1} × S_{2}, where S_{1} and S_{2} are strongly closed in F. Set F_{i} = F|_{≤S_{i}} (i = 1, 2), regarded as a fusion system over S_{i}. For each i, let F′_{i} ⊆ F_{i} be the fusion subsystem over S_{i} where for P, Q ≤ S_{i},

Hom_{F′_{i}}(P, Q) = {φ ∈ Hom_{F_{i}}(P, Q) |

(φ, Id_{S_{3−i}}) ∈ Hom_{F}(P S_{3−i}, QS_{3−i})} .

Then for each i = 1, 2,

(a) F′_{i} and F_{i} are saturated fusion systems;

(b) F′_{i} is normal in F and in F_{i}; and

(c) F_{i} is weakly normal in F, and is normal in F only if F_{i} = F′_{i}.

Furthermore,

(d) F′_{1} × F′_{2} ⊆ F ⊆ F_{1} × F_{2}, and all three are equal if Aut_{F}(S) = Aut_{F′_{1}×F′_{2}}(S).

Proof. Points (a) and (d) were shown in [AOV, Proposition 3.3]. Points (b) and (c) were pointed out to us by David Craven.

We first claim that for i = 1, 2,

P, Q ≤ S_{i}, φ ∈ Hom_{F_{i}}(P, Q) ⟹ ∃ ψ ∈ Aut_{F}(S_{3−i})
and χ ∈ Aut_{F}(S_{i}) s.t. (φ, ψ) ∈ Hom_{F}(P S_{3−i}, QS_{3−i})
and χ|_{Q} ∘ φ ∈ Hom_{F′_{i}}(P, S_{i}) .

(2)

If φ(P) is fully centralized in F, the existence of ψ follows by the extension axiom, and since the S_{i} are all strongly closed in F. The general case then follows upon choosing α ∈ Iso_{F}(φ(P), R) where R ≤ S_{i} is fully centralized in F, and applying the extension axiom to α ∘ φ and to α. By the extension axiom again, this time applied to ψ, there is χ such that (χ^{−1}, ψ) ∈ Aut_{F}(S), and hence χ|_{Q} ∘ φ ∈ Hom_{F′_{i}}(P, S_{i}). This finishes the proof of (2).

(a) Fix i = 1, 2. Fix P ≤ S_{i}, and choose Q which is F-conjugate to P and fully normalized in F. By (2), there is Q^{∗} in the Aut_{F}(S_{i})-orbit of Q which is F′_{i}-conjugate to P. Then Q^{∗} is also fully normalized in F, and upon replacing Q by Q^{∗}, we can assume Q is F′_{i}-conjugate to P.

Since Aut_{S}(Q) ≤ Aut_{F′_{i}}(Q) ≤ Aut_{F_{i}}(Q) = Aut_{F}(Q) (and since Q is fully automized in F), Q is fully automized in F_{i} and in F′_{i}. In the notation of Definition 2.2, for all R ≤ S_{i} and φ ∈ Iso_{F}(R, Q), N_{φ}^{F_{i}} = N_{φ}^{F} ∩ S_{i}, and so Q is receptive in F_{i} since it is receptive in F. Finally, if φ ∈ Iso_{F′_{i}}(R, Q), then since QS_{3−i} is also fully normalized and hence receptive in F, (φ, Id_{S_{3−i}}) ∈ Iso_{F}(RS_{3−i}, QS_{3−i}) extends to an F-homomorphism on N_{φ}^{F′_{i}}S_{3−i}, this restricts to an F′_{i}-homomorphism on N_{φ}^{F′_{i}}, and thus Q is also receptive in F′_{i}. We now conclude that F_{i} and F′_{i} are both saturated fusion systems over S_{i}.

(b) By construction, for all φ ∈ Aut_{F_{i}}(S_{i}) = Aut_{F}(S_{i}) (i = 1, 2), ^{φ}F′_{i} = F′_{i}. Since F_{i} = ⟨Aut_{F}(S_{i}), F′_{i}⟩ by (2) (and since F′_{i} is saturated and S_{i} is strongly closed in F), F′_{i} is weakly normal in F and in F_{i}.

By definition of F′_{i}, each α ∈ Aut_{F′_{i}}(S_{i}) extends to some α ∈ Aut_{F}(S) such that α|_{S_{3−i}} = Id. Hence F′_{i} is normal in F and in F_{i} by Proposition 6.4.

(c) Since F_{i} is a full subcategory of F and S_{i} is strongly closed, F_{i} is F-invariant. Thus F_{i} is weakly normal in F since it is saturated by (a).

If F_{i} is normal in F, then each α ∈ Aut_{F}(S_{i}) extends to some α ∈ Aut_{F}(S) such that [α, Z(S_{i})S_{3−i}] ≤ Z(S_{i}). Since S_{3−i} is strongly closed in F, this means that α acts trivially on S_{3−i}, and hence that α ∈ Aut_{F′_{i}}(S_{i}). So in this case, Aut_{F′_{i}}(S_{i}) = Aut_{F}(S_{i}), and F′_{i} = F_{i} by what was shown in (b).

(d) Clearly, F contains F′_{1} × F′_{2}. We must show that F ⊆ F_{1} × F_{2}. Assume otherwise. Then by Theorem 3.5, there is a subgroup P ≤ S such that P = S or P is F-essential, and an automorphism α ∈ Aut_{F}(P) such that α ∉ Aut_{F_{1}×F_{2}}(P). Assume P is maximal subject to this constraint. For i = 1, 2, let pr_{i}: S ⟶ S_{i} be the projection, and set P_{i} = pr_{i}(P). We regard P as a subgroup of P̂ =^{def} P_{1} × P_{2}. Since α(S_{i} ∩ P) = S_{i} ∩ P for i = 1, 2 (the S_{i} being strongly closed), there are unique automorphisms α_{i} ∈ Aut(P_{i}), defined by setting α_{i}(pr_{i}(g)) = pr_{i}(α(g)) for g ∈ P. Thus, if g = (g_{1}, g_{2}) (i.e., g_{i} = pr_{i}(g)), then α(g) = (α_{1}(g_{1}), α_{2}(g_{2})); and we conclude that α = (α_{1}, α_{2})|_{P}.

Since P is fully normalized in F (hence receptive), α extends to an F-automorphism of NP̂(P). By the maximality assumption on P, this implies NP̂(P) = P, and so P̂ = P by Lemma A.1. Hence α = (α_{1}, α_{2}), so α_{i} = α|_{P_{i}} ∈ Aut_{F_{i}}(P_{i}), and this contradicts the original assumption that α ∉ Aut_{F_{1}×F_{2}}(P). Thus F ⊆ F_{1} × F_{2}.

If Aut_{F}(S) = Aut_{F′_{1}×F′_{2}}(S), then Aut_{F_{i}}(S_{i}) = Aut_{F′_{i}}(S_{i}) (i = 1, 2) by construction, and so F_{i} = ⟨F′_{i}, Aut_{F_{i}}(S_{i})⟩ = F′_{i} by (b). Hence F = F′_{1} × F′_{2} = F_{1} × F_{2}. □

As an example, fix a prime q ≠ p, and two pairs of finite groups H_{i} ⊴ G_{i} (i = 1, 2) such that |G_{i}/H_{i}| = q, and such that for S_{i} ∈ Syl_{p}(G_{i}), C_{G_{i}}(S_{i}) ≤ H_{i}. Let Γ ≤ G_{1} ×G_{2} be a subgroup of index q which contains H_{1} ×H_{2}, and contains neither G_{1} nor G_{2}. Set S = S_{1}×S_{2} ∈ Syl_{p}(Γ) and F = F_{S}(Γ), and let F_{i} and F′_{i} be as in Proposition 6.7. Then F_{i} = F_{S_{i}}(G_{i}), F′_{i} = F_{S_{i}}(H_{i}), and F′_{i} ⫋ F_{i}. The subgroups H_{i} are normal in Γ, corresponding to F′_{i} being normal in F, while G_{i} is not even contained in Γ (and F_{i} is only weakly normal in F).

7. Fusion subsystems of p-power index or of index prime to p

We next look at certain fusion subsystems of a saturated fusion system which play a role analogous to that of subgroups of a finite group G which contain O^{p}(G) or O^{p′}(G). Before we can define exactly what we mean by this, we need to define the focal and hyperfocal subgroups of a fusion system. These definitions are also motivated by those for the focal and hyperfocal subgroup of a finite group.

Definition 7.1. For any saturated fusion system F over a p-group S, the focal subgroup 𝔣𝔬𝔠(F) and the hyperfocal subgroup 𝔥𝔶𝔭(F) are defined by setting

𝔣𝔬𝔠(F) = ⟨g^{−1}h | g, h ∈ S are F-conjugate⟩
= ⟨g^{−1}α(g) | g ∈ P ≤ S, α ∈ Aut_{F}(P)⟩
𝔥𝔶𝔭(F) = ⟨g^{−1}α(g) | g ∈ P ≤ S, α ∈ O^{p}(Aut_{F}(P))⟩ .

The two definitions of 𝔣𝔬𝔠(F) are equivalent by Theorem 3.5. One easily sees that 𝔣𝔬𝔠(F) and 𝔥𝔶𝔭(F) are not only normal in S, but are also invariant under the action of each α ∈ Aut_{F}(S). The next lemma describes the relationship between these two subgroups.

Lemma 7.2. For any saturated fusion system F over a p-group S,

𝔣𝔬𝔠(F) = 𝔥𝔶𝔭(F)·[S, S] .

In particular, 𝔣𝔬𝔠(F) = S if and only if 𝔥𝔶𝔭(F) = S.

Proof. By Theorem 3.5,

𝔣𝔬𝔠(F) = ⟨s^{−1}α(s) | s ∈ P ≤ S, P is F-essential, α ∈ Aut_{F}(P)⟩ .

By the Frattini argument (Proposition A.4(a)),

Aut_{F}(P) = O^{p}(Aut_{F}(P))·Aut_{S}(P)

whenever Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)); in particular, whenever P is Fessential. Also, s^{−1}α(s) ∈ [S, S] when s ∈ P and α ∈ Aut_{S}(P), and thus 𝔣𝔬𝔠(F) = 𝔥𝔶𝔭(F)·[S, S].

If 𝔥𝔶𝔭(F) < S, then it is contained in a subgroup P < S of index p. But then [S, S] ≤ P, and hence 𝔣𝔬𝔠(F) ≤ P < S. □

The hyperfocal subgroup is also called H_{F} by Puig [P6, § 7.1], and is denoted O_{F}^{p}(S) in [5a2]. If F = F_{S}(G) is the fusion system of a finite group G with respect to S ∈ Syl_{p}(G), then the focal subgroup theorem (cf. [G1, Theorem 7.3.4]) says that 𝔣𝔬𝔠(F) = S ∩ [G, G], while the hyperfocal subgroup theorem of Puig [P5, § 1.1] says that 𝔥𝔶𝔭(F) = S ∩ O^{p}(G).

Definition 7.3. Fix a saturated fusion system F over a p-group S, and a fusion subsystem E in F over T ≤ S.

• The subsystem E has p-power index in F if T ≥ 𝔥𝔶𝔭(F), and Aut_{E}(P) ≥ O^{p}(Aut_{F}(P)) for each P ≤ S.

• The subsystem E has index prime to p in F if T = S, and Aut_{E}(P) ≥ O^{p′}(Aut_{F}(P)) for each P ≤ S.

The saturated fusion subsystems of p-power index are described by the following theorem.

Theorem 7.4. Let F be a saturated fusion system over a p-group S. Then for each subgroup T ≤ S which contains 𝔥𝔶𝔭(F), there is a unique saturated fusion subsystem

F_{T} = ⟨Inn(T), O^{p}(Aut_{F}(P)) | P ≤ T ⟩  (1)

over T of p-power index in F. Also, F_{T} ⊴ F if T ⊴ S, and F_{T} ⊆ F_{U} if T ≤ U ≤ S. In particular, there is a unique minimal saturated fusion subsystem O^{p}(F) ⊴ F of p-power index, over the subgroup 𝔥𝔶𝔭(F) ⊴ S.

Proof. Except for the precise description of F_{T}, and the claim that F_{T} ⊴ F if T ⊴ S, this was shown in [5a2, Theorem 4.3]. A different proof of this was given in [A6, § 7].

Let F_{T} ⊆ F denote the unique saturated fusion subsystem over T of p-power index. When P is fully automized in F_{T}, then Aut_{F_{T}} (P) must be generated by O^{p}(Aut_{F}(P)) and Aut_{T} (P), since by assumption, it contains the former and contains the latter as Sylow p-subgroup. Hence F_{T} is generated by the O^{p}(Aut_{F}(P)) and Aut_{T} (P) for all P ≤ S by Theorem 3.5 (Alperin’s fusion theorem). This proves (1).

If T ⊴ S, then T is strongly closed in F by definition of 𝔥𝔶𝔭(F), and α(F_{T}) = F_{α(T)} = F_{T} for all α ∈ Aut_{F}(T) by (1). Also, for P ≤ S, restriction sends Aut_{F}(P) into Aut_{F}(P ∩T) and hence sends O^{p}(Aut_{F}(P)) into O^{p}(Aut_{F}(P ∩ T)). So by (1) again,

F|_{≤T} = ⟨Aut_{S}(T), O^{p}(Aut_{F}(P)) | P ≤ T ⟩
= ⟨Aut_{S}(T), F_{T} ⟩ = ⟨Aut_{F}(T), F_{T} ⟩ .

Thus F_{T} is weakly normal in F by Proposition 6.4(b ⇒ a).

To show that F_{T} ⊴ F, it remains to prove the extension condition. Fix α ∈ Aut_{F_{T}} (T). Then α = β ∘ γ, where β has order prime to p and γ has p-power order, and β, γ ∈ ⟨α⟩. Since Inn(T) is a normal Sylow p-subgroup of Aut_{F_{T}} (T), γ = c_{g} for some g ∈ T. Since T is receptive in F (since it is strongly closed), there is β ∈ Aut_{F}(T ·C_{S}(T)) such that β|_{T} = β, and (upon replacing β by an appropriate power, if necessary) we can assume β has order prime to p. Thus β ∘ c_{g} ∈ Aut_{F}(T ·C_{S}(T)) extends α, [g, C_{S}(T)] = 1, so

[β ∘ c_{g}, C_{S}(T)] ≤ [β, C_{S}(T)] ≤ 𝔥𝔶𝔭(F) ∩ C_{S}(T) ≤ Z(T) ,

and the extension condition holds. □

The following corollary follows immediately from Theorem 7.4 and Lemma 7.2.

Corollary 7.5. For any saturated fusion system F over a p-group S,

O^{p}(F) = F ⟺ 𝔥𝔶𝔭(F) = S ⟺ 𝔣𝔬𝔠(F) = S .

We now turn our attention to fusion subsystems of index prime to p. In the following lemma, by the “Frattini condition” on a pair of fusion systems E ⊆ F over the same p-group S, we mean the condition of Definition 6.1: for each P ≤ S and φ ∈ Hom_{F}(P, S), there are φ_{0} ∈ Hom_{E}(P, S) and α ∈ Aut_{F}(S) such that φ = α ∘ φ_{0}.

Lemma 7.6. Fix a saturated fusion system F over a p-group S, and let E ⊆ F be a fusion subsystem over S. Then the following hold.

(a) If Aut_{E}(P) ≥ O^{p′}(Aut_{F}(P)) for each P ∈ F^{c}, then E has index prime to p in F, the Frattini condition holds for E ⊆ F, and F^{c} = E^{c}.

(b) Assume E has index prime to p in F. Then E is saturated if and only if E is F^{c}-generated, and for each P, Q ∈ F^{c} and φ ∈ Hom_{F}(Q, S) such that P ≤ Q and φ|_{P} ∈ Hom_{E}(P, S), φ ∈ Hom_{E}(Q, S).

Proof. (a) Set E_{0} = ⟨O^{p′}(Aut_{F}(P)) | P ∈ F^{c}⟩; i.e., the smallest fusion system over S which contains the automorphism groups O^{p′}(Aut_{F}(P)) for F-centric subgroups P ≤ S. For each F-essential subgroup P ≤ S, Aut_{F}(P) = O^{p′}(Aut_{F}(P))·N_{Aut_{F} (P)}(Aut_{S}(P)) by the Frattini argument (Proposition A.4(a)). Since P is receptive, each morphism φ ∈ N_{Aut_{F} (P)}(Aut_{S}(P)) extends to some φ ∈ Aut_{F}(N_{S}(P)). So by Theorem 3.5 and a downwards induction on |P |, we see that

F = ⟨Aut_{F}(S), O^{p′}(Aut_{F}(P)) | P F-essential⟩ = ⟨Aut_{F}(S), E_{0}⟩ .

Also, ^{α}E_{0} = E_{0} for each α ∈ Aut_{F}(S) by construction of E_{0}. So by Proposition 6.4(b ⇒ a,c), E_{0} is F-invariant (hence the Frattini condition holds for E_{0} ⊆ F), and Aut_{E_{0}}(P) ⊴ Aut_{F}(P) for each P ≤ S. The last condition in turn implies that Aut_{E_{0}}(P) ≥ O^{p′}(Aut_{F}(P)) for each fully normalized subgroup P ≤ S (since Aut_{E_{0}}(P) contains the Sylow subgroup Aut_{S}(P)), and hence for all P ≤ S. Thus E_{0} has index prime to p in F.

By assumption, Aut_{E}(P) ≥ O^{p′}(Aut_{F}(P)) for all P ∈ F^{c}, and hence E ⊇ E_{0}. So E has index prime to p in F, and the Frattini condition holds for E ⊆ F.

Clearly, F^{c} ⊆ E^{c}: each F-centric subgroup P is also E-centric since P^{E} ⊆ P^{F}. Conversely, if P ∈ E^{c}, then by the Frattini condition, each subgroup in P^{F} has the form α(Q) for some Q ∈ P^{E} and α ∈ Aut_{F}(S), and C_{S}(α(Q)) ≤ α(Q) since C_{S}(Q) ≤ Q. Thus F^{c} = E^{c}.

(b1) Assume E is saturated. Then E is E^{c}-generated by Theorem 3.5, and hence is F^{c}-generated by (a).

Assume P, Q ∈ F^{c} and φ ∈ Hom_{F}(Q, S) are such that P ≤ Q and φ|_{P} ∈ Hom_{E}(P, S); we must show that φ ∈ Hom_{E}(Q, S). Since P is a subnormal subgroup of Q, it suffices by iteration to do this when P ⊴ Q. Set φ_{0} = φ|_{P}. Since there is some extension of φ_{0} to a homomorphism defined on Q, N_{φ_{0}}^{E} ≥ Q in the notation of Definition 2.2. Also, P ∈ F^{c} = E^{c}, and hence P is receptive in E. So there is φ′ ∈ Hom_{E}(Q, S) such that φ′|_{P} = φ|_{P}. By Lemma 5.6, there is x ∈ Z(P) such that φ = φ′ ∘ c_{x}, and so φ ∈ Hom_{E}(Q, S).

(b2) Now assume E is F^{c}-generated, and for each P, Q ∈ F^{c} and ψ ∈ Hom_{F}(Q, S) such that P ≤ Q and ψ|_{P} ∈ Hom_{E}(P, S), ψ ∈ Hom_{E}(Q, S). We must show that E is saturated.

If P ≤ S is fully automized and receptive in F, then the same holds for α(P) for each α ∈ Aut_{F}(S). So by the Frattini condition (which holds by (a)), each subgroup of S is E-conjugate to one which is fully automized and receptive in F.

Assume P ∈ F^{c}, and choose Q which is E-conjugate to P and fully automized and receptive in F. Thus Aut_{S}(Q) ∈ Syl_{p}(Aut_{F}(Q)), so Aut_{S}(Q) ∈ Syl_{p}(Aut_{E}(Q)), and Q is also fully automized in E. Fix an isomorphism φ ∈ Iso_{E}(R, Q), and let N_{φ} = N_{φ}^{E} = N_{φ}^{F} be as in Definition 2.2. Since Q is receptive in F, φ extends to some φ̂ ∈ Hom_{F}(N_{φ}, S). Thus φ̂|_{R} ∈ Hom_{E}(R, Q), R ∈ F^{c}, so by assumption, φ̂ ∈ Hom_{E}(N_{φ}, S). Thus Q is receptive in E.

We have now shown, for each P ∈ F^{c}, that there is Q ∈ P^{E} which is fully automized and receptive in E. Thus E is F^{c}-saturated in the sense of Definition 3.9. Since E is F^{c}-generated by assumption, and F^{c} = E^{c} by (a), E is a saturated fusion system by Theorem 3.10. □

We are now ready to describe all saturated fusion subsystems of index prime to p in a saturated fusion system. For any fusion system F over a p-group S, let Γ_{p′}(F) be the free group on the set Mor(F^{c}), modulo the relations induced by composition, and by dividing out by O^{p′}(Aut_{F}(P)) for all P ∈ F^{c}. In other words, the natural map Mor(F^{c}) ⟶ Γ_{p′}(F) is universal among all maps from Mor(F^{c}) to a group which send composites to products, and send O^{p′}(Aut_{F}(P)) to the identity for each P ∈ F^{c}. We will see later (Theorem III.4.19) that Γ_{p′}(F) is the fundamental group of the geometric realization of the category F^{c}.

The following theorem is essentially the same as [5a2, Theorem 5.4], although formulated somewhat differently. See also [P6, Theorem 6.11].

Theorem 7.7. Fix a saturated fusion system F over a p-group S. Let θ: Mor(F^{c}) ⟶ Γ_{p′}(F) be the canonical map. Then the following hold.

(a) Γ_{p′}(F) is finite of order prime to p, and θ(Aut_{F}(S)) = Γ_{p′}(F).

(b) For each saturated fusion subsystem E ⊆ F of index prime to p, there is H ≤ Γ_{p′}(F) such that E = ⟨θ^{−1}(H)⟩. Also, Hom_{E}(P, Q) = Hom_{F}(P, Q) ∩ θ^{−1}(H) for each P, Q ∈ F^{c}.

(c) For each H ≤ Γ_{p′}(F), set E_{H} = ⟨θ^{−1}(H)⟩. Then E_{H} is a saturated fusion subsystem of index prime to p in F, and E_{H} ⊴ F if and only if H ⊴ Γ_{p′}(F).

(d) Set E_{0} = ⟨O^{p′}(Aut_{F}(P)) | P ≤ S⟩, as a fusion system over S. Then θ induces an isomorphism Γ_{p′}(F) ≅ Aut_{F}(S)/Aut_{F}^{0}(S), where

Aut_{F}^{0}(S) =^{def} Ker(θ|_{Aut_{F} (S)}) =

⟨α ∈ Aut_{F}(S) | α|_{P} ∈ Hom_{E_{0}}(P, S), some P ∈ F^{c}⟩ .

In particular, there is a unique minimal saturated fusion subsystem O^{p′}(F) = ⟨θ^{−1}(1)⟩ ⊴ F of index prime to p, and Aut_{O^{p′} (F)}(S) = Aut_{F}^{0}(S).

Proof. Let E_{0} ⊆ F be the fusion system defined in (d). Since θ sends inclusions to the identity, θ(φ) = θ(ψ) whenever ψ is a restriction of φ. Since θ(O^{p′}(Aut_{F}(P))) = 1 for each P ∈ F^{c} by definition of Γ_{p′}(F), θ(Mor(E_{0})) = 1.

By Lemma 7.6(a), E_{0} is the minimal fusion subsystem of index prime to p in F, and the Frattini condition holds for E_{0} ⊆ F.

(a) By the Frattini condition, for each φ ∈ Mor(F^{c}), φ = α ∘ φ_{0} for some α ∈ Aut_{F}(S) and some φ_{0} ∈ Mor(E_{0}). Then θ(φ_{0}) = 1, so θ(φ) = θ(α) ∈ θ(Aut_{F}(S)). Thus θ(Aut_{F}(S)) = θ(Mor(F^{c})) = Γ_{p′}(F).

In particular, since θ(O^{p′}(Aut_{F}(S))) = 1, Γ_{p′}(F) has order prime to p.

(d) Set Γ = Aut_{F}(S) for short, and set

Γ_{0} = ⟨α ∈ Γ | α|_{P} ∈ Hom_{E_{0}}(P, S), some P ∈ F^{c}⟩ .

Thus θ(Γ_{0}) = 1 since θ(Mor(E_{0})) = 1. We must show that Γ_{0} = Ker(θ|_{Γ}). To see this, define

θ̂: Mor(F^{c}) ⟶ Γ/Γ_{0}

as follows. For each P ∈ F^{c} and each φ ∈ Hom_{F}(P, S), φ = α ∘φ_{0} for some φ_{0} ∈ Hom_{E_{0}}(P, S) and some α ∈ Γ (Lemma 7.6), and we set θ̂(φ) = αΓ_{0}. To see this is well defined, assume φ = β ∘ φ_{1} is another such factorization, and set Q_{i} = φ_{i}(P) (i = 0, 1). Thus φ_{1} ∘φ_{0}^{−1} = (β ∘α^{−1})|_{Q_{0}} ∈ Iso_{F}(Q_{0}, Q_{1}), so β ∘ α^{−1} ∈ Γ_{0}, and βΓ_{0} = αΓ_{0}.

Clearly, θ̂ sends composites to products, and θ̂(O^{p′}(Aut_{F}(P))) = 1 for each P ∈ F^{c}. Since θ was defined to be universal among such maps, θ̂ factors through θ, and thus Ker(θ|_{Γ}) ≤ Γ_{0}. We conclude that Aut_{F}^{0}(S) =^{def} Ker(θ|_{Γ}) = Γ_{0}. Since θ|_{Γ} is onto by (a), this also proves that Γ_{p′}(F) ≅ Aut_{F}(S)/Aut_{F}^{0}(S).

(b) Fix a saturated fusion subsystem E of index prime to p in F, and set H = θ(Aut_{E}(S)). If α ∈ Aut_{F}(S) is such that α|_{P} ∈ Hom_{E_{0}}(P, S) for some P ∈ F^{c}, then α|_{P} ∈ Mor(E), and hence α ∈ Aut_{E}(S) by Lemma 7.6. Thus Γ_{0} ≤ Aut_{E}(S). Hence for arbitrary α ∈ Aut_{F}(S), α ∈ Aut_{E}(S) if and only if θ(α) ∈ H.

Fix any φ ∈ Mor(F^{c}), and let φ_{0} ∈ Mor(E_{0}) and α ∈ Aut_{F}(S) be such that φ = α ∘ φ_{0}. Thus φ_{0} ∈ Mor(E) and θ(φ_{0}) = 1. So θ(φ) = θ(α), and φ ∈ Mor(E) if and only if α ∈ Aut_{E}(S) by Lemma 7.6. We just saw that α ∈ Aut_{E}(S) if and only if θ(α) ∈ H, and hence φ ∈ Mor(E) if and only if θ(φ) ∈ H. This proves the last statement in (b).

By Lemma 7.6 again, E^{c} = F^{c}. Hence θ^{−1}(H) = Mor(E^{c}), and so E = ⟨θ^{−1}(H)⟩ by Theorem 3.5.

(c) Now fix H ≤ Γ_{p′}(F), and set E = ⟨θ^{−1}(H)⟩ ⊇ E_{0}. Thus E has index prime to p in F, is F^{c}-generated, satisfies the Frattini condition, and satisfies the extension condition of Lemma 7.6(b). Hence by the lemma, E is a saturated fusion system of index prime to p in F.

If H ⊴ Γ_{p′}(F), then for α ∈ Aut_{F}(S), ^{α}E = E, since θ^{−1}(H) is invariant under conjugation by α. Thus E is weakly normal in F, and is normal since the extension condition of Definition 6.1 is vacuous when the fusion subsystem is over the same p-group. Conversely, if E ⊴ F, then Aut_{E}(S) ⊴ Aut_{F}(S), and hence H ⊴ Γ_{p′}(F). □

We can now state the following theorem of David Craven, which gives a new way of viewing the difference between weakly normal and normal fusion subsystems.

Theorem 7.8 ([Cr3]). Let F be a saturated fusion system over a p-group S, and let E ˙⊴ F be a weakly normal subsystem. Then O^{p′}(E) is normal in F.

As one immediate consequence of Theorem 7.8, a saturated fusion system F is simple (has no proper nontrivial normal subsystems) if and only if it has no proper nontrivial weakly normal subsystems.

8. The transfer homomorphism for saturated fusion systems

We describe here an injective transfer homomorphism, defined from S/𝔣𝔬𝔠(F) to S/[S, S] for any saturated fusion system F over a p-group S, which is useful for getting information about the focal subgroup of F. This is, in fact, a special case of transfer homomorphisms in homology and in cohomology, which can be defined (at least in the case of mod p cohomology) using [BLO2, Proposition 5.5]. However, to keep the exposition elementary, we deal here only with the special case H_{1}(S) = S/[S, S] and H_{1}(F) = S/𝔣𝔬𝔠(F). Some brief remarks on the general case are given at the end of the section.

For any group G, we write G^{ab} = G/[G, G] to denote the abelianization of G. We take as starting point the transfer as defined, for example, in [A4, § 37] and [G1, § 7.3]. In both of those references, when H ≤ G is a pair of finite groups, the transfer is defined as a map from Hom(H, A) to Hom(G, A) for any abelian group A. When A = H^{ab}, this map sends the canonical surjection H −↠ A to a homomorphism from G to A = H^{ab}, and that in turn factors through a homomorphism trf_{H}^{G}: G^{ab} ⟶ H^{ab}. This is what we refer to here as the transfer homomorphism for the pair H ≤ G. In fact, if we identify Hom(G, A) ≅ Hom(G^{ab}, A) and Hom(H, A) ≅ Hom(H^{ab}, A) (now for arbitrary abelian A), then the transfer defined in [A4] and [G1] is just composition with trf_{H}^{G}.

When G is any finite group, S ∈ Syl_{p}(G), and m = [G:S], then trf_{S}^{G} is an injective homomorphism from G^{ab} to S^{ab}, with the property that for each g ∈ S, trf_{S}^{G}([g]) = [h^{m}] for some h such that g ≡ h (mod [G, G]) (cf. formula (6) in [CE, § XII.8]). Since G∩[S, S] = 𝔣𝔬𝔠(F) when F = F_{S}(G) by the focal subgroup theorem, S/𝔣𝔬𝔠(F) is the p-power torsion subgroup of G^{ab}. Hence it is natural, when generalizing trf_{S}^{G} to abstract fusion systems, to construct a homomorphism from S/𝔣𝔬𝔠(F) to S^{ab}.

To construct such a map, we must first look at bisets. For any pair of groups G and H, an (H, G)-biset is a set X on which H acts on the left, G acts on the right, and the two actions commute. In other words, for each g ∈ G, h ∈ H, and x ∈ X, (hx)g = h(xg). This is equivalent to a set with left action of H × G, where a pair (h, g) acts by sending x ∈ X to hxg^{−1}. If the left action of H is free (i.e., hx = x implies h = 1), and {t_{i} | i ∈ I} is a set of representatives for the H-orbits in X, then each element of X has the form ht_{i} for some unique h ∈ H and i ∈ I.

When X is an (H, G)-biset and Y is a (K, H)-biset, we define Y ×_{H}X = (Y ×X)/H: the set of orbits of the H-action on Y ×X defined by (y, x)h = (yh, h^{−1}x). Let [y, x] ∈ Y ×_{H}X denote the orbit of (y, x) ∈ Y × X. We make Y ×_{H}X into a (K, G)-biset by setting k[y, x]g = [ky, xg] for k ∈ K and g ∈ G.

Lemma 8.1. For any pair of finite groups G and H, and any finite (H, G)- biset X whose left H-action is free, there is a homomorphism

X_{∗}: G^{ab} ⟶ H^{ab}

with the following property. For any set {t_{1}, …, t_{n}} ⊆ X of representatives for the orbits of the left H-action, and any element g ∈ G, if σ ∈ S_{n} and h_{1}, …, h_{n} ∈ H are the unique permutation and elements such that t_{i}g = h_{i}t_{σ(i)} for each i, then

X_{∗}([g]) = ∏_{i=1}^{n}[h_{i}] .

Furthermore, the following hold.

(a) For any homomorphism φ ∈ Hom(G, H), let H_{G,φ} be the (H, G)-biset where H acts on itself by left multiplication, and G acts on the right via (h, g) ↦ hφ(g). Then (H_{G,φ})_{∗}([g]) = [φ(g)] for each g ∈ G. Thus (H_{G,φ})_{∗} = φ_{∗} is the homomorphism induced by φ.

(b) If H ≤ G, and X = G is the (H, G)-biset with the actions defined by left and right multiplication, then X_{∗} is the usual transfer homomorphism trf_{H}^{G} from G^{ab} to H^{ab}.

(c) If X and Y are two (H, G)-bisets with free left H-actions, then for each g ∈ G, (X ⨿ Y)_{∗}([g]) = X_{∗}([g])·Y_{∗}([g]).

(d) If X is an (H, G)-biset and Y is a (K, H)-biset, where both left actions are free, then

(Y ×_{H}X)_{∗} = Y_{∗} ∘ X_{∗}: G^{ab} ⟶ K^{ab} .

(e) Fix H ≤ G of index m, and let incl_{H}^{G}: H ⟶ G be the inclusion. Then the composite

G^{ab} ⟶^{trf_{H}^{G}} H^{ab} ⟶^{(incl_{H}^{G})_{∗}} G^{ab}

sends [g] to [g^{m}] for each g ∈ G.

Proof. For any set T = {t_{1}, …, t_{n}} ⊆ X of representatives for the orbits of the left H-action, define

X_{∗}^{T} : G ⟶ H^{ab}

as above: X_{∗}^{T} (g) = ∏_{i=1}^{k}[h_{i}] if t_{i}g = h_{i}t_{σ(i)} for each i. If g′ ∈ G is another element, and t_{i}g′ = h′_{i}t_{τ(i)} for each i, then t_{i}gg′ = h_{i}h′_{σ(i)}t_{τσ(i)}. Thus

X_{∗}^{T} (gg′) = ∏_{i=1}^{n}[h_{i}h′_{σ(i)}] = ∏_{i=1}^{n}[h_{i}]· ∏_{i=1}^{n}[h′_{i}] = X_{∗}^{T} ([g])X_{∗}^{T} ([g′]) .

This proves that X_{∗}^{T} is a homomorphism.

Let T′ = {t′_{1}, …, t′_{n}} be another set of H-orbit representatives. We can assume the t′_{i} are ordered such that t′_{i} ∈ Ht_{i} for each i. Let η_{i} ∈ H be such that t′_{i} = η_{i}t_{i}. Fix g ∈ G, and let σ ∈ S_{n} and h_{i} ∈ H be such that t_{i}g = h_{i}t_{σ(i)} for each i. Then

t′_{i}g = η_{i}t_{i}g = η_{i}h_{i}t_{σ(i)} = η_{i}h_{i}η_{σ(i)}^{−1}t′_{σ(i)}

for each i. So

X_{∗}^{T ′}(g) = ∏_{i=1}^{n}[η_{i}h_{i}η_{σ(i)}^{−1}] = ∏_{i=1}^{n}[h_{i}] = X_{∗}^{T} (g) ∈ H^{ab} .

Thus X_{∗}^{T ′} = X_{∗}^{T} is independent of the choice of set of orbit representatives, and induces a unique homomorphism X_{∗}: G^{ab} ⟶ H^{ab}.

It remains to prove points (a)–(e).

(a) Fix φ ∈ Hom(G, H), and assume X = H_{G,φ}: the (H, G)-biset with underlying set H, and with actions defined by (h, x, g) ↦ hxφ(g). Then {1} is a set of orbit representatives for the H-action. For g ∈ G, 1·g = φ(g) = φ(g)·1, so X_{∗}([g]) = [φ(g)], and X_{∗} is just the usual homomorphism from G^{ab} to H^{ab} induced by φ.

(b) Assume H ≤ G, and let X = G be the (H, G)-biset with the actions defined by left and right multiplication. Let g_{1}, …, g_{n} ∈ G be any set of representatives for the right cosets Hg. Then for each g ∈ G, X_{∗}([g]) = ∏_{i=1}^{n}[h_{i}], where h_{i} ∈ H and σ ∈ S_{n} are such that g_{i}g = h_{i}g_{σ(i)} for each i. Thus X_{∗} is the usual transfer trf_{H}^{G} as defined, for example, in [A4, § 37] and [G1, Theorem 7.3.2]. More precisely, in the notation of [A4] and [G1], the composite G ⟶ G^{ab} ⟶^{X_{∗}} H^{ab} is the transfer of the natural homomorphism H ⟶ H^{ab}.

(c) If X and Y are two (H, G)-bisets with free left H-action, then by definition of X_{∗}, (X ⨿ Y)_{∗}([g]) = X_{∗}([g])·Y_{∗}([g]) for each g ∈ G.

(d) Assume X is an (H, G)-biset and Y is a (K, H)-biset, where both left actions are free. Let t_{1}, …, t_{n} be representatives for the left H-orbits in X, and let u_{1}, …, u_{m} be representatives for the left K-orbits in Y. For each (y, x) ∈ Y × X, let [y, x] ∈ Y ×_{H} X be its equivalence class. There are unique h ∈ H and j such that x = ht_{j}, so [y, x] = [yh, t_{j}], and yh = ku_{i} for some unique i and k ∈ K. Thus the K-action on Y ×_{H} X is free, and the pairs [u_{i}, t_{j}] form a set of representatives for its orbits.

For fixed g ∈ G, set t_{j}g = h_{j}t_{σ(j)} (where h_{j} ∈ H and σ ∈ S_{n}), and set u_{i}h_{j} = k_{ij}u_{τ_{j}(i)} (k_{ij} ∈ K and τ_{j} ∈ S_{m}). Then [u_{i}, t_{j}]g = k_{ij}[u_{τ_{j}(i)}, t_{σ(j)}], and so

(Y ×_{H}X)_{∗}([g]) = ∏_{j=1}^{n}(∏_{i=1}^{m} k_{ij}) = ∏_{j=1}^{n} Y_{∗}([h_{j}]) = Y_{∗}(X_{∗}([g])) .

(e) Fix H ≤ G of index m, and let {g_{1}, …, g_{m}} be representatives for the right cosets Hg. Fix g ∈ G, and let σ ∈ S_{m} and h_{i} ∈ H be such that g_{i}g = h_{i}g_{σ(i)}. Then ∏_{i=1}^{m}[g_{i}g] = ∏_{i=1}^{m}[h_{i}g_{σ(i)}] in G^{ab}, and hence ∏_{i=1}^{m}[h_{i}] = [g]^{m} in G^{ab}. So

(incl_{H}^{G})_{∗} ∘ trf_{H}^{G}([g]) = ∏_{i=1}^{m}[h_{i}] = [g^{m}] ∈ G^{ab}.  □

The following special case of the transfer homomorphism for groups will be needed.

Lemma 8.2. Fix a prime p, and a p-group P. Then for each proper subgroup Q < P and each element g ∈ Ω_{1}(Z(P)), trf_{Q}^{P}([g]) = 1.

Proof. By Lemma 8.1(b,d), for Q < R < P, trf_{Q}^{P} = trf_{Q}^{R} ∘ trf_{R}^{P}. It thus suffices to prove the lemma when [P:Q] = p. If g ∈ Q, let {x_{1}, x_{2}, …, x_{p}} be any set of coset representatives. Then x_{i}g = gx_{i} for each i, and so trf_{Q}^{P}([g]) = ∏_{i=1}^{p}[g] = [g]^{p} = 1.

If g ∉ Q, we take {1, g, g^{2}, …, g^{p−1}} as our set of coset representatives. Then g^{i}·g = 1·g^{i+1} if 0 ≤ i < p − 1, and g^{p−1}·g = 1·1. So trf_{Q}^{P}([g]) = [1] again in this case. □

More generally, by similar arguments, for any pair of finite groups H ≤ G and any g ∈ Z(G), trf_{H}^{G}([g]) = [g^{m}] where m = [G:H] (cf. [A4, (37.3)] or [G1, Theorem 7.3.3]).

The construction of the transfer homomorphism for saturated fusion systems over S depends on the existence of an (S, S)-biset satisfying certain conditions. Since this biset seems to have many applications, not just for constructing transfer homomorphisms, we describe it in the following lemma.

Lemma 8.3 ([BLO2, Proposition 5.5]). Let F be a saturated fusion system over a finite p-group S. Then there is an (S, S)-biset Ω with the following properties:

(i) Each (S, S)-orbit of Ω has the form (S_{P,φ})×_{P} S for some P ≤ S and some φ ∈ Hom_{F}(P, S). (Thus the S-action on each side is free.)

(ii) For each P ≤ S and each φ ∈ Hom_{F}(P, S), Ω×_{S}(S_{P,Id}) ≅ Ω×_{S}(S_{P,φ}) as (S, P)-bisets.

(iii) |Ω|/|S| is prime to p.

Recall that an (S, S)-biset X can be regarded as a set with left (S × S)- action by setting (s, t)·x = sxt^{−1}. In terms of this action, point (i) says that for each x ∈ Ω, the stabilizer subgroup at x of the (S × S)-action has the form {(φ(g), g) | g ∈ P } for some P ≤ S and some φ ∈ Hom_{F}(P, S). In particular, since each such subgroup has trivial intersection with 1 × S and S × 1, this implies that the left and right actions of S on Ω are both free.

Note, in (ii), that as (S, P)-bisets, Ω×_{S}(S_{P,Id}) is isomorphic to Ω with its right action restricted to P, and Ω×_{S}(S_{P,φ}) is isomorphic to Ω with the right action “twisted” by φ (g ∈ P acts by sending x ∈ Ω to x·φ(g)).

A biset satisfying conditions (i)–(iii) in Lemma 8.3 for the fusion system F will be called a characteristic biset for F. These conditions were originally formulated by Linckelmann and Webb, who conjectured the existence of such a biset. As the motivating example, when F = F_{S}(G) for a finite group G, we can take Ω = G with the left and right S-actions defined by multiplication. By a theorem of Puig [P7, Proposition 21.9] (discovered independently by Ragnarsson and Stancu [RSt, Theorem A]), a fusion system has a characteristic biset only if it is saturated.

As an example of another application of characteristic bisets, we note [5a2, Proposition 1.16]: for any saturated fusion system F over a p-group S, and any subgroup P of S, the set of S-conjugacy classes of fully normalized subgroups in P^{F} has order prime to p. Yet another example will be given in Theorem III.4.23.

We are now ready to construct a transfer for fusion systems. This will be done, for a fusion system F over a p-group S, by applying Lemma 8.1 to a certain (S, S)-biset constructed in [BLO2].

Proposition 8.4. Fix a p-group S, and a saturated fusion system F over S. Then there is an injective homomorphism

trf_{F}: S/𝔣𝔬𝔠(F) ⟶ S^{ab} =^{def} S/[S, S]

which has the following properties.

(a) There are proper subgroups P_{1}, …, P_{m} < S, and morphisms φ_{i} ∈ Hom_{F}(P_{i}, S) (i = 1, …, m), such that for g ∈ S,

trf_{F}([g]) = ∏_{[α]∈Out_{F} (S)}[α(g)] · ∏_{i=1}^{m} φ_{i}_{∗}(trf_{P_{i}}^{S}([g])).

Here, [g] denotes the class of g in S/𝔣𝔬𝔠(F) or in S^{ab} = S/[S, S], and the terms on the right are regarded as lying in S^{ab}.

(b) There is m prime to p such that for arbitrary g ∈ G, trf_{F}([g]) = [h^{m}] for some h ≡ g (mod 𝔣𝔬𝔠(F)).

(c) If g ∈ Ω_{1}(Z(S)), then trf_{F}([g]) = ∏_{[α]∈Out_{F} (S)}[α(g)].

(d) If g ∈ Ω_{1}(Z(S)) is Out_{F}(S)-invariant, then trf_{F}([g]) = [g]^{k} where k = |Out_{F}(S)| is prime to p.

Proof. Fix a characteristic biset

Ω = ∐_{i=1}^{k}(S_{P_{i},φ_{i}})×_{P_{i}}S

for F, as in Lemma 8.3. Thus P_{i} ≤ S and φ_{i} ∈ Hom_{F}(P_{i}, S) for each i. By Lemma 8.1(a–d), for each g ∈ S,

Ω_{∗}([g]) = ∏_{i=1}^{k} φ_{i}_{∗}(trf_{P_{i}}^{S}([g])) .

Also, for each P ≤ S, each φ ∈ Hom_{F}(P, S), and each g ∈ P,

Ω_{∗}([g]) = Ω_{∗} ∘ (S_{P,Id})_{∗}([g]) = Ω_{∗} ∘ (S_{P,φ})_{∗}([g]) = Ω_{∗}([φ(g)])

by condition (ii) and Lemma 8.1(a,d), and hence Ω_{∗}([g^{−1}φ(g)]) = 1. Thus 𝔣𝔬𝔠(F)/[S, S] ≤ Ker(Ω_{∗}), and Ω_{∗} factors through a homomorphism T: S/𝔣𝔬𝔠(F) ⟶ S^{ab}.

For all g ∈ G,

Ω_{∗}([g]) = ∏_{i=1}^{k} φ_{i}_{∗}(trf_{P_{i}}^{S}([g])) ≡ ∏_{i=1}^{k}(incl_{P_{i}}^{S})_{∗}(trf_{P_{i}}^{S}([g]))
(mod 𝔣𝔬𝔠(F)/[S, S])

= ∏_{i=1}^{k}[g^{[S:P_{i}]}] = [g^{|Ω|/|S|}] ,

(1) where the next-to-last equality follows from Lemma 8.1(e), and the last one since

|S_{P_{i},φ_{i}} ×_{P_{i}} S|/|S| = |S|/|P_{i}| = [S:P_{i}] for each i .  (2)

In particular, if Ω_{∗}([g]) = 1 ∈ S^{ab}, then g^{|Ω|/|S|} ∈ 𝔣𝔬𝔠(F), and g ∈ 𝔣𝔬𝔠(F) since |Ω|/|S| is prime to p by (iii). Thus Ker(Ω_{∗}) = 𝔣𝔬𝔠(F)/[S, S], and so T is injective.

When P = S and α, β ∈ Aut_{F}(S), then an isomorphism from S_{S,α} to S_{S,β} is a bijection f: S ⟶ S such that f(sx) = sf(x) and f(xα(s)) = f(x)β(s) for each s, x ∈ S. The first equality implies that f(x) = xg for some fixed g ∈ S (and all x), and the second then implies α(s)g = gβ(s) and hence α = c_{g} ∘ β. In other words, S_{S,α} ≅ S_{S,β} as (S, S)-bisets if and only if α ≡ β (mod Inn(S)).

By (ii), for each α ∈ Aut_{F}(S), there is an isomorphism of bisets f: Ω ⟶^{≅} Ω ×_{S} S_{S,α}. For each P ≤ S and φ ∈ Hom_{F}(P, S), if we set Q = α^{−1}(P), then

(S_{P,φ} ×_{P} S) ×_{S} S_{S,α} ≅ S_{Q,φα} ×_{Q} S

via the isomorphism which sends [s, t, u] to [s, α^{−1}(tu)]. In particular, condition (ii) in Lemma 8.3 implies that for each β ∈ Aut_{F}(S), terms isomorphic to S_{S,β} (≅ S_{S,β} ×_{S} S) occur in Ω with the same multiplicity as those isomorphic to S_{S,βα}. Since this holds for each α, β ∈ Aut_{F}(S), there is r such that for each ξ ∈ Out_{F}(S), there are exactly r orbits in Ω isomorphic to S_{S,α} for [α] = ξ. Thus if we let I = {1 ≤ i ≤ k | P_{i} < S}, then for all g ∈ S,

T ([g]) = ∏_{[α]∈Out_{F} (S)}[α(g)]^{r} · ∏_{i∈I} φ_{i}_{∗}(trf_{P_{i}}^{S}([g])).  (3)

For each i ∈ I, |S_{P_{i},φ_{i}} ×_{P_{i}} S|/|S| = [S:P_{i}] ≡ 0 (mod p) by (2). Hence |Ω|/|S| ≡ r·|Out_{F}(S)| (mod p), and so p∤r by (iii). Choose n > 0 such that rn ≡ 1 (mod |S|), and define trf_{F} by setting trf_{F}([g]) = T ([g])^{n}. By (3), trf_{F} has the form given in (a).

Point (b) follows from (1) (with m = n·|Ω|/|S|). If g ∈ Ω_{1}(Z(S)), then for each P_{i} < S, trf_{P_{i}}^{S}([g]) = 1 by Lemma 8.2. Hence trf_{F}([g]) = ∏_{[α]∈Out_{F} (S)}[α(g)], and this proves (c). Point (d) is a special case of (c). □

Note that the transfer trf_{F} as constructed in Proposition 8.4 depends a priori on the choice of characteristic biset for F. This choice is, in fact, not unique. This uniqueness question has been studied by Ragnarsson, who described in [Rg, Proposition 5.6] a way to choose a virtual, p-completed biset which is unique and “canonical” in a certain sense. Also, in [DGPS, Lemma 2.6], the transfer is shown to be independent (up to a scalar multiple) of the choice of characteristic biset. But none of this is needed here: we just need to have one transfer homomorphism with the right properties.

The following is one easy example of how Proposition 8.4 can be applied.

Corollary 8.5. Let F be a saturated fusion system over a p-group S, and set S_{0} = Ω_{1}(Z(S)) and S_{1} = S_{0} ∩ [S, S]. Then no element of S_{0}∖S_{1} whose coset in S_{0}/S_{1} is fixed by the action of Out_{F}(S) is in 𝔣𝔬𝔠(F). In particular, if S_{0} > S_{1} and Out(S) is a p-group, or if p = 2 and rk(S_{0}/S_{1}) = 1, then O^{p}(F) ⫋ F for every saturated fusion system F over S.

Proof. Assume g ∈ S_{0}∖S_{1} is such that [g] ∈ S_{0}/S_{1} is fixed by the action of Out_{F}(S). Set

g′ = ∏_{[α]∈Out_{F} (S)} α(g) ∈ S_{0} .

Set k = |Out_{F}(S)|. Then g′ ∈ g^{k}S_{1}, and g′ ∉ S_{1} since p∤k. Hence trf_{F}([g]) ≠ 1 by Proposition 8.4(c), and so g ∉ 𝔣𝔬𝔠(F).

The other statements now follow immediately. Recall (Corollary 7.5) that 𝔣𝔬𝔠(F) < S implies O^{p}(F) ⫋ F. □

We finish this section with some remarks on more general forms of the transfer. When H ≤ G is a pair of groups such that [G:H] < ∞, then the transfer trf_{H}^{G} is a homomorphism from the cohomology of H to that of G, or from the homology of G to that of H. In other words, it goes in the opposite direction of the usual homomorphism induced by the inclusion. A special role is played by the transfer homomorphisms trf_{S}^{G} when G is finite and S ∈ Syl_{p}(G): this is useful when describing the (co)homology of G (after localization at p) in terms of that of S (cf. [CE, Theorem XII.10.1]).

When F is a saturated fusion system over a p-group S, we define the (co)homology of F by setting, when n ≥ 0 and A is a ℤ_{(p)}-module,

H^{n}(F; A) = \underleftarrow{lim}_{F} H^{n}(−; A)  and  H_{n}(F; A) = colim_{F} H_{n}(−; A) .

For example, when A is finite, H^{∗}(F; A) is isomorphic to the cohomology of a “classifying space” for F by [BLO2, Theorem B] (when A = 𝔽_{p}) or [5a2, Lemma 6.12] (see also Theorem III.4.23). The transfer maps for F are homomorphisms H^{n}(S; A) ⟶ H^{n}(F; A) or H_{n}(F; A) ⟶ H_{n}(S; A). The transfer of Proposition 8.4 is the transfer for H_{1}(−; ℤ_{(p)}).

9. Other definitions of saturation

We now look at two other, equivalent definitions of saturation: the original definition given by Puig (who calls saturated fusion systems “Frobenius categories” in [P6, Definition 2.9]), and an alternative definition given by Stancu (see [Li3, Definition 1.4] or [KS, Definition 2.4]).

The following terminology is useful when stating Puig’s definition, and will be used only in this section.

Definition 9.1. Let F be any fusion system over S. Fix a subgroup P ≤ S and a group of automorphisms K ≤ Aut(P).

• P is totally K-normalized in F if for each φ ∈ Hom_{F}(P ·N_{S}^{K}(P), S), φ(N_{S}^{K}(P)) = N_{S}^{^{φ}K}(φ(P)).

• P is K-receptive in F if for each Q ≤ S and φ ∈ Iso_{F}(Q, P), there are homomorphisms φ ∈ Hom_{F}(Q·N_{S}^{K^{φ}}(Q), S) and χ ∈ Aut_{F}^{K}(P) such that φ|_{Q} = χ ∘ φ.

Both of these concepts are equivalent to being fully K-normalized when the fusion system F is saturated. This is why we need separate names for them only when working in fusion systems which might not be saturated; in particular, when comparing different definitions of saturation. For example, “K-receptive” is just the condition of Proposition 5.2(c).

The term “totally K-normalized” is what Puig calls “fully K-normalized” in [P6]; we give it a different name to distinguish between these two concepts. They are equivalent in a saturated fusion system, as shown in the next lemma, but need not be equivalent in general.

Lemma 9.2. Fix a fusion system F over a p-group S. Let P ≤ S be any subgroup, and let K ≤ Aut(P) be a group of automorphisms. If P is fully K-normalized, then it is totally K-normalized. If F is saturated, then P is fully K-normalized if and only if it is totally K-normalized.

Proof. If P is fully K-normalized, then |N_{S}^{K}(P)| ≥ |N_{S}^{^{φ}K}(φ(P))| for each φ ∈ Hom_{F}(P ·N_{S}^{K}(P), S), and hence φ(N_{S}^{K}(P)) = N_{S}^{^{φ}K}(φ(P)). Thus P is totally K-normalized.

Now assume F is saturated and P is totally K-normalized. Choose Q which is F-conjugate to P, and φ ∈ Iso_{F}(P, Q), such that Q is fully φK-normalized. Set L = ^{φ}K for short. By Proposition 5.2, there are χ ∈ Aut_{F}^{L}(Q) and φ ∈ Aut_{F}(P ·N_{S}^{K}(P), S) such that φ|_{P} = χ ∘ φ. In particular, φ(P) = Q and ^{χφ}K = L. Since P is totally K-normalized, φ(N_{S}^{K}(P)) = N_{S}^{L}(Q), so the two have the same order, and P is fully Knormalized since Q is fully L-normalized. □

We are now ready to show the equivalence of our definitions of saturation (Definition 2.2 and Proposition 2.5) with those of Puig and of Stancu. The equivalence with Puig’s definition is sketched in the appendix of [BLO2], and the equivalence with Stancu’s is proven in [Li3, Propositions 1.5–1.7].

Proposition 9.3. The following three conditions are equivalent for a fusion system F over a p-group S.

(a) F is saturated in the sense of Definition 2.2.

(b) (Puig’s definition) Inn(S) ∈ Syl_{p}(Aut_{F}(S)); and for each P ≤ S and each K ≤ Aut(P), if P is totally K-normalized in F, then P is Kreceptive in F.

(c) (Stancu’s definition) Inn(S) ∈ Syl_{p}(Aut_{F}(S)); and for each P ≤ S, if P is fully normalized in F, then P is receptive in F.

Proof. (a ⇒ b) Assume F is saturated. In particular, we have Inn(S) ∈ Syl_{p}(Aut_{F}(S)). If P ≤ S and K ≤ Aut(P) are such that P is totally Knormalized in F, then P is fully K-normalized by Lemma 9.2, and hence is K-receptive by Proposition 5.2.

(b ⇒ c) Assume F satisfies the conditions in (b). In particular, Inn(S) ∈ Syl_{p}(Aut_{F}(S)).

Assume P ≤ S is fully normalized in F. Then P is totally Aut(P)- normalized by Lemma 9.2, and so P is Aut(P)-receptive. Thus for each Q F-conjugate to P, there is φ ∈ Hom_{F}(N_{S}(Q), N_{S}(P)) such that φ(Q) = P, and in particular, φ(C_{S}(Q)) ≤ C_{S}(P). Hence P is fully centralized in F, and is totally {1}-normalized by Lemma 9.2 again.

Fix Q ≤ S and φ ∈ Iso_{F}(Q, P). Set

N_{φ} = {g ∈ N_{S}(Q) |^{φ}c_{g} ≤ Aut_{S}(P)}  and  K = Aut_{N_{φ}}(Q) ,

and set L = ^{φ}K ≤ Aut_{S}(P). For each morphism ψ ∈ Hom_{F}(P ·N_{S}^{L}(P), S), ψ(C_{S}(P)) = C_{S}(ψ(P)) since P is totally {1}-normalized. Hence ψ(N_{S}^{L}(P)) = N_{S}^{^{ψ}L}(ψ(P)) since

|N_{S}^{L}(P)/C_{S}(P)| = |L| ≥ |N_{S}^{^{ψ}L}(ψ(P))/C_{S}(ψ(P))| .

Thus P is totally L-normalized, and hence is L-receptive by (b). There are thus χ ∈ L and φ ∈ Hom_{F}(Q·N_{S}^{K}(Q), S) such that φ|_{Q} = χ ∘ φ. Also, χ^{φ} = c_{g} for some g ∈ N_{φ}, and so φ extends to φ ∘ c_{g}^{−1} ∈ Hom_{F}(N_{φ}, S).

(c ⇒ a) Assume Inn(S) ∈ Syl_{p}(Aut_{F}(S)), and each fully normalized subgroup of S is receptive. Since each F-conjugacy class contains a fully normalized subgroup, we need only show that each fully normalized subgroup is fully automized. Assume otherwise: let P be a fully normalized subgroup which is not fully automized, chosen such that |P | is maximal among such subgroups. Since S is fully automized, P < S.

Fix a subgroup T ∈ Syl_{p}(Aut_{F}(P)) such that Aut_{S}(P) < T. Choose α ∈ N_{T} (Aut_{S}(P)) not in Aut_{S}(P) (see Lemma A.1). Since P is receptive and α normalizes Aut_{S}(P), α extends to an automorphism α ∈ Aut_{F}(N_{S}(P)).

Upon replacing α by α^{r} for some appropriate r ≡ 1 (mod |α|), we can assume α has p-power order. Choose R ≤ S which is F-conjugate to N_{S}(P) and fully normalized in F, and fix φ ∈ Iso_{F}(N_{S}(P), R). Since |R| = |N_{S}(P)| > |P | (Lemma A.1 again), R is fully automized by assumption. Hence there is χ ∈ Aut_{F}(R) such that ^{χφ}α ∈ Aut_{S}(R) ∈ Syl_{p}(Aut_{F}(R)). Let g ∈ N_{S}(R) be such that c_{g} = ^{χφ}α. In particular, g ∉ R.

Set Q = χφ(P) ⊴ R. Then g ∈ N_{S}(Q) since α(P) = P, and so ⟨R, g⟩ ≤ N_{S}(Q). But then |N_{S}(Q)| > |R| = |N_{S}(P)|, which contradicts the assumption P was fully normalized. Thus each fully normalized subgroup of S is fully automized. □

Part II. The local theory of fusion systems

Michael Aschbacher

Part II of the book is intended to be an introduction to the local theory of saturated fusion systems. By the “local theory of fusion systems” we mean an extension of some part of the local theory of finite groups to the setting of saturated fusion systems on finite p-groups.

Recall the notions of a p-local finite group and its associated linking system defined in Definitions III.4.1 and III.4.4. One can ask: Why deal with saturated fusion systems rather than p-local finite groups? There are two reasons for this choice. First, it is not known whether to each saturated fusion system there is associated a unique p-local finite group. Thus it remains possible that the class of saturated fusion systems is larger than the class of p-local finite groups. But more important, to date there is no accepted notion of a morphism of p-local finite groups, and hence no category of p-local groups. (The problem arises already for fusion systems and linking systems of groups, since a group homomorphism α from G to H need not send p-centric subgroups of G to p-centric subgroups of H, so it is not clear how to associate to α a map which would serve as a morphism of the p-local group of G with the p-local group of H.) The local theory of finite groups is inextricably tied to the notion of group homomorphism and factor group, so to extend the local theory of finite groups to a different category, we must at the least be dealing with an actual category.

Part I of the book records most of the basic definitions, notation, and notions from the theory of saturated fusion systems necessary for Part II. For emphasis, we repeat some of those results from Part I (which we use most often) here in Part II. For example in Theorem 4.2 in Section 4 we again record the deeper result of [5a1] that if F is saturated and constrained on S, then the set G(F) of models of F is nonempty. Here G ∈ G(F) if G is a finite group with S ∈ Syl_{p}(G), C_{G}(O_{p}(G)) ≤ O_{p}(G), and F = F_{S}(G). This fact is the basis for much of the local theory of fusion systems, and allows us to translate suitable statements from the local theory of groups into the setting of fusion systems.

Exercise 2.4 shows that if α: F → F̃ is a morphism of fusion systems, then the kernel ker(α) of the group homomorphism α: S → S̃ is strongly closed in S with respect to F. In Section 5, we see how to construct a factor system F/T of F over a strongly closed subgroup T of F. Moreover when F is saturated, we find that there is a surjective homomorphism Θ: F → F/T, and a bijection T ↦ F/T between strongly closed subgroups T of S and the set of isomorphism classes of homomorphic images of F.

In [P6], Puig defines the same factor systems and proves that Θ is a surjective morphism of fusion systems, although via a different approach.

Proceeding by analogy with the situation for groups, we would like to show there exists a “normal subsystem” E of F on T which is saturated, and hence realize F as an “extension” of E by F/E = F/T. Unfortunately such a subsystem need not exist, but it is still possible to develop a theory of normal subsystems of saturated fusion systems which is fairly satisfactory. Section I.6 in Part I contains a discussion of various possible definitions of the notion of a “normal subsystem”; our normal subsystems are defined in Definition I.6.1 in Part I, and that definition is repeated in Definition 7.1 of this Part. A theory of normal subsystems is described in Sections 6 and 7, where a few examples are also introduced to indicate some of the places where the theory diverges from the corresponding theory for groups.

We need effective conditions to verify when a subsystem of F on T is normal, and to produce normal subsystems. Moreover in most situations, these conditions should be local; that is we should be able to check them in local subsystems, and indeed even in constrained local subsystems. In Section 8 we record some such conditions from [A5]. Then in Section 9 we record some of the theorems about normal subsystems from [A6] which can be proved using the conditions. In particular we define the generalized Fitting subsystem F^{∗}(F) of F, and recall the (obvious) notion of a simple system, which appears already in Definition I.6.1. Of course F is simple if it has no nontrivial normal subsystems.

In Section 10 we define the notion of a composition series for saturated fusion systems, and prove a Jordan-Hölder Theorem for such systems. For example this makes possible the definition of a solvable system F: All composition factors are of order p, or more precisely of the form F_{R}(R) for R a group of order p. There is a second natural definition of solvability for fusion systems, due to Puig. We compare the two definitions in Section 12.

Section 13 investigates the composition factors of systems F_{S}(G), where G is a finite simple group and S ∈ Syl_{p}(G). Often such systems are simple, but not always. Occasionally F_{S}(G) may even have an exotic composition factor which is not obtainable from a finite group.

The last two sections contain some speculation about the possibility of, first, classifying simple saturated fusion systems (or some suitable subclass of such systems), particularly at the prime 2, and then, second, using results on fusion systems to simplify the proof of the theorem classifying the finite simple groups. We also include a few existing results of that sort, together with some open problems which may be of interest.

Finally we note that in Part II we often abbreviate F-centric by centric.

1. Notation and terminology on groups

Our convention in Part II will be to write many functions (particularly functions which may be composed, like group homomorphisms) on the right.

We adopt the notation and terminology in [A4] when discussing groups; some of this can also be found at the end of the Introduction to this volume. For example let G be a group and x, y ∈ G. Then x^{y} = y^{−1}xy is the conjugate of x under y, and

c_{y}: G → G
x ↦ x^{y}

is conjugation by y. This notation differs (because we are applying maps on the right) from that in the introduction, where c_{y}(x) = ^{y}x. Of course c_{y} ∈ Aut(G) is an automorphism of G. For X ⊆ G, write X^{y} for the conjugate Xc_{y} of X under y, and set X^{G} = {X^{y}: y ∈ G}, the conjugacy class of X in G.

Let K ≤ H ≤ G. We say K is strongly closed in H with respect to G if for all k ∈ K, k^{G} ∩ H ⊆ K.

See Definition A.9 for the definition of the commutator notation, [x, y], [X, Y], for x, y ∈ G, X, Y ≤ G, and see Section 8 of [A4] for more discussion of these notions.

See Definition A.11 for the definition of quasisimple groups and the generalized Fitting subgroup F^{∗}(G) of a finite group G, and see Section 31 in [A4] for further discussion of these notions.

Our notation for the finite simple groups is defined in section 47 of [A4], and there is a much deeper discussion of the simple groups in [GLS3].

Notation. Suppose C is a category and α: A → B is an isomorphism in C. Let Aut_{C}(A) = Mor_{C}(A, A) be the group of automorphisms of A in C. Observe that for C a subobject of A, we have the bijection Mor_{C}(C, A)) → Mor_{C}(Cα, B) defined by β ↦ α^{−1}βα = β^{α}, which, in the case C = A, induces an isomorphism of Aut_{C}(A) with Aut_{C}(B).

2. Fusion systems

Part I contains the basic notation, terminology, and concepts for fusion systems. In this short section we record a few more facts and observations.

In this section, p is a prime and F is a fusion system on a finite p-group S. Write F^{f} for the set of nontrivial fully normalized subgroups of S.

Let P ≤ S. Recall from Definition I.5.3 that C_{F}(P) is the category whose objects are the subgroups of C_{S}(P), and for U, V ≤ C_{S}(P), Hom_{C_{F} (P)}(U, V) consists of those ϕ ∈ Hom_{F}(U, V) which extend to ϕ̂ ∈ Hom_{F}(P U, P V) with ϕ̂ = 1 on P. The system C_{F}(P) is the centralizer in F of P, and is a fusion system.

Similarly N_{F}(P) is the category on N_{S}(P) whose objects are the subgroups of N_{S}(P), and for U, V ≤ N_{S}(P), Hom_{N_{F} (P)}(U, V) consists of those ϕ ∈ Hom_{F}(U, V) which extend to ϕ̂ ∈ Hom_{F}(P U, P V) with ϕ̂ acting on P.

As a consequence of Theorem I.5.5:

Theorem 2.1. If F is saturated and P is fully normalized, then C_{F}(P) and N_{F}(P) are saturated fusion systems.

Thus the systems N_{F}(P), for P ∈ F^{f}, play the role of the local subsystems of F, analogous to the local subgroups in finite group theory. For example if F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G), then N_{F}(P) = F_{N_{S}(P)}(N_{G}(P)). These local subsystems are the focus of interest in the local theory of fusion systems.

Recall from Definition I.3.1 that P ≤ S is F-centric (or just centric) if C_{S}(Q) ≤ Q for each Q ∈ P^{F}. We write F^{c} for the set of centric subgroups of S.

Definition 2.2. A morphism α: F → F̃ of fusion systems from F to a system F̃ on S̃, is a family (α, α_{P,Q}:P, Q ∈ F) such that α: S → S̃ is a group homomorphism, and α_{P,Q}: Hom_{F}(P, Q) → HomF̃(P α, Qα) is a function, such that for all P, Q, ϕα = α(ϕα_{P,Q}). Notice if the family is a morphism, then the maps α_{P,Q} are uniquely determined by the group homomorphism α and the last property.

The kernel ker(α) of the morphism α is the kernel of the group homomorphism α: S → S̃. Thus ker(α) is a normal subgroup of S.

The morphism α is surjective if α: S → S̃ is surjective, and for all P, Q ≤ S,

α_{P_{0},Q_{0}} : Hom_{F}(P_{0}, Q_{0}) → HomF̃(P α, Qα)

is surjective, where for X ≤ S, X_{0} is the preimage in S of Xα under α.

Exercises for Section 2

2.1. Assume G is a finite group, p is a prime, and S ∈ Syl_{p}(G). Set Ḡ = G/O_{p′}(G). Prove F_{S}(G) ≅ FS̄(Ḡ).

2.2. Assume G is a finite group, p is a prime, and S ∈ Syl_{p}(G) is abelian. Let H = N_{G}(S) and prove F_{S}(G) ≅ FS̄(H).

2.3. Assume G is a finite group, p is a prime, and S ∈ Syl_{p}(G). Prove

(a) Let T ≤ S. Then T is strongly closed in S with respect to F_{S}(G) iff T is strongly closed in S with respect to G.

(b) Let H ⊴ G. Then H ∩ S is strongly closed in S with respect to F_{S}(G).

2.4. Let α: F → F̃ be a morphism of fusion systems. Prove ker(α) is strongly closed in S with respect to F.

3. Saturated fusion systems

In this section we assume F is a saturated fusion system on a finite p-group S.

Lemma 3.1. Let P ≤ S and Q ∈ P^{F} with Q fully normalized. Then there exists φ ∈ Hom_{F}(N_{S}(P), N_{S}(Q)) with P φ = Q.

Proof. This is Lemma I.2.6.c. □

Recall from Definition I.4.1 that a subgroup R of S is normal in F if F = N_{F}(R), and we write R ⊴ F to indicate that R is normal in F. Further F is constrained if there exists a normal centric subgroup of F.

Example 3.2. Let G be a finite group, S ∈ Syl_{p}(G), and F = F_{S}(G). If R is a normal p-subgroup of G, then by Exercise 3.1, R ⊴ F. Further if C_{G}(O_{p}(G)) ≤ O_{p}(G), then by Exercise 3.1, F is constrained.

Lemma 3.3. Let R ≤ S. Then the following are equivalent:

(a) R ⊴ F.

(b) R is strongly closed in S with respect to F and contained in all radical centric subgroups of F

(c) There exists a series 1 = R_{0} ≤ R_{1} ≤ ⋯ ≤ R_{n} = R such that

(i) for each 1 ≤ i ≤ n, R_{i} is strongly closed in S with respect to F, and

(ii) for each 1 ≤ i < n, [R, R_{i+1}] ≤ R_{i}.

Proof. This is a restatement of Propositions I.4.5 and I.4.6. □

By Exercise 3.2, there is a largest subgroup of S normal in F. Recall from Definition I.4.3 that we write O_{p}(F) for that subgroup.

Lemma 3.4. (a) An abelian subgroup R of S is normal in F iff R is strongly closed in S with respect to F.

(b) O_{p}(F) ≠ 1 iff there is a nontrivial abelian subgroup of S strongly closed in S with respect to F.

Proof. See I.4.7. □

Theorem 3.5. (Alperin’s Fusion Theorem) F = ⟨Aut_{F}(R) : R ∈ F^{frc}⟩, where F^{frc} is the set of fully normalized radical centric subgroups of F.

Proof. This special case of I.3.5 is the version of Alperin’s theorem which we will use most often. □

Exercises for Section 3

3.1. Let G be a finite group, S ∈ Syl_{p}(G), and F = F_{S}(G). Prove

(a) If R is a normal p-subgroup of G, then R ⊴ F.

(b) If C_{G}(O_{p}(G)) ≤ O_{p}(G), then F is constrained.

3.2. Assume F is a saturated fusion system on the finite p-group S. Prove:

(a) If R, Q ⊴ F then RQ ⊴ F.

(b) There is a largest subgroup of S normal in F.

3.3. Prove R ⊴ F iff for each P ≤ R and each ϕ ∈ hom_{F}(P, S), ϕ extends to a member of Aut_{F}(R).

4. Models for constrained saturated fusion systems

In this section we assume F is a saturated fusion system on a finite p-group S.

Definition 4.1. Write G(F) for the class of finite groups G such that S ∈ Syl_{p}(G), C_{G}(O_{p}(G)) ≤ O_{p}(G), and F = F_{S}(G). Call the members of G(F) the models of F.

By Exercise 3.1.b, if G(F) ≠ ∅ then F is constrained. By Theorem I.4.9 (see also Theorem III.5.10), the converse is true, while Lemma 4.4 says all models for a constrained system are isomorphic in a strong sense.

Theorem 4.2. If F is constrained then G(F) ≠ ∅.

Proof. This is part (a) of Theorem I.4.9, and is proven as Theorem III.5.10. □

Lemma 4.3. Assume F is constrained, F̃ is a fusion system over S̃, and α: F → F̃ is an isomorphism of fusion systems. Let G ∈ G(F) and G̃ ∈ G(F̃). Then (a) The set E(α) of isomorphisms α̌: G → G̃ extending α: S → S̃ is

nonempty.

(b) Let α̌ ∈ E(α). Then E(α) = {c_{z}α̌: z ∈ Z(S)}, where c_{z} ∈ Aut(G) is the conjugation map.

Proof. Part (a) is essentially I.4.9.b. Namely by transfer of structure, there is a group H with S̃ ∈ Syl_{p}(H) and F̃ = FS̃(H), and an isomorphism β: G → H extending α. By I.4.9.b, there exists an isomorphism γ: H → G̃ extending the identity map ι on S̃. Then α̌ = βγ: G → G̃ is an isomorphism extending α = αι.

Similarly part (b) follows from III.5.10.c. Namely if ζ ∈ E(α) then ρ = α̌ζ^{−1} ∈ Aut(G) is the identity on S, so by III.5.10.c, ρ = c_{z} for some z ∈ Z(S), and hence ζ = c_{z}α̌. □

Lemma 4.4. Assume F is constrained and let G_{1}, G_{2} ∈ G(F). Then there exists an isomorphism φ: G_{1} → G_{2} which is the identity on S.

Proof. Apply 4.3.1 with F̃ = F, α the identity map on F, G = G_{1}, and G̃ = G_{2}. Or appeal to I.4.9.b. □

Example 4.5. Let U ∈ F^{f} with C_{S}(U) ≤ U, and set D = N_{F}(U). By 2.1, D is a saturated fusion system on D = N_{S}(U). By Exercise 4.1, U ∈ F^{fc}. As D = N_{F}(U), U ⊴ D. Then as C_{D}(U) = C_{S}(U) ≤ U, D is constrained, so by 4.2 there exists G = G_{F}(U) ∈ G(D). Thus D ≅ F_{D}(G). By 4.4, G is unique up to isomorphism.

Example 4.6. Assume T is strongly closed in S with respect to F and let U ≤ T with U ∈ F^{f} and C_{T} (U) ≤ U. Set V = UC_{S}(U) and D = N_{F}(V). By Exercise 4.2, V ∈ F^{f} with C_{S}(V) ≤ V, D ≤ N_{F}(U), and D = N_{S}(V) ≤ N_{S}(U). Hence applying Example 4.5 to V in the role of U, we conclude that D is a saturated constrained fusion system on D, so there exist G = G_{F,T} (U) ∈ G(D), and G is unique up to isomorphism.

Exercises for Section 4

4.1. Assume U ≤ S be fully centralized with C_{S}(U) ≤ U. Prove U ∈ F^{c}.

4.2. Assume T is strongly closed in S with respect to F. Assume U ≤ T with U ∈ F^{f} and C_{T} (U) ≤ U. Set V = UC_{S}(U), and prove:

(a) U = T ∩ V ⊴ N_{F}(V).

(b) V ∈ F^{f}.

(c) C_{S}(V) ≤ V.

5. Factor systems and surjective morphisms

In this section F is a fusion system over the finite p-group S.

Definition 5.1. Assume S_{0} ≤ S is strongly closed in S with respect to F and set N = N_{F}(T).

Set S^{+} = S/S_{0} and let θ: S → S^{+} be the natural map θ: x ↦ x^{+} = xS_{0}. We define a category F^{+} and a morphism θ: N → F^{+}. The objects of F^{+} are the subgroups of S^{+}. For P ≤ S and α ∈ Hom_{N} (P, S) define α^{+} : P^{+} → S^{+} by x^{+}α^{+} = (xα)^{+}. This is well defined as S_{0} is strongly closed in S with respect to F. Now define

Hom_{F^{+}}(P^{+}, S^{+}) = {β^{+} : β ∈ Hom_{N} (P S_{0}, S)},

and define θ_{P} : Hom_{N} (P, S) → Hom_{F^{+}}(P^{+}, S^{+}) by αθ_{P} = α^{+}. For α ∈ Hom_{N} (P, S), α extends to α̂ ∈ Hom_{N} (P S_{0}, S) and α̂^{+} = α^{+}, so θ_{P} is well defined and surjective.

Lemma 5.2. (a) F^{+} is a fusion system on the finite p-group S^{+}.

(b) θ: N → F^{+} is a surjective morphism of fusion systems.

Proof. Observe x^{+} ∈ ker(α^{+}) iff 1 = x^{+}α^{+} = (xα)^{+} iff xα ∈ T iff x ∈ T iff x^{+} = 1. So the members of Hom_{F^{+}}(P^{+}, Q^{+}) are monomorphisms.

Suppose α ∈ Hom_{N} (P, Q) and β ∈ Hom_{N} (Q, S). Then for x ∈ P,

(x^{+}α^{+})β^{+} = (xα)^{+}β^{+} = ((xα)β)^{+} = (x(αβ))^{+} = x^{+}(αβ)^{+},

so

α^{+}β^{+} = (αβ)^{+}  (!)

By (!), F^{+} is a category.

For s ∈ S, c_{s^{+}} : S^{+} → S^{+} is the map c_{s^{+}}, so Hom_{S^{+}}(P^{+}, Q^{+}) ⊆ Hom_{F^{+}}(P^{+}, Q^{+}).

If ϕ ∈ Hom_{F^{+}}(P^{+}, Q^{+}) then ϕ = α^{+} for some α ∈ Hom_{F}(P S_{0}, QS_{0}). As F is saturated, α: P S_{0} → (P S_{0})α = P αS_{0} is in Hom_{F}(P S_{0}, P αS_{0}), so ϕ: P^{+} → P^{+}ϕ is in Hom_{F^{+}}(P^{+}, P^{+}ϕ), since P^{+}ϕ = (P αS_{0})^{+}. Suppose ϕ is an isomorphism. Then again as F is saturated, α^{−1} ∈ Hom_{F}(QS_{0}, P S_{0}), so ϕ^{−1} = (α^{−1})^{+} ∈ Hom_{F^{+}}(Q^{+}, P^{+}), completing the proof of (1).

Let ϕ ∈ Hom_{F}(P, Q) and x ∈ P. Then

xϕθ = (xϕ)^{+} = x^{+}ϕ^{+} = xθ(ϕθ_{P,Q}),

so from 2.2, θ: F → F^{+} is a morphism of fusion systems. We observed in 5.1 that θ is surjective. □

Definition 5.3. Suppose S_{0} is strongly closed in S with respect to F. Then appealing to 5.2.a, we can form the fusion system F^{+} as in 5.1. We write F/S_{0} for this fusion system, and call it the factor system of F modulo S_{0}.

By 5.2.b, θ: N → F/S_{0} is a surjective morphism of fusion systems, which we denote by θ_{F/S_{0}}.

Lemma 5.4. Assume α: F → F̃ is a surjective morphism of fusion systems and F is saturated. Then F̃ is saturated.

Proof. This was first proved by Puig; cf. Proposition 12.3 in [P7]. See also Lemma 8.5 in [A5], where another proof appears. Here is a sketch of a different proof, based on the Roberts-Shpectorov definition of saturation.

Let F̃ be a system on S̃ and let ker(α) ≤ P ≤ S be fully automized and receptive. It suffices to show that P̃ = P α is fully automized and receptive. Recall from 2.2 that α induces a surjective homomorphism α_{P} : Aut_{F}(P) → AutF̃(P̃) defined by (xα)(ϕα_{P}) = xϕα for ϕ ∈ Aut_{F}(P) and x ∈ P. Check that Aut_{S}(P)α_{P} = AutS̃(P̃). But as Aut_{S}(P) is Sylow in Aut_{F}(P), its image is Sylow in the image of Aut_{F}(P), so P̃ is fully automized.

Let φ̃ ∈ IsoF̃(Q̃, P̃). Then there is a subgroup Q of S containing ker(α) and φ ∈ Iso_{F}(Q, P) with Qα = Q̃ and φα_{Q,P} = φ̃. Check that for g ∈ N_{S}(Q), c_{g}^{φ}α_{P} = cφ̃ ; gα. Then as Aut_{S}(P)α_{P} = AutS̃(P̃), if g ∈ N_{φ} then

cφ̃ ; gα = C_{g}^{φ}α_{P} ∈ Aut_{S}(P)α_{P} = AutS̃(P̃), so that gα ∈ Nφ̃. Hence N_{φ}α ≤

Nφ̃.

Next let N be the preimage in S of Nφ̃ under α. Then

Aut_{N}(Q)^{φ}α_{P} ≤ Aut_{Nφ̃}(Q̃)φ̃ ≤ AutS̃(P̃) = Aut_{S}(P)α_{P} ,

and as P is fully automized, Aut_{S}(P) is Sylow in Aut_{S}(P) ker(α_{P}), so there is χ ∈ ker(α_{P}) with Aut_{N}(Q)^{φχ} ≤ Aut_{S}(P). Then (φχ)α_{Q,P} = φ̃, so replacing φ by φχ and appealing to the previous paragraph, Nφ̃ = N_{φ}α.

As P is receptive, φ lifts to φ̄ ∈ Hom_{F}(N_{φ}, S). Then φ̄α_{N_{φ},S} is a lift of φ̃ to HomF̃(Nφ̃, S̃), so P̃ is receptive. □

Lemma 5.5. Assume F is saturated and S_{0} is strongly closed in S with respect to F. Then F/S_{0} is saturated.

Proof. As S_{0} is strongly closed in S, S_{0} ⊴ S, so S_{0} ∈ F^{f}. Therefore N is saturated by 2.1. Therefore F/S_{0} is saturated by 5.2.b and 5.4. □

Example 5.6. Assume F = F_{S}(G), H ⊴ G, and set S_{0} = S ∩ H. Let M = N_{G}(S_{0}) and M^{∗} = M/N_{H}(S_{0}). By Exercise 2.3.2, S_{0} is strongly closed in S. In 8.8 in [A5] is is shown that:

(a) F/S_{0} ≅ F_{S^{∗}}(M^{∗}).

(b) F_{S^{∗}}(M^{∗}) ≅ F_{SH/H}(G/H).

Thus F/S_{0} ≅ F_{SH/H}(G/H).

In the remainder of the section, assume F is saturated and S_{0} is strongly closed in S with respect to F.

Definition 5.7. For P, Q ≤ S, write P_{0} for P ∩ S_{0} and define

Φ(P, Q) = {ϕ ∈ Hom_{F}(P, Q): [P, ϕ] ≤ S_{0}},

where for x ∈ P, [x, ϕ] = x^{−1}·xϕ ∈ S, and [P, ϕ] = ⟨[x, ϕ]: x ∈ P ⟩ ≤ S.

For α ∈ Hom_{F}(P, S) define 𝔉(α) to be the set of pairs (φ, ϕ) such that φ ∈ Hom_{F}(P S_{0}, S), ϕ ∈ Φ(P φ, S), and α = φϕ.

Form N = N_{F}(S_{0}) and the factor system F^{+} = F/S_{0} on S^{+} = S/S_{0} and θ: N → F^{+} as in 5.1.

Lemma 5.8. Let α ∈ Hom_{F}(P, S), β ∈ Hom_{F}(P α, S), (φ, ϕ) ∈ 𝔉(α), and (Ψ, ψ) ∈ 𝔉(β). Then

(a) If Q, R ≤ S, μ ∈ Φ(P, Q), and η ∈ Φ(Q, R), then μη ∈ Φ(P, R).

(b) ϕ^{Ψ} ∈ Φ(P φΨ, S).

(c) (φΨ, ϕ^{Ψ}ψ) ∈ 𝔉(αβ).

Proof. Exercise 5.1. □

Theorem 5.9. For each P ≤ S and α ∈ hom_{F}(P, S), 𝔉(α) ≠ ∅.

Proof. This is [A6, 12.5]. Here is the idea of the proof. Choose a counter example with m = |S:P | minimal. Observe P_{0} ≠ S_{0} as in that case (α, 1) ∈ 𝔉(α); in particular m > 1. By minimality of m, α does not extend to a proper overgroup of P in S. Then, using Alperin’s Fusion Theorem 3.5 and 5.8, we reduce to the case where P ∈ F^{frc} and α ∈ Aut_{F}(P). Hence by 4.2 there is a model G for N_{F}(P) and α = c_{g|P} for some g ∈ G.

Set Q = N_{S}(P); as P_{0} ≠ S_{0}, P_{0} < Q_{0}. Set K = ⟨Q_{0}^{G}⟩. As Q_{0} centralizes P/P_{0}, so does K. Therefore for each k ∈ K, c_{k|P} ∈ Φ(P, P), so (1, c_{k|P}) ∈ 𝔉(c_{k|P}). But by a Frattini argument, G = KN_{G}(Q_{0}), so g = kh for some h ∈ N_{G}(Q_{0}) and k ∈ K. As P_{0} < Q_{0}, P < P Q_{0}, so 𝔉(c_{h|P}) ≠ ∅ by minimality of m. Since also 𝔉(c_{k|P}) ≠ ∅, 5.8 completes the proof. □

Lemma 5.10. Let P ≤ S, α ∈ Hom_{F}(P, S), and (φ, ϕ) ∈ 𝔉(α). Then

(a) φ ∈ Hom_{N} (P, S) and φ^{+} ∈ Hom_{F^{+}}(P^{+}, S^{+}).

(b) For x ∈ P, (xα)^{+} = x^{+}φ^{+}.

Proof. As (φ, ϕ) ∈ 𝔉(α), φ ∈ N, so by definition of the +-notation, x^{+}φ^{+} = (xφ)^{+}. That is (a) holds. Further

(xα)^{+} = (xφϕ)^{+} = (xφ·[xφ, ϕ])^{+} = (xφ)^{+}[xφ, ϕ]^{+} = (xφ)^{+},

as ϕ ∈ Φ(P φ, S), so [xφ, ϕ] ∈ S_{0}, the kernel of θ: S → S^{+}, where θ: x ↦ x^{+} = xS_{0}. Thus (b) holds. □

Definition 5.11. For P ≤ S and α ∈ Hom_{F}(P, S), define αΘ ∈ Hom_{F^{+}}(P^{+}, S^{+}) by αΘ = φ^{+} and (φ, ϕ) ∈ 𝔉(α). Observe that Θ is well defined: Namely by 5.10.a, φ^{+} ∈ Hom_{F^{+}}(P^{+}, S^{+}). Further if (Ψ, ψ) ∈ 𝔉(α) and x ∈ P, then by 5.10.b, x^{+}φ^{+} = (xα)^{+} = x^{+}Ψ^{+}, so the definition of αΘ is independent of the choice of (φ, ϕ) in 𝔉(α).

Next define Θ: S → S^{+} to be the natural map Θ: s ↦ s^{+}.

Write Θ_{F,S_{0}} for this map from F to F/S_{0}.

Theorem 5.12. (a) Θ = Θ_{F,S_{0}} : F → F/S_{0} is a surjective morphism of fusion systems.

(b) θ is the restriction of Θ to N = N_{F}(S_{0}).

Proof. Let P ≤ S. For γ ∈ Hom_{N} (P, S), (γ, 1) ∈ 𝔉(γ), so γΘ = γ^{+} = γθ. Then as θ = Θ as a map of groups on S, (b) holds.

Let α ∈ Hom_{F}(P, S) and β ∈ Hom_{F}(P α, S). By 5.8.c, (αβ)Θ = αΘ·βΘ. Let (φ, ϕ) ∈ 𝔉(α) and x ∈ P. Then by 5.10.b,

(xα)Θ = (xα)^{+} = x^{+}φ^{+} = (xΘ)(αΘ),

so Θ is a morphism of fusion systems. By definition, Θ: S → S^{+} is surjective. By (b), Θ extends θ, so as θ is surjective, so is Θ. □

Lemma 5.13. Assume ρ: F → F̃ is a surjective morphism of fusion systems, with S_{0} = ker(ρ). Then

(a) For P, Q ≤ S and ϕ ∈ Φ(P, Q), ϕρ = 1.

(b) For P ≤ S, α ∈ Hom_{F}(P, S), and (φ, ϕ) ∈ 𝔉(α), αρ = φρ.

(c) Define π: F^{+} → F̃ by x^{+}π = xρ for x ∈ S, and φ^{+}π = φρ, for φ an N -map. Then π: F^{+} → F̃ is an isomorphism of fusion systems such that Θπ = ρ.

Proof. First assume the setup of (a), and let x ∈ P. Then

(xρ)(ϕρ) = (xϕ)ρ = (x·[x, ϕ])ρ = xρ·[x, ϕ]ρ = xρ,

as [x, ϕ] ∈ S_{0} and S_{0}ρ = 1. Thus (a) holds.

Next assume the setup of (b). Then by (1), αρ = (φϕ)ρ = (φρ)(ϕρ) = φρ, establishing (b).

As S_{0} = ker(ρ), π: S^{+} → S̃ is a well defined group isomorphism, with Θπ = ρ as a map of groups.

Let P ≤ S and η, μ ∈ Hom_{N} (P, S). Then ηρ = μρ iff for all x ∈ P, xηρ = (xρ)(ηρ) = (xρ)(μρ) = (xμ)ρ iff xη ∈ xμS_{0}. Thus if η^{+} = μ^{+} then as S_{0} = ker(ρ), ηρ = μρ, so π: Hom_{N} (P^{+}, S^{+}) → HomF̃(P ρ, S̃) = HomF̃(P^{+}π, S̃) is well defined. Further η^{+} = μ^{+} iff ηρ = μρ, so π is injective.

For x ∈ P,

(x^{+}π)(η^{+}π) = (xρ)(ηρ) = (xη)ρ = (xη)^{+}π = (x^{+}η^{+})π.

For ξ ∈ hom_{N} (P η, S),

(η^{+}ξ^{+})π = (ηξ)^{+}π = (ηξ)ρ = ηρ·ξρ = η^{+}π·ξ^{+}π.

Thus π: F^{+} → F̃ is a morphism of fusion systems. Further by (b),

αΘπ = φ^{+}π = φρ = αρ.

Then as ρ is a surjection, so is π. We saw π: S^{+} → S̃ is an isomorphism, as is π: Hom_{N} (P^{+}S^{+}) → HomF̃(P^{+}π, S̃), so (c) follows. □

Theorem 5.14. The map S_{0} ↦ F/S_{0} is a bijection between the set of subgroups S_{0} of S, strongly closed in S with respect to F, and the set of isomorphism classes of homomorphic images of F.

Proof. This is a consequence of 5.12.a and 5.13.c. □

Exercises for Section 5

5.1. Prove 5.8.

6. Invariant subsystems of fusion systems

In this section F is a saturated fusion system over the finite p-group S. We saw in the previous section that there is a 1-1 correspondence between the homomorphic images of F and strongly closed subgroups T of S, and we may take the factor system F/T to be the image corresponding to T. Proceeding by analogy with the situation for groups, we would like to show there exists a “normal subsystem” E of F on T which is saturated, and hence realize F as an “extension” of E by F/T. Unfortunately such a subsystem may not exist, but in many instances one does. We begin to investigate the situation.

Let E is a subsystem of F on a subgroup T of S. Write F_{T}^{f} for the set of nontrivial subgroups P of T such that P is fully normalized in F.

Definition 6.1. Define E to be F-invariant if:

(I1) T is strongly closed in S with respect to F.

(I2) For each P ≤ Q ≤ T, ϕ ∈ Hom_{E}(P, Q), and α ∈ Hom_{F}(Q, S), ϕ^{α} ∈ Hom_{E}(P α, T).

Notice that Definition 6.1 does not quite agree with the definition of Finvariance in Definition I.6.1, but the two notions are equivalent by I.6.4. The following definition also appears in I.6.1:

Definition 6.2. The subsystem E is F-Frattini if for each P ≤ T and γ ∈ Hom_{F}(P, S), there exists φ ∈ Aut_{F}(T) and ϕ ∈ Hom_{E}(P φ, S), such that γ = φϕ on P.

Lemma 6.3. Assume T is a subgroup of S which is strongly closed in S with respect to F and E is a subsystem of F on T. Then the following are equivalent:

(a) E is F-invariant.

(b) Aut_{F}(T) ≤ Aut(E) and E is F-Frattini.

(c) Aut_{F}(T) ≤ Aut(E) and condition (I2A) holds: (I2A) For each U ∈ F_{T}^{f} there exists a normal subgroup A(U) of Aut_{F}(U), such that for each U′ ≤ T, and each β ∈ Iso_{F}(U, U′), Aut_{T} (U′) ≤ A(U)^{β} ≤ Aut_{E}(U′).

Proof. Conditions (a) and (b) are equivalent by Proposition I.6.4. Condition (c) is shown to be equivalent to the first two conditions in [A5, 3.3]. Condition (I2A) presages the notion of an F-invariant map in Definition 8.1, which leads in turn to the normal maps in Definition 8.6. □

Invariant subsystems are fairly natural and have many nice properties. For example:

Lemma 6.4. Assume E is F-invariant and D is a subsystem of F on the subgroup D of S. Then

(a) E ∩ D is a D-invariant subsystem of D on T ∩ D.

(b) If D is F-invariant then E ∩ D is F-invariant on T ∩ D.

Proof. Exercise 6.1. □

On the other hand invariant subsystems have the big drawback that they need not be saturated.

Example 6.5. Assume T is strongly closed in S with respect to F. Define E to be the subsystem of F on T such that for each P, Q ≤ T, Hom_{E}(P, Q) = Hom_{F}(P, Q); that is E is the full subcategory of F whose objects are the subgroups of T. Then trivially E is F-invariant. But in most circumstances, E is not saturated. For example for P ≤ T, Aut_{E}(P) = Aut_{F}(P). In particular if P ∈ F_{T}^{f} then as F is saturated, Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) = Aut_{E}(T). However (cf. 3.4.5 in [A5]) P ∈ E^{f}, so if Aut_{S}(P) ≠ Aut_{T} (P) then E is not saturated.

While we are primarily interested in saturated fusion systems, in investigating such objects it is often useful to argue in the larger category of (possibly unsaturated) fusion systems, where the notion of F-invariance seems to be the “right” analogue of the normal subgroup relation from group theory. For example the intersection of normal subgroups H and K of a group G is again a normal subgroup of G, and indeed the greatest lower bound for the pair in the lattice of subgroups of G. By 6.4, the same property holds for the lattice of invariant subsystems of F. But the intersection of normal subsystems of F need not be normal in F, as we find in Example 7.3. Instead the greatest lower bound for normal subsystems (in the lattice of normal subsystems of F partially ordered by the normality relation) is the more subtle object appearing in Theorem 9.1, whose definition is rather complicated.

Exercises for Section 6

6.1. Prove 6.4.

7. Normal subsystems of fusion systems

In this section F is a saturated fusion system over the finite p-group S and E is a subsystem of F on a subgroup T of S. We continue the notation and terminology from the previous sections.

We saw in the previous section that if E is F-invariant, then E need not be saturated. One way to repair this problem is to only consider saturated F-invariant subsystems. Recall from Definition I.6.1 that a saturated Finvariant subsystem of F is said to be weakly normal in F. It turns out that to obtain a class of subsystems of a saturated fusion system F which have properties analogous to the class of normal subgroups of a finite group, one more condition must be added to the definition of weak normality. In any event we are lead to the following definition:

Definition 7.1. The subsystem E is normal in F if E is F-invariant and saturated, and the following condition holds:

(N1) Each ϕ ∈ Aut_{E}(T) extends to ϕ̂ ∈ Aut_{F}(T C_{S}(T)) such that [ϕ̂, C_{S}(T)] ≤ Z(T).

We write E ⊴ F to indicate that E is normal in F.

Example 7.2. Assume F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G). Let H ⊴ G, T = S ∩H, and E = F_{T} (H). Then from Proposition I.6.2, E ⊴ F.

Example 7.3. Let H = H_{1} × H_{2} × H_{3} be the direct product of three copies H_{i}, 1 ≤ i ≤ 3, of A_{4}. Let X_{i} = ⟨x_{i}⟩ ∈ Syl_{3}(H_{i}), S_{i} = O_{2}(H_{i}), and S = S_{1} × S_{2} × S_{3} ∈ Syl_{2}(H). Let X = ⟨x_{1}x_{2}, x_{1}x_{3}⟩ ≤ H and set G = XS. Then G_{1} = ⟨x_{1}x_{2}, S_{1}, S_{2}⟩ and G_{2} = ⟨x_{1}x_{3}, S_{1}, S_{3}⟩ are normal subgroups of G with Sylow 2-subgroups T_{1} = S_{1}S_{2} and T_{2} = S_{1}S_{3}, respectively. Let F_{i} = F_{T_{i}}(G_{i}), for i = 1, 2. As G_{i} ⊴ G, F_{i} ⊴ F by 7.2.

Let E = F_{1} ∩ F_{2}. Then E = F_{S_{1}}(H_{1}) as

ϕ = c_{x_{1}x_{2}|S_{1}} = c_{x_{1}x_{3}|S_{1}} = c_{x_{1}|S_{1}} ∈ Aut_{E}(S_{1}).

In particular E is a saturated fusion system, and by 6.4.a, E is F-invariant; that is E is weakly normal in F. On the other hand E is not normal in F, as (N1) is not satisfied. Namely S_{1}C_{S}(S_{1}) = S, but ϕ does not extend to ϕ̂ ∈ Aut_{F}(S) with [ϕ̂, S] ≤ S_{1}. That is E is weakly normal in F, but not normal in F.

This shows, first, that there exist weakly normal subsystems of saturated fusion systems which are not normal, and, second, that the intersection of normal subsystems is not in general normal. Moreover this also shows, third, that (N1) is a necessary hypothesis if the converse of 7.2 is to hold for constrained fusion systems, since E has no model normal in G. Put another way, if we are to extend arguments from the local theory of finite groups to the domain of saturated fusion systems, it is crucial to have the property that, when F is constrained, G ∈ G(F), and E ⊴ F, then there exist H ∈ G(E) with H ⊴ G. Thus it is necessary that the definition of “normal subsystem” contain some condition such as (N1).

See also Proposition I.5.6 for more examples illustrating differences between weakly normal and normal subsystems.

Recall from Definition 4.1 that for F saturated, the set G(F) of models of F consists of groups G with S ∈ Syl_{p}(G) and F = F_{S}(G), such that C_{G}(O_{p}(G)) ≤ O_{p}(G). From the discussion of the generalized Fitting subgroup F^{∗}(G) of a group G in A.9 and A.10, this last condition is equivalent to the condition that F^{∗}(G) = O_{p}(G), and that is the form in which we usually state the condition.

Lemma 7.4. Assume F is constrained and T is strongly closed in S with respect to F. Then

(a) There exists G ∈ G(F). Set L = ⟨T^{G}⟩.

(b) If E is a normal subsystem of F on T, then E = F_{T} (H) for the unique normal subgroup H of G such that T ∈ Syl_{p}(H), L = O^{p′}(H), and Aut_{H}(T) = Aut_{E}(T).

Proof. Part (a) is Theorem 4.2. The proof of (b) is a bit complicated and takes a number of lemmas, culminating in 6.7 in [A5]. But here is a discussion of a few of the high points.

Let R = O_{p}(G), Q = T ∩R, and X = C_{S∩L}(Q). As E ⊴ F, Aut_{T} (Q) is Sylow in B = ⟨Aut_{T} (Q)^{Aut_{F} (Q)}⟩. But as G is a model for F, B = Aut_{L}(Q), so

(*) T X ∈ Syl_{p}(L).

Next [T, R] ≤ Q, so L = ⟨T^{G}⟩ centralizes R/Q. Thus as F^{∗}(G) = R, O^{p}(C_{L}(Q)) ≤ O^{p}(C_{G}(R/Q) ∩ C_{G}(Q)) = R, so X ≤ R and hence X ⊴ L. Now [T, X] ≤ C_{T ∩R}(Q) = Z(Q), so [L, X] ≤ Z(Q). Then the strong closure of T together with (*) and a transfer argument show that T ∈ Syl_{p}(L).

As T ∈ Syl_{p}(L), G = LN_{G}(T) by a Frattini argument. As E ⊴ F, Σ = Aut_{E}(T) ⊴ Aut_{F}(T) = Aut_{G}(T) and Inn(T) ∈ Syl_{p}(Σ). Further by (N1), for each p′-element σ in Σ, there is a p′-element g ∈ N_{G}(T) with c_{g|T} = σ and [C_{S}(T), g] ≤ Z(T). As Σ ⊴ Aut_{G}(T), [R, g] ≤ T C_{S}(T), so as [C_{S}(T), g] ≤ Z(T) and g is a p′-element, [R, g] ≤ T ∩ R = Q. Thus g ∈ K = C_{G}(R/Q) ∩ N_{G}(T). Let K_{0} be the preimage in K of Σ and J = O^{p}(K). Then J ⊴ N_{G}(T), so H = LJ ⊴ G. Further argument shows that H is a model for E. The uniqueness of H, subject to the constraints in (b), is essentially a consequence of I.6.5. □

Theorem 7.5. Assume G is a finite group, S ∈ Syl_{p}(G), and F = F_{S}(G). Assume C_{G}(O_{p}(G)) ≤ O_{p}(G) and E ⊴ F on T ≤ S. Then there exists a unique normal subgroup H of G such that T ∈ Syl_{p}(H) and E = F_{T} (H).

Proof. As F = F_{S}(G), F is saturated and G ∈ G(F). Let R = O_{p}(F). As F = F_{S}(G), R = O_{p}(G). Thus C_{S}(R) ≤ R, and hence F is a constrained saturated fusion system. Now 7.4.2 completes the proof. □

Remark 7.6. Observe that Theorem 7.5 shows that our definition of “normal subsystem” has the desirable property discussed in Example 7.3. On the other hand our next example shows that one cannot remove the condition in 7.5 that C_{G}(O_{p}(G)) ≤ O_{p}(G).

Example 7.7. Here is an example which shows the hypothesis in 7.5 that F^{∗}(G) = O_{p}(G) cannot be removed. Assume G is a finite group, S ∈ Syl_{2}(G), and T is a strongly closed abelian subgroup of S contained in Z(S). Then M = N_{G}(T) controls fusion in T. Set E = F_{T} (T). Then E ≤ F and as T is abelian, E consists only of inclusion maps. Therefore E is F-invariant. Let P ≤ T. Then Aut_{E}(P) = 1, so trivially E is saturated and (N1) is satisfied, and hence E ⊴ F.

However there are plenty of examples of this set up in which T is not normal in G. For example take G to be simple and S = T an abelian Sylow group of G. To get examples where T is proper in S, take G simple and S cyclic with |S| > p, and T = Ω_{1}(S). Or take p = 2, G = Sz(2^{n}) or U_{3}(2^{n}), and T = Ω_{1}(S).

On the other hand these examples are a bit deceiving, since F is isomorphic to F_{S}(M), and T is normal in M. Here is a different sort of example:

Example 7.8. Take Ĝ to be the extension of the natural module V for G = Sz(2^{n}) or U_{3}(2^{n}), take T as in Example 7.7, and set T̂ = T V. Then T̂ is strongly closed in Ŝ = SV with respect to Ĝ, but is not Sylow in any normal subgroup of Ĝ. Thus by 7.5, there is no normal subsystem of F̂ = FŜ(Ĝ) on T̂. This example shows that even when F̂ is a saturated constrained fusion system, there can be a strongly closed subgroup T̂ of Ŝ in F̂ such that there exists no normal subsystem of F̂ on T̂.

Further V = O_{2}(Ĝ), so V ⊴ F̂ by 7.2. Moreover F̂/V ≅ F_{S}(G) by 5.6, and, as we saw above, T ⊴ F. However the preimage T̂ of T under Θ_{F̂,V} : F̂ → F̂/V is not normal in F̂, and indeed by 7.5, there is no normal subsystem of F̂ on T̂. This shows that the standard result in group theory fails for morphisms of saturated fusion systems: A normal subsystem of F̂/V need not lift under Θ_{F̂,V} to a normal subsystem of F̂.

8. Invariant maps and normal maps

In this section F is a saturated fusion system over the finite p-group S, and T is a subgroup of S strongly closed in S with respect to F.

In order to work with the normal subsystems defined in the previous section, we need effective conditions to verify when a subsystem of F on T is normal, and to produce normal subsystems on T. Moreover in most situations, these conditions should be local; that is we should be able to check them in local subsystems, and indeed even in local constrained subsystems.

This section contains a brief overview of some such conditions.

Recall from section 6 that F_{T}^{f} denotes the set of nontrivial subgroups P of T with P fully normalized in F. Write F_{T}^{fc} for the set of U ∈ F_{T}^{f} such that C_{T} (U) ≤ U, and set

F_{T}^{c} = ⋃_{U∈F_{T}^{fc}} U^{F}.

By Exercise 8.1,

F_{T}^{c} = {P ≤ T: C_{T} (P ϕ) ≤ P ϕ for all ϕ ∈ hom_{F}(P, S)}.

Definition 8.1. Define a F-invariant map on T to be a function A on the set of subgroups of T such that:

(IM1) For each P ≤ T and α ∈ Hom_{F}(P, S), A(P)^{α} = A(P α) ≤ Aut_{F}(P α).

(IM2) For each P ∈ F_{T}^{f}, Aut_{T} (P) ≤ A(P).

Given an F-invariant map A on T, set 𝔈(A) = ⟨A(P):P ≤ T ⟩, regarded as a fusion system on T.

Example 8.2. Pick a set U of representatives in F_{T}^{f} for the orbits of F on the subgroups of T. For U ∈ U, pick a normal subgroup A(U) of Aut_{F}(U) containing Aut_{T} (U). For α ∈ Hom_{F}(U, T), define A(Uα) = A(U)^{α}. As A(U) ⊴ Aut_{F}(U), the function A is well defined, and by construction A is a F-invariant map on T. Thus each such map on U extends uniquely to an F-invariant map A, and it is easy to check that each invariant map can be obtained via this construction.

Here is a special case: Define B(U) = ⟨Aut_{T} (U)^{Aut_{F} (U)}⟩ as in the proof of 7.4.b. Then B defines an invariant map.

Lemma 8.3. Let A be a F-invariant map on T. Then 𝔈(A) is a Finvariant subsystem of F on T.

Proof. See 5.5 in [A5]. □

Example 8.4. Pick U as in Example 8.2, let V = U ∩ F_{T}^{fc} and suppose A is a map on V satisfying Aut_{T} (V) ≤ A(V) ⊴ Aut_{F}(V) for each V ∈ V. Then as in 8.2, we can extend A to a map on F_{T}^{c} via A(V α) = A(V)^{α}. Then for U ∈ U, set A(U) = Aut_{A(UC_{T} (U))}(U). One can show that A satisfies the conditions of 8.2, and hence defines an invariant map.

Definition 8.5. Define an F-invariant map A to be constricted if A(U) = Aut_{A(UC_{T} (U))}(U) for each U ∈ F_{T}^{f}. Thus the maps A constructed in Example 8.4 are constricted.

Our object is to use invariant maps A to construct normal subsystems of F. The corresponding normal subsystem should be 𝔈(A), which should satisfy Aut_{𝔈(A)}(U) = A(U) for U ≤ T. A necessary condition for 𝔈(A) to be saturated is that A be constricted. But still more conditions are required. This leads to the following definition:

Definition 8.6. Define a constricted F-invariant map A on T to be normal if for each U ∈ F_{T}^{fc}:

(SA1) Aut_{T} (U) ∈ Syl_{p}(A(U)).

(SA2) For each U ≤ P ≤ Q = N_{T} (U) with P fully normalized in N_{F}(UC_{S}(U)), Aut_{A(P)}(U) = N_{A(U)}(Aut_{P} (U)).

(SA3) Each ϕ ∈ N_{A(Q)}(U) extends to ϕ̂ ∈ Aut_{F}(QC_{S}(Q)) with [C_{S}(Q), ϕ̂] ≤ Z(Q).

Conditions (SA1) and (SA2) are necessary to insure that 𝔈(A) is saturated. Of course condition (SA3) is included to insure that the pair F, 𝔈(A) satisfies condition (N1) in the definition of normal subsystem. In any event one can show:

Theorem 8.7. Let A be a normal map on T. Then 𝔈(A) is a normal subsystem of F on T such that for each P ≤ T, Aut_{E(A)}(P) = A(P).

Theorem 8.7 follows from a more general result which we state in a moment.

If U ∈ F_{T}^{fc} then by Example 4.6, D(U) = N_{F}(UC_{S}(U)) is saturated and constrained, so there exists G(U) ∈ G(D(U)); that is G(U) is a finite group with N_{S}(U) ∈ Syl_{p}(G(U)) and D(U) = F_{N_{S}(U)}(G(U)).

Theorem 8.8. Let E be a subsystem of F on T. Then the following are equivalent:

(a) E ⊴ F.

(b) There exists a normal map A′ on T such that E = 𝔈(A′).

(c) For each U ∈ F_{T}^{fc} there exists a normal subgroup H(U) of G(U) such

that N_{T} (U) ∈ Syl_{p}(H(U)) and

(i) for each P ∈ D(U)^{f} with U ≤ P, and for each α ∈ Hom_{F}(N_{S}(P), S) with P α ∈ F_{T}^{fc}, Aut_{H(P α)}(Uα) = Aut_{N_{H(U)}(P)}(U)^{α}, and

(ii) E = ⟨A(Uφ): U ∈ F_{T}^{fc}, φ ∈ Aut_{F}(T)⟩, where A is the constricted invariant map defined by A(U) = Aut_{H(U)}(U) as in 8.4.

Proof. This is 7.18 in [A5]. □

Remark 8.9. One interpretation of 8.8 is the following: Given a strongly closed subgroup T of S, we look for normal subsystems E of F on T. To do so we consider the members U of F_{T}^{fc}, the associated constrained saturated systems D(U) = N_{F}(UC_{S}(U)), and their models G(U). We look for a set {H(U): U ∈ F_{T}^{fc}} of subgroups H(U) ⊴ G(U) with N_{T} (U) ∈ Syl_{p}(H(U)), satisfying the compatibility conditions in 8.8.c.i. Given such a collection, E is essentially the subsystem of F generated by the systems Aut_{H(U)}(U), and Aut_{E}(U) is Aut_{H}(U).

Exercises for Section 8

8.1. Prove F_{T}^{c} = {P ≤ T: C_{T} (P ϕ) ≤ P ϕ for all ϕ ∈ hom_{F}(P, S)}.

9. Theorems on normal subsystems

In this section F is a saturated fusion system on a finite p-group S.

We list various results about normal subsystems, and extensions of theorems in the local theory of finite groups to the setting of saturated fusion systems. These result are proved in [A6]. The proofs use results from the previous section. Most of the proofs are moderately difficult.

In Example 7.3, we saw that the intersection E_{1}∩E_{2} of normal subsystems E_{i} of F need not be normal in F. However this is not a serious problem, since it develops that E_{1} ∩ E_{2} is not quite the right object to consider. Rather:

Theorem 9.1. Let E_{i} be a normal subsystem of F on a subgroup T_{i} of S, for i = 1, 2. Then there exists a normal subsystem E_{1} ∧ E_{2} of F on T_{1} ∩ T_{2} normal in E_{1} and E_{2}. Moreover E_{1} ∧ E_{2} is the largest normal subsystem of F on T_{1} ∩ T_{2} normal in E_{1} and E_{2}.

The next result may already appear in the literature in the special case where F = F_{S}(G) is the system of a finite group G on a Sylow p-subgroup S of G, but we are not aware of such a result.

Theorem 9.2. Assume T_{i}, i = 1, 2, are strongly closed in S with respect to F. Then T_{1}T_{2} is strongly closed in S with respect to F.

If H_{1} and H_{2} are normal subgroups of a group G, then H_{1}H_{2} ⊴ G. The analogue of this result may hold for saturated fusion systems, but in [A6] there is a proof only in a very special case; this case suffices for our most immediate applications.

That special case is treated in Theorem 9.3; it bears some resemblance to earlier theorems about finite groups due to Gorenstein-Harris in [GH], and Goldschmidt in [Gd2]. Namely in each of these papers, the authors prove the existence of certain normal subgroups of a group G under the hypothesis that for S ∈ Syl_{2}(G), there are subgroups T_{i} of S for i = 1, 2, such that [T_{1}, T_{2}] = 1 and T_{i} is strongly closed in S with respect to G.

Theorem 9.3. Assume E_{i} ⊴ F on T_{i} for i = 1, 2, and that [T_{1}, T_{2}] = 1. Then there exists a normal subsystem E_{1}E_{2} of F on T_{1}T_{2}. Further if T_{1} ∩ T_{2} ≤ Z(E_{i}) for i = 1, 2, then E_{1}E_{2} is a central product of E_{1} and E_{2}.

Definition I.6.5 contains the definition of the direct product F_{1} × F_{2} of fusion systems F_{1} and F_{2}. A central product F_{1} ×_{Z} F_{2} is a factor system (F_{1} × F_{1})/Z, for some Z ≤ Z(F_{1} × F_{2}) such that Z ∩ F_{i} = 1 for i = 1, 2.

Let E ⊴ F. In [A6] the centralizer in F of E is defined, and is denoted by C_{F}(E). Unfortunately the definition is rather complicated, so we will not reproduce it here. However it is proved that:

Theorem 9.4. If E ⊴ F is a fusion system on T, then the set of subgroups Y of C_{S}(T) such that E ≤ C_{F}(Y) has a largest member C_{S}(E), C_{F}(E) is a normal subsystem of F on C_{S}(E), and for X ∈ C_{F}(E)^{fc}, Aut_{C_{F} (E)}(X) = O^{p}(Aut_{C_{F} (T)}(X))Aut_{C_{S}(E)}(X).

Recall the hyperfocal subgroup 𝔥𝔶𝔭(F) is defined in Definition I.7.1, and from Theorem I.7.4, there is a unique saturated subsystem O^{p}(F) of F on 𝔥𝔶𝔭(F) such that O^{p}(Aut_{F}(P)) ≤ Aut_{O^{p}(F)}(P) for each P ≤ 𝔥𝔶𝔭(F). Moreover O^{p}(F) ⊴ F. For example if F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G), then O^{p}(F) = F_{S∩O^{p}(G)}(O^{p}(G)).

In [A6] there is a different proof of the existence of O^{p}(F). Very roughly the proof goes as follows: First prove T = 𝔥𝔶𝔭(F) is strongly closed in S. Then for U ∈ F_{T}^{fc}, let G(U) be the model for D(U) = N_{F}(UC_{S}(U)), H^{p}(U) = O^{p}(G(U))N_{T} (U), and A^{p}(U) = Aut_{H^{p}(U)}(U). Let A^{p} be the constricted F-invariant map determined by the groups A^{p}(U) as in Example 8.4, and set O^{p}(F) = 𝔈(A^{p}). Verify the compatibility conditions in 8.8.c.i, and then appeal to Theorem 8.8 to conclude O^{p}(F) ⊴ F.

In addition it is shown in [A6] that:

Theorem 9.5. Let E ⊴ F on T, and T ≤ R ≤ S. Then there exists a unique saturated fusion subsystem RE of F on R such that O^{p}(RE) = O^{p}(E). In particular F = SO^{p}(F).

Recall from Definition I.6.1 that F is simple if F has no proper nontrivial normal subsystems. Define F to be quasisimple if F = O^{p}(F) and F/Z(F) is simple. Define the components of F to be the subnormal quasisimple subsystems of F. Recall O_{p}(F) is the largest subgroup of S normal in F. We can view R = O_{p}(F) as the normal subsystem F_{R}(R) of F.

Define E(F) to be the normal subsystem of F generated by the set Comp(F) of components of F (which exists by Theorem 9.1), and set F^{∗}(F) = E(F)O_{p}(F). We call F^{∗}(F) the generalized Fitting subsystem of F. Of course all of these notions are similar to the analogous notions for groups (cf. Definition A.11), and the following theorem is a list of the standard properties of the generalized Fitting subgroup of a finite group, only stated for fusion systems.

Theorem 9.6. (a) E(F) is a characteristic subsystem of F.

(b) E(F) is the central product of the components of F.

(c) O_{p}(F) centralizes E(F).

(d) C_{F}(F^{∗}(F)) = Z(F^{∗}(F)).

Next [A6] establishes a version of the Gorenstein-Walter theorem on so called L-balance [GW2]:

Theorem 9.7. For each fully normalized subgroup X of S, E(N_{F}(X)) ≤ E(F).

It is worth noting that the proof of L-balance for a group G requires that the components of G/O_{p′}(G) satisfy the Schreier conjecture, or when p = 2, a weak version of the Schreier conjecture due to Glauberman. Our proof of Theorem 9.7 requires no deep results. The theorem does not quite imply L-balance for groups, since there is not a nice one to one correspondence between quasisimple groups and quasisimple fusion systems. The proof can be translated into the language of groups, but even there at some point one seems to need some result like Theorem A of Goldschmidt in [Gd2], which is only proved for p = 2 without the classification. Still, something is going on here, which suggests that in studying fusion systems, one may be lead to new theorems or better proofs of old theorems about finite groups.

Finally by Theorem 9.1 there is a smallest normal subsystem of F on S. Denote this subsystem by O^{p′}(F). For example if F = F_{S}(G) for some finite group G with F^{∗}(G) = O_{p}(G) and S ∈ Syl_{p}(G), then O^{p′}(F) = F_{S}(O^{p′}(G)). The system O^{p′}(F) is the smallest subsystem of index prime to p, in the sense of Definition I.7.3, and as such is described in Theorems I.7.7 and III.4.19.

There is a slightly different approach to these notions in [A6]. For P ≤ S, define B(P) = O^{p′}(Aut_{F}(P)). Then B is an F-invariant map, as defined in Section 8, so by 8.3, B = 𝔈(B) is an F-invariant subsystem of F. Essentially as in [5a2], define

Aut_{F}^{0}(S) = ⟨α ∈ Aut_{F}(S): α_{|P} ∈ Hom_{B}(P, S) for some P ∈ F^{c}⟩.

As B is F-invariant, Aut_{F}^{0}(S) ⊴ Aut_{F}(S), so we can define

Γ_{p′}(F) = Aut_{F}(S)/Aut_{F}^{0}(S).

Then Theorem 8 in [A6] establishes the following version of Theorems I.7.7 and III.4.19:

Theorem 9.8. (a) π_{1}(F^{c}) ≅ Γ_{p′}(F).

(b) The map E ↦ Aut_{E}(S)/Aut_{F}^{0}(S) is a bijection between the set of normal subsystems of F on S and the set of normal subgroups of Γ_{p′}(F).

(c) F = O^{p′}(F) iff Aut_{F}(S) = Aut_{F}^{0}(S) iff π_{1}(F^{c}) = 1.

(d) F is simple iff the following hold:

(i) For each normal subsystem D of F on a subgroup D of S, we have D = S.

(ii) Aut_{F}(S) = Aut_{F}^{0}(S).

Recall that parts (a)-(c) of Theorem 9.8 are proved in Theorems I.7.7 and III.4.19. Observe that (d) is an easy consequence of (b), since by (b), F has no proper normal subsystems on S iff (di) holds.

10. Composition series

In this section F is a saturated fusion system over the finite p-group S.

We wish to define and explore analogues of the notions of “composition series” and “composition factors” from finite group theory in the category of saturated fusion systems. In the category of groups, a composition series is a maximal subnormal series and the simple factor groups from such a series are independent of the series and defined to be the composition factors of the group. Moreover given a finite group G, its composition series 1 = G_{n} ◁ ⋯ ◁ G_{0} = G can all be obtained from the following algorithm: Pick a maximal normal subgroup G_{1} of G; then pick a maximal normal subgroup G_{2} of G_{1}; and so on.

Unfortunately this approach does not quite work even for constrained fusion systems, as the following example shows:

Example 10.1. Let L be a finite simple group with a abelian Sylow psubgroup P of order p^{e} > p. Let V be a faithful irreducible F_{p}L-module of dimension d > 1. Form the semidirect product G = LV and observe S = P V ∈ Syl_{p}(G). Let F = F_{S}(G). Then F^{∗}(G) = V, so F is constrained and from 7.2 and 7.5 the map H ↦ F_{S}(H) is a bijection between the normal subgroups of G and the normal subsystems of F. As L is simple and irreducible on V, V is the unique nontrivial proper normal subgroup of G, so V = F_{V} (V) is the unique nontrivial proper normal subsystem of F. Then choosing a chain 1 = V_{0} < V_{1} < ⋯ < V_{d} = V of subspaces with dim(V_{i}) = i, we get a subnormal series 1 = V_{0} < ⋯ < V_{d} = V, where V_{i} = F_{V_{i}}(V_{i}), with simple factors of order p; that is each factor is of the form F_{U}(U), where U is a group of order p. Moreover, adjoining F to this series, we have a maximal subnormal series for F of length d + 1.

However while V is a maximal normal subsystem of F, F/V is not simple. Namely as S/V ≅ P is abelian, F/V ≅ F_{P} (N_{L}(P)) has a normal Sylow group P. Thus we obtain a series 1 = P_{0} < ⋯ < P_{e} = P of subgroups with |P_{i}| = p^{i}, giving rise to the subnormal series 1 = P_{0} < ⋯ < P_{e} of subsystems P_{i} = F_{P_{i}}(P_{i}), again with factors of order p. As defined below in Definition 10.6, this construction gives rise to a “supranormal series” for F, and the factors in this series are the composition factors for F. So in this example, F has e + d composition factors, each of order p.

We now begin to develop a theory of composition series and composition factors for saturated fusion systems, adjusted to allow for systems like those in the previous example and based on a more complicated algorithm than the one for groups: If O^{p′}(F) is not simple, pick a nontrivial proper normal subsystem E of O^{p′}(F) and, proceeding recursively, continue by applying this process to E and O^{p′}(F)/E. Notice the difficulties arise from the fact that if T is strongly closed in S with respect to F, and D is the preimage in F of a normal subsystem of F/T, it need not be the case that D is normal in F.

Definition 10.2. By 9.1, for each subgroup T of S such that T is Sylow in a normal subsystem of F, there is a smallest normal subsystem [T^{F}] of F on T. For example from Exercise 10.1, [S^{F}] is the system O^{p′}(F).

Example 10.3. Let G be a finite group with S ∈ Syl_{p}(G), C_{G}(O_{p}(G)) ≤ O_{p}(G), and F = F_{S}(G). Then by 7.2 and 7.5 the map H ↦ F_{S∩H}(H) is a bijection between the normal subgroups H of G and the normal subsystems of F. Therefore T ≤ S is Sylow in a normal subsystem of F iff T is Sylow in some normal subgroup of G, and in that event [T^{F}] is F_{T} (⟨T^{G}⟩). In particular O^{p′}(F) = F_{S}(O^{p′}(G)), since O^{p′}(G) is the smallest normal subgroup H of G such that G/H is a p′-group; equivalently O^{p′}(G) = ⟨S^{G}⟩.

Definition 10.4. Set

D = [0, 1] ∩ ℤ[\frac{1}{2}] = \{\frac{k}{2^{n}} | n ≥ 0, 0 ≤ k ≤ 2^{n}\} .

(a) Define l, u: D ⟶ D by l(0) = l(1) = 0, u(0) = u(1) = 1, and when n > 0 and 0 < k < 2^{n} is odd:

l(\frac{k}{2^{n}} ) = \frac{k−1}{2^{n}} and u(\frac{k}{2^{n}} ) = \frac{k+1}{2^{n}}

(b) Define h: D ⟶ ℕ by

h(0) = h(1) = 0, h(\frac{k}{2^{n}} ) = n for all n ≥ 1, k odd.

(c) A finite subset X ⊆ D is admissible if 0, 1 ∈ X, l(X) ⊆ X, and u(X) ⊆ X.

(d) Define bijections ι: D ⟶ D ∩ [0, \frac{1}{2}] and κ: D ⟶ D ∩ [\frac{1}{2}, 1] by ι(a) = \frac{a}{2} and κ(a) = \frac{a+1}{2}.

(e) For any X ⊆ D, set

X′ = ι^{−1}(X) = \{2a | a ∈ X, a ≤ \frac{1}{2}\} and X′′ = κ^{−1}(X) = \{2a−1 | a ∈ X, a ≥ \frac{1}{2}\}.

For a ∈ D, h(a) will be called the height of a. If 0 < a < 1, then l(a) < a < u(a), and l(a) and u(a) are the closest elements to a in D with smaller height.

Lemma 10.5. Let X ⊆ D be a finite subset. Then X is admissible if and only if either X = \{0, 1\}, or \frac{1}{2} ∈ X and X′ and X′′ are both admissible.

Proof. Assume X ≠ {0, 1}. If 0 ∉ X, then X is not admissible and 0 ∉ X′ so X′ is not admissible. Similarly if 1 ∉ X, then neither X nor X′′ is admissible. So we can assume {0, 1} ⫋ X.

Fix a ∈ X, 0 < a < 1, such that n = h(a) is minimal. If l(a), u(a) ∈ X, then since they have height less than n, we must have l(a) = a − \frac{1}{2^{n}} = 0 and u(a) = a + \frac{1}{2^{n}} = 1, and hence a = \frac{1}{2}. Thus if X is admissible then \frac{1}{2} ∈ X. So we can also assume \frac{1}{2} ∈ X.

Let X_{0} = X ∩ [0, \frac{1}{2}] and X_{1} = X ∩ [\frac{1}{2}, 1]. By Exercise 10.2 the bijections ι: X′ → X_{0} and κ: X′′ → X_{1} commute with l and u. Thus l(X′) ⊆ X′ if and only l(X_{0}) ⊆ X and u(X′) ⊆ X′ if and only if u(X_{0}) ⊆ X, and similarly for X′′. Thus X is admissible if and only if X′ and X′′ are both admissible. □

Let 𝔖𝔲𝔟(S) denote the set of subgroups of the group S.

Definition 10.6. We recursively define the set S = S(F) of supranormal series of F. A member of S is a pair (X, η), where

(a) X ⊆ D is a finite admissible subset;

(b) η: X ⟶ 𝔖𝔲𝔟(S) is a strictly order preserving map;

(c) η(0) = 1 and η(1) = S; and

(d) if \frac{1}{2} ∈ X, then there is a normal subsystem of O^{p′}(F) over T =^{def} η(1/2), (X′, η′) is a supranormal series for [T^{F}], and (X′′, η′′) is a supranormal series for O^{p′}(F)/T where η′ = ιη and η′′: X′′ ⟶ 𝔖𝔲𝔟(S/T) is defined by η′′(a) = η(κ(a))/T.

We call η(X) the set of pivots of (X, η) and call T the primary pivot of the pair. The length of the series is |X| − 1. For (X, η) and (Y, θ) in S, define (X, η) ≤ (Y, θ) if η ⊆ θ, where we regard η and θ as sets of ordered pairs. In particular this implies that X ⊆ Y. Define a composition series for F to be a maximal member of the partially ordered set (S, ≤).

In the remainder of the section let λ = (X, η) ∈ S(F) with \frac{1}{2} ∈ X. Set T = η(\frac{1}{2}), λ′ = (X′, η′), and λ′′ = (X′′, η′′).

Example 10.7. In Example 10.1, the only nontrivial proper normal subsystem of F = O^{p′}(F) is F_{V} (V). So when n > 1 (that is when \frac{1}{2} ∈ X), η(\frac{1}{2}) = V is the primary pivot and λ_{1/2} = F_{V} (V). One series of maximal length n = d + e is 1 = V_{0} < ⋯ < V_{d} < V P_{1} < ⋯ < V P_{e} = S. For this series there are many choices for the admissible set X. One choice is to take

X = \{0, \frac{1}{2^{i}} , \frac{1}{2} + \frac{1}{2^{j}} | 1 ≤ i ≤ d, 1 ≤ j ≤ e\},

and set η(\frac{1}{2^{i}}) = V_{d+1−i} and η(\frac{1}{2} + \frac{1}{2^{j}}) = V P_{1+e−j}.

As the notion of a supranormal series λ = (X, η) is defined recursively, one can view the members of the admissible set X and the pivots of λ as becoming visible over time, as measured by the height h(a) of a ∈ X and its pivot η(a). Most results will be established by induction on the length of the series or on the height of members of X and their pivots. The next few lemmas record some elementary observations useful in such proofs.

Lemma 10.8. Assume a ∈ D. Then l(u(a)) ≤ l(a); u(a) ≤ u(l(a)); and at least one of these inequalities is an equality.

Proof. This is clear if h(a) ≤ 1 (ie. when a ∈ \{0, \frac{1}{2}, 1\}), so we assume n = h(a) ≥ 2. Set a = \frac{k}{2^{n}} where k is odd. By Exercise 10.3, h(l(a)) ≤ n − 1 ≥ h(u(a)), so

l(u(a)) = a+\frac{1}{2^{n}} −\frac{1}{2^{h(u(a))}} ≤ a−\frac{1}{2^{n}} = l(a), u(l(a)) = a−\frac{1}{2^{n}} +\frac{1}{2^{h(l(a))}} ≥ a+\frac{1}{2^{n}} = u(a);

further by Exercise 10.3, either h(l(a)) or h(u(a)) is n − 1, so one of the inequalities is an equality. □

Lemma 10.9. Fix an admissible subset X of D, and a consecutive pair of elements a < b in X. Then either

• X = {0, 1}, a = 0, and b = 1; or

• h(a) > h(b), b = u(a), and l(b) < a; or

• h(a) < h(b), l(b) = a, and b < u(a).

Proof. We may assume X ≠ {0, 1}. Thus a > 0 or b < 1. Set n = max\{h(a), h(b)\} > 0. Write a = \frac{j}{2^{n}} and b = \frac{k}{2^{n}} for j < k in ℤ. If n = h(a), then j is odd, u(a) = \frac{j+1}{2^{n}} ∈ X, and hence b = u(a). Also, in this case, h(b) < n since k = j + 1 is even, so l(b) = b − \frac{1}{2^{h(b)}} < a. Similarly, if n = h(b), then j = k − 1, a = l(b), h(a) < n, and u(a) > b. □

Associated to each a ∈ X and a ≤ b ≤ u(a) are various p-groups T_{a} and T_{a,b} and fusion systems λ_{l(a),a}, λ_{l(a),a}, and λ_{l(a),u(a)}. We begin with the p-groups T_{a} = η(a) and T_{0,b} = T_{b}, and the fusion systems λ_{0,0} = λ_{1,1} = 1, λ_{0,1} = O^{p′}(F), λ_{0, \frac{1}{2}} = [T^{F}], and λ_{\frac{1}{2} ,1} = O^{p′}(λ_{0,1}/T_{j}). The next few lemmas introduce groups and systems associated to pairs of elements of higher height, defined recursively.

Lemma 10.10. We make the following recursive definition. For a ∈ X, set T_{0,a} = T_{a} = η(a), and for a ≤ b ≤ u(a) set T_{a,b} = T_{l(a),b}/T_{l(a),a}. Then

(a) For a ≤ b ≤ u(a), T_{l(a),a} ⊴ T_{l(a),b}, so T_{a,b} is well defined. Moreover, T_{a} ⊴ T_{b} and T_{a,b} ≅ T_{b}/T_{a}.

(b) Let T′_{a}, T′_{a,b} be the groups defined analogously for the series λ′ = (X′, η′). For c ≤ d ≤ u(c) in X′, T_{ι(c)} = T′_{c} and T_{ι(c),ι(d)} = T′_{c,d}.

(c) Let T′′_{a,b} be the groups defined analogously for the series λ′′ = (X′′, η′′). For c < 1 and c ≤ d ≤ u(c) in X′′, T_{κ(d)} = T′′_{d} and T_{κ(c),κ(d)} = T′′_{c,d}.

Proof. By definition of the notation, T′_{c} = T_{ι(c)} and T′′_{d} = T_{κ(d)}. If a = \frac{1}{2} then T_{l(a),a} = T_{0,a} = T_{a} and T_{a} ⊴ S since T_{1/2} must be strongly closed in O^{p′}(F), hence T_{a} ⊴ T_{b} = T_{0,b} and T_{0,b}/T_{0,a} = T_{b}/T_{a}. So (a) and (c) hold in this case.

Suppose a = ι(c) < \frac{1}{2} and b = ι(d). Then T_{l(a),a} = T′_{l(c),c} and T_{l(a),b} = T′_{l(c),d} by induction on h(a) and Exercise 10.2. Also by induction on |X| and Exercise 10.2, T′_{c} ⊴ T′_{d} and T′_{l(c),c} ⊴ T′_{l(c),d}. Therefore T_{a} = T′_{c} ⊴ T′_{d} = T_{b} and T_{a,b} = T_{l(a),b}/T_{l(a),a} = T′_{c,d} is well defined. So (b) holds, as does (a) in this case.

Finally assume a = κ(c) > \frac{1}{2} where c ∈ X′′. By induction on |X|, for r ≤ s ≤ u(a) in X′′, T′′_{r,s} is well defined and isomorphic to T′′_{s}/T′′_{r}, so to complete the proof of (a) it remains to prove (c). We proved (c) when c = 0 in paragraph one. If c > 0 then by induction on h(c),

T′′_{c,d} = T′′_{l(c),d}/T′′_{l(c),c} = T_{κ(l(c)),κ(d)}/T_{κ(l(c)),κ(c)} = T_{l(κ(c)),κ(d)}/T_{l(κ(c)),κ(c)} = T_{κ(c),κ(d)},

completing the proof. □

Lemma 10.11. We make the following recursive definition: Define λ_{0,0} = λ_{1,1} = 1, λ_{0,1} = λ_{1} = O^{p′}(F), λ_{0,1/2} = [T^{F}] and λ_{1/2,1} = O^{p′}(λ_{1}/T). If 0 < a < \frac{1}{2} where a = ι(c), set λ_{l(a),a} = λ′_{l(c),c} and λ_{a,u(a)} = λ′_{c,u(c)}. If \frac{1}{2} < a < 1 where a = κ(d), set λ_{l(a),a} = λ′′_{l(d),d} and λ_{a,u(a)} = λ′′_{d,u(d)}. Then for each a ∈ X, λ_{l(a),u(a)}, λ_{a,u(a)}, and λ_{l(a),a} are well defined saturated fusion systems over T_{l(a),u(a)}, T_{a,u(a)}, and T_{l(a),a}, respectively, and λ_{l(a),a} = [T_{l(a),a}^{λ_{l(a),u(a)}}] and λ_{a,u(a)} = O^{p′}(λ_{l(a),u(a)}/T_{l(a),a}).

Proof. This is immediate if a ∈ {0, 1}, and follows from Definition 10.6.d if a = \frac{1}{2}. So we can assume h(a) ≥ 2. Set l = l(a) and u = u(a). By 10.8, either l = l(u) or u = u(l). Thus (l, u) = (l(u), u) or (l, u(l)), so λ_{l,u} is well defined and saturated over T_{l,u} by induction on h(a).

Let T′_{a,b} and T′′_{a,b} be the groups defined analogously to T_{a,b}, for the series λ′ and λ′′, respectively. Thus by 10.10, T′_{a,b} = T_{ι(a),ι(b)} and T′′_{a,b} = T_{κ(a),κ(b)} (when defined). We can assume by induction on |X| that the lemma holds for λ′ and λ′′. By this assumption and the recursive definition, λ_{l(a),a} and λ_{a,u(a)} are saturated fusion systems over T_{l(a),a} and T_{a,u(a)} for each a ∈ X. If 0 < a < \frac{1}{2} and a = ι(c), then

λ_{a,u(a)} = λ′_{c,u(c)} = O^{p′}(λ′_{l(c),u(c)}/T′_{l(c),c}) = O^{p′}(λ_{l(a),u(a)}/T_{l(a),a})

(recall ι(l(c)) = l(a) and ι(u(c)) = u(a) by Exercise 10.2); and

λ_{l(a),a} = λ′_{l(c),c} = [T′_{l(c),c}^{λ′_{l(c),u(c)}}] = [T_{l(a),a}^{λ_{l(a),u(a)}}].

The proof when \frac{1}{2} < a < 1 is similar. □

Lemma 10.12. Let a < b be consecutive elements in X. Then λ_{a,b} = O^{p′}(λ_{a,b}) is a saturated fusion system on T_{a,b}.

Proof. By Lemma 10.9, either b = u(a) or a = l(b). The lemma is thus a special case of 10.11. □

Definition 10.13. For λ = (X, η) ∈ S, let

F (λ) = (λ_{a,b} | a < b are consecutive elements of X.)

By 10.12, the above definition makes sense and the factors in F (λ) are saturated.

Proposition 10.14. For any saturated fusion system F, a series λ ∈ S(F) is a composition series for F iff all factors of λ are simple.

Proof. We prove this by induction on |X|. Assume first X = {0, 1}. If μ = (Y, θ) > (X, η) = λ, then \frac{1}{2} ∈ Y by Lemma 10.5, 1 < θ(\frac{1}{2}) < θ(1) = S, and μ_{0,1/2} is a normal subsystem of λ_{0,1} = O^{p′}(F) over θ(\frac{1}{2}) by Lemma 10.11. Thus λ_{0,1} is not simple in this case.

Conversely, assume λ_{0,1} = O^{p′}(F) is not simple, and let E ⊴ O^{p′}(F) be a normal subsystem over a nontrivial proper subgroup T ⊴ S. Set Y = \{0, \frac{1}{2}, 1\}, and define θ by setting θ(0) = 1, θ(\frac{1}{2}) = T, and θ(1) = S. Then (Y, θ) ∈ S, (Y, θ) > λ, and so λ is not maximal.

Now assume |X| > 2, so that \frac{1}{2} ∈ X by Lemma 10.5 again. If λ is not maximal in S(F), then either λ′ is not maximal in S(λ_{0,1/2}), or λ′′ is not maximal in S(λ_{1/2,1}). Hence at least one of these has a factor which is not simple, and so the same holds for λ.

Conversely, if λ has a factor which is not simple, then the same is true of λ′ or λ′′. By the induction hypothesis, there are series λ̂′ = (X̂′, η′) ≥ λ′ and λ̂′′ = (X̂′′, η′) ≥ λ′′ such that at least one of the inequalities is strict. Set Y = ι(X̂′) ∪ κ(X̂′′). Then Y′ = X̂′ and Y′′ = X̂′′, and Y is admissible by 10.5. Let θ be such that θ′ = η′ and θ′′ = η′′. Then (Y, θ) ∈ S(F) and by Definition 10.6, (Y, θ) > λ, so that λ is not maximal. □

Theorem 10.15. (Jordan-Hölder Theorem for fusion systems) If λ and μ are composition series for a saturated fusion system F, then λ and μ have the same length and F (λ) = F (μ).

Proof. Assume the theorem is false and pick a counter example λ = (X, η), μ = (Y, θ), F over S with F minimal and |X| of minimal order n. By minimality of F, F = O^{p′}(F). If n = 2 then by 10.14, F = F/1 is simple, so S = {λ} and the theorem holds. Thus n > 2 < |Y |.

For any saturated fusion system E smaller than F, let CF (E) denote the set of composition factors for any composition series for E. This set is unique by the minimality assumption on F.

Set T = η(\frac{1}{2}) and U = θ(\frac{1}{2}). Thus λ_{1/2} and μ_{1/2} are proper nontrivial normal subsystems of F over T and U, respectively. If T ∩ U ≠ 1, then by 9.1, F_{0} = λ_{1/2} ∧ μ_{1/2} is a fusion system over S_{0} = T ∩ U, and is normal in F, λ_{1/2}, and μ_{1/2}. Thus

F (λ) = F (λ′) ⨿ F (λ′′) = CF (λ_{1/2}) ⨿ CF (F/T) = CF (F_{0}) ⨿ CF (λ_{1/2}/S_{0}) ⨿ CF (F/T)

= CF (F_{0}) ⨿ CF (F/S_{0}) = CF (F_{0}) ⨿ CF (μ_{1/2}/S_{0}) ⨿ CF (F/U)

= CF (μ_{1/2}) ⨿ CF (F/U) = F (μ′) ⨿ F (μ′′) = F (μ).

Now assume T ∩ U = 1. By 9.3, there is a normal subsystem E ⊴ F over T U, and E ≅ λ_{1/2} × μ_{1/2}. If T U < S, then F (λ) = F (μ) by an argument similar to that used when T ∩ U ≠ 1. Otherwise, if T U = S, then F = E ≅ λ_{1/2} × μ_{1/2}, and

F (λ) = F (λ′) ⨿ F (λ′′) = CF (λ_{1/2}) ⨿ CF (μ_{1/2}) = F (μ′′) ⨿ F (μ′) = F (μ)

for our final contradiction. □

Definition 10.16. By 10.15, we may define the family CF (F) of composition factors of F to be the set F (λ) of factors of any composition series λ of F.

Lemma 10.17. For each normal subsystem E of F, CF (F) = CF (E) ⨿ CF (F/E).

Proof. Let E be a Sylow group for E. Replacing E by O^{p′}(E) we may assume E = O^{p′}(E). Then we may choose a composition series λ for F with primary pivot E and associated normal system E. Now CF (F) = F (λ′) ⨿ F (λ′′) with F (λ′) = CF (E) and F (λ′′) = CF (F/E). □

Exercises for Section 10

10.1. If F is a saturated fusion system on a p-group S and E is a normal subsystem of F on T ≤ S then [T^{F}] = O^{p′}(E).

10.2. Prove, for a ∈ D with 0 < a < 1, that ι(l(a)) = l(ι(a)), ι(u(a)) = u(ι(a)), κ(l(a)) = l(κ(a)), and κ(u(a)) = u(κ(a)).

10.3. Prove, for a ∈ D with 0 < a < 1, that h(a) = max{h(l(a)), h(u(a))}+

11. Constrained systems

In this section F is a saturated fusion system over the finite p-group S.

In this section we consider constrained systems. If F is constrained, then by 4.2, F = F_{S}(Ḡ) for some finite group Ḡ with S ∈ Syl_{p}(Ḡ) and CḠ(O_{p}(Ḡ)) ≤ O_{p}(Ḡ). (From Theorem A.13, this last condition is equivalent to F^{∗}(Ḡ) = O_{p}(Ḡ).) Moreover Ḡ is unique up to isomorphism by 4.4. On the other hand there can be finite groups G such that S ∈ Syl_{p}(G) and F = F_{S}(G), but F^{∗}(G) ≠ O_{p}(G).

Lemma 11.1. The following are equivalent:

(a) F is constrained.

(b) F^{∗}(F) = O_{p}(F).

(c) E(F) = 1.

Proof. Let R = O_{p}(F). Suppose (a) holds, so that C_{S}(R) ≤ R. By 9.6.c, E(F) centralizes R, so E = S ∩ E(F) ≤ C_{S}(R) = Z(R), so E is abelian. Then E(F) = 1 by Exercise 11.1, so (b) holds.

By definition of F^{∗}(F), (b) and (c) are equivalent. Assume (b) holds. By 9.6.d, C_{S}(F^{∗}(F)) ≤ F^{∗}(F), so C_{S}(R) ≤ R, and hence (a) holds. □

Lemma 11.2. Assume F is constrained. Then

(a) Each subnormal subsystem of F is constrained.

(b) Assume G is a finite group with S ∈ Syl_{2}(G) and F = F_{S}(G). Then: (a) For each H ⊴ ⊴ G, F_{S∩H}(H) is constrained.

(b) If L is a component of G, T = S ∩ L, and L̄ = L/Z(L), then FT̄(L̄) is constrained.

Proof. Exercise 11.2. □

Definition 11.3. A Bender group is a finite simple group which is of Lie type of characteristic 2 and Lie rank 1. A Goldschmidt group is a nonabelian finite simple group with a nontrivial strongly closed abelian subgroup.

The Bender groups are the groups L_{2}(q), Sz(q), U_{3}(q), where q is a suitable power of 2. By the work of Bender and Suzuki, these are the simple groups with a strongly embedded subgroup. In each case the strongly embedded subgroup is the normalizer of a Sylow 2-subgroup T of the Bender group L. Equivalently T is a TI-subgroup of L; that is distinct conjugates of T in L intersect trivially.

By a theorem of Goldschmidt in [Gd3], a nonabelian finite simple group G is a Goldschmidt group iff G is a Bender group or a Sylow 2-subgroup S of G is abelian. The groups in the latter case are L_{2}(q), q ≡ ±3 mod 8, 2G_{2}(q), and J_{1}. This follows from Walter’s original classification of simple groups with an abelian Sylow 2-subgroup, or the later simplified treatment of Bender.

Lemma 11.4. Assume G is a finite group, S ∈ Syl_{p}(G), and F = F_{S}(G). Assume in addition any one of the following hold:

(a) S is abelian, or

(b) S is a TI-subgroup of G, or

(c) S is of class 2 and Z(S) is strongly closed in S with respect to G.

Then F = F_{S}(N_{G}(S)) and S = O_{p}(F).

Proof. By Exercise 11.3, F = F_{S}(N_{G}(S)) iff S = O_{p}(F) iff N_{G}(S) controls fusion in S.

By Burnside’s Fusion Theorem (cf. A.8), if S is abelian then N_{G}(S) controls fusion in S. If (b) holds then S ∩ S^{g} = 1 for g ∈ G − N_{G}(S), so N_{G}(S) controls fusion in S. Finally suppose that (c) holds. As Z(S) is strongly closed in S with respect to G, and as F = F_{S}(G), Z(S) is strongly closed in S with respect to F. Then as S is of class 2, the series 1 < Z(S) < S satisfies condition (b) of 3.3, so S = O_{p}(F) by 3.3. Thus the lemma holds. □

Lemma 11.5. Assume p = 2, G is a nonabelian finite simple group, S ∈ Syl_{2}(G), and F = F_{S}(G). Then the following are equivalent:

(a) F is constrained.

(b) S ⊴ F.

(c) G is a Goldschmidt group. In particular either S is abelian or G is a Bender group.

Proof. If G is Bender then S is a TI-subgroup of G. Thus (c) implies (b) by 11.4. Trivially, (b) implies (a). Finally suppose (a) holds. Then there is a nontrivial abelian subgroup of S, strongly closed in S with respect to G by 3.4.b. Thus the first statement in (c) holds by definition of Goldschmidt groups, while the second statement follows from the theorem of Goldschmidt in [Gd3]. □

Exercises for Section 11

11.1. Prove that if 1 ≠ F is quasisimple then S is nonabelian.

11.2. Prove Lemma 11.2.

11.3. Assume G is a finite group, S ∈ Syl_{p}(G), and F = F_{S}(G). Prove the following are equivalent:

(a) F = F_{S}(N_{G}(S)).

(b) S = O_{p}(F).

(c) N_{G}(S) controls fusion in S.

12. Solvable systems

In this section F is a saturated fusion system over the finite p-group S.

In this section we will define the notion of a “solvable saturated fusion system”. There are at least two natural definitions of such systems, which turn out to be inequivalent. Indeed the second notion is stronger and implies the first. We will focus on the weaker of the two notions.

Definition 12.1. Define F to be solvable if all composition factors of F are of the form F_{G}(G) for G of order p. That is all composition factors of F are of order p.

As a finite group is solvable iff all its composition factors are of prime order, Definition 12.1 provides a natural extension of the notion of solvability for groups to fusion systems.

Definition 12.2. Following Puig in Chapter 19 of [P7], set S_{0} = S, F_{0} = F, and proceeding recursively, for m > 0 define S_{2m+1} = S_{2m}, F_{2m+1} = O^{p′}(F_{2m}), S_{2m+2} = 𝔥𝔶𝔭(F_{2m+1}), and F_{2m+2} = O^{p}(F_{2m+1}). Then define F to be Puig solvable if F_{n} = 1 for some positive integer n.

Notice the series (F_{n}: n) is analogous to the derived series of a group (cf. Definition A.9), and hence Puig solvability is also a natural extension of the notion of solvability for groups to fusion systems.

Example 12.3. Recall Example 10.1, where L is a nonabelian simple group with an abelian Sylow p-subgroup P of order p^{e} > p, V is an irreducible F_{p}L-module of dimension d > 1, G = LV is the semidirect product of L and V, S = V P ∈ Syl_{p}(G), and F = F_{S}(G). As we saw in Example 10.7, F has d + e composition factors, each of order p, so F is solvable. On the other hand G = O^{p}(G) = O^{p′}(G) and F^{∗}(G) = V = O_{p}(G), so F = O^{p′}(F) = O^{p}(F), and hence F is not Puig solvable.

Recall that a finite group G is p-solvable iff each of its composition factors is of order p or of order prime to p.

Theorem 12.4. (Puig) The following are equivalent:

(a) F is Puig solvable.

(b) F is constrained and for G ∈ G(F), G is p-solvable.

Proof. First suppose F is constrained and let G ∈ G(F). Set G_{0} = G and proceeding recursively, for m > 0 set G_{2m+1} = O^{p′}(G) and G_{2m+2} = O^{p}(G_{2m+1}). Then F_{n} = F_{S∩G_{n}}(G_{n}), so F is Puig solvable iff G_{n} = 1 for some n iff G is p-solvable.

Thus we may assume that F is Puig solvable, and it remains to show that F is constrained. Now F_{i} ≠ F for i = 1 or 2; let k be the least such i and E = F_{k}. Then E is constrained, so Q = O_{p}(E) is centric, so C_{S_{k}}(Q) ≤ Q. As E ⊴ F, also Q ⊴ F. If k = 1 then S = S_{k}, so C_{S}(Q) ≤ Q and hence F is constrained. Thus we may assume k = 2, so that E = O^{p}(F) and S_{k} = 𝔥𝔶𝔭(F).

As Q ⊴ F, C = C_{F}(Q) ⊴ F by Theorem 9.4. If P ≤ T = C_{S}(Q) and α ∈ Aut_{C}(P) is a p′-element, then by definition of O^{p}(F), [P, α] ≤ S_{k}, so [P, α] ≤ P ∩ S_{k} = P ∩ Z(Q) ≤ Z(P), contradicting Lemma A.2. Therefore C = F_{T} (T), so T ⊴ F. Thus QT ≤ O_{p}(F), so as QT is centric, F is constrained. □

Lemma 12.5. (a) For each normal subsystem E of F, F is solvable iff E and F/E are solvable.

(b) If F is solvable then F is constrained.

Proof. Part (a) follows from 10.17, which says the composition factors for F are the disjoint union of the factors for E and F/E.

Suppose F is solvable but not constrained. Then by 11.1, E(F) ≠ 1, so we may take F = E(F) by (a). Next each component C of F is solvable by (a), so we may take F = C. Replacing F by F/Z(F) and appealing to (a), we may take F simple. Thus F is of order p, and hence constrained. □

Definition 12.6. Define the series O_{p}^{n}(F) of subgroups of S recursively by O_{p}^{0}(F) = 1, and for n > 0, O_{p}^{n}(F) is the preimage in S of O_{p}(F/O_{p}^{n−1}(F)).

Observe that the fusion system F in Example 12.3 is a system with O_{p}(F) = V and O_{p}^{2}(F) = S. In particular O_{p}(F) < O_{p}^{2}(F).

Theorem 12.7. The following are equivalent:

(a) F is solvable.

(b) O_{p}^{n}(F) = S for some positive integer n.

(c) There exists a series 1 = S_{0} ≤ ⋯ ≤ S_{m} = S of subgroups of S such that for each 0 ≤ i < m, S_{i} is strongly closed in S with respect to F and S_{i+1}/S_{i} is abelian.

(d) F is constrained and for G ∈ G(F), F_{T} (H) is solvable for each composition factor H of G and T ∈ Syl_{p}(H).

Proof. See 15.3 in [A6]. For example suppose (a) holds. Then F is constrained by 12.5.2, so Q = O_{p}(F) ≠ 1. By 12.5.a, F/Q is solvable, so by induction on |S|, O_{p}^{n}(F/Q) = S/Q for some n. Thus O_{p}^{n+1}(F) = S, so (b) holds. □

Lemma 12.8. Assume F is solvable. Then each saturated subsystem of F is solvable.

Proof. By 12.7 there is a series 1 = S_{0} ≤ ⋯ ≤ S_{m} = S satisfying the conditions in 12.7.c. Let E be a saturated subsystem of F on T ≤ S and set T_{i} = T ∩ S_{i}. Then the series (T_{i}:0 ≤ i ≤ m) satisfies the hypotheses of 12.7.c with respect to E, so the lemma follows from 12.7. □

In the remainder of the section we work toward a determination of the simple groups whose fusion systems are solvable. This is achieved in the last theorem in the section. The proof uses the classification of the finite simple groups.

We begin by extending the definition of “Goldschmidt groups” to odd primes, by defining the notion of a p-Goldschmidt group. In this case the relevant property of Goldschmit groups G with Sylow 2-group S is that S ⊴ F_{S}(G); recall that this is one of the three equivalent conditions defining Goldschmidt groups by 11.5, so the Goldschmidt groups are indeed the 2-Goldschmidt groups. From 12.12, when p = 3 the groups G_{2}(q) with q ≡ ±1 mod 9 have a strongly closed abelian 3-subgroup, but are not 3-Goldschmidt. So when p = 3, the three conditions in 11.5 are not equivalent.

Definition 12.9. Define a nonabelian finite simple group G with p ∈ π(G) to be a p-Goldschmidt group if for S ∈ Syl_{p}(G), S ⊴ F_{S}(G).

The following two results follow from work of Foote and Flores and Foote in [F] and [FF]; their proofs use the classification of the finite simple groups.

Theorem 12.10. Let G be a nonabelian finite simple group with p ∈ π(G) and S ∈ Syl_{p}(G). Then G is a p-Goldschmidt group iff one of the following hold:

(a) S is abelian.

(b) L is of Lie type in characteristic p of Lie rank 1.

(c) p = 5 and L ≅ Mc.

(d) p = 11 and L ≅ J_{4}.

(e) p = 3 and L ≅ J_{2}.

(f) p = 5 and G ≅ HS, Co_{2}, or Co_{3}.

(g) p = 3 and G ≅ G_{2}(q) for some prime power q prime to 3 such that q is not congruent to ±1 modulo 9.

(h) p = 3 and G ≅ J_{3}.

Remark 12.11. In cases (b)-(d), we say G is p-Bender. In those cases, S is a TI-subgroup of G, or equivalently, strongly p-embedded in G. In cases (c)-(g), S ≅ p^{1+2}.

Theorem 12.12. (Flores-Foote) Assume G is a nonabelian finite simple group and S ∈ Syl_{p}(G). Suppose T is a nontrivial proper subgroup of S strongly closed in S with respect to G. Then either

(a) G is p-Goldschmidt, or

(b) p = 3, G ≅ G_{2}(q) with q ≡ ±1 mod 9, and T = Z(S) is of order 3.

Theorem 12.13. Let G be a nonabelian finite simple group with p ∈ π(G) and S ∈ Syl_{p}(G), and set F = F_{S}(G). Then the following are equivalent:

(a) F is constrained.

(b) F is solvable.

(c) S = O_{p}(F).

(d) G is p-Goldschmidt.

(e) G satisfies one of conclusions (a)–(h) of Theorem 12.10.

Proof. Trivially, (c) and (d) are equivalent and (c) implies (b). Part (b) implies (a) by 12.5.2, and (d) and (e) are equivalent by 12.10. Thus it suffices to show that (a) implies (e), so assume F is constrained but (e) fails. Then by 3.3, there is a nontrivial abelian subgroup T of S strongly closed in S with respect to F. Therefore by Theorem 12.12, conclusion (b) of that theorem holds. Then from 16.11.5 in [A6], F is quasisimple, contradicting F constrained. □

13. Fusion systems in simple groups

In Section 14 we speculate on the possibility of classifying simple saturated p-fusion systems. Before undertaking such a task, one would like to have a good feeling for known examples of simple systems. An obvious source of examples are the composition factors of the systems of finite groups. If G is a finite group, S ∈ Syl_{p}(G), H is a proper normal subgroup of G, and G^{∗} = G/H, then by 10.17 the family of composition factors of F_{S}(G) is the union of the factors of F_{S∩H}(H) and F_{S^{∗}}(G^{∗}). Thus it suffices to consider the case where G is a nonabelian finite simple group and p ∈ π(G) is a prime divisor of the order of G.

So in this section p is a prime, G is a finite group, 1 ≠ S ∈ Syl_{p}(G), and F = F_{S}(G). Thus F is a saturated fusion system over the finite p-group S. Our first result gives a sufficient criterion for F to be simple.

Lemma 13.1. Assume

(a) there exists no nontrivial proper subgroup of S strongly closed in S with respect to G, and

(b) Aut_{G}(S) = ⟨Aut_{O^{p′} (N_{G}(R))}(S):R ∈ F^{frc}⟩.

Then F_{S}(G) is simple.

Proof. We apply part (d) of Theorem 9.8. Assume conditions (a) and (b) of the lemma hold. Then by 9.8.d, it suffices to verify condition (ii) of that result. However for X ≤ S, Aut_{F}(X) = Aut_{G}(X), and for R ∈ F^{frc}, Aut_{O^{p′} (N_{G}(R))}(S) consists of the maps α = c_{g} for g ∈ O^{p′}(N_{G}(R)∩N_{G}(S)). Further for each such α, α_{|R} ∈ Aut_{O^{p′} (N_{G}(R))}(R) = B(R) ≤ Aut_{B}(R), so α ∈ Aut_{F}^{0}(S). Therefore condition (ii) of 9.8.d does indeed hold, completing the proof. □

Remark 13.2. If G is a nonabelian finite simple group which does not satisfy condition (a) of 13.1, then by 12.12, either G is p-Goldschmidt, or p = 3 and G ≅ G_{2}(q) is described in 12.12.b. In the latter case F_{S}(G) ≅ F_{S}(H) is quasisimple, where S ∈ Syl_{3}(H) and H ≅ SL_{3}^{ϵ}(q) (cf. 16.11 in [A6]). Thus in the remainder of the discussion, we may assume condition (a) of 13.1 is satisfied, so in particular G is not p-Goldschmidt and G is not G_{2}(q) if p = 3.

We next record some results on various families of nonabelian finite simple groups from [A6].

Lemma 13.3. Assume G is simple of Lie type and characteristic p, or p = 2 and G = ^{2}F_{4}(2)′ is the Tits group. Assume the Lie rank of G is at least 2. Then F is simple.

Proof. A proof appears in 16.3 in [A6], but to illustrate how to apply 13.1, we give some indication of how the proof goes.

As the Lie rank of G is at least 2, G is not p-Goldschmidt, so from Remark 13.2 it remains to verify condition (b) of 13.1. From the Borel-Tits Theorem (cf. 3.1.3 and 3.1.5 in [GLS3]), F^{frc} is the set of unipotent radicals O_{p}(P) of the proper parabolics P of G containing S. Let B = SH be the Borel subgroup over S, with H a Cartan subgroup corresponding to a root system Σ and simple system π. For α ∈ Σ, let U_{α} be the root subgroup of α. The minimal parabolics over B are of the form P_{α} = ⟨B, r_{α}⟩ for α ∈ π, where r_{α} is the fundamental reflection in the Weyl group of G corresponding to α. Thus R_{α} = O_{p}(P_{α}) ∈ F^{frc} and B = N_{P_{α}}(S).

It is more convenient to work in the universal group of type G, so replacing G by that group, we may assume G us universal rather than simple. Then from the proof of Lemma 64 in [Stn],

H_{α} = H ∩ ⟨U_{α}, U_{−α}⟩ = H ∩ O^{p′}(P_{α}).

Now as G is universal, H is the direct product of the groups H_{α}, α ∈ π (cf. 2.4.7 in [GLS3]). Hence as H is a Hall p′-subgroup of B = N_{G}(S) = N_{P_{α}}(S) for each α ∈ π, condition (b) of 13.2 does indeed hold. □

In the next lemma we use the following notation. If G is a group of permutations on a set Ω and Δ ⊆ Ω, then G_{Δ} denotes the pointwise stabilizer in G of Δ, N_{G}(Δ) denotes the global stabilizer in G of Δ, and for X ≤ N_{G}(Δ), X^{Δ} denotes the image of X in Sym(Δ) under the restriction map.

Lemma 13.4. Assume G ≅ A_{n} is an alternating group on Ω = {1, …, n} with n ≥ 6 and S ≠ 1. Write n = ap+b with 0 ≤ b < p, let X = N_{G}(M(S)), and Y = G_{M(S)}, where M(S) is the set of points of Ω moved by S. Then

(a) F_{S}(G) = F_{S}(X) ≅ F_{S}(X^{M(S)}).

(b) p ≥ n.

(c) S is abelian iff n < p^{2}.

(d) If n ≥ p^{2} and b ≤ 1 then F_{S}(G) is simple and X^{M(S)} ≅ A_{pa}.

(e) If n ≥ p^{2} and b ≥ 2 then X^{M(S)} ≅ S_{pa}, F_{S}(Y^{M(S)}) ◁ F_{S}(X^{M(S)}), and Y^{M(S)} ≅ A_{pa}.

Proof. See 16.5 in [A6]. □

Lemma 13.5. Assume G is a sporadic simple group, and let Π = Π(G) be the set of odd primes p ∈ π(G) such that |G|_{p} > p^{2}. Then

(a) S is nonabelian iff either:

(i) p ∈ Π and (G, p) ≠ (O′N, 3), or

(ii) p = 2 and G is not J_{1}.

(b) If G is M_{11}, M_{22}, M_{23}, or J_{1}, then Π = ∅.

(c) If G is M_{12}, M_{24}, J_{2}, J_{3}, Suz, F_{22}, or F_{23}, then Π = {3}.

(d) If G is a Conway group, Mc, Ru, Ly, F_{5}, F_{3}, or F_{2}, then Π = {3, 5}.

(e) If G is HS then Π = {5}.

(f) If G is He, O′N, or F_{24} then Π = {3, 7}.

(g) If G is J_{4} then Π = {3, 11}.

(h) If G is F_{1} then Π = {3, 5, 7, 13}.

Proof. See 16.6 in [A6]. □

Lemma 13.6. Assume G is a sporadic simple group, but not J_{1}, and p = 2. Then F_{S}(G) is simple.

Proof. See 16.8 in [A6]. □

Lemma 13.7. Assume G is a sporadic simple group, but not p-Goldschmidt. Then either

(a) F_{S}(G) is simple, or

(b) (G, p) = (Ru, 3), (M_{24}, 3), (Ru, 5), (J_{4}, 3), or (Co_{1}, 5), and S ∈ Syl_{p}(L) where F_{S}(G) ≅ F_{S}(L) and L ≅ ^{2}F_{4}(2), Aut(M_{12}), Aut(L_{3}(5)), 2F_{4}(2), or P O_{5}(5), respectively.

Proof. See 16.10 in [A6]. □

Remark 13.8. We are addressing the following question: Let G be a nonabelian finite simple group, p a prime divisor of |G|, S ∈ Syl_{p}(G), and F = F_{S}(G). What are the composition factors of F? In particular, when is F simple? The results listed above, together with the classification of the finite simple groups, have reduced us to the case where G is of Lie type in characteristic r ≠ p.

When p = 2 we are aware of only isolated results addressing the question. Presumably the theory of fundamental subgroups of G contained in [A1] and [A3] would be useful here. In any event, during the remainder of the discussion assume that p is odd.

The structure of S is discussed in section 4.10 of [GLS3]. In particular one can decide when S is abelian. For example if p does not divide the order of the Weyl group of the associated algebraic group, then S is abelian.

Let G be a group over F_{q}. Theorem III.1.18 tells us that the isomorphism type of F does not depend on G, but on the Lie type of G and the p-adic evaluation of certain polynomials in q. Indeed there are other isomorphisms between fusion systems of groups of Lie type, particularly among the systems of classical groups. In particular from [BMO], if G is classical then F is isomorpic to the system of L_{n}(q′) for some integer m and prime power q′. In any event, before entering into a detailed analysis of F it would seem to be best to sort out such isomorphisms, which of course are of interest in their own right.

In preliminary work, Aschbacher has tentatively analyzed F when G is an exceptional group, and in particular determined when F is simple. In [Rz], Ruiz analyzes the case where G is L_{n}(q). This work is discussed briefly in section III.6.5. In particular it can be determined when F is simple. In some cases the unique nonsolvable composition factor is exotic; see for example section 17 in [A6], where the case G ≅ L_{20}(2) and p = 5 is considered as an example, and we find that O^{5′}(F) is simple and exotic of index 4 in F; that is π_{1}(F^{c}) ≅ C_{4}.

14. Classifying simple groups and fusion systems

Since the local theory of finite groups has proven to be a sufficiently powerful tool to classify the finite simple groups, one might hope that the local theory of fusion systems could be used to classify the simple saturated p-fusion systems. Evidence suggests that the behavior of simple systems for odd primes may be more wild than the behavior of simple 2-fusion systems, so perhaps the case p = 2 should be considered first.

There are other reasons for focusing on 2-fusion systems. For example certain arguments in local group theory seem to be easier to implement in saturated fusion systems than in groups. Thus there is some hope that the proof of the classification of the finite simple groups might be simplified by, first, proving suitable results about saturated 2-fusion systems, and then, second, using the theorems on fusion systems to prove results about finite simple groups. In this section and its sequel, we speculate a bit about the possibility of implementing these two programs.

Let F be a saturated fusion system on a finite p-group S.

Definition 14.1. Define F be be of characteristic p-type if for each U ∈ F^{f}, N_{F}(U) is constrained. Define F to be of component type if for some fully normalized subgroup X of S of order p, C_{F}(X) has a component.

Remark 14.2. Let G be a finite group. Then G is said to be of characteristic p-type if for each p-local subgroup H of G, C_{H}(O_{p}(H)) ≤ O_{p}(H). Further G is of component type if for some involution t in G, C_{G}(t)/O(C_{G}(t)) has a component.

It is easy to see that if G is of characteristic p-type than its p-fusion system is of characteristic p-type. If G is of component type the situation is a bit more subtle. If L is a component of C_{G}(t)/O(C_{G}(t)) with Sylow 2-group T, it may be the case that F_{T} (L) is solvable. Indeed from 12.13 this will be the case precisely when L is a Goldschmit group. If for some involution t and component L of C_{G}(t)/O(C_{G}(t)), L is not Goldschmidt, then F_{S}(G) is of component type.

Furthermore the classification of the finite simple groups proceeds by partitioning the simple groups into those of small 2-rank, those of component type, and those of characteristic 2-type. One of the first major steps in the classification of the finite simple groups is the proof that each finite simple group falls into one of these three classes (cf. [G2]). The proof of this result requires the Bender-Suzuki work on groups with a strongly embedded subgroup, signalizer functor theory, and the Gorenstein-Walter theorem on L-balance. On the other hand we see in a moment that the analogous result for fusions systems is much easier, requiring only the Lbalance theorem for fusion systems. This begins to show why one might hope to be able to classify simple 2-fusion systems, and use some parts of that proof to simplify the proof of the classification of the finite simple groups.

Theorem 14.3. Let F be a saturated p-fusion system. Then either

(a) F is of characteristic p-type, or

(b) F is of component type.

Proof. Assume F is not of characteristic p-type. Then there exists some U ∈ F^{f} such that N = N_{F}(U) is not constrained. Therefore by 11.1, E(N) ≠ 1. Let T = N_{S}(U) and X of order p in Z(T) ∩ U. Then X is fully normalized in N and centralizes E(N). Therefore (cf. 10.3 in [A6]) E(N) = E(C_{N} (X)). Set C = C_{F}(X). Then C_{N} (X) = N_{C}(U), so E(N_{C}(X)) ≠ 1. Conjugating in F and using 3.1, we may assume X ∈ F^{f} and U ∈ C^{f}. Therefore by L-balance, Theorem 9.7, 1 ≠ E(N_{C}(U)) ≤ E(C), completing the proof. □

Remark 14.4. Theorem 14.3, together with the proof of the classification of the finite simple groups, suggest that we should partition the problem of classifying simple saturated p-fusion systems into two subproblems:

Problem 14.4.1 Classify all simple saturated p-fusion systems of component type.

Problem 14.4.2 Classify all simple saturated fusion systems of characteristic p-type.

However to avoid difficulties in the first problem, associated with certain wreath products and extensions of groups of Lie type of characteristic p by field automorphisms of order p, there is reason to believe it might be better to modify the partition corresponding to the two problems. For example this is one of the guiding principals of the GLS approach to revising the classification of the finite simple groups. This leads to the following definitions, where we specialize to the case p = 2.

Definition 14.5. Let p = 2, P a 2-group, and recall the definition of the Thompson subgroup J(P) and the Baumann subgroup Baum(P) of P from Definition A.15.

Define F to be of even characteristic if N_{F}(U) is constrained for all 1 ≠ U ⊴ S. Define F to be of even component type if C_{F}(z) has a component for some involution z in the center of S,

Define F to be of Baumann characteristic 2 if N_{F}(U) is constrained for each U ∈ F^{f} with Baum(S) ≤ N_{S}(U). Define F to be of Baumann component type if there exists X ∈ F^{f} of order 2 such that X ≤ Z(Baum(S)) and C_{F}(X) contains a component.

Example 14.6. Let L be a group of Lie type in characteristic 2, t an involutory outer automorphism of L, and G = ⟨t⟩L the semidirect product of L by ⟨t⟩. Let t ∈ S ∈ Syl_{2}(G) and S_{L} = S ∩ L. By the Borel-Tits Theorem (cf. 3.1.3 in [GLS3]) L is of characteristic 2-type. On the other hand, with a few small exceptions, C_{L}(i) has a component for some involution i ∈ tG, so G is of component type. However, as is well known, Z(S) ≤ S_{L}, so F = F_{S}(G) is of even characteristic. Moreover, again with some small exceptions, Baum(S) = Baum(S_{L}), so G is also of Baumann characteristic 2.

Example 14.7. Let L be a nonabelian finite simple group and G the wreath product of L by C_{2}. Thus G is a semidirect product G = ⟨t⟩H, where H = L_{1} × L_{2} with L_{i} ≅ L, and t is an involution with L_{1}^{t} = L_{2}. As G is a wreath product, C_{L}(t) = {ll^{t}: l ∈ L_{1}} ≅ L is a component of C_{G}(t), so G is of component type, and if L is not a Goldschmidt group then F = F_{S}(G) is also of component type, where t ∈ S ∈ Syl_{2}(G).

Let S_{i} = S ∩ L_{i}, so that S = ⟨t⟩S_{1}S_{2} is also a wreath product. In particular Z(S) = {zz^{t}: z ∈ Z(S_{i})} is diagonally embedded in L_{1} × L_{2}, so F is of even characteristic iff F_{S_{1}}(L_{1}) is of even characteristic.

Finally set J_{i} = J(S_{i}) be the Thompson subgroup of S_{i} for i = 1, 2. As

m_{2}(C_{S}(t)) = 1 + m_{2}(L) < 2m_{2}(L) = m_{2}(H)

since m_{2}(L) > 1, it follows that J(S) = J_{1}×J_{2}, and then Baum(S) = B_{1}B_{2}, where B_{i} = Baum(S_{i}). In particular G is of Baumann component type as L_{1} is a component of C_{G}(z_{2}) for z_{2} an involution in Z(S_{2}). Thus F is of Baumann component type unless L is a Goldschmidt group.

As we will see in a while, these examples suggest that it may be best to partition the classification of the simple saturated 2-fusion systems either into those of even characteristic and those of even component type, or into those of Baumann characteristic 2 and those of Baumann component type.

In the remainder of the section assume p = 2. We engage in some speculation as to how to classify the simple 2-fusion systems of component type. The idea is of course to imitate the proof of the classification of the finite simple groups of component type. Here is an outline of the major steps in such a program:

Problem 14.8. Translate the notion of a “classical involution” [A1] from finite group theory into the realm of 2-fusion systems, and prove an analogue of the classical involution theorem for 2-fusion systems. This result would provide a characterization of almost all the 2-fusion systems of groups of Lie type in odd characteristic.

More specifically, consider the following class of fusion systems.

Definition 14.9. A quaternion fusion packet is a pair (F, Ω) where F is a saturated fusion system on a finite 2-group S, and Ω is a collection of subgroups of S such that Ω^{F} = Ω and

(a) There exists e ≥ 3 such that for all K ∈ Ω, K has a unique involution z(K) and K is nonabelian of order 2^{e}.

(b) For each pair of distinct K, J ∈ Ω, |K ∩ J| ≤ 2.

(c) If K, J ∈ Ω and v ∈ J − Z(J), then v^{F} ∩ C_{S}(z(K)) ⊆ N_{S}(K).

(d) If K, J ∈ Ω with z = z(K) = z(J), v ∈ K, and ϕ ∈ Hom_{C_{F} (z)}(⟨v⟩, S), then either vϕ ∈ J or vϕ centralizes J.

Then try to extend the various Theorems in [A1] to results about quaternion fusion packets. In particular if F is simple, show that (essentially) F is the fusion system of some group of Lie type and odd characteristic, coming from one of the following examples:

Example 14.10. Let p be an odd prime and G a group of Lie type over a finite field of characteristic p, other than L_{2}(q) or ^{2}G_{2}(q). Let Σ be a root system for G, and for α a long root in Σ, write U_{α} for the center of the root group of α and set L_{α} = ⟨U_{α}, U_{−α}⟩. The conjugates of L_{α} are the fundamental subgroups of G. Each fundamental subgroup L is isomorphic to SL_{2}(q) for some power q of p; in particular L has quaternion Sylow 2subgroups and a unique involution z = z(L). Moreover L is subnormal in C_{G}(z), so L is a component of C_{G}(z), unless q = 3 where L is a “solvable component” in the sense that SL_{2}(3) is almost quasisimple.

Pick a Sylow 2-subgroup S of G and let Ω consist of the groups S∩L such that L is a fundamental subgroup and S ∩ L ∈ Syl_{2}(L). Set F = F_{S}(G). Then (F, Ω) is a quaternion fusion packet and these Lie packets are the generic examples of quaternion fusion packets. There are however other infinite families of examples, many of which are constrained. There are also a few sporadic examples. Most of the examples are subpackets of Lie packets.

Aschbacher has done some preliminary work on this problem.

Problem 14.11. Prove an analogue of the theorem of Walter [W] classifying finite simple groups in which some involution centralizer has a 2component of Lie type in odd characteristic. The proof would proceed by reducing to Problem 14.8 in most cases.

More specifically, consider the following hypothesis (or something like it):

Hypothesis W. F is a saturated fusion system on a finite 2-group S with F^{∗}(F) quasisimple. Assume there exists an involution i ∈ S such that ⟨i⟩ ∈ F^{f} and C_{F}(i) has a component in Chev^{∗}(r) for some odd prime r.

Here Chev^{∗}(r) is essentially the class of fusion systems of quasisimple groups of Lie type and odd characteristic, distinct from L_{2}(r^{e}) and ^{2}G_{2}(3^{e}) when r = 3. One would like to show that if Hypothesis W holds, then, with known exceptions, there is some collection Ω of subgroups of S such that (F, Ω) is a quaternion fusion packet. Note that one class of exceptions are the exotic 2-fusion systems constructed by Levi and Oliver in [LO]. These arise in [W] during the proof of Proposition 4.3 of that paper.

Problem 14.12. Prove an analogue of the Component Theorem in [A2] from finite group theory for 2-fusion systems. The result would say that, modulo known exceptions, if F is a saturated fusion system on a finite 2-group S, and there exists an involution i ∈ S such that ⟨i⟩ ∈ F^{f} and C_{F}(i) is not constrained, then there exists a “standard component” in the centralizer of some involution.

Let G be a finite group. A subgroup K of G is tightly embedded in G if K is of even order but K ∩ K^{g} is of odd order for each g ∈ G − N_{G}(K). A quasisimple subgroup L of G is a standard component of G if K = C_{G}(L) is tightly embedded in G, N_{G}(K) = N_{G}(L), and L commutes with none of its conjugates.

One would need to settle on the “right” definition of a “standard component” of a 2-fusion system. This would involve finding an analogue of the notion of a tightly embedded subgroup from [A2], and constructing a theory of tightly embedded subsystems of 2-fusion systems.

Problem 14.13. Prove the various Standard Form Theorems. That is prove that if F is a saturated 2-fusion system with a standard component of a given isomorphism type, then the system F is isomorphic to a known system.

This is the step where it would be easier to deal with a standard component L either in the centralizer of an involution in the center of S, or normalized by the Baumann subgroup Baum(S) of S. Working with such a component would avoid treating cases leading to the following two situations:

Situation A. F is the 2-fusion system of the wreath product of a group L of Lie type and characteristic 2 by C_{2}.

Situation B. F is the 2-fusion system of a group of Lie type in characteristic 2 by an involutory field automorphism.

For we saw in Examples 14.6 and 14.7 that neither situation can occur in a fusion system of even component type, and only Situation A can occur in a system of Baumann component type. Even in Situation A with F = F_{S}(G) for G as in Example 14.7, L_{1} is not a standard component in the group G as L_{1} commutes with its conjugate L_{2}. In addition we also have the strong condition L_{1} ⊴ C_{G}(i) for each involution i ∈ L_{2} = C_{G}(L_{1}). Thus one can hope that Situation A also presents relatively few difficulties in the context of Problem 14.13.

If one takes this approach, then Problem 14.12 would also have to be modified to prove that if F is of even component type or of Baumann component type, then there exists a standard component in the centralizer of an involution in the center of S or normalized by Baum(S), respectively.

Remark 14.14. Observe that this program does not include any steps analogous to the treatment of the B-Conjecture or the Unbalanced Group Conjecture, two extremely difficult steps in the classification of the finite simple groups (cf. [G2]). We recall that the B-Conjecture asserts that, for a finite simple group G and an involutory automorphism t of G, L(C_{G}(t)) = E(C_{G}(t)), where L(C_{G}(t)) = O^{p′}(Ê) and Ê is the preimage in C_{G}(t) of E(C_{G}(t)/O(C_{G}(t))). Similarly the Unbalanced Group Conjecture supplies a list of the simple groups G and involutory automorphisms t such that O(C_{G}(t)) ≠ 1.

Such steps are unnecessary in the category of fusion systems since they involve the cores O(H) of 2-local subgroups H of a minimal counter example to the classification. But from Exercise 2.1, the 2-fusion systems of H and H/O(H) are the same, so cores of 2-locals are not obstructions when we work in the category of 2-fusion systems.

15. Systems of characteristic 2-type

In this section F is a saturated fusion system over the finite 2-group S. We will be primarily interested in the case where F is of characteristic 2-type.

Unlike the case of systems of component type, we have no grand program to offer aimed at classifying the simple saturated 2-fusion systems of characteristic 2-type. This is because the corresponding problem in finite group theory—the classification of the finite simple groups of characteristic 2-type—was achieved, in the generic case, by switching attention from 2-locals to p-locals, for suitable odd primes p. Of course such a change of point of view would appear to be impossible in the category of 2-fusion systems.

However the small groups of characteristic 2-type were treated via 2-local analysis, and there is an ongoing program of Meierfrankenfeld, Stellmacher, and Stroth [MSS] to treat all groups of characteristic 2-type 2-locally. Thus there are some techniques available to steal from finite group theory.

Therefore in this section we discuss some results which put in place machinery to begin the analysis of 2-fusion systems of characteristic 2type, and then consider two theorems which classify certain special classes of systems. These theorems are analogues of results from finite simple group theory which began the study of groups of characteristic 2-type, and served as test cases for the classification of such groups. Their analogues for 2-fusion systems can thus be regarded as test cases for classifying systems of characteristic 2-type.

A finite group H is a K-group if each simple section of H is on the list K of “known” simple groups appearing in the statement of the theorem classifying the finite simple groups. In a minimal counter example to that theorem, each proper subgroup is a K-group. Presumably we also wish to classify the simple 2-fusion systems using an inductive proof, so we are lead to:

Definition 15.1. Define a saturated fusion system D on a finite p-group D to be a local CK-system if for each nontrivial subgroup P of S, Aut_{F}(P) is a K-group.

For example if G is a minimal counter example to the classification and S ∈ Syl_{p}(G), then F_{S}(G) is a local CK-system. Since we hope our results can be used to simplify portions of the proof of the classification theorem, it is important not to use that theorem in their proofs, but instead to prove theorems about local CK-systems.

The following collection of 2-fusion systems share certain unusual behavior.

Definition 15.2. Define F to be an obstruction to pushing up at the prime 2 if F = F_{S}(G) for some finite group G with S ∈ Syl_{2}(G) such that one of the following holds:

(a) S is dihedral of order at least 16 and G ≅ L_{2}(q) or P GL_{2}(q) for some odd prime power q.

(b) S is semidihedral and G ≅ L_{2}(q)^{(1)} for some odd prime power q which is a square.

(c) S is semidihedral of order 16 and G ≅ L_{3}(3).

(d) |S| = 32 and G ≅ Aut(A_{6}) or Aut(L_{3}(3)).

(e) |S| = 2^{7} and G ≅ J_{3}.

(f) F^{∗}(G) ≅ Sp_{4}(q), where q = 2^{e} with e > 1 odd, |G:O^{2}(G)| = 2, and O^{2}(G) is an extension of F^{∗}(G) by a group of field automorphisms of odd order.

(g) F^{∗}(G) ≅ L_{3}(q), q = 2^{e}, e > 1, |O^{2}(G):F^{∗}(G)| is odd, and G is the extension of O^{2}(G) by a graph or graph-field automorphism.

Here L_{2}(q)^{(1)} is the extension of L_{2}(q) of degree 2 with semidihedral Sylow 2-subgroups.

What is pushing up? The term comes from the local theory of groups of characteristic 2-type, but we will reinterpret it for 2-fusion systems. Let U be the set of radical centric subgroups of S which are normal in S. By Alperin’s Fusion Theorem 3.5, F = ⟨Aut_{F}(R): R ∈ F^{frc}⟩; that is the automizers of fully normalized radical centric subgroups of S control fusion. Observe that if U ⊴ S then U is fully normalized. Moreover we would like to work, as much as possible, in local subsystems containing S. Thus we would like to show that F = ⟨N_{F}(U): U ∈ U⟩. Unfortunately this is not always the case.

In attempting to prove such a result, we are led to attempt to “push up” the local systems N_{F}(R), for R ∈ F^{frc} with R not normal in S. That is we try to show N_{F}(R) ≤ ⟨N_{F}(Q): Q ∈ Q⟩ for some suitable subset Q of N_{F}(R)^{f} such that |N_{S}(Q)| > |N_{S}(R)| for Q ∈ Q.

When F is of characteristic 2-type, “pushing up” techniques borrowed from local group theory allow us to show that either F = ⟨N_{F}(U): U ∈ U⟩ or F is one of the obstructions to pushing up appearing in 15.2. Then as a corollary we obtain the smaller list of finite groups G of characteristic 2-type in which 2-fusion is not controlled in the normalizers of normal subgroups of a Sylow 2-subgroup of G. This is a new theorem for groups, giving yet another example of how it is sometime easier to prove results about fusion systems as a stepping stone to theorems about finite groups.

The theorems we just discussed appear in [A7]. Indeed it is possible to prove an even stronger result. We now state the main theorem from [A7]. Recall the Baumann subgroup Baum(S) of S is defined in Definition A.15.

Theorem 15.3. Assume F is a saturated fusion system on a finite 2-group S, such that F is a local CK-system of characteristic 2-type. Then one of the following holds:

(a) F = ⟨C_{F}(Ω_{1}(Z(S))), N_{F}(Baum(S))⟩.

(b) Campbell pairs for F exist, and for each such pair (C_{1}, C_{2}), F = ⟨C_{F}(C_{1}), N_{F}(C_{2})⟩.

(c) There exists a finite group G with S ∈ Syl_{2}(G) such that F = F_{S}(G) and either G is M_{23} or G is an extension of L_{3}(2^{e}) or Sp_{4}(2^{e}) by a group of odd order for some positive integer e.

(d) F is an obstruction to pushing up at the prime 2.

Remark 15.4. An F-characteristic subgroup of a subgroup P of S is a subgroup C which is Aut_{F}(P)-invariant. A Baumann pair for F is a pair (C_{1}, C_{2}) of nontrivial F-characteristic subgroups of S, such that C_{1} ≤ Z and C_{2} is an F-characteristic subgroup of B, where Z = Ω_{1}(Z(S)) and B = Baum(S). Thus for example, (Z, B) is the “largest” Baumann pair. Campbell pairs are certain Baumann pairs; the precise definition is technical, so we will only give a rough idea here.

Assume F is of even characteristic, as defined in Definition 14.5, and, as earlier, let U be the set of radical centric subgroups of S which are normal in S. Pick U ∈ U and let G be a model for N_{F}(U). A minimal parabolic of G is an overgroup M of S in G such that S is not normal in M and S is contained in a unique maximal subgroup of M. By a result of McBride (cf. B.6.3 in [ASm]), G is generated by N_{G}(S) and its set 𝔐 = 𝔐(U) of minimal parabolics. Then by 2.5 and 7.12 in [A7],

N = N_{F}(U) = ⟨C_{N} (Z), N_{N} (B), F_{S}(M): M ∈ 𝔐⟩.

So if

(*) for each M ∈ 𝔐(U), either Z ≤ Z(M) or B ⊴ M,

then N = ⟨C_{N} (Z), N_{N} (B)⟩. Therefore if F = ⟨N_{F}(U): U ∈ U⟩ and (*) holds for each U ∈ U, then conclusion (1) of Theorem 15.3 holds. So suppose that (*) fails for some U ∈ U and M ∈ 𝔐. Then (cf. 8.4 in [A7]) there exists a subgroup H of M with B ∈ Syl_{2}(H) such that H is a minimal parabolic in the set G(B) of Definition 1.3 in [A7]. Write 𝔊(F) for the set of such groups H as U, M varies over pairs for which (*) fails.

A Glauberman-Niles pair is a Baumann pair (C_{1}, C_{2}) such that for each H ∈ 𝔊(F), either C_{1} ≤ Z(H) or C_{2} ⊴ H, or H has just one noncentral chief factor in O_{2}(H). By a theorem of Glauberman and Niles in [GN], Glauberman-Niles pairs exist. Moreover in his thesis [Cm], Campbell gave a constructive proof of the existence of such a pair. Namely given H ∈ 𝔊(F), set F = Ω_{1}(Z(J(O_{2}(H)))) and V = [O^{2}(H), V]. Then set C_{1} = Z ∩⟨C_{V} (B)^{Aut_{F} (B)}⟩ and C_{2} = ⟨F^{Aut_{F} (B)}⟩. The pair (C_{1}, C_{2}) is a Campbell pair.

What is the significance of Theorem 15.3? We wish to analyze some system F of characteristic 2-type. Applying Theorem 15.3, we may assume we are in case (1) or (2) of that theorem, so we have a pair (C_{1}, C_{2}) with C_{1} an F-characteristic subgroup of S contained in Ω_{1}(Z(S)), C_{2} F-characteristic in Baum(S), and F = ⟨C_{F}(C_{1}), N_{F}(C_{2})⟩. As F is of characteristic 2type there are models G_{1} for F_{1} = C_{F}(C_{1}) and G_{2} for F_{2} = N_{F}(C_{2}). Let F_{1,2} = C_{F_{2}}(C_{1}) and let G_{1,2} be a model for F_{1,2}. We can choose G_{1,2} = C_{G_{2}}(C_{1}). Further F_{1,2} = N_{F_{1}}(C_{2}), so N_{G_{1}}(C_{2}) is also a model for F_{1,2}, so by 4.4 there is an isomorphism α_{1} : G_{1,2} → N_{G_{1}}(C_{2}) extending the identity map on S. Form the amalgam

A = (G_{1} ⟵^{α_{1}} G_{1,2} ⟶^{α_{2}} G_{2}),

where α_{2} is the inclusion map. Let G be the universal completion of A; that is G is the free group on G_{1} ∪ G_{2} ∪ G_{1,2} modulo relations defining the groups and identifying elements of G_{1,2} with those in G_{i} via the maps α_{i}, for i = 1, 2. Then

Lemma 15.5. F = F_{S}(G), so the amalgam A determines the fusion system F.

It follows from 15.5 that, given a suitable set Φ of local conditions on F, if we can show that A is determined up to isomorphism by Φ, then also F is determined up to isomorphism by Φ. That is we can hope to characterize 2-fusion systems of characteristic 2-type via sets Φ of local conditions using Theorem 15.3.

Moreover Theorem 15.3 says that fusion in F is controlled by the normalizers of nontrivial normal subgroups U of S. Notice each such U is fully normalized, and hence N_{F}(U) has a model G(U) which has S as a Sylow 2-subgroup. There is a natural equivalence relation on the set of pairs (U, H) with 1 ≠ U ⊴ S and S ≤ H ≤ G(U); let H = H(F) be the set of equivalence classes of this equivalence relation and write [U, H] for the equivalence class of (U, H), or sometimes just H. There is also a natural partial order ≤ on H, making H into a poset. Indeed if F = F_{S}(G) for some finite group G, then H can be regarded as the set H(G) of overgroups H of S in G with O_{2}(H) ≠ 1, and the relation ≤ is the subgroup relation. This theory is developed in [A8].

One would like to prove a version of Theorem 15.3 for systems of Baumann characteristic 2, or even better, for systems of even characteristic. For if one partitions systems into those of Baumann characteristic 2 and Baumann component type, in order to simplify the analysis in the “component type” case, as discussed near the end of Section 14, then one is left with the problem of dealing with systems of Baumann characteristic 2, where the first step in the analysis would be to prove some extension of Theorem 15.3.

The 2-local theory of finite groups of characteristic 2-type focuses on the poset H(G), and many results in the theory are theorems about this poset. We can hope to prove analogues of those theorems for the poset H(F). For example Theorem 5.4 in [A8] proves such an analogue of the Stellmacher qrc-Lemma, which appears for example as Theorem D.1.5 in [ASm].

The statement of the qrc-Lemma is technical, so we will not give it here. Interpreted in the language of fusion systems, it considers the situation where [U_{i}, G_{i}], i = 1, 2, are equivalence classes in H = H(F) such that

(qrc1) no nontrivial subgroup of S is normal in both G_{1} and G_{2}; and

(qrc2) V is a nontrivial normal elementary abelian 2-subgroup of G_{1} such that O_{2}(Aut_{G_{1}}(V)) = 1 and R = C_{S}(V) ⊴ G_{1}; and

(qrc3) G_{2} is a minimal parabolic; that is S is not normal in G_{2} and S is contained in a unique maximal subgroup of G_{2}.

In this situation the qrc-Lemma says that the pair (G_{1}, G_{2}) lives in one of five cases, with the cases distinguished by properties of the representation of G_{1} on V and the embedding of V and R in G_{2}.

How does one find pairs [U_{i}, G_{i}], i = 1, 2, satisfying the hypotheses (qrc1)–(qrc3) of the qrc-Lemma? First, if H has a unique maximal member [M, U] then (qrc1) is never satisfied, as U ⊴ H for each H ∈ H. On the other hand in this situation, Theorem 15.3 says that either F is an obstruction to pushing up, or U ⊴ F. Thus one consequence of Theorem 15.3 is that we may assume that H has at least two maximal members.

Let H ∈ H, Z = Ω_{1}(Z(S)), and V (H) = ⟨Z^{H}⟩. Then as F is of characteristic 2-type, O_{2}(Aut_{H}(V (H))) = 1 (cf. B.2.13 in [ASm]). Set R = C_{S}(V (H)). By a Frattini argument, H = C_{H}(V (H))N_{H}(R) and hence [R, N_{H}(R)] ∈ H with V (H) = V (N_{H}(R)) and Aut_{H}(V (H)) = Aut_{N_{H}(R)}(V (H)). Moreover R = C_{S}(V (H)) ⊴ N_{H}(R). That is G_{1} = N_{H}(R) satisfies condition (qrc2).

Now specialize to the case where H is maximal in H and recall we may assume there exists a second maximal member M of H. It follows that condition (qrc1) is satisfied by the pair (H, M). However we wish to work with G_{1} rather than H. To do so we work with a second partial order ⪯ on H and choose H to also be maximal with respect to ⪯. As a consequence of the maximality of H with respect to ⪯, results in [A8] show that H is the unique member of H maximal under < containing G_{1}, so the pair (G_{1}, M) satisfies (qrc1), and indeed for each G_{2} ≤ M such that S ≤ G_{2} ≰ H, (G_{1}, G_{2}) satisfies (qrc1). Choose G_{2} minimal subject to these constraints.

By a result of McBride (cf. B.6.3 in [ASm]), M is generated by N_{M}(S) and the minimal parabolics of M. Thus if N_{M}(S) ≤ H then we may choose G_{2} to be a minimal parabolic of M, and we have achieved the hypotheses of the qrc-Lemma. The case where N_{F}(S) is not contained in H is even nicer; for example in that case no nontrivial characteristic subgroup of S is normal in H, so the structure of H and its action on V (H) is determined by the local C(G, T)-Theorem (cf. C.1.29 in [ASm]).

Using the poset H(F) and the qrc-Lemma, it is possible to prove an analogue of Glauberman’s theorem classifying S_{4}-free finite simple groups, for 2-fusion systems. Indeed the theorem for fusion systems is more natural and attractive than the theorem for groups, and can be used to give a simple proof of Glauberman’s theorem. We begin to describe this constellation of ideas.

Let G be a finite group. A section of G is a group of the form H/K, where K ⊴ H ≤ G. Given a positive integer n, write S_{n} for the symmetric group on a set of order n. The group G is S_{n}-free if G has no section isomorphic to S_{n}.

One of the striking results in finite group theory from the decade before the classification of the finite simple groups, was the classification of the S_{4}- free finite simple groups. The first result in this direction was Thompson’s proof (cf. [Th2]) that the Suzuki groups Sz(2^{2m+1}) are the only nonabelian finite simple groups of order prime to 3. Next in [Gl2], Glauberman proved a triple factorization theorem for constrained S_{4}-free groups, and used this result to show each S_{4}-free group has a nontrivial strongly closed abelian 2-subgroup. As a corollary to this result and Goldschmidt’s theorem in [Gd3] classifying finite groups with such a subgroup, Glauberman classified the S_{4}-free and S_{3}-free nonabelian finite simple groups. Later in [St1], Stellmacher showed that in a constrained S_{4}-free group G, there is a nontrivial characteristic subgroup of a Sylow 2-subgroup of G normal in G; this theorem can be used to give an alternate treatment of S_{4}-free groups.

What are the analogues of these results for 2-fusion systems?

Definition 15.6. F is S_{3}-free if for each subgroup U of S, the group Aut_{F}(U) is S_{3}-free.

Theorem 15.7. Let F be a saturated S_{3}-free fusion system on a finite 2-group S. Then F is constrained.

Proof. See [A8]. □

Onofrei and Stancu [OS] have proved a result on S_{3}-free 2-fusion systems stronger than Theorem 15.7, using Stellmacher’s theorem on constrained S_{4}-free groups.

Once the general machinery for fusion systems of characteristic 2-type is in place, particularly Stellmacher’s qrc-Lemma for fusion systems, the fusion system theoretic proof of Theorem 15.7 is only a few paragraphs, and hence to our minds, more attractive than the group theoretic proofs of Glauberman’s theorem. This is another example of how certain results about finite groups are best proved fusion system theoretically.

Moreover Theorem 15.7, together with Goldschmidt’s Theorem, leads almost immediately to the following corollaries for groups:

Corollary 15.8. Assume G is a finite S_{4}-free nonabelian finite simple group. Then G is a Goldschmidt group.

Corollary 15.9. Assume G is a finite S_{3}-free nonabelian finite simple group. Then G is Sz(2^{2m+1}) or L_{2}(3^{2m+1}), with m ≥ 1.

Corollary 15.10. Assume G is a finite nonabelian simple group of order prime to 3. Then G ≅ Sz(2^{2m+1}) for some m ≥ 1.

Finally we come to the second of our seminal papers on groups of characteristic 2-type. Recall that an N-group is a nonabelian finite simple group G such that all local subgroups of G are solvable. In a monumental series of papers [Th1], Thompson determined all N-groups. Thompson’s work on N-groups was a test case for the classification of the finite simple groups, and served as a blue print for the classification.

In [J], [Sm], and [GL], Janko, F. Smith, and Gorenstein and Lyons extended Thompson’s work to determine all finite groups in which all 2local subgroups are solvable. Later in [St2], Stellmacher gave an alternate approach to treating finite simple groups of characteristic 2-type in which all 2-locals are solvable, based on the amalgam method.

It should be noted that, while N-groups need not be of characteristic 2-type, from the modern point of view, N-groups which are not of characteristic 2-type are relatively easy to treat; this is basically the reduction described in Remark 14.2. Of course the machinery required to make that reduction was not available to Thompson when he did his work on Ngroups. Rather Gorenstein and Walter were led to the reduction by an analysis of Thompson’s treatment of that case in the N-group paper.

Our goal is to carry out an analysis of groups in which all 2-locals are solvable in the category of 2-fusion systems. This leads to the following definition:

Definition 15.11. Define F to be an N-system if for each P ∈ F^{f}, N_{F}(P) is constrained and Aut_{F}(P) is solvable.

For example if G is a finite group in which all 2-local subgroups are solvable, and S ∈ Syl_{2}(G), then F_{S}(G) is an N-system. The converse is not true however, as we see in a moment. Such examples give yet another hint that some theorems about finite groups have more attractive statements and proofs in the language of fusion systems.

Remark 15.12. Observe that F is an N-system iff for each P ∈ F^{f}, N_{F}(P) is Puig solvable. This follows from Theorem 12.4 and the fact that groups of odd order are solvable. Since Puig solvability is one of the two natural notions of solvability in the category of 2-fusion systems, our notion of an N-system is a natural translation of the notion of an N-group into the category of 2-fusion systems. Thus the classification of N-systems serves as a test case for classifying all simple 2-fusion systems of characteristic 2-type, just as the N-group paper was a test case for classifying simple groups of characteristic 2-type.

Of course instead of working with N-systems, one could work with systems F in which, for each P ∈ F^{f}, N_{F}(P) is solvable. However this class of systems is much larger than the class of N-systems, and not as closely related to the 2-fusions systems of N-groups. For example for each group G of Lie type and Lie rank 2 over a field of even order, the 2-fusion system of G is in the larger class.

Theorem 15.13. Assume F is an N-system. Then there exists a finite group G with S ∈ Syl_{2}(G) such that F = F_{S}(G), and one of the following holds:

(a) G is a solvable group with F^{∗}(G) = O_{2}(G).

(b) S is dihedral of order at least 16, and G ≅ L_{2}(q) or P GL_{2}(q) for some odd prime power q.

(c) S is semidihedral and G ≅ L_{2}(q)^{(1)} for some odd prime power q which is a square.

(d) S is semidihedral of order 16 and G ≅ L_{3}(3).

(e) |S| = 32 and G ≅ Aut(A_{6}) or Aut(L_{3}(3)).

(f) G ≅ L_{3}(2) or Sp_{4}(2).

(g) G is isomorphic to U_{3}(3), G_{2}(2), M_{12}, Aut(M_{12}), ^{2}F_{4}(2)′, or ^{2}F_{4}(2).

Proof. This appears in the preprint [A9]. □

Notice that M_{12} is not a N-group; indeed the centralizer of a non-2-central involution of M_{12} is isomorphic to C_{2} × S_{5}. However the 2-fusion system of M_{12} is an N-system. As a corollary to Theorem 15.13, we obtain an alternate approach to the treatment of finite groups in which all 2-local subgroups are solvable:

Theorem 15.14. (Thompson, Janko, Smith, Gorenstein, Lyons) Assume G is a finite group in which all 2-local subgroups are solvable. Let S ∈ Syl_{2}(G) and set G^{∗} = G/O(G) and H^{∗} = O^{2′}(G^{∗}). Then one of the following holds:

(a) G is solvable.

(b) S is dihedral and H^{∗} ≅ A_{7}, L_{2}(q), or P GL_{2}(q) for some odd prime power q.

(c) S is semidihedral and H^{∗} ≅ L_{2}(q)^{(1)} for some odd prime power q which is a square.

(d) S is semidihedral of order 16 and H^{∗} ≅ L_{3}(3) or M_{11}.

(e) |S| = 32 and H^{∗} ≅ Aut(A_{6}) or Aut(L_{3}(3)).

(f) H^{∗} ≅ Sp_{4}(2) or S_{7}.

(g) H^{∗} is isomorphic to U_{3}(3), G_{2}(2), ^{2}F_{4}(2)′, or ^{2}F_{4}(2).

(h) H^{∗} is a Bender group.

The proof of Theorem 15.14, based on Theorem 15.13, also appears in the preprint [A9]. That proof uses the classification of simple groups with dihedral Sylow 2-subgroups by Gorenstein and Walter in [GW1], or the more modern treatment of such group by Bender and Glauberman in [Ben] and [BG]. It also requires a classification of simple groups whose Sylow 2-subgroup is semidihedral of order 16, or wreathed of order 32, in which the centralizers of involutions are solvable (cf. [ABG] or [GLS6]). The proof also appeals to Goldshmidt’s theorem classifying simple groups with a strongly closed abelian 2-subgroup [Gd3], the solvable 2-signalizer functor theorem. (cf. Chapter 15 in [A4]), the Feit-Thompson Theorem [FT], and suitable characterizations of M_{12} and the Tits group.

This shows the strengths and limitations of any approach to proving theorems about finite simple groups in the category of fusion systems. If the 2-local structure of a group G is reasonably rich, then it seems often to be relatively easy to retrieve the group from its fusion system. On the other hand if the 2-local structure is sparse, the fusion system seems to give relatively little information. For example if S is dihedral, standard elementary transfer and fusion easily pin down the fusion system F_{S}(G) (cf. Examples I.2.7 and I.3.8), but the hard work in [GW1] or [Ben] and [BG] is still required to go on and determine G.

Exercises for Section 15

15.1. Let F be a constrained, saturated 2-fusion system and G a model for F. Prove F is S_{3}-free iff G/O_{2}(G) is an S_{3}-free group.

15.2. Let G be a finite group with O_{3′}(G) = 1 such that G is S_{3}-free. Prove O^{2′}(G) = E(G) and E(G) = 1 or E(G) = L_{1} ×⋯×L_{n} where L_{i} ≅ L_{2}(3^{e_{i}}) for some odd integer e_{i}.

15.3. Let F be a saturated 2-fusion system. Prove the following are equivalent:

(a) F is S_{3}-free.

(b) F is constrained, and if G is a model for F and H = O^{2′}(G), either H is a 3′-group or H/O_{3′}(H) = L_{1} × ⋯ × L_{n} with L_{i} ≅ L_{2}(3^{e_{i}}) for some odd integer e_{i} > 1.

Part III. Fusion and homotopy theory

Bob Oliver

To each group G (discrete or topological), one associates a topological space BG, called the classifying space of G. Originally, these were defined by Milnor to classify certain fiber bundles with a structure determined by the given group. Much later, the proof in the early 1980’s by Lannes, Miller, and Carlsson of the Sullivan conjecture made it possible to attack certain problems involving classifying spaces which earlier had been considered impossible. One consequence was the discovery that the classifying space of a group G has homotopy theoretic properties much more rigid, and much more closely connected to the structure of G itself, than was previously thought possible.

For each topological space X and each prime p, Bousfield and Kan, in the 1970’s, defined the p-completion X_{p}^{∧} of X: a space which allows us to focus on the properties of X “at the prime p”. There is a map X ⟶ X_{p}^{∧}, functorial in X, which in “nice” cases is universal among all maps X ⟶ Y which induce an isomorphism in mod p homology (see Section 1.4 for more details). It turns out that there is a very close connection between the homotopy theoretic properties of BG_{p}^{∧}, when G is a finite group or a compact Lie group, and the p-local structure of G. Here, by the p-local structure of G is meant roughly the structure of its fusion category over a Sylow p-subgroup S (see Section I.1); i.e., the structure of S together with the conjugacy relations between its subgroups.

More concretely, the Martino-Priddy conjecture stated that for any prime p and any pair G_{1}, G_{2} of finite groups, G_{1} and G_{2} have the same p-local structure (in a sense which will be made precise in Section 1.5) if and only if their p-completed classifying spaces BG_{1}_{p}^{∧} and BG_{2}_{p}^{∧} are homotopy equivalent. While working on this conjecure, and also trying to understand the group of self homotopy equivalences of BG_{p}^{∧}, Broto, Levi, and Oliver [BLO1] were led to investigate the centric linking category L_{S}^{c}(G) associated to G (where S ∈ Syl_{p}(G)). They discovered that the p-completed geometric realization |L_{S}^{c}(G)|_{p}^{∧} of this category has the homotopy type of BG_{p}^{∧}, and also that many of the homotopy properties of BG_{p}^{∧} can be described in terms of properties of L_{S}^{c}(G). For example, the group Out(BG_{p}^{∧}) of homotopy classes of self homotopy equivalences of BG_{p}^{∧} is isomorphic to a certain group of “outer” automorphisms of the category L_{S}^{c}(G).

One implication in the Martino-Priddy conjecture — G_{1} and G_{2} have the same p-local structure if their p-completed classifying spaces are homotopy equivalent — was shown by Martino and Priddy, via a proof based on the Sullivan conjecture and the work of Lannes [La]. As one application of this result, Broto, Møller, and Oliver [BMO] listed many examples of families of finite groups of Lie type in characteristic different from p all of whose members have the same p-local structure (described in Section 1.7). The opposite implication in the conjecture was proven in [O2, O3], but via a proof which depends on the classification of finite simple groups. This is not very satisfactory, and there are still hopes that a “classification-free” proof could lead to a better understanding of why this relationship should be true.

The fusion category F_{S}(G) of a finite group G (when S ∈ Syl_{p}(G) for some prime p), and abstract saturated fusion systems over a p-group, have already been described at length in Part I. Motivated by this and by work of Ron Solomon, Dave Benson (in [Be3] and in unpublished work) predicted that there should be a way of associating classifying spaces to saturated fusion systems, which would generalize the association between the p-fusion category of a finite group and its p-completed classifying space. This was carried out by Broto, Levi, and Oliver [BLO2], who defined a certain class of related categories, abstract linking systems, which can be associated to fusion systems. A classifying space for a fusion system F can then be defined to be the space |L|_{p}^{∧} — the p-completion of the geometric realization of L — for any linking system L associated to F.

It is still not known whether or not a classifying space can be associated to each fusion system (nor whether it is unique), although this is the case in all examples which we have looked at. This problem of determining whether or not each saturated fusion system has a unique associated linking system and classifying space is closely related to the Martino-Priddy conjecture, and provides an additional reason for wanting to find a proof of that conjecture which does not use the classification of finite simple groups.

The p-completed classifying spaces of finite groups have some very remarkable homotopy theoretic properties, and classifying spaces of abstract fusion systems share many of these same properties. For example, the group of homotopy classes of self equivalences of a classifying space |L|_{p}^{∧} can be described explicitly in terms of certain automorphisms of the (finite) category L; it is very unusual to find spaces whose self equivalences can be described so explicitly. The mod p cohomology H^{∗}(|L|_{p}^{∧}; 𝔽_{p}) can be described explicitly as a “ring of stable elements” depending on the fusion system, in a way analogous to the description by Cartan and Eilenberg of H^{∗}(G; 𝔽_{p}) ≅ H^{∗}(BG_{p}^{∧}; 𝔽_{p}). These and other, similar properties are described in Section 4.6.

The fundamental group of any classifying space |L|_{p}^{∧} of a saturated fusion system F is always a finite p-group. The connected covering spaces of |L|_{p}^{∧} are all classifying spaces of fusion subsystems of F: the subsystems of ppower index in F (Definition I.7.3). For example, the universal covering space of |L|_{p}^{∧} is the classifying space of the fusion subsystem O^{p}(F) ⊴ F described in Theorem I.7.4. There is a similar geometric description of O^{p′}(F) (see Theorem I.7.7) via the universal cover of a different space associated to F. All of these connections between covering spaces and fusion subsystems are described in Section 4.5.

A saturated fusion system is called “realizable” if it is the fusion system of a finite group, and is called “exotic” otherwise. Classifying spaces of exotic fusion systems are thus spaces which have many of the very nice properties of p-completed classifying spaces of finite groups, but which are not equivalent to BG_{p}^{∧} for any finite G. This provides part of the motivation for describing in Section 6 some of the known examples of exotic fusion systems.

We would like to thank Kasper Andersen, Carles Broto, Paul Goerss, Jesper Grodal, Ran Levi, Kári Ragnarsson, and Albert Ruiz for their many suggestions and corrections made while preparing this survey.

Notation: When C is a (small) category, C^{op} denotes the opposite category (morphisms going in the opposite direction), N (C) is the nerve of C (as a simplicial set), and |C| is the geometric realization of C (i.e., of N (C)). For any functor F: C ⟶ D between categories, we let

F_{c,c′}: Mor_{C}(c, c′) ⟶ Mor_{D}(F (c), F (c′))  and  F_{c} = F_{c,c}

denote the induced maps between morphism sets.

For each n ≥ 0, D^{n} denotes the closed ball of radius one in ℝ^{n}; i.e., the space of points x such that ∥x∥ ≤ 1. Also, S^{n−1} ⊆ D^{n} denotes the unit sphere: the space of those x such that ∥x∥ = 1. When X is any topological space and A is a subspace (usually closed), X/A denotes the quotient space where all points of A have been identified to a point. Thus, for example, D^{n}/S^{n−1} is homeomorphic to S^{n} for n ≥ 1.

Some of the basic concepts in algebraic topology, such as the fundamental group π_{1}(X), covering spaces, and CW complexes and their (co)homology, are surveyed briefly in Section 1.1–1.2. However, a reader not already somewhat familiar with these topics will probably have to supplement what is written there by referring to the book of Hatcher [Ht], or that of Benson [Be2]. Note that for a pair of spaces X and Y, X ≅ Y means that X and Y are homeomorphic, while X ≃ Y means that they are homotopy equivalent (see Section 1.1).

Throughout this part, p is assumed to be a fixed prime, and all p-groups are finite.

1. Classifying spaces, p-completion, and the Martino-Priddy conjecture

This section is intended to give an introduction and general motivation for the connection between fusion systems and topology. The main goal is to present the material needed to state precisely the Martino-Priddy conjecture (Theorem 1.17). This conjecture is what provided the original motivation for this author and some of his collaborators to begin studying the connections between fusion in finite groups and homotopy theory.

Throughout this section, all maps between topological spaces are assumed to be continuous.

1.1. Homotopy and fundamental groups.

We begin with a very brief description of some of the basic concepts in algebraic topology. For more detail, but still in a concise formulation, we refer to Benson’s book [Be2, Chapter 1]. Hatcher’s textbook [Ht] provides a much more lengthy introduction to these topics.

Let I = [0, 1] denote the unit interval. When X and Y are two spaces, and f, f′: X ⟶ Y are (continuous) maps from X to Y, then f and f′ are homotopic, written f ≃ f′, if there is a map F: X × I ⟶ Y such that F (x, 0) = f(x) and F (x, 1) = f′(x) for all x ∈ X. In other words, f and f′ are homotopic if there is a “continuous deformation” of f into f′. One easily sees that homotopy defines an equivalence relation among the set of all maps from X to Y.

A map f: X ⟶ Y is a homotopy equivalence if it has a homotopy inverse: a map g: Y ⟶ X such that f ∘ g ≃ Id_{Y} and g ∘ f ≃ Id_{X}. When there is a homotopy equivalence between X and Y, one says that X has the homotopy type of Y. A space X is contractible, written X ≃ ∗, if X has the homotopy type of a point; equivalently, if Id_{X} is homotopic to a constant map.

If X is a space, and x, y ∈ X, then a path in X from x to y is a (continuous) map γ: I ⟶ X such that φ(0) = x and γ(1) = y. A loop in X based at x is a path from x to itself. The space X is path connected if there is a path in X between any two points. Two paths γ and γ′ from x to y are homotopic (relative to their endpoints) if there is a homotopy Γ: I × I ⟶ X such that for all s, t ∈ I, Γ(t, 0) = γ(t), Γ(t, 1) = γ′(t), Γ(0, s) = x, and Γ(1, s) = y. Homotopy relative to endpoints defines an equivalence relation among paths between any two given points of X.

If γ and δ are two paths in X, and γ(1) = δ(0), then the composite path δ·γ from γ(0) to δ(1) is defined by setting

(δ·γ)(t) = \begin{cases}γ(2t) & if 0 ≤ t ≤ \frac{1}{2} \\ δ(2t − 1) & if \frac{1}{2} ≤ t ≤ 1. \end{cases}

Paths are thus composed from right to left throughout this survey (but note that composition from left to right is quite frequently found in the literature).

Composition of paths is associative up to homotopy. Also, for each path γ, we define the “inverse” path γ by setting γ(t) = γ(1 − t). This is an inverse up to homotopy, in the sense that γ·γ and γ·γ are both homotopic to constant paths.

For any choice of basepoint x_{0} ∈ X, the fundamental group of X at x_{0} is the group π_{1}(X, x_{0}) of homotopy classes of loops based at x_{0} (homotopy relative to the endpoints), with multiplication defined by composition of paths. When X is path connected, then for any pair of points x_{0}, x_{1} ∈ X, π_{1}(X, x_{0}) and π_{1}(X, x_{1}) are isomorphic. More precisely, if γ is a path from x_{0} to x_{1}, then there is an isomorphism

γ_{#} : π_{1}(X, x_{1}) ⟶^{≅} π_{1}(X, x_{0})

which sends the class of a loop ϕ based at x_{1} to the class of the composite loop γ·ϕ·γ based at x_{0} (cf. [Ht, Proposition 1.5] and [Be2, pp.6–7]). This isomorphism depends on the choice of path γ, but if δ is another path from x_{0} to x_{1}, then γ_{#} and δ_{#} differ by an inner automorphism of π_{1}(X, x_{0}) (conjugation by [δ·γ]).

A path connected space is simply connected if its fundamental group is the trivial group. We just saw that this is independent of the choice of basepoint.

When f: X ⟶ Y is a map between spaces, x_{0} ∈ X, and y_{0} = f(x_{0}) ∈ Y, then we let

f_{#} : π_{1}(X, x_{0}) ⟶ π_{1}(Y, y_{0})

denote the homomorphism of groups defined by setting f_{#}([ϕ]) = [f ∘ ϕ]. If f is a homotopy equivalence, then f_{#} is an isomorphism of groups (cf. [Ht, Proposition 1.18]).

The fundamental group of a path connected space X is closely related to its covering spaces.

Definition 1.1. A covering space of a space X is a space X̃, together with a surjective map f: X̃ ⟶ X, such that each x ∈ X has an open neighborhood U ⊆ X for which f^{−1}(U) is a disjoint union of open sets, each of which is sent homeomorphically to U. If X is path connected, then X̃ is a universal covering space if it is path connected and simply connected.

The simplest nontrivial example of a universal covering space is the map f: ℝ ⟶ S^{1}, defined by f(t) = e^{2πit}. Here, for simplicity, we identify S^{1} as the unit circle in the complex plane. Also, for each n ∈ ℤ with n ≠ 0, the map f_{n}: S^{1} ⟶ S^{1} defined by f_{n}(z) = z^{n} is a covering space over S^{1}. One can show that each connected covering space X̃ ⟶ S^{1}, where X̃ is path connected, is homeomorphic to one of these covering spaces (ℝ, f) or (S^{1}, f_{n}).

We refer to [Ht, §1.3] for more details about covering spaces.

Assume f: X̃ ⟶ X is a universal covering space for X. Assume also that X is locally path connected: each point has a family of arbitrarily small neighborhoods which are path connected. Let G be the group of all homeomorphisms h: X̃ ⟶^{≅} X̃ such that f ∘ h = f: the group of “deck transformations” of X̃. By one of the fundamental theorems about covering spaces (cf. [Ht, Proposition 1.39]), G ≅ π_{1}(X), and G acts freely and transitively on f^{−1}(x) for each x ∈ X.

The following theorem describes a way to construct covering spaces with a given group of deck transformations.

Proposition 1.2. Fix a topological space X which is path connected and locally path connected, and a discrete group G which acts continuously on X. Assume G acts freely and properly, in the sense that for each x ∈ X, there is an open neighborhood U of x such that g(U) ∩ U = ∅ for each 1 ≠ g ∈ G. Let X/G be the orbit space of the action (with the quotient topology), and let f: X ⟶ X/G be the natural projection. Then f is a covering space. For any x_{0} ∈ X, with G-orbit y_{0} = f(x_{0}), the induced homomorphism f_{#} from π_{1}(X, x_{0}) to π_{1}(X/G, y_{0}) is injective, its image is normal in π_{1}(X/G, y_{0}), and π_{1}(X/G, y_{0})/Im(f_{#}) ≅ G. In particular, if X is simply connected, then π_{1}(X/G, y_{0}) ≅ G.

Proof. That f: X ⟶ X/G is a covering space follows immediately from the assumption that G acts freely and properly.

There is a surjective homomorphism χ: π_{1}(X/G, y_{0}) ⟶ G defined as follows. For each loop γ in X/G based at y_{0}, there is a unique path γ̃ in X such that γ̃(0) = x_{0} and f ∘ γ̃ = γ (cf. [Ht, Propositions 1.33 & 1.34]). Define χ([γ]) to be the (unique) element g ∈ G such that γ̃(1) = g(x_{0}). For any g ∈ G, there is a path ϕ from x_{0} to g(x_{0}), and g = χ([f ∘ ϕ]). Thus χ is surjective. Also, Ker(χ) = f_{#}(π_{1}(X, x_{0})) by construction, and hence χ induces an isomorphism π_{1}(X/G, y_{0})/Im(f_{#}) ≅ G. The injectivity of f_{#} follows by lifting homotopies between paths, using [Ht, Propositions 1.33] again. □

Returning to the example of a universal covering space f: ℝ ⟶ S^{1} defined above, f is the orbit map of the free ℤ-action on ℝ by translation.

Hence this is the group of deck transformations of f, and defines an isomorphism ℤ ≅ π_{1}(S^{1}). Under this isomorphism, each n ∈ ℤ is sent to the homotopy class of f ∘ ϕ, where ϕ: I ⟶ ℝ is any path from 0 to n (or from k to k + n for k ∈ ℤ).

We will occasionally need to mention higher homotopy groups of spaces. For n ≥ 1, let I^{n} be the product of n copies of the interval I = [0, 1]. Let ∂I^{n} be its boundary: the subspace of all (t_{1}, …, t_{n}) ∈ I^{n} such that for some i, t_{i} ∈ ∂I = {0, 1}. When X is a space and x_{0} ∈ X, π_{n}(X, x_{0}) is defined as the set of homotopy classes of maps ϕ: I^{n} ⟶ X such that ϕ(∂I^{n}) = {x_{0}}; more precisely, the homotopy classes via homotopies which are constant on ∂I^{n}. This set has the structure of a group via juxtaposition with respect to the first coordinate:

ϕ·ψ(t_{1}, …, t_{n}) = \begin{cases}ϕ(2t_{1}, t_{2}, …, t_{n}) & if t_{1} ≤ \frac{1}{2} \\ ψ(2t_{1}−1, t_{2}, …, t_{n}) & if t_{1} ≥ \frac{1}{2} . \end{cases}

Since I^{n}/∂I^{n} ≅ S^{n}, elements of π_{n}(X, x_{0}) can also be regarded as homotopy classes of maps S^{n} ⟶ X which send a basepoint to x_{0}.

When n ≥ 2, π_{n}(X, x_{0}) is an abelian group. Also, if f: X ⟶ Y is a covering space, x_{0} ∈ X, and y_{0} = f(x_{0}), then the homomorphism f_{#} : π_{n}(X, x_{0}) ⟶ π_{n}(Y, y_{0}) is an isomorphism for all n ≥ 2. See, e.g., [Ht, pp. 338–340] or [Be2, § 1.2,1.6] for more details, and for proofs of these facts.

1.2. CW complexes and cellular homology.

Intuitively, a CW complex is a space built up by starting with a discrete set (the 0-cells), and attaching higher dimensional cells (copies of the closed unit ball D^{n} ⊆ ℝ^{n} for n > 0) via their boundaries. The following definition is one way of making this more precise.

Definition 1.3. A CW complex consists of a space X, together with a sequence of closed subspaces ∅ = X^{(−1)} ⊆ X^{(0)} ⊆ X^{(1)} ⊆ X^{(2)} ⊆ ⋯ whose union is X (the “skeleta” of X), indexing sets J_{n} for all n ≥ 0, and “characteristic maps”

ρ_{j}^{n}: D^{n} ⟶ X^{(n)} for all n ≥ 0 and j ∈ J_{n}

which satisfy the following conditions:

(i) For each n, each j ∈ J_{n}, and each t ∈ D^{n}, ρ_{j}^{n}(t) ∈ X^{(n−1)} if and only if t ∈ S^{n−1}.

(ii) For each n ≥ 0 and each x ∈ X^{(n)}∖X^{(n−1)}, there are unique j ∈ J_{n} and t ∈ D^{n}∖S^{n−1} such that ρ_{j}^{n}(t) = x.

(iii) For each n ≥ 0, a subset U ⊆ X^{(n)} is open in X^{(n)} if and only if (ρ_{j}^{m})^{−1}(U) is open in D^{m} for each m ≤ n and each j ∈ J_{m}.

(iv) A subset U ⊆ X is open in X if and only if U ∩ X^{(n)} is open in X^{(n)} for each n ≥ 0.

For more details (and slightly different but equivalent formulations), we refer to [Ht, pp.5–8 & Appendix] and to [Be2, §1.5].

When X is a CW complex with skeleta and structural maps as above, let C_{n}(X), for each n ≥ 0, be the free group with basis the set J_{n} of cells of dimension n. Let x_{j} ∈ C_{n}(X) be the generator corresponding to j ∈ J_{n}.

A boundary homomorphism ∂: C_{n}(X) ⟶ C_{n−1}(X) is defined as follows. Fix a pair of cells j ∈ J_{n} and k ∈ J_{n−1}. For any m, let D^{m}/S^{m−1} be the quotient space of D^{m} where all points of the boundary S^{m−1} are identified to a point. Then D^{m}/S^{m−1} is homeomorphic to the m-sphere S^{m}, and we fix some identification. Consider the following map

∂_{jk}: S^{n−1} ⟶^{ρ_{j}^{n}|_{S^{n−1}}} X^{(n−1)} ⟶^{ψ_{k}} D^{n−1}/S^{n−2} ≅ S^{n−1},

where ψ_{k}(x) = (ρ_{k}^{n−1})^{−1}(x) if x ∈ ρ_{k}^{n−1}(D^{n−1}/S^{n−2}), and ψ_{k}(x) is the basepoint (the identification point of all elements of S^{n−2}) otherwise. Then ψ_{k} is a continuous map from X^{(n−1)} to S^{n−1}. Let n_{jk} ∈ ℤ be the degree of ∂_{jk}. By the compactness of S^{n−1}, for each j ∈ J_{n}, ρ_{j}^{n}(S^{n−1}) has nonempty intersection with ρ_{k}^{n−1}(D^{n−1}/S^{n−2}) for at most finitely many k ∈ J_{n−1}, so n_{jk} is nonzero for at most finitely many k. We now define, for each j ∈ J_{n}, ∂(x_{j}) = ∑_{k∈J_{n−1}} n_{jk}x_{k}.

One can show that the composite of any two successive boundary maps is zero, and hence that (C_{∗}(X), ∂) is a chain complex. Then for any abelian group A and any q ≥ 0,

H_{q}(X; A) ≅ H_{q}(C_{∗}(X) ⊗_{ℤ} A, ∂)
H^{q}(X; A) ≅ H^{q}(Hom_{ℤ}(C_{∗}(X), A), ∂^{∗}) .

We thus get a direct connection between the cell structure of X and its homology and cohomology.

For a more detailed construction of the celluar chain complex, we refer to [Be2, § 1.5] and [Ht, § 2.2].

1.3. Classifying spaces of discrete groups.

We begin with the definition of an abstract classifying space of a discrete group G. A more explicit definition of a space which will be called the classifying space of G will be given in Section 2.3.

Definition 1.4. A classifying space for a discrete group G is a CW complex BG which satisfies the following conditions:

(a) π_{1}(BG) ≅ G (for any choice of basepoint); and (b) BG has a universal covering space EG which is contractible.

The assumption that a classifying space for G is a CW complex has been put here for convenience, but is not really necessary. In most situations, this can be replaced by the condition that it is paracompact and has certain local properties.

The simplest example of a classifying space is the case where G ≅ ℤ: the circle S^{1} is a classifying space, since its universal covering space is the real line, and its fundamental group is isomorphic to ℤ.

To construct a slightly more complicated example, let S^{∞} be the union of the finite dimensional spheres: the space of all sequences (x_{0}, x_{1}, x_{2}, …) such that x_{i} = 0 for i sufficiently large, and ∑_{i=0}^{∞} x_{i}^{2} = 1. This space is, in fact, contractible; we describe here one way to see that. Let Δ^{n} ⊇ ∂Δ^{n} denote the “standard n-simplex” and its boundary:

Δ^{n} = {(x_{0}, …, x_{n}) ∈ ℝ^{n+1} | x_{i} ≥ 0 for each i, ∑_{i=0}^{n} x_{i} = 1}
∂Δ^{n} = {(x_{0}, …, x_{n}) ∈ Δ^{n} | x_{i} = 0 for some i} .

Homeomorphisms S^{n−1} ⟶^{≅} ∂Δ^{n} can be chosen for all n in such a way as to be consistent with the inclusions, and hence S^{∞} is homeomorphic to the union of the ∂Δ^{n}. This last space is convex, hence contractible, and so S^{∞} is contractible.

Since S^{∞} is the universal covering space of the infinite projective space ℝP^{∞} (the union of the ℝP^{n}), ℝP^{∞} = S^{∞}/(x ∼ −x) is a classifying space for the cyclic group of order two. More generally, if we identify S^{∞} with the unit sphere in ℂ^{∞}, then for any n > 1, the group μ_{n} of n-th roots of unity acts freely on S^{∞} via scalar multiplication. So by Proposition 1.2, the quotient space S^{∞}/μ_{n} of this action has fundamental group μ_{n} and universal covering space S^{∞}, and is a classifying space for the cyclic group μ_{n}.

We next describe a “universal property” of classifying spaces, which helps to explain the name and the origin of their interest to topologists. Very roughly, it says that for any path connected space X, maps from X to BG are “classified” by homomorphisms from π_{1}(X) to G. We state the property here for CW complexes. A slightly different version will be given in Section 2.5 (Proposition 2.10), with a sketch of the proof. There are also versions which hold for paracompact spaces with certain local properties.

When X and Y have chosen basepoints x_{0} ∈ X and y_{0} ∈ Y, a pointed map from X to Y is a map f: X ⟶ Y such that f(x_{0}) = y_{0}. If f and f′ are two pointed maps from X to Y, then they are pointed homotopic if there is a homotopy F: X ×I ⟶ Y such that F (x, 0) = f(x), F (x, 1) = f′(x), and F (x_{0}, t) = y_{0} for all x ∈ X and t ∈ I. The set of all pointed homotopy classes of pointed maps from X to Y will be denoted here [X, Y]_{∗}. By comparison, the set of unpointed homotopy classes of (all) maps from X to Y will be denoted [X, Y].

For any pair of groups G, H, define

Rep(H, G) = Hom(H, G)/Inn(G).

Proposition 1.5. Fix a discrete group G, and a classifying space BG for G with basepoint b_{0} ∈ BG. Identify π_{1}(BG, b_{0}) with G. Then the following hold for any path connected CW complex X with basepoint x_{0} ∈ X.

(a) The natural map of sets

[X, BG]_{∗} ⟶^{≅} Hom(π_{1}(X, x_{0}), G),

which sends the class of a pointed map f: X ⟶ BG to its induced homomorphism f_{#}, is a bijection.

(b) The natural map of sets

[X, BG] ⟶^{≅} Rep(π_{1}(X, x_{0}), G)

which sends the homotopy class of f: X ⟶ BG to the class of f_{#} (modulo Inn(G)) is a bijection. In particular, two pointed maps f, f′: X ⟶ BG are (freely) homotopic if and only if the induced homomorphisms f_{#} and f′_{#} are G-conjugate.

Proof. See, e.g., [Wh, Theorems V.4.3–4]. Point (a) is also shown in [Ht, Proposition 1B.9], and (b) follows easily from that.

To see the “only if” part of the last statement, let F: X ×I ⟶ BG be a homotopy from f to f′. Thus F (x, 0) = f(x) and F (x, 1) = f′(x) for all x ∈ X. Let ϕ: I ⟶ BG be the loop ϕ(t) = F (x_{0}, t), and set g = [ϕ] ∈ π_{1}(BG, b_{0}) = G. Then for each loop γ in X based at x_{0}, the composite F ∘ (γ × Id_{I}): I × I ⟶ BG defines a homotopy ϕ·(f ∘ γ) ≃ (f′ ∘ γ)·ϕ, so that c_{g}(f_{#}([γ])) = f′_{#}([γ]). □

A proof of Proposition 1.5 will be sketched in Section 2.5 (Proposition 2.10), in the case where X is the geometric realization of a simplicial set. In addition, some information will be given there about the individual connected components of the space of maps.

One important special case of Proposition 1.5 is that of maps between classifying spaces of discrete groups.

Corollary 1.6. If G and H are two discrete groups, with classifying spaces BG and BH, then [BG, BH]_{∗} ≅ Hom(G, H) and [BG, BH] ≅ Rep(G, H).

In particular, Corollary 1.6 implies that two classifying spaces of the same group (or of isomorphic groups) have the same homotopy type.

There is also a close connection between classifying spaces and group (co)homology.

Proposition 1.7. Let BG be a classifying space of the discrete group G. Then H_{∗}(BG; A) ≅ H_{∗}(G; A) (and H^{∗}(BG; A) ≅ H^{∗}(G; A)) for all abelian groups A.

Proof. See, e.g., [Be2, Theorem 2.2.3]. We give here a very rough sketch of a proof.

Let EG be the universal covering space of BG, and let C_{∗}(EG) be its cellular chain complex (see Section 1.2). The sequence

⋯ ⟶^{∂} C_{1}(EG) ⟶^{∂} C_{0}(EG) ⟶ ℤ ⟶ 0

is exact (the homology of EG is isomorphic to that of a point since EG is contractible), and each group C_{q}(EG) has a natural basis which is permuted freely by G. Thus C_{∗}(EG) is a ℤ[G]-free resolution of ℤ. In addition, C_{∗}(BG) ≅ C_{∗}(EG) ⊗_{ℤ[G]} ℤ, and hence for each q ≥ 0,

H_{q}(BG; A) ≅ H_{q}(C_{∗}(EG) ⊗_{ℤ[G]} A, ∂)
H^{q}(BG; A) ≅ H^{q}(Hom_{ℤ[G]}(C_{∗}(EG), A), ∂^{∗}) .

The result now follows from the definition of group (co)homology. □

In fact, Proposition 1.7 also holds when A is an arbitrary ℤ[G]-module, once one defines appropriately H_{∗}(X, A) and H^{∗}(X; A) for a path connected space X and a ℤ[π_{1}(X)]-module A (“homology with twisted coefficients”).

1.4. The p-completion functor of Bousfield and Kan.

The Bousfield-Kan p-completion functor is a functor from spaces to spaces, denoted here X ↦ X_{p}^{∧}, together with a natural transformation ϕ: Id ⟶ (−)_{p}^{∧}. More precisely, it is a functor from the category of simplicial sets to itself (see Section 2.1), but we suppress such details in this presentation. We refer to [BK] for the precise definition of this functor, and for most of the properties referred to here. Another source for the definition of p-completion and some of its properties is [GJ, § VIII.3].

A map of spaces f: X ⟶ Y will be called a p-equivalence if f induces an isomorphism from H_{∗}(X; 𝔽_{p}) to H_{∗}(Y; 𝔽_{p}). Note that since 𝔽_{p} is a field, mod p cohomology is the dual of mod p homology, and so f is a pequivalence if and only if it induces an isomorphism in mod p cohomology.

Many of the important properties of p-completions hold only on certain classes of spaces. The next proposition describes one of the few properties which holds for all spaces.

Proposition 1.8 ([BK, Lemma I.5.5]). A map f: X ⟶ Y induces a homotopy equivalence f_{p}^{∧}: X_{p}^{∧} ⟶ Y _{p}^{∧} if and only if it is a p-equivalence.

When working with p-completion, the classes of p-complete spaces and p-good spaces play a central role.

Definition 1.9. A space X is

• p-complete if ϕ_{X}: X ⟶ X_{p}^{∧} is a homotopy equivalence; and is

• p-good if ϕ_{X} is a p-equivalence.

By [BK, Proposition I.5.2], a space X is p-good if and only if X_{p}^{∧} is p-complete (i.e., X_{p}^{∧} ≃ (X_{p}^{∧})_{p}^{∧}).

One important family of examples of p-complete spaces are the classifying spaces of (finite) p-groups.

Proposition 1.10. The classifying space of any p-group is p-complete.

Proof. This is very well known, but does not seem to be stated explicitly anywhere in [BK]. If P is an abelian p-group, then by [BK, Example VI.5.2], π_{n}(BP _{p}^{∧}) ≅ ℤ_{p} ⊗_{ℤ} π_{n}(BP) for each n ≥ 1. Thus BP _{p}^{∧} is again a classifying space for P, and so BP is p-complete.

Now let P be an arbitrary p-group, and set Z = Z(P). We can assume by induction on |P | that B(P/Z) is p-complete. There is a fibration BP ⟶ B(P/Z) with fiber BZ. By the mod-𝔽_{p} fiber lemma of Bousfield and Kan (see Lemma II.5.1 and Example II.5.2(iv) in [BK]), and since π_{1}(B(P/Z)) ≅ P/Z is a finite p-group, the p-completed map BP _{p}^{∧} ⟶ B(P/Z)_{p}^{∧} is a fibration with fiber homotopy equivalent to BZ_{p}^{∧}. (More generally, by [BK, Lemma II.5.1], if E ⟶ B is a fibration with fiber F, where all three spaces are connected and the action of π_{1}(B) on H_{i}(F; 𝔽_{p}) is nilpotent for all i, then E_{p}^{∧} ⟶ B_{p}^{∧} is also a fibration and has fiber homotopy equivalent to F _{p}^{∧}.) Since BZ and B(P/Z) are both pcomplete, the map ϕ_{BP} : BP ⟶ BP _{p}^{∧} must be a homotopy equivalence, and thus BP is p-complete. □

If a space is p-bad (i.e., not p-good), then in fact, all of its iterated pcompletions are also p-bad [BK, Proposition I.5.2]. The simplest example of a p-bad space is S^{1} ∨S^{1}: the one point union of two circles [Bf, Theorem 11.1]. In fact, S^{1} ∨ S^{n} (the one-point union of a circle and an n-sphere) is p-bad for all p and all n ≥ 2 [Bf, Theorem 10.1]. So even a space with very nice fundamental group (the integers) can be p-bad.

The most important criterion for checking that a space is p-good, at least for the spaces of the type we work with here, is given in the following proposition. As usual, a group G is called p-perfect if H_{1}(G; 𝔽_{p}) = 0; equivalently, if G is generated by its commutators and p-th powers.

Proposition 1.11. A connected space X is p-good if π_{1}(X) is finite; in particular, if X is simply connected. More generally, X is p-good if π_{1}(X) contains a p-perfect subgroup of finite index. In this situation, if K ⊴ π_{1}(X) is the maximal p-perfect subgroup of π_{1}(X), then π_{1}(X_{p}^{∧}) ≅ π_{1}(X)/K.

Proof. The first statement is shown in [BK, Proposition VII.5.1]. The other two statements are implicit in [BK], but do not seem to be proven there explicitly. So we sketch a proof here.

Since any group generated by p-perfect subgroups is p-perfect, there is a unique maximal p-perfect subgroup K ⊴ π_{1}(X) which is normal. Set π = π_{1}(X)/K for short. Since π is finite and contains no nontrivial pperfect subgroups, it must be a p-group.

Let X̃ be the connected covering space of X with fundamental group K. Since π_{1}(X̃) is p-perfect, X̃ is p-good and X̃_{p}^{∧} is simply connected by [BK, VII.3.2].

Since π is a p-group, the augmentation ideal in the group ring 𝔽_{p}[π] is nilpotent, and hence every 𝔽_{p}[π]-module is nilpotent. In particular, H_{i}(X̃; 𝔽_{p}) is nilpotent as an 𝔽_{p}[π]-module for all i. Hence the mod-𝔽_{p} fiber lemma of Bousfield and Kan ([BK, II.5.1]) implies that the fibration sequence X̃ ⟶ X ⟶ Bπ is still a fibration sequence after p-completion. Since Bπ is p-complete by Proposition 1.10, this means that X_{p}^{∧} ⟶ Bπ is a fibration with fiber X̃_{p}^{∧} and total space X_{p}^{∧}, and that π_{1}(X_{p}^{∧}) ≅ π. Upon applying the mod-𝔽_{p} fiber lemma to this new sequence, we see that X_{p}^{∧} ≃ (X_{p}^{∧})_{p}^{∧} is p-complete, and hence that X is p-good. Also, the universal covering space of X_{p}^{∧} is X̃_{p}^{∧}. □

Thus, for example, when π_{1}(X) is finite, π_{1}(X_{p}^{∧}) ≅ π_{1}(X)/O^{p}(π_{1}(X)). Most importantly for our purposes here, for any finite group G and any classifying space BG of G, BG is p-good and π_{1}(BG_{p}^{∧}) ≅ G/O^{p}(G).

The next proposition states that when X is p-good, ϕ_{X}: X ⟶ X_{p}^{∧} is universal among all p-equivalences X ⟶ Y.

Proposition 1.12. For any p-good space X, and any p-equivalence f: X ⟶ Y, there is a map g: Y ⟶ X_{p}^{∧}, unique up to homotopy, such that g ∘ f ≃ ϕ_{X}. Thus ϕ_{X}: X ⟶ X_{p}^{∧} is a final object among homotopy classes of mod p equivalences defined on X.

Proof. By [BK, Lemma I.5.5], any p-equivalence f: X ⟶ Y induces a homotopy equivalence f_{p}^{∧}: X_{p}^{∧} ⟶ Y _{p}^{∧}. So g is defined to be ϕ_{Y} followed by a homotopy inverse to f_{p}^{∧}. □

As one consequence of Proposition 1.12, we have the following condition for the p-completions of two spaces to be homotopy equivalent. Note that this condition does not in itself involve p-completion.

Corollary 1.13. If X and Y are two spaces, and one of them is p-good, then their p-completions are homotopy equivalent if and only if there exists some space Z, and maps X ⟶^{f} Z ⟵^{g} Y, such that f and g are both p-equivalences.

1.5. Equivalences between fusion systems of finite groups.

Recall that for any finite group G and any S ∈ Syl_{p}(G), the fusion category of G over S is the category F_{S}(G), whose objects are the subgroups of S, and where for each P, Q ≤ S,

Mor_{F_{S}(G)}(P, Q) = Hom_{G}(P, Q)

=^{def} {φ ∈ Hom(P, Q) | φ = c_{g}, some g ∈ G such that ^{g}P ≤ Q} .

Our goal here is to study the relationship between this category F_{S}(G) and the homotopy type of the p-completed classifying space BG_{p}^{∧}. In order to state precise results, we first need some terminology for describing equivalences between fusion categories.

Definition 1.14. Fix a prime p, a pair of finite groups G_{1} and G_{2}, and Sylow p-subgroups S_{i} ∈ Syl_{p}(G_{i}). An isomorphism φ: S_{1} ⟶^{≅} S_{2} is fusion preserving if for all P, Q ≤ S_{1} and all α ∈ Hom(P, Q),

α ∈ Hom_{G_{1}}(P, Q) ⟺ ^{φ}α =^{def} (φ|_{Q})α(φ|_{P})^{−1} ∈ Hom_{G_{2}}(φ(P), φ(Q)).

In other words, in the above situation, an isomorphism φ: S_{1} ⟶ S_{2} is fusion preserving if and only if it induces an isomorphism of categories from F_{S_{1}}(G_{1}) to F_{S_{2}}(G_{2}) by sending P to φ(P) and α to ^{φ}α. The next proposition says, roughly, that there is a fusion preserving isomorphism between Sylow p-subgroups of G_{1} and G_{2} if and only if there is a bijection between conjugacy classes of p-subgroups of G_{1} and G_{2}. Recall that for any pair of groups H and G, Rep(H, G) = Hom(H, G)/Inn(G).

Proposition 1.15. Fix a pair of finite groups G_{1} and G_{2}, a prime p and Sylow p-subgroups S_{i} ≤ G_{i}. Then there is a fusion preserving isomorphism φ: S_{1} ⟶^{≅} S_{2} if and only if there are bijections

β_{P} : Rep(P, G_{1}) ⟶^{≅} Rep(P, G_{2}),

defined for all p-groups P and natural with respect to homomorphisms.

Proof. This is due mostly to Martino and Priddy [MP2]. We sketch a proof here. See also [BMO, Proposition 1.3] for more details.

Assume first φ: S_{1} ⟶^{≅} S_{2} is a fusion preserving isomorphism. For each p-group P and each α ∈ Hom(P, G_{1}), choose g ∈ G_{1} such that c_{g}(α(P)) ≤ S_{1}, and set β_{P} ([α]) = [φ∘c_{g} ∘α] ∈ Rep(P, G_{2}). If h ∈ G_{1} is another choice of element such that c_{h}(α(P)) ≤ S_{1}, then c_{hg^{−1}} ∈ Iso_{G_{1}}(^{g}(α(P)),^{h}(α(P))), so there is x ∈ G_{2} such that^{φ}(c_{hg^{−1}}) = c_{x} ∈ Iso_{G_{2}}(φ(c_{g}(α(P))), φ(c_{h}(α(P)))), and hence [φ ∘ c_{g} ∘ α] = [φ ∘ c_{h} ∘ α] in Rep(P, G_{2}). Thus β_{P} is well defined, and it is a bijection since an inverse can be defined via composition with φ^{−1}. Clearly, the β_{P} are natural in P.

Now assume that β_{P} are natural bijections, defined for all p-groups P. Choose φ ∈ Hom(S_{1}, S_{2}) and ψ ∈ Hom(S_{2}, S_{1}) such that [φ] = β_{S_{1}}([Id_{S_{1}}]) and [ψ] = β_{S_{2}}([Id_{S_{2}}]). If φ is not injective, then it factors through φ ∈ Hom(S_{1}/K, S_{2}) for some 1 ≠ K ⊴ S_{1}, and by the naturality of the β with respect to S_{1} ⟶ S_{1}/K, Id_{S_{1}} must also factor through S_{1}/K. Since this is not the case, we conclude that φ is injective; and also (by similar reasoning) that ψ is injective. In particular, |S_{1}| = |S_{2}|, and so φ and ψ are both isomorphisms.

Now fix P, Q ≤ S_{1} and α ∈ Hom(P, Q). By the naturality of the β, β_{P} sends [α] ∈ Rep(P, G_{1}) to [φ ∘ α] ∈ Rep(P, G_{2}). Similarly, by naturality with respect to the inclusion incl_{P}^{S_{1}}, β_{P} ([incl_{P}^{G_{1}}]) = [φ|_{P}]. Since α ∈ Hom_{G_{1}}(P, Q) if and only if [α] = [incl_{P}^{G_{1}}] in Rep(P, G_{1}), this is the case exactly when [φ ∘ α] = [φ|_{P}] in Rep(P, G_{2}), or equivalently, ^{φ}α ∈ Hom_{G_{2}}(φ(P), φ(Q)). Thus φ is fusion preserving. □

1.6. The Martino-Priddy conjecture.

The Martino-Priddy conjecture states roughly that two finite groups have the same p-local structure if and only if their classifying spaces have the same p-local structure. Here, the “p-local structure” of a group G translates to mean its fusion at p, and the p-local structure of a space means the homotopy type of its p-completion.

The first hint that some such result might be true came from the following theorem of Mislin.

Theorem 1.16 ([Ms, pp. 457–458]). For any prime p, any p-group Q, and any finite group G, the natural map

Rep(Q, G) ⟶^{≅} [BQ, BG_{p}^{∧}]

is a bijection.

Upon comparing Theorem 1.16 with Corollary 1.6, we see that [BQ, BG_{p}^{∧}] ≅ [BQ, BG] in this situation. Mislin’s theorem is, however, a much deeper result than Corollary 1.6, and depends for its proof on the Sullivan conjecture in the form shown by Carlsson [Ca], Lannes [La], and Miller [Mi].

If G_{1} and G_{2} are two finite groups such that BG_{1}_{p}^{∧} ≃ BG_{2}_{p}^{∧}, then by Mislin’s theorem, there are bijections

Rep(Q, G_{1}) ⟶^{≅} Rep(Q, G_{2}),

defined for all p-groups Q, and natural with respect to group homomorphisms. Hence by Proposition 1.15, there is a fusion preserving isomorphism between Sylow p-subgroups of G_{1} and of G_{2}. In fact, the converse to this is also true.

Theorem 1.17 (Martino-Priddy conjecture). For any prime p, and any pair G_{1}, G_{2} of finite groups, BG_{1}_{p}^{∧} ≃ BG_{2}_{p}^{∧} if and only if there is a fusion preserving isomorphism S_{1} ⟶_{≅}^{φ} S_{2} between Sylow p-subgroups S_{i} ≤ G_{i}.

The “if” part of Theorem 1.17 was proven for odd primes p in [O2, Theorem B], and for p = 2 in [O3, Theorem B]. It follows from a combination of several results stated later in this survey. We describe in Theorem 3.7 how it is reduced to a question about uniqueness of linking categories (as defined in Section 3.1). The obstruction groups for answering that question are described in Proposition 5.11; and some of the techniques used to prove they vanish are discussed in Section 5.3 (see, e.g., Theorem 5.13). It is the proof that the obstruction groups vanish which fills most of [O2] and [O3], and which depends on the classification theorem for finite simple groups.

Theorem 1.17 can be thought of as a refinement of a classical theorem of Cartan and Eilenberg in group cohomology (see [CE, Theorem XII.10.1]). Their theorem describes how for any finite group G, the cohomology ring H^{∗}(BG; 𝔽_{p}) is determined by S ∈ Syl_{p}(G) and fusion in S. By comparison, Theorem 1.17 says that if G_{1} and G_{2} have isomorphic Sylow p-subgroups and the same fusion at p, then not only is H^{∗}(G_{1}; 𝔽_{p}) ≅ H^{∗}(BG_{1}; 𝔽_{p}) abstractly isomorphic to H^{∗}(G_{2}; 𝔽_{p}) ≅ H^{∗}(BG_{2}; 𝔽_{p}), but this isomorphism is realized by maps of spaces BG_{1} ⟶ Z ⟵ BG_{2} (see Corollary 1.13).

1.7. An application: fusion in finite groups of Lie type.

We describe here one application of the “only if” part of the Theorem 1.17: the part proven by Martino and Priddy. So it does depend on Mislin’s theorem and through that on the proofs of the Sullivan conjecture, but it does not depend on the classification of finite simple groups.

The following is the main result in a paper by Broto, Møller, and Oliver [BMO].

Theorem 1.18 ([BMO, Theorem A]). Fix a prime p, a connected reductive integral group scheme 𝔾, and a pair of prime powers q and q′ both prime to p.

(a) If ⟨q⟩ = ⟨q′⟩ as closed subgroups of ℤ_{p}^{×}, then there is a fusion preserving isomorphism between Sylow p-subgroups of 𝔾(q) and 𝔾(q′).

(b) If 𝔾 is of type A_{n}, D_{n}, or E_{6}, τ is a graph automorphism of 𝔾, and ⟨q⟩ = ⟨q′⟩ as closed subgroups of ℤ_{p}^{×}, then there is a fusion preserving isomorphism between Sylow p-subgroups of ^{τ}𝔾(q) and ^{τ}𝔾(q′).

(c) If the Weyl group of 𝔾 contains an element which acts on the maximal torus by inverting all elements, and ⟨−1, q⟩ = ⟨−1, q′⟩ as closed subgroups of ℤ_{p}^{×}, then there is a fusion preserving isomorphism between Sylow p-subgroups of 𝔾(q) and 𝔾(q′) (and between Sylow p-subgroups of ^{τ}𝔾(q) and ^{τ}𝔾(q′) if τ is as in (b)).

(d) If 𝔾 is of type A_{n}, D_{n} for n odd, or E_{6}, τ is a graph automorphism of 𝔾 of order two, and ⟨−q⟩ = ⟨q′⟩ as closed subgroups of ℤ_{p}^{×}, then there is a fusion preserving isomorphism between Sylow p-subgroups of τ𝔾(q) and 𝔾(q′).

When p = 2, q and q′ generate the same closed subgroup of ℤ_{2}^{×} if and only if q ≡ q′ (mod 8), and v_{2}(q^{2} − 1) = v_{2}(q′^{2} − 1). Thus, for example, if q, q′ ≡ 7 (mod 16) and q′′ ≡ 9 (mod 16), then for n ≥ 2, SL_{n}(q), SL_{n}(q′), and SU_{n}(q′′) all have the same 2-fusion. We refer to [BMO] for more examples (also at odd primes), and also for more precise details on how and when this theorem can be applied.

We know of no purely algebraic proof of Theorem 1.18. The proof in [BMO] is carried out by showing in each case that the p-completed classifying spaces of the two groups are homotopy equivalent. The starting point for doing this is a theorem of Friedlander [Fr, Theorem 12.2], which describes B𝔾(q)_{p}^{∧} or B^{τ}𝔾(q)_{p}^{∧} as the “homotopy fixed space” of a certain self-equivalence of B𝔾(ℂ)_{p}^{∧}. This was then combined with, among other things, a general result [BMO, Theorem 2.4] stating that under certain conditions on a space X, the homotopy fixed spaces of two self-equivalences are homotopy equivalent if they generate the same closed subgroup in the group Out(X) of homotopy classes of all self-equivalences of X.

2. The geometric realization of a category

Roughly speaking, the geometric realization of a category C is the space obtained starting with a disjoint set of vertices, one for each object in C, then attaching one edge for each morphism (where the endpoints of the edge are attached to the source and target of the morphism), then one 2-simplex (triangle) for each commutative triangle of morphisms in C, etc. Also, the edge corresponding to an identity morphism Id_{c} for c ∈ Ob(C) is identified to the vertex corresponding to c, and similarly for higher dimensional simplices which involve identity morphisms.

In order to make this construction more precise, we begin by defining simplicial sets and their geometric realizations.

2.1. Simplicial sets and their realizations.

Let Δ be the simplicial category: the category whose objects are the sets [n] = {0, 1, …, n} for n ≥ 0, and whose morphisms are the order preserving maps between objects. For each n, there are n + 1 face morphisms d_{n}^{i} ∈ Mor_{Δ}([n − 1], [n]) (0 ≤ i ≤ n), where d_{n}^{i} is the (unique) injective morphism whose image does not contain i. Also, there are n degeneracy morphisms s_{n}^{i} ∈ Mor_{Δ}([n], [n − 1]) (0 ≤ i ≤ n − 1), where s_{n}^{i} is the unique surjective morphism such that s_{n}^{i}(i) = s_{n}^{i}(i + 1) = i. One easily sees that each morphism in Δ is a composite of such face and degeneracy morphisms.

A simplicial set is a functor K: Δ^{op} ⟶ Sets. If K is a simplicial set, we often write K_{n} = K([n]), which is regarded as the set of “n-simplices” of K. For each n-simplex σ ∈ K_{n}, the codimension one faces of σ are the (n−1)-simplices d_{n}^{i}^{∗}(σ) ∈ K_{n−1}. If σ ∈ Im(s_{n}^{i}^{∗}) for some i, then σ is called a degenerate simplex. Equivalently, σ is degenerate if σ ∈ Im(χ^{∗}) for any χ ∈ Mor(Δ) which is not injective.

A map of simplicial sets or a simplicial map from K to L is a natural transformation of functors from K to L; i.e., a sequence φ = {φ_{n}}_{n=0}^{∞} of maps φ_{n}: K_{n} ⟶ L_{n} which commute with all face and degeneracy morphisms. We let S denote the category of simplicial sets, and write Mor_{S}(K, L) for the set of all simplicial maps from K to L.

Let {e_{0}, …, e_{n}} be the canonical basis for ℝ^{n+1}. Let Δ^{n} be the nsimplex spanned by these elements; i.e.,

Δ^{n} = {(t_{0}, …, t_{n}) ∈ ℝ^{n+1} | 0 ≤ t_{i} ≤ 1, ∑_{i=0}^{n} t_{i} = 1}.

There is an obvious functor Δ ⟶ Top which sends each object [n] to Δ^{n}, and which sends a morphism φ ∈ Mor_{Δ}([n], [m]) to the affine map φ_{∗}: Δ^{n} ⟶ Δ^{m} which sends a vertex e_{i} ∈ Δ^{n} to e_{φ(i)} ∈ Δ^{m}. Equivalently, φ_{∗}(t_{0}, …, t_{n}) = (s_{0}, …, s_{m}), where s_{i} = ∑_{j∈φ^{−1}(i)} t_{j} (and s_{i} = 0 if φ^{−1}(i) = ∅).

The geometric realization |K| of a simplicial set K is now defined by setting

|K| = (∐_{n=0}^{∞} K_{n} × Δ^{n})/∼

(with the quotient topology), where

(σ, φ_{∗}(τ)) ∼ (φ^{∗}(σ), τ) ∀ σ ∈ K_{m}, τ ∈ Δ^{n}, and φ ∈ Mor_{Δ}([n], [m]).

Thus |K| has the structure of a CW complex with one vertex for each σ ∈ K_{0}, an edge Δ^{1} for each nondegenerate element of K_{1}, a 2-cell for each nondegenerate element of K_{2}, etc.

We can regard the geometric realization |K| of a simplicial set K as a type of “generalized” simplicial complex, but not as a simplicial complex in the strict sense. For example, the two endpoints of a (nondegenerate) edge can be equal, and there can be several different edges between the same pair of vertices.

For example, let Δ_{m}, for any m ≥ 0, be the simplicial set defined by setting Δ_{m}([n]) = Mor_{Δ}([n], [m]). Let ∂Δ_{m} ⊆ Δ_{m} denote its boundary: ∂Δ_{m}([n]) is the set of all maps in Mor_{Δ}([n], [m]) which are not surjective. Let Δ_{m}/∂Δ_{m} be the quotient simplicial set, which sends [n] to the quotient set Δ_{m}([n])/∂Δ_{m}([n]) (i.e., all elements of the subset are identified to a point). Then |Δ_{m}| is an m-simplex, as a topological space in the usual sense, |∂Δ_{m}| is its boundary (homeomorphic to S^{m−1}), and |Δ_{m}/∂Δ_{m}| is homeomorphic to D^{m}/S^{m−1} ≅ S^{m}. Thus ∂Δ_{m+1} and Δ_{m}/∂Δ_{m} are two very different simplicial sets whose geometric realizations are homeomorphic to the sphere S^{m}.

For any simplicial set K, let C_{n}(K), for each n ≥ 0, be the free abelian group with basis K_{n} = K([n]). There are “boundary maps”

∂: C_{n}(K) ⟶ C_{n−1}(K) ,

defined by setting ∂(σ) = ∑_{i=0}^{n}(−1)^{i}d_{n}^{i}^{∗}(σ) for each σ ∈ K_{n}. The homology H_{∗}(K; A) of K with coefficients in an abelian group A is defined to be the homology of the chain complex

⟶ C_{n}(K) ⊗_{ℤ} A ⟶^{∂⊗Id_{A}} C_{n−1}(K) ⊗_{ℤ} A ⟶^{∂⊗Id_{A}}

⋯ ⟶^{∂⊗Id_{A}} C_{0}(K) ⊗_{ℤ} A ⟶ 0.

The cohomology H^{∗}(K; A) of K with coefficients in A is the homology of the dual chain complex with terms Hom_{ℤ}(C_{n}(K), A).

The (co)homology groups of the simplicial set K are naturally isomorphic to the cellular (co)homology groups of its geometric realization |K|, when regarded as a CW complex. To see this, let D_{n} ⊆ C_{n}(K) be the subgroup generated by all degenerate n-simplices. The natural map of chain complexes (C_{∗}(K), ∂) −↠ (C_{∗}(K)/D_{∗}, ∂) induces an isomorphism in homology (cf. [McL, Theorem VIII.6.1]), and so H_{∗}(K; A) ≅ H_{∗}(|K|; A) (and likewise in cohomology) since the cells in |K| as a CW complex correspond to the nondegenerate simplices in K.

The singular simplicial set S.(X) of a topological space X is defined by sending [n] to the set S_{n}(X) of all continuous maps Δ^{n} ⟶ X, and by sending a morphism φ to composition by φ_{∗}. The geometric realization |S.(X)| has the same homotopy and homology groups as X, and has the homotopy type of X whenever X is sufficiently “nice” (for example, when it has the structure of a CW complex). By definition, the singular (co)homology of X is just the (co)homology of this simplicial set S.(X).

It is an easy exercise to construct a bijection

Mor_{S}(K, S.(X)) ⟶^{≅} map(|K|, X) ,

for any simplicial set K and any space X. Thus the functor S.(−) from spaces to simplicial sets is right adjoint to the realization functor.

Simplicial sets behave very nicely with respect to products (and this is one way in which the “degenerate simplices” play a key role). The product K × L of two simplicial sets K and L is defined simply by setting (K × L)_{n} = K_{n} × L_{n}, and similarly for induced morphisms. One of the basic theorems in the subject says that the map of spaces

|K × L| ⟶ |K| × |L|

induced by the obvious projections of simplicial sets is always a continuous bijection, and is a homeomorphism under certain conditions (for example, if K or L has finitely many nondegenerate simplices). See [GZ, Section III.3] for more details about this and a proof. For our purposes here, we use this only in the case where L = Δ_{m} for some m ≥ 1, and hence where |L| is a simplex. Showing that |K × Δ_{m}| is homeomorphic to |K| × Δ^{m} is an easy exercise, and gives insight into why this also works in the general case — and also into the importance of degenerate simplices in a simplicial set.

2.2. The nerve of a category as a simplicial set.

Now let C be a small category. The nerve of C is a simplicial set N (C), defined by

N (C)_{n} = N (C)([n]) =^{def}

{c_{0} ⟶^{α_{1}} c_{1} ⟶^{α_{2}} ⋯ ⟶^{α_{n}} c_{n} | c_{i} ∈ Ob(C), α_{i} ∈ Mor(C)},

and where φ ∈ Mor_{Δ}([n], [m]) sends N (C)_{m} to N (C)_{n} by composing morphisms or inserting identity morphisms as appropriate. The geometric realization |C| of the category C is defined to be the geometric realization |N (C)| of its nerve.

For example, let (X, ≤) be a poset: a set with a partial ordering ≤. We regard X as a category, with object set X and with a unique morphism x → y whenever x ≤ y. Then N (X) is the simplicial set where for n ≥ 0, N (X)_{n} is the set of all order preserving maps [n] ⟶ X. Hence there is a nondegenerate n-simplex for each strictly increasing chain x_{0} < x_{1} < ⋯ < x_{n} of elements of X. The faces of such a simplex are obtained by removing elements from the chain in the obvious way. Note that in this case, |X| is a simplicial complex in the strict sense.

As one special case, if we regard the poset [m] = {0, 1, …, m} as a category, then its nerve N ([m]) is the simplicial set Δ_{m} defined above, and hence |[m]| is an m-simplex.

If f: C ⟶ D is any functor between small categories, then it follows immediately from the definition that f induces a simplicial map N (f) ∈ Mor_{S}(N (C), N (D)), and hence a map of spaces |f|: |C| ⟶ |D| between the geometric realizations. In fact, as first shown by Segal, one can say much more.

Proposition 2.1 ([Sg, Proposition 2.1]). Fix two categories C and D, and let f and g be a pair of functors from C to D. Assume there is a natural transformation of functors u from f to g. Then the two maps |f| and |g| between the geometric realizations are homotopic.

Proof. Recall that [1] denotes the category with object set {0, 1}, and with a unique nonidentity morphism 0 → 1. By definition, the natural transformation u consists of morphisms u(c) ∈ Mor_{D}(f(c), g(c)), for each c ∈ Ob(C), such that g(α)∘u(c) = u(c′)∘f(α) for each α ∈ Mor_{C}(c, c′). This determines a functor

û: C × [1] ⟶ D,

on objects by setting û(c, 0) = f(c) and û(c, 1) = g(c), and on morphisms by setting (for each c, c′ ∈ Ob(C) and α ∈ Mor_{C}(c, c′)):

û((c, 0) ⟶^{(α,Id_{0})} (c′, 0)) = f(α) ,  û((c, 1) ⟶^{(α,Id_{1})} (c′, 1)) = g(α)

and

û((c, 0) ⟶^{(α,0→1)} (c′, 1)) = g(α) ∘ u(c) = u(c′) ∘ f(α) .

Since |[1]| is the unit interval, |C ×[1]| is homeomorphic in a natural way to |C| × I. So the geometric realization of û is a homotopy between |f| and |g|. □

We now list some easy consequences of Proposition 2.1. By an equivalence of categories we mean a functor which induces a bijection on isomorphism classes of objects, and bijections on all morphism sets. Alternatively, a functor f: C ⟶ D is an equivalence if and only if there is a functor g: D ⟶ C such that both composites g ∘ f and f ∘ g are naturally isomorphic to the identity functors.

Corollary 2.2. (a) If a category C has an initial or a final object, then |C| is contractible.

(b) If f: C ⟶ D is an equivalence of categories, then |f| is a homotopy equivalence from |C| to |D|.

Proof. If c_{0} ∈ Ob(C) is an initial object, and F_{c_{0}} is the constant functor which sends all objects to c_{0} and all morphisms to the identity, then there is a natural transformation of functors from F_{c_{0}} to Id_{C} which sends an object c to the unique morphism from c_{0} to c. Hence |F_{c_{0}}| ≃ Id_{|C|}, and so |C| is contractible. The argument when c_{0} is a final object is similar.

Point (b) follows immediately from the above remarks about equivalences of functors. □

We also note the following lemma, which is among the most elementary of the many results which are useful for comparing higher limits over different categories.

Lemma 2.3 ([BLO1, Lemma 1.3]). Fix a prime p. Let f: C ⟶ D be a functor between small categories which satisfies the following conditions:

(i) f is bijective on isomorphism classes of objects and is surjective on morphism sets.

(ii) For each c ∈ Ob(C), the subgroup

K(c) = Ker[Aut_{C}(c) ⟶ Aut_{D}(f(c))]

is finite of order prime to p.

(iii) For each pair of objects c, d ∈ Ob(C) and each φ, ψ ∈ Mor_{C}(c, d), f(φ) = f(ψ) if and only if there is some α ∈ K(c) such that ψ = φ∘α (i.e., Mor_{D}(f(c), f(d)) ≅ Mor_{C}(c, d)/K(c)).

Then |f|: |C| ⟶ |D| is a p-equivalence of spaces, and hence |f|_{p}^{∧} is a homotopy equivalence from |C|_{p}^{∧} to |D|_{p}^{∧}.

2.3. Classifying spaces as geometric realizations of categories.

Classifying spaces of groups and their universal covering spaces provide two very useful examples of geometric realizations of categories.

Definition 2.4. For any discrete group G, define categories B(G) and E(G) as follows.

• Ob(B(G)) = {o_{G}} and Mor_{B(G)}(o_{G}, o_{G}) = G. Thus B(G) has a unique object, morphisms from this object to itself are identified with the elements of G, and composition in B(G) is defined by group multiplication.

• Ob(E(G)) = G, and there is a unique morphism (g → h) for each pair (g, h) ∈ G × G.

• p_{G}: E(G) ⟶ B(G) denotes the functor which sends each object of E(G) to the unique object o_{G} in B(G), and which sends a morphism (g → h) to the morphism hg^{−1} in B(G).

Since each object of E(G) is both an initial and a final object, |E(G)| is contractible by Corollary 2.2. Also, G acts on E(G) via right multiplication on objects and on morphisms, this induces a free G-action on the geometric realization |E(G)|, and |p_{G}| is the orbit map of this action. Hence by Proposition 1.2, |E(G)| is the universal cover of |B(G)|, G is the group of deck transformations of this covering space, and π_{1}(|B(G)|) ≅ G. In other words:

Proposition 2.5 ([Sg, § 3]). For any discrete group G, π_{1}(|B(G)|) ≅ G, |B(G)| is a classifying space for G, and |E(G)| is its universal covering space.

For example, when G = {1, g} is cyclic of order 2, then for each n ≥ 0, N (E(G))_{n} contains exactly two nondegenerate simplices, represented by the sequences of objects (1, g, 1, g, …) and (g, 1, g, 1, …) (each containing n+1 entries). Each of them is attached to the two nondegenerate (n−1)- simplices via one face each, while all of its other faces are degenerate. Using this description, it is not hard to see that |E(G)| is homeomorphic to S^{∞} =^{def} ⋃_{n=1}^{∞} S^{n}, with its cell structure containing S^{n} as the n-skeleton for each n. Thus |B(G)| is homeomorphic to the infinite dimensional real projective space ℝP^{∞}.

In contrast, |B(ℤ)| is not homeomorphic to the circle S^{1}, although they have the same homotopy type. Since N (B(ℤ)) contains nondegenerate simplices of arbitrarily large dimension, its geometric realization is an infinite dimensional space.

From now on, whenever we write the classifying space of G, or BG, we mean the geometric realization of the category B(G). Similarly, we write EG = |E(G)|.

2.4. Fundamental groups and coverings of geometric realizations.

The fundamental group of the geometric realization of a category or a simplicial set is defined topologically, but it can also be described purely algebraically in terms of generators and relations. We show here one way to do this.

Notation. We will use the following notation for the vertices and edges of a simplex in a simplicial set K. Fix n ≥ 1. For each 0 ≤ i ≤ n, let v_{i} ∈ Mor_{Δ}([0], [n]) be the morphism with v_{i}(0) = i ∈ [n]. For each 0 ≤ i < j ≤ n, let e_{ij} ∈ Mor_{Δ}([1], [n]) be the morphism with image {i, j}.

Thus for σ ∈ K_{n}, v_{i}^{∗}(σ) is the i-th vertex of σ, and for i < j, e_{ij}^{∗}(σ) is the edge from v_{i}^{∗}(σ) to v_{j}^{∗}(σ). Note that when σ ∈ K_{1}, e_{01}^{∗}(σ) = σ.

Lemma 2.6. For any simplicial set K and any discrete group G, restriction to 1-simplices defines a bijection

R_{K,G}: Mor_{S}(K, N (B(G))) ⟶^{≅} {α: K_{1} → G | ∀ σ ∈ K_{2},

α(e_{02}^{∗}(σ)) = α(e_{12}^{∗}(σ))α(e_{01}^{∗}(σ))} .

Proof. Clearly, the condition φ_{1}(e_{02}^{∗}(σ)) = φ_{1}(e_{12}^{∗}(σ))φ_{1}(e_{01}^{∗}(σ)) holds for each simplicial map φ and each σ ∈ K_{2}. Hence R_{K,G} is well defined. It is injective since a simplicial map to N (B(G)) is determined by its restriction to 1-simplices.

Now fix α: K_{1} ⟶ G such that α(e_{02}^{∗}(σ)) = α(e_{12}^{∗}(σ))α(e_{01}^{∗}(σ)) for all σ ∈ K_{2}. In particular, if v ∈ K_{0} is a vertex, and σ_{v}^{n} ∈ K_{n} is the degenerate n-simplex over v, then e_{ij}^{∗}(σ_{v}^{2}) = σ_{v}^{1} for each 0 ≤ i < j ≤ 2, and hence α(σ_{v}^{1}) = 1. In other words, α sends all degenerate 1-simplices to the identity.

Let φ_{0} : K_{0} ⟶ N (B(G))_{0} be the map which sends each element to the unique vertex of N (B(G)), and set φ_{1} = α. For n ≥ 2, define φ_{n}: K_{n} ⟶ N (B(G))_{n} by setting

φ_{n}(σ) = (α(e_{01}^{∗}(σ)), α(e_{12}^{∗}(σ)), …, α(e_{n−1,n}^{∗}(σ))) .

The φ_{n} clearly commute with all face and degeneracy maps, and hence define a simplicial map φ ∈ Mor_{S}(K, N (B(G))). Then R_{K,G}(φ) = α, and thus R_{K,G} is onto. □

We are now ready to describe the fundamental group of the realization of a simplicial set. When doing this, it will be useful to consider trees as special types of simplicial sets. A (directed) graph Γ is a simplicial set all of whose nondegenerate simplices are in Γ_{0} or Γ_{1}. Thus the realization |Γ| is 1-dimensional. A circuit in a graph Γ is a simplicial subset whose geometric realization is homeomorphic to the circle. A tree is a connected graph which contains no circuits.

Proposition 2.7. Fix a simplicial set K, a vertex x_{0} ∈ K_{0}, and a tree T ⊆ K which contains all vertices of K (i.e., T_{0} = K_{0}). We also regard x_{0} as a vertex in |K|. For each v ∈ K_{0}, let ι_{v} be a path in |T | (unique up to homotopy) from x_{0} to v. Define

θ: K_{1} ⟶ π_{1}(|K|, x_{0})

by sending e ∈ K_{1} to the class of the loop ι_{v_{1}^{∗}(e)}·e·ι_{v_{0}^{∗}(e)}, where as usual we compose paths from right to left. Then θ induces an isomorphism of π_{1}(|K|, x_{0}) with the free group on generators [e] for each e ∈ K_{1}, modulo the relations [e] = 1 for e ∈ T_{1}, and [e_{02}^{∗}(σ)] = [e_{12}^{∗}(σ)][e_{01}^{∗}(σ)] for each σ ∈ K_{2}.

Equivalently, for any group G, and any map α: K_{1} ⟶ G such that α(T_{1}) = 1 and α(e_{02}^{∗}(σ)) = α(e_{12}^{∗}(σ))α(e_{01}^{∗}(σ)) for each σ ∈ K_{2}, there is a unique homomorphism

α: π_{1}(|K|, x_{0}) ⟶ G

such that α = α ∘ θ.

Proof. This is an easy consequence of standard facts about fundamental groups of cell complexes (cf. [Ht, Proposition 1.26]), and is shown explicitly in [OV1, Proposition A.3(a)]. We outline a proof here.

We first prove the last statement. For each map α: K_{1} ⟶ G such that α(e_{02}^{∗}(σ)) = α(e_{12}^{∗}(σ))α(e_{01}^{∗}(σ)) for all σ ∈ K_{2}, there is a unique simplicial map φ: K ⟶ N (B(G)) such that φ_{1} = α (Lemma 2.6). Then |φ| is a map of spaces from |K| to |B(G)| = BG, and hence induces a homomorphism of groups

α =^{def} |φ|_{#} : π_{1}(|K|, x_{0}) ⟶ π_{1}(|B(G)|, ∗) = G .

If in addition, α(T_{1}) = 1, then |φ| sends |T | to the basepoint by Lemma 2.6 again, and hence α ∘ θ = α by definition of θ.

The uniqueness of α is clear once we know that π_{1}(|K|, x_{0}) is generated by Im(θ). Essentially, this is a consequence of [Ht, Proposition 1.26], which implies (as a special case) that every element in the fundamental group of a cell complex X can be represented by a loop following only the edges of X.

Thus θ is universal among all maps K_{1} ⟶ G which send T_{1} to the identity and satisfy the relation for 2-simplices. It follows that π_{1}(|K|, ∗) is isomorphic to the free group on K_{1}, modulo relations given by T_{1} and 2-simplices. □

When K is the nerve of a category C, Proposition 2.7 takes the following form.

Proposition 2.8. Fix a small category C and an object c_{0} in C. Assume we can choose morphisms ι_{c} ∈ Mor_{C}(c, c_{0}), for each c ∈ Ob(C), where ι_{c_{0}} = Id_{c_{0}}. Let ∗ ∈ |C| be the vertex which represents the object c_{0}. Define

θ: Mor(C) ⟶ π_{1}(|C|, ∗)

by sending α ∈ Mor_{C}(c, d) to the class of the loop ι_{d}·α·ι_{c}. Then θ induces an isomorphism of π_{1}(|C|, ∗) with the free group on generators [α] for each α ∈ Mor(C), modulo the relations [ι_{c}] = 1 for c ∈ Ob(C) and [β ∘α] = [β][α] for any composable pair β, α ∈ Mor(C). Hence for any group G, and any functor F: C ⟶ B(G) such that F (ι_{c}) = 1 for each c ∈ Ob(C), there is a unique homomorphism

F: π_{1}(|C|, ∗) ⟶ G

such that Mor(F) = F ∘ θ.

One can, of course, give a presentation of π_{1}(|C|) in terms of morphisms in C without the assumption that some object is the target of morphisms from all other objects, but the above formulation suffices for our purposes.

The next lemma gives a procedure, in certain cases, for constructing covering spaces (up to homotopy type) of |C| in terms of subcategories of C.

Proposition 2.9. Fix a small category C, an object c_{0} in C, a group G, and a functor F: C ⟶ B(G). Assume, for each c ∈ Ob(C) and each g ∈ G, there are d ∈ Ob(C) and ψ ∈ Iso_{C}(c, d) such that F (ψ) = g. Let ∗ ∈ BG = |B(G)| be the unique vertex, and let

F = |F |_{#} : π_{1}(|C|, ∗) ⟶ π_{1}(BG) = G

be the homomorphism of groups induced by F. For each H ≤ G, let C_{H} ⊆ C be the subcategory with the same objects, where for all ψ ∈ Mor(C), ψ ∈ Mor(C_{H}) if and only if F (ψ) ∈ H. Then for each H ≤ G, |C_{H}| is homotopy equivalent to the covering space of |C| whose fundamental group is F^{−1}(H).

Proof. This is shown in [OV1, Proposition A.4], and generalizes constructions in [5a2]. We sketch here the proof when H = 1.

Let C̃ be the pullback category in the following square:

\xymatrix{C̃ \ar[r] \ar[d] & E(G) \ar[d]^{p_{G}} \\ C \ar[r]^{F} & B(G) ,}

where p_{G}: E(G) ⟶ B(G) is as in Definition 2.4. Thus Ob(C̃) = Ob(C) × G, and

MorC̃((c, g), (d, h)) = {φ ∈ Mor_{C}(c, d) | F (φ) = hg^{−1}} .

Identify C_{1} with the full subcategory of C̃ whose objects are the pairs (c, 1) for c ∈ Ob(C).

By assumption, for each c ∈ Ob(C) and each g ∈ G, there are d ∈ Ob(C) and ψ ∈ Iso_{C}(c, d) such that F (ψ) = g^{−1}. Then ψ ∈ IsoC̃((c, g), (d, 1)). Since c and g were arbitrary, this shows that each object of C̃ is isomorphic to an object in the subcategory C_{1}, and hence that the inclusion C_{1} ⊆ C̃ is an equivalence of categories. So by Corollary 2.2(b), it induces a homotopy equivalence |C_{1}| ≃ |C̃|.

By construction, G acts freely on the sets of objects and morphisms in C̃: a ∈ G sends (c, g) ∈ Ob(C̃) to (c, ga^{−1}) and sends φ ∈ MorC̃((c, g), (d, h)) to φ ∈ MorC̃((c, ga^{−1}), (d, ha^{−1})). The orbit sets of those actions are Ob(C) and Mor(C). This free action extends to each of the sets N (C̃)_{n} which makes up the nerve of C̃, and from that to a free action of G on |C̃|. The projection functor C̃ ⟶ C thus induces a covering space |C̃| ⟶ |C| of geometric realizations. Since |C̃| ≃ |C_{1}|, this finishes the proof of the proposition. □

2.5. Spaces of maps.

This section plays a relatively minor role in the rest of this survey, and it can easily be skipped. But since some definitions and some of the arguments can be more easily understood or motivated when one knows something about certain spaces of maps between classifying spaces or their p-completions, we include here a short discussion, with references, about spaces of maps between a pair of topological spaces. This information will be referred to later, in Section 5.7, when sketching the proofs of some theorems about homotopy classes of maps.

If we want to discuss homotopy groups of the connected components of a space map(X, Y), we first must explain which topology we choose for the set of all maps. The usual choice, which works when X and Y are CW complexes, is the compact-open topology (cf. [Ht, Appendix]). However, what we really need to know, for certain constructions which will be described later (e.g., in Section 5.7) are not the homotopy groups of the space of maps itself, but the groups of homotopy classes of maps S^{k} × X ⟶ Y relative to a given map on ∗ × X (for some ∗ ∈ S^{k}). It is simply more convenient to describe these in terms of homotopy groups of a mapping space map(X, Y).

In other words, we can replace map(X, Y) by any topological space M, together with a continuous “evaluation map” ev: M × X ⟶ Y, such that ev defines a bijection from the path components of M to the homotopy classes of maps X ⟶ Y, and such that for each k > 0 and each m_{0} ∈ M, ev defines an isomorphism between π_{k}(M, m_{0}) and the set of homotopy classes of maps S^{k} × X ⟶ Y whose restriction to ∗ × X (for a fixed basepoint ∗ ∈ S^{k}) is the map ev(m_{0}, −). When X and Y are realizations of simplicial sets K and L, respectively, and L is a “Kan complex” (cf. [Cu, Definition 1.12] or [GJ, p.12]), then this can be done combinatorially, in terms of simplicial maps (cf. [GJ, § I.5]).

The following is the main result in this subsection. Note that it includes Proposition 1.5 in the case where X is the realization of a simplicial set.

Proposition 2.10. Fix a group G, and a simplicial set K such that |K| is connected. Choose a base vertex x_{0} ∈ K_{0}, also regarded as a vertex in |K|, and let ∗ ∈ BG = |B(G)| be the unique vertex. We identify π_{1}(BG, ∗) = G.

(a) There is a bijection χ: [|K|, BG] ⟶ Rep(π_{1}(|K|, x_{0}), G) with the property: for any map f: |K| ⟶ BG such that f(x_{0}) = ∗, χ([f]) = [f_{#}].

(b) Fix f: |K| ⟶ BG such that f(x_{0}) = ∗. Set H = f_{#}(π_{1}(|K|, x_{0})). Then there is an isomorphism π_{1}(map(|K|, BG), f) ≅ C_{G}(H) defined by restriction to basepoint. Also, π_{n}(map(|K|, BG), f) = 0 for all n ≥ 2.

Proof. We use here the notation defined before Lemma 2.6 for the vertices and edges of a simplex in K.

Step 1: We prove here that each continuous map f: |K| ⟶ BG is homotopic to a one which is realized by a simplicial map. Moreover, this holds in a relative version: if L ⊆ K is any simplicial subset, f: |K| ⟶ BG is a continuous map, and ψ: L ⟶ N (B(G)) is a simplicial map such that |ψ| = f|_{|L|}, then there is a simplicial map φ: K ⟶ N (B(G)), and a homotopy Φ: |K| × I ⟶ BG from |φ| to f which is constant on |L|.

This is a special case of the “simplicial extension theorem” in [Cu, Theorem 12.1], where it is proved under the assumption that the target is a Kan complex. That N (B(G)) is a Kan complex is shown, for example, in [GJ, Lemma 3.5]. Since it is much easier to work with maps to N (B(G)) than to arbitrary Kan complexes, we sketch the proof here.

For each vertex v ∈ K_{0}, choose a path ϕ_{v}: I ⟶ BG such that ϕ_{v}(0) = ∗ and ϕ_{v}(1) = f(v), and such that ϕ_{v} is constant if f(v) = ∗. For each edge e ∈ K_{1}, we regard f(e) as a path in BG from f(v_{0}^{∗}(e)) to f(v_{1}^{∗}(e)), and set

α(e) = [ϕ_{v_{1}^{∗}(e)}·f(e)·ϕ_{v_{0}^{∗}(e)}] ∈ π_{1}(BG, ∗) = G .

Then for each 2-simplex σ ∈ K_{2},

α(e_{12}^{∗}(σ))α(e_{01}^{∗}(σ)) = [ϕ_{v_{2}^{∗}(σ)}·f(e_{12}^{∗}(σ))·ϕ_{v_{1}^{∗}(σ)}]·[ϕ_{v_{1}^{∗}(σ)}·f(e_{01}^{∗}(σ))·ϕ_{v_{0}^{∗}(σ)}]

= [ϕ_{v_{2}^{∗}(σ)}·f(e_{12}^{∗}(σ))·f(e_{01}^{∗}(σ))·ϕ_{v_{0}^{∗}(σ)}] = α(e_{02}^{∗}(σ)) .

So by Lemma 2.6, there is a (unique) simplicial map φ: K ⟶ N (B(G)) such that φ_{1} = α. Furthermore, since α|_{L_{1}} = ψ_{1} by construction, φ|_{L} = ψ.

Define X_{0} ⊆ X_{1} ⊆ X_{2} ⊆ ⋯ ⊆ X = |K| × I, by setting

X_{n} = (|K| × {0, 1}) ∪ (|L| × I) ∪ (|K|^{(n)} × I)

for each n ≥ 0. Here, |K|^{(n)} ⊆ |K| is the union of the realizations of all simplices in K of dimension ≤ n. Set

Φ_{0} = (f × 0) ∪ (|φ| × 1) ∪ (f ∘ pr_{|L|}) ∪ {ϕ_{v}}: X_{0} ⟶ BG .

We will construct maps Φ_{n}: X_{n} ⟶ BG such that Φ_{n}|_{X_{n−1}} = Φ_{n−1} for each n ≥ 0.

Assume, for some n ≥ 1, that Φ_{n−1} has been defined. To define Φ_{n}, we must define a map Φ_{σ}: Δ^{n} × I ⟶ BG for each σ ∈ K_{n}, which extends a given map (the restriction of Φ_{n−1}) on

(Δ^{n} × {0, 1}) ∪ (∂Δ^{n} × I) ≅ S^{n} .

When n = 1, this is possible since by definition of α, the two paths |α(σ)| and ϕ_{v_{1}^{∗}(σ)}·f(σ)·ϕ_{v_{0}^{∗}(σ)} in BG are homotopic. When n ≥ 2, this can be done since π_{n}(BG) ≅ π_{n}(EG) is the trivial group (so each map S^{n} ⟶ BG extends to D^{n+1}). Alternatively, when n = 2, since S^{n} is simply connected, the restriction of Φ_{n−1} to (Δ^{n} × {0, 1}) ∪ (∂Δ^{n} × I) lifts to a map to EG = |E(G)|, the lifting extends to a map defined on Δ^{n} × I since EG is contractible, and hence the original map extends to one defined on Δ^{n} × I.

In this way, we construct Φ_{n} for each n. Then Φ = ⋃_{n=0}^{∞} Φ_{n} is a homotopy from |φ| to f which is constant on |L|, and this proves the above claim.

Step 2: Choose a tree T ⊆ K such that T_{0} = K_{0}. That this always can be done is shown, for example, in [Se2, Proposition 11], applied to the pair (K_{0}, K_{1}) regarded as a graph.

Assume φ, ψ: K ⟶ N (B(G)) are simplicial maps. By Step 1, |φ| is homotopic to |ψ| as maps from |K| to BG if and only if φ ≃ ψ; i.e., homotopic via a simplicial homotopy K×Δ_{1} ⟶ N (B(G)). Using Lemma 2.6, we see that

φ ≃ ψ ⟺ ∃ β: K_{0} ⟶ G such that

ψ(e) = β(v_{1}^{∗}(e))φ(e)β(v_{0}^{∗}(e))^{−1} ∀ e ∈ K_{1} . (1)

Fix φ ∈ Mor_{S}(K, N (B(G))). For each v ∈ K_{0}, choose a path ϕ_{v} in |T | from x_{0} to v (unique up to homotopy), and set β(v) = [|φ| ∘ ϕ_{v}]^{−1} ∈ π_{1}(BG, ∗) = G. By Lemma 2.6, there is ψ ∈ Mor_{S}(K, N (B(G))) such that ψ(e) = β(v_{1}^{∗}(e))φ(e)β(v_{0}^{∗}(e))^{−1} for each e ∈ K_{1}, and ψ ≃ φ by (1). Since ψ(T_{1}) = 1 by construction, this proves that each map from K to BG is homotopic to one induced by a simplicial map which sends |T | to the basepoint.

We identify Mor_{S}(K/T, N (B(G))) with the set of all morphisms φ ∈ Mor_{S}(K, N (B(G))) such that φ(T_{1}) = 1. By Lemma 2.6, and the description of π_{1}(|K|) in Proposition 2.7, there is a bijection

Mor_{S}(K/T, N (B(G))) ⟶ Hom(π_{1}(|K|, x_{0}), G)

which sends φ to |φ|_{#}. Also, φ, ψ ∈ Mor_{S}(K/T, N (B(G))) are simplicially homotopic if and only if there is a constant function β which satisfies the condition in (1); i.e., there is g ∈ G such that |φ|_{#} = c_{g} ∘ |ψ|_{#}. Together with Step 1, this finishes the proof that [|K|, BG] ≅ Rep(π_{1}(|K|, x_{0}), G).

Again assume φ ∈ Mor_{S}(K/T, N (B(G))). By Step 1, each loop in map(|K|, BG) based at |φ| is homotopic to the realization of a simplicial homotopy from φ to itself, which by (1) is determined by an element g ∈ G which commutes with φ(e) for each e ∈ K_{1}. Thus g ∈ C_{G}(H), where H = Im(|φ|_{#}). Two such loops are homotopic (relative to the endpoints) only if they are equal as simplicial homotopies (Lemma 2.6 again), and thus π_{1}(map(|K|, BG), |φ|) ≅ C_{G}(H).

For n ≥ 2, by Step 1 again, each element of π_{n}(map(|K|, BG), |φ|) is represented by a simplicial map Φ: K × Δ_{n} ⟶ N (B(G)) whose restriction to K × ∂Δ_{n} is φ ∘ pr_{K}. Since all 1-simplices of K × Δ_{n} are contained in K × ∂Δ_{n}, Lemma 2.6 implies that there is only one such map, and thus that π_{n}(map(|K|, BG), |φ|) = 1. □

Thus, in the situation of Proposition 2.10(b), the connected component of f in map(|K|, BG) has the weak homotopy type of BC_{G}(H) (i.e., they have the same homotopy groups).

The following, much deeper theorem describes mapping spaces in certain special cases when we replace the target BG by its p-completion. This result is in part a special case of Theorem 4.21, whose proof will be sketched in Section 5.7.

Theorem 2.11. Fix a prime p, a p-group Q, and a finite group G. Then the space of maps map(BP, BG_{p}^{∧}) has the (weak) homotopy type of the pcompletion of map(BP, BG). In particular, [BP, BG_{p}^{∧}] ≅ Rep(P, G), and for each ρ ∈ Hom(Q, G),

π_{1}(map(BP, BG_{p}^{∧}), Bρ) ≅ π_{1}(BC_{G}(ρ(P))_{p}^{∧}) ≅ C_{G}(ρ(P))/O^{p}(C_{G}(ρ(P))) .

Proof. The description of map(BP, BG) follows as a special case of Proposition 2.10. The description of map(BQ, BG_{p}^{∧}) as the p-completion of map(BQ, BG) is shown in [BL, Proposition 2.1], and also follows as a special case of [BLO2, Theorem 6.3]. The formula for the fundamental group of a component then follows from Proposition 1.11. □

3. Linking systems and classifying spaces of finite groups

The p-fusion category of a finite group G was just one of a family of categories defined by Puig [P1], which he called the “p-localités”. These categories allowed him to give a more uniform treatment of certain results in group theory, and to interpret them in category theoretical language. The close connection between these categories and p-completed classifying spaces was discovered much later by Broto, Levi, and Oliver in [BLO1], while attempting to prove the Martino-Priddy conjecture.

3.1. The linking category of a finite group.

For any group G and any pair of subgroups H, K ≤ G, we let T_{G}(H, K) denote the transporter set:

T_{G}(H, K) =^{def} {g ∈ G |^{g}H ≤ K}.

In other words, T_{G}(H, K) is the set of all elements of G which induce homomorphisms in Hom_{G}(H, K). Clearly, T_{G}(H, H) = N_{G}(H) is the normalizer of H in G.

Definition 3.1. Fix a finite group G and a Sylow subgroup S ∈ Syl_{p}(G).

• The transporter category of G is the category T (G) whose objects are the subgroups of G, and whose morphism sets are the transporter sets:

Mor_{T (G)}(H, K) = T_{G}(H, K) .

Let T_{S}(G) ⊆ T (G) be the full subcategory whose objects are the subgroups of S. More generally, if H is any set of subgroups of G, then T_{H}(G) ⊆ T (G) denotes the full subcategory with object set H.

• A p-subgroup P ≤ G is p-centric in G if Z(P) ∈ Syl_{p}(C_{G}(P)); equivalently, if C_{G}(P) = Z(P) × C′_{G}(P) for some (unique) subgroup C′_{G}(P) of order prime to p.

• The centric linking category of G over S is the category L_{S}^{c}(G) whose objects are the subgroups of S which are p-centric in G, and whose morphism sets are given by

Mor_{L_{S}^{c}(G)}(P, Q) = T_{G}(P, Q)/C′_{G}(P)

for all P, Q ≤ S which are p-centric in G.

In addition, when needed, we let T_{S}^{c}(G) ⊆ T_{S}(G) and F_{S}^{c}(G) ⊆ F_{S}(G) be the full subcategories whose objects are the subgroups of S which are p-centric in G. Thus in general, the superscript “c” means restriction to subgroups which are p-centric in G.

The transporter category T_{S}(G) was called by Puig [P1] the 1-localité of G, and the centric linking category L_{S}^{c}(G) is a full subcategory of his Olocalité. In the same way, the fusion category F_{S}(G) is called the C-localité in his terminology, because it is defined by dividing out by centralizers.

There are obvious functors

B(G) ⟶^{Φ} T_{S}(G) ⟶^{Ψ} B(G),

where Φ sends the object o_{G} to the trivial subgroup 1 (and sends g ∈ Aut_{B(G)}(o_{G}) to g ∈ Aut_{T_{S}(G)}(1)), and Ψ sends each morphism set T_{G}(P, Q) to G via the inclusion. Then Ψ ∘ Φ = Id_{B(G)}, and there is a natural transformation of functors from Φ ∘ Ψ to Id_{T_{S}(G)}. So by Proposition 2.1, |T_{S}(G)| ≃ |B(G)| = BG.

Thus the transporter category completely describes the classifying space BG. The problem is that it contains too much information about G itself. What we need is a category which only contains the information about the p-local structure of G, and which can be used as a tool for describing the homotopy properties of the space BG_{p}^{∧}. This is what the linking system L_{S}^{c}(G) does for us, by providing a good intermediate category between the transporter and fusion categories. It contains enough information to completely determine the p-completed homotopy type of BG, but contains only that information and no more.

The following theorem, proven in [BLO1], was the first indication of the importance of the linking categories when working with classifying spaces.

Theorem 3.2. For any finite group G and any S ∈ Syl_{p}(G),

BG_{p}^{∧} ≃ |L_{S}^{c}(G)|_{p}^{∧}.

Proof. We outline the proof here; the details are given in [BLO1, Section 1].

Consider the functors

L_{S}^{c}(G) ⟵^{ρ_{1}} T_{S}^{c}(G) ⊆ T_{S}(G) ⟶^{ρ_{2}} B(G),

where ρ_{1} is the identity on objects and sends g ∈ T_{G}(P, Q) to its class modulo C′_{G}(P); and where ρ_{2} sends all objects to the unique object o_{G} of B(G) and sends a morphism g ∈ T_{G}(P, Q) to g as a morphism in B(G). Consider the maps between spaces induced by their geometric realizations:

|L_{S}^{c}(G)| ⟵^{|ρ_{1}|} |T_{S}^{c}(G)| ⊆ |T_{S}(G)| ⟶^{|ρ_{2}|} |B(G)| = BG .

Then |ρ_{1}| is a p-equivalence by Lemma 2.3, while |ρ_{2}| is a homotopy equivalence by the above remarks. The inclusion of |T_{S}^{c}(G)| into |T_{S}(G)| is a p-equivalence by a theorem of Dwyer [Dw, Theorem 8.3] (suitably interpreted), and the proof of this is sketched in Lemma 5.30 below. Thus all of these maps become homotopy equivalences after p-completion, and in particular, |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧}. □

3.2. Fusion and linking categories of spaces.

Fusion and linking categories can also be defined for spaces. The definition of the fusion category of a space is fairly straightforward, while the definition of its linking category is more subtle. The latter was first defined in [BLO1], while the idea of the fusion category of a space has been around much longer.

Recall that we now define BG =^{def} |B(G)| for each (finite) group G. In particular, this allows us to regard B as a functor from groups to spaces: each φ ∈ Hom(G, H) induces a map Bφ: BG ⟶ BH. Also, it means that when P ≤ Q, we regard BP as a subspace of BQ.

There are several possible definitions of fusion and linking systems for spaces, which differ in their precise choices of objects. The following definition, taken from [BLO2], is the most convenient for our purposes here.

Definition 3.3. Fix a space X, a p-group S, and a map f: BS ⟶ X. The fusion category of X with respect to (S, f) is the category F = F_{S,f}(X), whose objects are the subgroups of S, and where

Mor_{F}(P, Q) = {φ ∈ Inj(P, Q) | f|_{BQ} ∘ Bφ ≃ f|_{BP} }.

Here, Inj(P, Q) denotes the set of all injective homomorphsms from P to Q.

By comparison, for any space X, F_{p}(X) was defined in [BLO1] to be the category whose objects are the pairs (P, f), where P is a p-group and f: BP ⟶ X is a “monomorphism” in a certain sense made precise there. Morphisms in F_{p}(X) were defined in a way analogous to those of F_{S,f}(X). It was in part to avoid the technicalities of defining and working with monomorphisms between topological spaces that we decided to use the above definition. In the cases which interest us (when X = BG or BG_{p}^{∧} for a finite group G), the categories F_{S,f}(X) and F_{p}(X) are equivalent.

As an immediate consequence of Proposition 1.5 and Theorem 1.16, we have

Proposition 3.4. Fix a finite group G and a Sylow subgroup S ∈ Syl_{p}(G). Let f: BS ⟶ BG be the inclusion, and let f̂: BS ⟶ BG_{p}^{∧} be its composite with p-completion. Then

F_{S,f}(BG) ≅ F_{S,f̂}(BG_{p}^{∧}) ≅ F_{S}(G).

Proof. For each subgroup P ≤ S, [BP, BG] ≅ [BP, BG_{p}^{∧}] ≅ Rep(P, G) by Proposition 2.10(a) and Theorem 2.11. Using this, one sees immediately, for φ ∈ Inj(P, Q), that f|_{BQ} ∘ Bφ ≃ f|_{BP} if and only if φ ∈ Hom_{G}(P, Q); and similarly for f̂|_{BQ} ∘ Bφ ≃ f̂|_{BP}. □

The definition of the linking category of a space is more complicated. It is again motivated by the cases X = BG (for a finite group G) and X = BG_{p}^{∧}. When S ∈ Syl_{p}(G) and P, Q ≤ S are p-centric in G, we need to extend Mor_{F_{S,f} (BG)}(P, Q) by Z(P) or by C_{G}(P). Since C_{G}(P) ≅ π_{1}(map(BP, BG)_{incl}) by Proposition 2.10(b) and

Z(P) = C_{G}(P)/O^{p}(C_{G}(P)) ≅ π_{1}(map(BP, BG_{p}^{∧})_{incl})

by Theorem 2.11, this suggests that we add paths in map(BP, X) to the data which determines a morphism in F_{S,f}(X).

Definition 3.5. Fix a space X, a p-group S, and a map f: BS ⟶ X. The linking category of X with respect to (S, f) is the category L_{S,f}(X), whose objects are the subgroups of S, and where

Mor_{L_{S,f} (X)}(P, Q) = {(φ, [H]) | φ ∈ Inj(P, Q), H: BP × I ⟶ X,

H(−, 0) = f|_{BP} , H(−, 1) = f|_{BQ} ∘ Bφ} .

Here, [H] denotes the class of H modulo homotopies which leave fixed BP × {0, 1}. The identity morphism of an object P ≤ S is the pair (Id_{P} , [C_{f|_{BP}}]), where C_{f|_{BP}} is the constant homotopy at f|_{BP}. The composite of two morphisms

P ⟶^{(φ,[H])} Q ⟶^{(ψ,[K])} R

is the morphism (ψ ∘ φ, [(K ∘ (Bφ × Id_{I}))·H]), where (−)·(−) denotes the composite (from right to left) of the two homotopies.

In other words, a morphism in L_{S,f}(X) from P to Q consists of a pair (φ, [H]), where φ ∈ Mor_{F_{S,f} (X)}(P, Q), H is a path from f|_{BP} to f|_{BQ} ∘ Bφ in the mapping space map(BP, X), and [H] is its class modulo homotopies which fix the endpoints of the paths. Let

π_{S,f}: L_{S,f}(X) ⟶ F_{S,f}(X)

denote the forgetful functor which sends each object to itself and sends a morphism (φ, [H]) to φ. A functor

δ_{S,f}: T_{S}(S) ⟶ L_{S,f}(X)

can also be constructed, induced by the isomorphism T_{S}(S) ≅ L_{S,Id}(BS) of the next proposition, followed by composition with f.

For f: BS ⟶ X as above, L_{S,f}^{c}(X) ⊆ L_{S,f}(X) denotes the full subcategory whose objects are the F_{S,f}(X)-centric subgroups of S: those P ≤ S such that each P′ ≤ S which is isomorphic to P in the fusion category F_{S,f}(X) is centric in S. All of these definitions were motivated by the following proposition.

Proposition 3.6 ([BLO1, Proposition 2.6]). Fix a prime p, a finite group G, and a Sylow subgroup S ∈ Syl_{p}(G). Let f: BS ⟶ BG be the inclusion, and let f̂: BS ⟶ BG_{p}^{∧} be its composite with p-completion. Then

L_{S,f}(BG) ≅ T_{S}(G)  and  L_{S,f̂}^{c}(BG_{p}^{∧}) ≅ L_{S}^{c}(G).

Proof. We sketch here a slightly different argument from that given in [BLO1].

Let ∗ ∈ BG = |B(G)| denote the basepoint: the vertex corresponding to the object o_{G} in B(G). Thus ∗ ∈ BP for all P ≤ S ≤ G. Define a functor

κ: L_{S,f}(BG) ⟶^{≅} T_{S}(G)

by setting κ(P) = P for all P ≤ S, and

κ(φ, [H]) = [H|_{∗×I}] ∈ π_{1}(BG, ∗) = G

for all P, Q ≤ S and (φ, [H]) ∈ Mor_{L_{S,f} (BG)}(P, Q). For each g ∈ P, if we let ρ_{g}: I ⟶ BP be the inclusion of the edge corresponding to g ∈ Mor(B(P)), then the composite

I × I ⟶^{ρ_{g}×Id} BP × I ⟶^{H} BG

defines a homotopy (H|_{∗×I})·(f ∘ ρ_{g}) ≃ (f ∘ Bφ ∘ ρ_{g})·(H|_{∗×I}) of loops in BG. So if we set h = [H|_{∗×I}] ∈ G, then hg = φ(g)h in π_{1}(BG) = G, hence φ = c_{h} ∈ Hom(P, Q), and κ(φ, [H]) = h ∈ T_{G}(P, Q). This proves that κ is a well defined functor.

By definition, κ induces a bijection on objects, and commutes with the projections to F_{S,f}(BG) = F_{S}(G). For each pair of objects P, Q ≤ S, if (φ, [H_{1}]) and (φ, [H_{2}]) are two morphisms in L_{S,f}(BG) from P to Q which are both sent to φ ∈ Mor_{F_{S,f} (BG)}(P, Q), then H_{1} and H_{2} can be regarded as paths in map(BP, BG) from f|_{BP} to f|_{BQ} ∘ Bφ. Hence they differ by the loop H_{2}·H_{1} based at f|_{BP}, which represents an element of π_{1}(map(BP, BG)_{incl}). In other words, in the following commutative diagram

[figure: Mor_{L_{S,f} (BG)}(P, Q); κ_{P,Q}; (π_{S,f})_{P,Q}; Mor_{T_{S}(G)}(P, Q) = T_{G}(P, Q); g↦c_{g}; Mor_{F_{S,f} (BG)}(P, Q); Mor_{F_{S}(G)}(P, Q) = Hom_{G}(P, Q)]

the two vertical maps are orbit maps for free actions of the two groups π_{1}(map(BP, BG)_{incl}) and C_{G}(P), respectively. Since κ_{P} sends π_{1}(map(BP, BG)_{incl}) isomorphically to C_{G}(P) by Proposition 2.10(b), this proves that κ_{P,Q} is a bijection, and finishes the proof that κ is an isomorphism of categories.

It is straightforward to check that the objects of L_{S,f}^{c}(BG) and of L_{S,f}^{c}(BG_{p}^{∧}) are precisely the subgroups of S which are p-centric in G (cf. [BLO1, Lemma A.5]). Let κ^{c} be the restriction of κ to the full subcategories with these objects.

The isomorphism L_{S,f}^{c}(BG_{p}^{∧}) ≅ L_{S}^{c}(G) follows upon showing that the natural functor from L_{S,f}^{c}(BG) to L_{S,f}^{c}(BG_{p}^{∧}) is surjective on morphisms, and that κ^{c} factors through an isomorphism of categories from L_{S,f}^{c}(BG_{p}^{∧}) to L_{S}^{c}(G). By definition, for each pair of objects P and Q, the map T_{G}(P, Q) ⟶ Mor_{L_{S}^{c}(P)}(P, Q) is the orbit map of a free action of C′_{G}(P) = O^{p}(C_{G}(P)) on T_{G}(P, Q). By comparison, the map

Mor_{L_{S,f}^{c} (BG)}(P, Q) ⟶ Mor_{L_{S,f}^{c} (BG_{p}^{∧})}(P, Q)

is the orbit map of a free action of the group

Ker[π_{1}(map(BP, BG), incl) ⟶ π_{1}(map(BP, BG_{p}^{∧}), incl)].

This kernel is also isomorphic to O^{p}(C_{G}(P)) (see Theorem 2.11), and this finishes the proof that L_{S,f}^{c}(BG_{p}^{∧}) ≅ L_{S}^{c}(G). □

The argument outlined in the proof of Proposition 3.6 shows, in fact, that for all subgroups P, Q ≤ S (p-centric or not), Mor_{L_{S,f} (BG_{p}^{∧})}(P, Q) ≅ T_{G}(P, Q)/O^{p}(C_{G}(P)). That helps to motivate this as the definition of morphisms in a category L_{S}(G), whose objects consist of all subgroups of S. However, from the homotopy theoretic point of view, this category is much less interesting than the centric linking category. For example, when G = O^{p}(G), the trivial subgroup is an initial object in L_{S}(G), and hence |L_{S}(G)| is contractible. More generally, when G is an arbitrary finite group and S ∈ Syl_{p}(G), it is not hard to show that |L_{S}(G)| ≃ B(G/O^{p}(G)).

There is, however, a category intermediate between L_{S}^{c}(G) and L_{S}(G) which has all of the homotopy theoretic properties of the former. Define a p-subgroup P ≤ G to be p-quasicentric if O^{p}(C_{G}(P)) has order prime to p. Let L_{S}^{q}(G) ⊆ L_{S}(G) be the full subcategory whose objects are the subgroups of S which are p-quasicentric in G. This is a linking system in the sense of Definition 4.1 (while L_{S}(G) is not); and the inclusion L_{S}^{c}(G) ⊆ L_{S}^{q}(G) induces a homotopy equivalence |L_{S}^{q}(G)| ≃ |L_{S}^{c}(G)| by [5a1, Theorem 3.5] (see Theorem 4.20 below).

3.3. Linking systems and equivalences of p-completed classifying spaces.

If the centric linking categories L_{S_{1}}^{c}(G_{1}) and L_{S_{2}}^{c}(G_{2}) of two finite groups are equivalent as categories, then by Corollary 2.2, their geometric realizations are homotopy equivalent. So BG_{1}_{p}^{∧} ≃ BG_{2}_{p}^{∧} by Theorem 3.2.

Conversely, if two p-completed classifying spaces BG_{1}_{p}^{∧} and BG_{2}_{p}^{∧} are homotopy equivalent, then as noted above, Mislin’s theorem (Theorem 1.16) implies that they have isomorphic Sylow p-subgroups S_{i} ≤ G_{i}. From this, together with Proposition 3.6, it follows that there are equivalences of categories

L_{S_{1}}^{c}(G_{1}) ≅ L_{S_{1},f}^{c}(BG_{1}_{p}^{∧}) ≅ L_{S_{2},f′}^{c}(BG_{2}_{p}^{∧}) ≅ L_{S_{2}}^{c}(G_{2}).

We can thus conclude:

Theorem 3.7 ([BLO1]). For any prime p and any pair G_{1}, G_{2} of finite groups, BG_{1}_{p}^{∧} ≃ BG_{2}_{p}^{∧} if and only if for some S_{i} ∈ Syl_{p}(G_{i}), L_{S_{1}}^{c}(G_{1}) and L_{S_{2}}^{c}(G_{2}) are equivalent as categories.

Hence to prove the Martino-Priddy conjecture for a pair of finite groups G_{1} and G_{2}, with S_{i} ∈ Syl_{p}(G_{i}), it remains to show that any fusion preserving isomorphism S_{1} ⟶ S_{2} lifts to an equivalence of categories L_{S_{1}}^{c}(G_{1}) ≃ L_{S_{2}}^{c}(G_{2}). There is a fairly straightforward obstruction theory for doing this, analogous to the obstruction theory for the uniqueness of group extensions. This will be described in Section 5.4.

4. Abstract fusion and linking systems

So far, we have described the fusion and linking systems F_{S}(G) and L_{S}^{c}(G) of a finite group G, and their close connection with homotopy properties of the p-completed classifying space BG_{p}^{∧}. We now turn our attention to abstract versions of these structures. Abstract fusion systems have already been defined in Section I.2, and many of their properties were described throughout the rest of Part I.

Taking as starting point Puig’s original ideas about abstract fusion systems, Benson predicted, in [Be3] and in unpublished work, that there should be linking systems and classifying spaces associated to abstract fusion systems. His predictions were quite similar to what was actually constructed much later in [BLO2], and which will be described here.

In this section, we first define abstract linking systems, p-local finite groups, and their classifying spaces. We then look at fundamental groups and covering spaces of the geometric realizations of these categories, and their connection with certain types of fusion or linking subsystems. Afterwards, we list some of the other homotopy theoretic properties of the classifying spaces, showing that they keep many of the properties already stated for the spaces BG_{p}^{∧}.

4.1. Linking systems, centric linking systems and p-local finite groups.

We first consider linking systems associated to abstract fusion systems. For a p-group S and a set H of subgroups of S, T_{H}(S) denotes the full subcategory of the transporter category T_{S}(S) (Definition 3.1) whose object set is H.

We recall some definitions from Part I (Definition I.3.2). When F is a fusion system over a p-group S and P ≤ S, we say P is F-centric if C_{S}(Q) = Z(Q) for each Q ∈ P^{F}, and P is F-radical if O_{p}(Out_{F}(P)) = 1. We let F^{cr} ⊆ F^{c} ⊆ F denote the full subcategories with objects the Fcentric-radical subgroups and the F-centric subgroups, respectively. We also write F^{cr} ⊆ F^{c} to denote the sets of F-centric-radical and F-centric subgroups of S.

Definition 4.1 ([BLO2, Definition 1.7], [5a1, Definition 3.3], [O4, Definition 3]). Let F be a fusion system over a p-group S. A linking system associated to F is a finite category L, together with a pair of functors

T_{Ob(L)}(S) ⟶^{δ} L ⟶^{π} F ,

satisfying the following conditions:

(A1) Ob(L) is a set of subgroups of S closed under F-conjugacy and overgroups, and contains F^{cr}. Each object in L is isomorphic (in L) to one which is fully centralized in F.

(A2) δ is the identity on objects, and π is the inclusion on objects. For each P, Q ∈ Ob(L) such that P is fully centralized in F, C_{S}(P) acts freely on Mor_{L}(P, Q) via δ_{P,P} and right composition, and

π_{P,Q}: Mor_{L}(P, Q) ⟶ Hom_{F}(P, Q)

is the orbit map for this action.

(B) For each P, Q ∈ Ob(L) and each g ∈ T_{S}(P, Q), π_{P,Q} sends δ_{P,Q}(g) ∈ Mor_{L}(P, Q) to c_{g} ∈ Hom_{F}(P, Q).

(C) For all ψ ∈ Mor_{L}(P, Q) and all g ∈ P, the diagram

\xymatrix{P \ar[r]^{ψ} \ar[d]_{δ_{P,P} (g)} & Q \ar[d]^{δ_{Q,Q}(π(ψ)(g))} \\ P \ar[r]^{ψ} & Q}

commutes in L.

A centric linking system associated to F is just a linking system L such that Ob(L) = F^{c}. For centric linking systems, the only difference between the above definition and that in [BLO2] is that we define here δ as a functor on the transporter category of S. That δ can be defined on T_{Ob(L)}(S) follows as a consequence of the earlier definitions (see [BLO2, Proposition 1.11]), and simplifies certain constructions.

For any finite group G with Sylow p-subgroup S, a subgroup P ≤ S is F_{S}(G)-centric if and only if it is p-centric in G (see Definition 3.1). This follows easily from the fact that P is fully centralized in F_{S}(G) if and only if C_{S}(P) ∈ Syl_{p}(C_{G}(P)). The category L_{S}^{c}(G) (Definition 3.1 again) is easily seen to satisfy conditions (A1), (A2), (B), and (C) above, and hence is a centric linking system associated to F_{S}(G).

The following proposition describes some of the basic properties of linking systems. Most of these results go back to [5a1, § 3], but we refer to [O4] where they are stated and proven more explicitly.

Proposition 4.2 ([O4, Proposition 4]). The following hold for any linking system L associated to a saturated fusion system F over a p-group S.

(a) For each P, Q ∈ Ob(L), the subgroup

E(P) =^{def} Ker[Aut_{L}(P) ⟶ Aut_{F}(P)]

acts freely on Mor_{L}(P, Q) via right composition, and π_{P,Q} induces a bijection

Mor_{L}(P, Q)/E(P) ⟶^{≅} Hom_{F}(P, Q) .

(b) A morphism ψ ∈ Mor(L) is an isomorphism if and only if π(ψ) is an isomorphism in F.

(c) If P ∈ Ob(L) is fully normalized in F, then

δ_{P} (N_{S}(P)) ∈ Syl_{p}(Aut_{L}(P)) .

(d) All morphisms in L are monomorphisms and epimorphisms in the categorical sense.

Proof. The only part not stated explicitly in [O4, Proposition 4] is (b), and this follows easily from (a). □

When L is a linking system associated to a saturated fusion system F over S, then Proposition 4.2(a) implies that the structural functor π: L ⟶ F is surjective on morphism sets (between objects of L). Also, by [O4, Proposition 4(c)], the structural functor δ is injective on all morphism sets.

When P ≤ Q are objects in a linking system L, we define ι_{P,Q} = δ_{P,Q}(1) ∈ Mor_{L}(P, Q), and regard this as the inclusion morphism. This terminology is motivated by axiom (B), which says that π: L ⟶ F sends ι_{P,Q} to the (ordinary) inclusion of P in Q as a morphism in F. Once inclusions have been defined, it is natural to consider restrictions and extensions of morphisms. Since all morphisms in a linking system are monomorphisms and epimorphisms, restrictions and extensions are unique whenever they exist. The following proposition describes the conditions under which they do exist.

Proposition 4.3 ([O4, Proposition 4(b,e)]). The following hold for any linking system L associated to a saturated fusion system F over a p-group S.

(a) For every morphism ψ ∈ Mor_{L}(P, Q), and every P_{0}, Q_{0} ∈ Ob(L) such that P_{0} ≤ P, Q_{0} ≤ Q, and π(ψ)(P_{0}) ≤ Q_{0}, there is a unique morphism ψ|_{P_{0},Q_{0}} ∈ Mor_{L}(P_{0}, Q_{0}) (the “restriction” of ψ) such that ψ ∘ ι_{P_{0},P} = ι_{Q_{0},Q} ∘ ψ|_{P_{0},Q_{0}}.

(b) Let P, Q, P , Q ∈ Ob(L) and ψ ∈ Mor_{L}(P, Q) be such that P ⊴ P, Q ≤ Q, and for each g ∈ P there is h ∈ Q such that ι_{Q,Q} ∘ ψ ∘ δ_{P} (g) =

δ_{Q,Q}(h) ∘ ψ. Then there is a unique morphism ψ ∈ Mor_{L}(P , Q) such

that ψ|_{P,Q} = ψ.

We are now ready to define a p-local finite group.

Definition 4.4. A p-local finite group is defined to be a triple (S, F, L), where S is a p-group, F is a saturated fusion system over S, and L is a centric linking system associated to F. The classifying space of the triple (S, F, L) is the p-completed nerve |L|_{p}^{∧}.

In particular, for any finite group G with Sylow p-subgroup S ≤ G, (S, F_{S}(G), L_{S}^{c}(G)) is a p-local finite group, with classifying space |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧} (see Theorem 3.2).

It is not yet known whether every saturated fusion system has an associated linking system, nor whether it is possible for it to have more than one of them. In other words, for all we know now, a saturated fusion system might not give rise to any p-local finite group, or it might give rise to several of them. This problem of the existence and uniqueness of associated linking systems will be discussed in detail in Section 5.3.

Linking systems are a special case of a more general type of structure called “transporter systems”, which were defined and studied in [OV1]. These categories are modelled on the transporter category of a finite group (Definition 3.1). The most important difference between linking systems and transporter systems is that in a transporter system T associated to a saturated fusion system F, the condition that the kernel subgroups E(P) =^{def} Ker[Aut_{T} (P) ⟶ Aut_{F}(P)] always be p-groups (and equal to C_{S}(P) when P is fully centralized) is relaxed. Transporter systems were first defined for use in studying extensions of linking systems with kernel a p-group, but they are also useful in many other situations where one wants to weaken the much more rigid conditions imposed on a linking system.

4.2. Quasicentric subgroups and quasicentric linking systems.

As long as we are working with linking systems associated to one fusion system at a time, it is usually simplest to work with centric linking systems. But as soon as we are working with a pair of linking systems associated to distinct fusion systems (as in Section 4.4), or with a functor between linking systems, it is often necessary to consider linking systems with more objects than just those which are centric. This is why the quasicentric subgroups play an important role: they provide an upper bound for the set of possible objects in a linking system.

Definition 4.5. Let F be a saturated fusion system over a p-group S. A subgroup Q ≤ S is F-quasicentric if for each P ∈ Q^{F} which is fully centralized in F, the centralizer fusion system C_{F}(P) is the fusion system of the p-group C_{S}(P). Let F^{q} ⊆ F be the full subcategory whose objects are the F-quasicentric subgroups of S. We also let F^{q} = Ob(F^{q}) denote the set of F-quasicentric subgroups of S.

Note in particular that each F-centric subgroup is F-quasicentric. The following lemma lists some of the important properties of quasicentric subgroups.

Lemma 4.6. Let F be a saturated fusion system over a p-group S.

(a) A subgroup P ≤ S fully centralized in F is F-quasicentric if and only if for each Q ≤ P ·C_{S}(P) containing P, and each α ∈ Aut_{F}(Q) such that α|_{P} = Id, α has p-power order.

(b) An arbitrary subgroup of S is F-quasicentric if and only if some fully centralized subgroup in its F-conjugacy class is F-quasicentric.

(c) Assume Q ≤ P ≤ S are such that Q ∈ F^{q}. Let φ, φ′ ∈ Hom_{F}(P, S) be such that φ|_{Q} = φ′|_{Q}, and φ(Q) = φ′(Q) is fully centralized in F. Then there is x ∈ C_{S}(φ(Q)) such that φ = c_{x} ∘ φ′.

(d) If Q ≤ P ≤ S and Q ∈ F^{q}, then P ∈ F^{q}. If Q is also fully centralized in F, then so is P.

(e) If F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G), then a subgroup P ≤ S is F-quasicentric if and only if O^{p}(C_{G}(P)) has order prime to p.

Proof. (a) If P and R are both fully centralized in F and R ∈ P^{F}, then by the extension axiom, there is φ ∈ Iso_{F}(P ·C_{S}(P), R·C_{S}(R)) such that φ(P) = R. For each Q_{1}, Q_{2} ≤ P ·C_{S}(P), φ induces a bijection from Hom_{F}(Q_{1}, Q_{2}) to Hom_{F}(φ(Q_{1}), φ(Q_{2})) which sends ψ to (φ|_{Q_{2}})ψ(φ|_{Q_{1}})^{−1}. In particular, φ induces an isomorphism of categories from C_{F}(P) to C_{F}(R), and so the first is the fusion system of C_{S}(P) if and only if the second is the fusion system of C_{S}(R).

Thus a fully centralized subgroup P is F-quasicentric if and only if C_{F}(P) is the fusion system of C_{S}(P). This clearly implies the condition in (a), and the converse follows from Corollary I.3.7.

(b) This is immediate from the definition.

(c_{0}) We first prove this under the additional assumption that Q ⊴ P. It was shown in [5a1, Lemma 3.8], but we give here a different, shorter proof. Upon replacing P by φ′(P), Q by φ(Q) = φ′(Q), and φ by φ ∘ (φ′)^{−1}, we can assume that φ′ = incl_{P}^{S} and φ|_{Q} = incl_{Q}^{S}. We are thus reduced to the case where Q is fully centralized and φ|_{Q} = Id, and must show that φ = c_{x} for some x ∈ C_{S}(Q).

Set K = Aut_{P} (Q). Since Aut_{F}^{K}(Q) = Aut_{S}^{K}(Q) = K and Q is fully centralized, Q is fully K-normalized by Proposition I.5.2. Hence by Theorem I.5.5, the normalizer subsystem N_{F}^{K}(Q) over N_{S}^{K}(Q) = P ·C_{S}(Q) is saturated. Also, since φ|_{Q} = Id, Aut_{φ(P)}(Q) = Aut_{P} (Q) = K. Thus φ(P) ≤ N_{S}^{K}(Q), and φ ∈ Mor(N_{F}^{K}(Q)).

It thus suffices to prove that N_{F}^{K}(Q) is the fusion system of N_{S}^{K}(Q). To show this, it suffices by Corollary I.3.7 to show that all automorphism groups in N_{F}^{K}(Q) are p-groups. Assume otherwise: fix a subgroup R ≤ N_{S}^{K}(Q) and α ∈ Aut_{F}(R) such that Q ≤ R, α(Q) = Q, α|_{Q} ∈ K, and α ≠ Id_{R} has order prime to p. Since α|_{Q} ∈ K and K is a p-group, α^{p^{k}} is the identity on Q for some k. But then α^{p^{k}}|_{C_{R}(Q)} ∈ Mor(C_{F}(Q)), α^{p^{k}}|_{C_{R}(Q)} has p-power order since C_{F}(Q) is the fusion system of C_{S}(Q), and thus α^{p^{ℓ}}|_{Q·C_{R}(Q)} = Id for some ℓ ≥ k. Hence for g ∈ R, c_{α(g)} = c_{g} ∈ Aut(Q·C_{R}(Q)), so g^{−1}α(g) ∈ C_{R}(Q·C_{R}(Q)) ≤ Q·C_{R}(Q), and thus α induces the identity on R/QC_{R}(Q). So α = Id_{R} by Lemma A.2, and this is a contradiction.

(d) It suffices to show this when Q < P. Under this assumption, since Q < N_{P} (Q) ≤ P (see Lemma A.1), it suffices (by iteration) to prove this when Q ⊴ P. By Lemma I.2.6(c), there is a morphism χ ∈ Hom_{F}(N_{S}(Q), S) such that χ(Q) is fully normalized in F. It thus suffices to consider the case where Q is fully centralized (and Q ∈ F^{q}).

We first prove that P is fully centralized. Fix P^{∗} ∈ P^{F} which is fully centralized in F. Choose φ ∈ Iso_{F}(P^{∗}, P), and set Q^{∗} = φ^{−1}(Q). Since Q is fully centralized and φ|_{Q^{∗}} ∈ Iso_{F}(Q^{∗}, Q) extends to P^{∗}, the extension axiom implies there is ψ ∈ Hom_{F}(P^{∗}·C_{S}(Q^{∗}), S) such that ψ|_{Q^{∗}} = φ|_{Q^{∗}}.

By (c_{0}), there is x ∈ C_{S}(Q) such that φ = c_{x} ∘ ψ|_{P^{∗}} in Iso_{F}(P^{∗}, P). Set ψ′ = c_{x} ∘ ψ ∈ Hom_{F}(P^{∗}·C_{S}(Q^{∗}), S). Since ψ′(P^{∗}) = P, ψ′(C_{S}(P^{∗})) ≤ C_{S}(P), and so P is fully centralized in F since P^{∗} is.

Now fix R ≤ P ·C_{S}(P) which contains P, and α ∈ Aut_{F}(R) such that α|_{P} = Id_{P}. Set R_{0} = R ∩ (Q·C_{S}(Q)) = Q·C_{R}(Q), and α_{0} = α|_{R_{0}}. Then R = P R_{0}, α_{0}(R_{0}) = R_{0}, and α_{0} and α have the same order as automorphisms. Since Q ∈ F^{q}, α_{0} has p-power order in Aut_{F}(R_{0}), and hence α also has p-power order. Thus P ∈ F^{q}.

(c) Now assume Q < P ≤ S are such that Q ∈ F^{q}, where Q need not be normal in P. Using Lemma A.1 again, we can choose a chain of subgroups Q = P_{0} ⊴ P_{1} ⊴ ⋯ ⊴ P_{k} = P connecting Q to P, each normal in the following one. By (d), the P_{i} are all F-quasicentric. If φ, φ′ ∈ Hom_{F}(P, S) are such that φ|_{Q} = φ′|_{Q} and φ(Q) is fully centralized in F, then each φ(P_{i}) is fully centralized by (d) again. So by (c_{0}), φ|_{P_{1}} = c_{x_{1}} ∘ φ′|_{P_{1}} for some x_{1} ∈ C_{S}(φ(Q)), φ|_{P_{2}} = c_{x_{2}} ∘ c_{x_{1}} ∘ φ′_{P_{2}} for some x_{2} ∈ C_{S}(φ(P_{1})), etc. Since C_{S}(P_{i}) ≤ C_{S}(Q) for each i, we conclude that φ = c_{x} ∘ φ′ for some x ∈ C_{S}(Q).

(e) It suffices to prove this when P is fully centralized in F = F_{S}(G). By (a), P ∈ F^{q} if and only if for each Q ≤ C_{S}(P) and each α ∈ Aut_{C_{G}(P)}(Q), α has p-power order. By a theorem of Frobenius (cf. [A4, Theorem 39.4]), this is the case if and only if C_{G}(P) has a normal p-complement; equivalently, O^{p}(C_{G}(P)) has order prime to p. □

When G is a finite group, we can define a p-quasicentric subgroup of G to be a p-subgroup P ≤ G such that O^{p}(C_{G}(P)) has order prime to p. Thus Lemma 4.6(e) says that if P ≤ S ∈ Syl_{p}(G), then P is p-quasicentric in G if and only if P is F_{S}(G)-quasicentric.

We are now ready to describe the role played by quasicentric subgroups as upper bounds to the set of objects in a linking system.

Proposition 4.7 ([O4, Proposition 4(g)]). For any linking system L associated to a saturated fusion system F over a p-group S, Ob(L) ⊆ F^{q}.

Proof. We sketch briefly the proof. It suffices to consider P ∈ Ob(L) which is fully centralized in F. Fix Q ≤ P ·C_{S}(P) which contains P, and fix α ∈ Aut_{F}(Q) such that α|_{P} = Id_{P}; we must show α has p-power order. Choose β ∈ Aut_{L}(Q) such that π(β) = α. Then β|_{P,P} ∈ Aut_{L}(P) lies in Ker(π_{P}) = δ_{P} (C_{S}(P)), hence has p-power order. Thus β^{p^{k}}|_{P,P} = Id for some k ≥ 0. But then β^{p^{k}} = Id in Aut_{L}(Q) by Proposition 4.2(d), so β and α both have p-power order. □

A quasicentric linking system associated to F is a linking system L such that Ob(L) = F^{q}. For quasicentric linking systems, the difference between the above definition and that in [5a1, Definition 3.3] is that we define here δ as a functor on the transporter category of S, and drop the axiom (D)_{q} needed there. Our definition here is easily seen to imply that of [5a1], while the definition in [5a1] implies this one by [5a1, Lemma 3.7] (and since the set of quasicentrics is closed under overgroups by Lemma 4.6(d)).

Fix a finite group G, S ∈ Syl_{p}(G). Let H be a set of subgroups of S which is closed under G-conjugacy and overgroups, contains all subgroups which are centric and radical in F_{S}(G), and contains only quasicentric subgroups. Let L_{H}(G) be the category with object set H, and where

Mor_{L_{H}(G)}(P, Q) = T_{G}(P, Q)/O^{p}(C_{G}(P)) .

Then L_{H}(G) is a linking system associated to F_{S}(G). The only thing which has to be checked is that this does, in fact, define a category (that composition is well defined on these quotient sets). By Proposition 4.7, when H is the set of all F_{S}(G)-quasicentric subgroups, then L_{S}^{q}(G) =^{def} L_{H}(G) is the largest linking system associated to F_{S}(G), in the sense of having the largest possible set of objects.

When L_{1} and L_{2} are two linking systems associated to the same fusion system F, with structure functors δ_{i} and π_{i} and with Ob(L_{1}) = Ob(L_{2}), an isomorphism of linking systems from L_{1} to L_{2} is a functor χ: L_{1} ⟶ L_{2} such that π_{1} = π_{2} ∘ χ and δ_{2} = χ ∘ δ_{1}. In particular, χ is the identity on objects. Axiom (A2) implies that any such functor is an isomorphism of categories. The next proposition says, for example, that every centric linking system can be extended to a quasicentric linking system, unique up to isomorphism.

Proposition 4.8. Fix a saturated fusion system F over a p-group S. Let F^{cr} ⊆ H ⊆ Ĥ ⊆ F^{q} be sets of subgroups closed under F-conjugacy and overgroups. Then each linking system L associated to F with Ob(L) = H is contained in a linking system L̂ with Ob(L̂) = Ĥ. Also, L̂ is unique: if L^{∗} is another linking system associated to F with Ob(L^{∗}) = Ĥ, then any isomorphism L ≅ L^{∗}|_{H} extends to an isomorphism L̂ ≅ L^{∗}.

Proof. When H = F^{c} and Ĥ = F^{q}, this was shown in [5a1, Proposition 3.4], and also in [P7, Chapter 20]. The proof in [5a1] is homotopy theoretic — L̂ is constructed as a full subcategory of the linking system of the space |L|_{p}^{∧} (see Definition 3.5) — while the proof in [P7] is algebraic.

We give here a different algebraic proof. Via induction on the number of F-conjugacy classes in Ĥ∖H, it suffices to prove the proposition when Ĥ∖H = P for some F-conjugacy class P. Since Ĥ is closed under overgroups, Q > P ∈ P implies Q ∈ H.

Step 1: Fix a subgroup P_{∗} ∈ P which is fully automized and receptive in F. For each P ∈ P, we fix morphisms

φ_{P} ∈ Hom_{F}(N_{S}(P), N_{S}(P_{∗})), ψ_{P} ∈ Mor_{L}(N_{S}(P), N_{S}(P_{∗})),

and φ_{P} ∈ Iso_{F}(P, P_{∗})

such that

φ_{P} = π(ψ_{P}),  φ_{P} = φ_{P} |_{P} ,  and  ψ_{P_{∗}} = Id_{P_{∗}} .

To arrange this, first choose φ_{P} such that φ_{P} (P) = P^{∗} using Lemma I.2.6(c). Then set φ_{P} = φ_{P} |_{P}, and choose any ψ_{P} ∈ π^{−1}(φ_{P}) (a nonempty set by Proposition 4.2(a)).

Set

P̂ = {g ∈ N_{S}(P_{∗}) | c_{g} ∈ O_{p}(Aut_{F}(P_{∗}))} .

Since O_{p}(Aut_{F}(P_{∗})) ≤ Aut_{S}(P_{∗}) ∈ Syl_{p}(Aut_{F}(P_{∗})), we have AutP̂(P_{∗}) = O_{p}(Aut_{F}(P_{∗})). Since P_{∗} ∉ H ⊇ F^{cr}, either O_{p}(Aut_{F}(P_{∗})) > Inn(P_{∗}) or C_{S}(P_{∗}) ≰ P_{∗}. In either case, P̂ > P_{∗}, and hence P̂ ∈ H = Ob(L). Set

Γ = {γ ∈ Aut_{L}(P̂) | π(γ)(P_{∗}) = P_{∗}} .

Since AutP̂(P_{∗}) = O_{p}(Aut_{F}(P_{∗})) is normal in Aut_{F}(P_{∗}) (and since P_{∗} is receptive),

for each α ∈ Aut_{F}(P_{∗}) there is ψ ∈ Γ such that π(ψ)|_{P_{∗}} = α . (1)

Intuitively, to construct L̂, we will identify AutL̂(P_{∗}) with Γ (each element will be the restriction to P_{∗} of a unique element of Γ), identify each element of IsoL̂(P, R) (for P, R ∈ P) with a composite of restrictions of ψ_{P}, ψ_{R}, and an element of Γ, and then regard each morphism with source group in P and target in H as an isomorphism followed by an inclusion. In order to define composition, we must show how to restrict morphisms in L to isomorphisms between subgroups in P (when appropriate), and this is what will be done in Step 2.

Step 2: For each P_{1}, P_{2} ∈ P, set

Ψ_{P_{1},P_{2}} = {(Q_{1}, ψ, Q_{2}) | P_{i} ⊴ Q_{i} ∈ H, ψ ∈ Mor_{L}(Q_{1}, Q_{2}), π(ψ)(P_{1}) = P_{2}} .

Let τ_{P_{1},P_{2}} : Ψ_{P_{1},P_{2}} ⟶ Ψ_{P_{∗},P_{∗}} be the “translation map”, defined by

τ_{P_{1},P_{2}}(Q_{1}, ψ, Q_{2}) = (R_{1}, ψ_{P_{2}}|_{Q_{2},R_{2}} ∘ ψ ∘ (ψ_{P_{1}}|_{Q_{1},R_{1}})^{−1}, R_{2})

where R_{i} = φ_{P_{i}}(Q_{i}).

Set Ψ = Ψ_{P_{∗},P_{∗}}. We claim that for each (Q_{1}, ψ, Q_{2}) ∈ Ψ, there is ψ̂ ∈ Mor_{L}(Q_{1}P̂, Q_{2}P̂) such that ψ̂|_{Q_{1},Q_{2}} = ψ. To see this, set φ_{0} = π(ψ)|_{P_{∗}} ∈ Aut_{F}(P_{∗}). Since P_{∗} is receptive (and since φ_{0} extends to π(ψ) ∈ Hom_{F}(Q_{1}, Q_{2})), φ_{0} extends to some φ_{1} ∈ Hom_{F}(Q_{1}P̂, S). Since φ_{1}|_{P_{∗}} = π(ψ)|_{P_{∗}}, Lemma 4.6(c) implies there is x ∈ C_{S}(P_{∗}) ≤ P̂ such that c_{x} ∘ φ_{1}|_{Q_{1}} = π(ψ). Thus c_{x} ∘ φ_{1}(Q_{1}) = Q_{2}, and c_{x} ∘ φ_{1}(P̂) = P̂ by definition of P̂ (and since c_{x} ∘ φ_{1}(P_{∗}) = P_{∗}). Choose any ψ_{1} ∈ Mor_{L}(Q_{1}P̂, Q_{2}P̂) such that π(ψ_{1}) = c_{x} ∘ φ_{1}. By axiom (A2) for the linking system L, and since all subgroups containing P_{∗} are fully centralized in F by Lemma 4.6(d),

ψ = ψ_{1}|_{Q_{1},Q_{2}} ∘ δ_{Q_{1}}(y) for some y ∈ C_{S}(Q_{1}) ≤ P̂. Thus ψ̂ =^{def} ψ_{1} ∘ δ_{Q_{1}P̂}(y)

lies in Mor_{L}(Q_{1}P̂, Q_{2}P̂) and extends ψ.

We can thus define a “restriction” map

ρ: Ψ = Ψ_{P_{∗},P_{∗}} ⟶ Γ ,

by sending (Q_{1}, ψ, Q_{2}) ∈ Ψ to ψ̂|_{P̂,P̂} ∈ Γ, where ψ̂ is the extension of ψ constructed above (and is unique by Proposition 4.2(d)). For each (Q_{1}, ψ, Q_{2}) and (Q_{2}, χ, Q_{3}) in Ψ,

ρ(Q_{2}, χ, Q_{3}) ∘ ρ(Q_{1}, ψ, Q_{2}) = ρ(Q_{1}, χψ, Q_{3})

and π(ρ(Q_{1}, ψ, Q_{2}))|_{P_{∗}} = π(ψ)|_{P_{∗}} . (2)

Upon combining the first equality with the definition of the τ_{P,Q}, we see that for P_{1}, P_{2}, P_{3} ∈ P, and any composable pair (Q_{1}, ψ, Q_{2}) ∈ Ψ_{P_{1},P_{2}} and (Q_{2}, χ, Q_{3}) ∈ Ψ_{P_{2},P_{3}},

ρτ_{P_{2},P_{3}}(Q_{2}, χ, Q_{3}) ∘ ρτ_{P_{1},P_{2}}(Q_{1}, ψ, Q_{2}) = ρτ_{P_{1},P_{3}}(Q_{1}, χψ, Q_{3}) ∈ Γ . (3)

We claim that

P ∈ P, P ⊴ Q ∈ H, g ∈ Q ⟹ ρτ_{P,P} (Q, δ_{Q}(g), Q) = δP̂(φ_{P} (g)). (4)

Set R = φ_{P} (Q), so that P_{∗} ⊴ R. Then

τ_{P,P} (Q, δ_{Q}(g), Q) = (R, ψ_{P} |_{Q,R}∘δ_{Q}(g)∘(ψ_{P} |_{Q,R})^{−1}, R) = (R, δ_{R}(φ_{P} (g)), R) ,

where the last equality follows from axiom (C) for the linking system L (and since π(ψ_{P}) = φ_{P}). Also, δ_{R}(φ_{P} (g)) extends to δ_{RP̂}(φ_{P} (g)), and hence ρ(R, δ_{R}(φ_{P} (g)), R) = δP̂(φ_{P} (g)).

Step 3: We are now ready to construct the linking system L̂ with Ob(L̂) = Ĥ. For P, Q ∈ Ĥ, set MorL̂(P, Q) = Mor_{L}(P, Q) if P, Q ∈ H, and set MorL̂(P, Q) = ∅ if P ∈ H and Q ∈ P. When P ∈ P, set

MorL̂(P, Q) = {γ_{P}^{P_{2},Q} | γ ∈ Γ, Q ≥ P_{2} ∈ P} .

When Q ∈ P (so P_{2} = Q), we write γ_{P}^{Q} = γ_{P}^{Q,Q} ∈ IsoL̂(P, Q). Intuitively, γ_{P}^{P_{2}} is the image of γ|_{P_{∗},P_{∗}} ∈ AutL̂(P_{∗}) under the bijection AutL̂(P_{∗}) ⟶^{≅} IsoL̂(P, P_{2}) induced by composition with (ψ_{P_{2}}|_{P_{2},P_{∗}})^{−1} and with ψ_{P} |_{P,P_{∗}}; and γ_{P}^{P_{2},Q} is then γ_{P}^{P_{2}} followed by the inclusion ι_{P_{2},Q}.

Define composition in L̂ as follows. Composition of morphisms in L is unchanged. For P_{1}, P_{2}, P_{3} ∈ P, P_{3} ≤ Q_{3} ∈ Ĥ, γ_{P_{1}}^{P_{2}} ∈ MorL̂(P_{1}, P_{2}), and β_{P_{2}}^{P_{3},Q_{3}} ∈ MorL̂(P_{2}, Q_{3}), set

β_{P_{2}}^{P_{3},Q_{3}} ∘ γ_{P_{1}}^{P_{2}} = (βγ)_{P_{1}}^{P_{3},Q_{3}} ∈ MorL̂(P_{1}, Q_{3}) .  (5)

For P_{1}, P_{2} ∈ P, Q_{2} > P_{2}, Q_{3} ∈ H, γ_{P_{1}}^{P_{2},Q_{2}} ∈ MorL̂(P_{1}, Q_{2}), and ψ ∈ Mor_{L}(Q_{2}, Q_{3}), write P_{3} = π(ψ)(P_{2}) < Q_{3} and N_{i} = N_{Q_{i}}(P_{i}) > P_{i} (i = 2, 3), and set

ψ ∘ γ_{P_{1}}^{P_{2},Q_{2}} = (ρτ_{P_{2},P_{3}}(N_{2}, ψ|_{N_{2},N_{3}}, N_{3}) ∘ γ)_{P_{1}}^{P_{3},Q_{3}} ∈ MorL̂(P_{1}, Q_{3}) . (6)

The only tricky point when proving that composition is associative comes when taking a composite of three morphisms of which the last two are in L, and in this case, associativity follows from (3). Thus L̂ is a category.

\underline{Definition of π̂:} Define π̂: L̂ ⟶ F as follows. When P, Q ∈ H, set π̂_{P,Q} = π_{P,Q}. Thus π̂|_{L} = π. When P ∈ P, Q ∈ Ĥ, Q ≥ P_{2} ∈ P, and γ ∈ Γ, set

π̂(γ_{P}^{P_{2},Q}) = incl_{P_{2}}^{Q} ∘ φ_{P_{2}}^{−1} ∘ π(γ)|_{P_{∗},P_{∗}} ∘ φ_{P} ∈ Hom_{F}(P, Q) .  (7)

To see that this defines a functor on L̂, we must check that it preserves composition as defined in (5) and (6). The first case is clear, while in the situation of (6),

π̂(ψ∘γ_{P_{1}}^{P_{2},Q_{2}}) = incl_{P_{3}}^{Q_{3}} ∘ φ_{P_{3}}^{−1} ∘ ((φ_{P_{3}} ∘ π(ψ)|_{P_{2},P_{3}} ∘ φ_{P_{2}}^{−1}) ∘ π(γ)|_{P_{∗},P_{∗}}) ∘ φ_{P_{1}}

= π_{Q_{2},Q_{3}}(ψ) ∘ incl_{P_{2}}^{Q_{2}} ∘ φ_{P_{2}}^{−1} ∘ π(γ)|_{P_{∗},P_{∗}} ∘ φ_{P_{1}} = π(ψ) ∘ π̂(γ_{P_{1}}^{P_{2},Q_{2}}) .

\underline{Definition of δ̂:} For each P ∈ P and g ∈ S, define

γ(P, g) = ρτ_{P,^{g}P} (N_{S}(P), δ_{N_{S}(P),N_{S}(^{g}P)}(g), N_{S}(^{g}P)) ∈ Γ .  (8)

Define δ̂: TĤ(S) ⟶ L̂ by setting δ̂|_{T_{H}(S)} = δ; and for P ∈ P, Q ∈ Ĥ, and g ∈ S such that ^{g}P ≤ Q,

δ̂_{P,Q}(g) = γ(P, g)_{P}^{^{g}P,Q} .

In particular, by (4),

P ∈ P, g ∈ N_{S}(P) ⟹

γ(P, g) = δP̂(φ_{P} (g)) and δ̂_{P} (g) = (δP̂(φ_{P} (g)))_{P}^{P} . (9)

We must show δ̂ is a functor; i.e., it commutes with composition. Fix P_{1}, Q_{2}, Q_{3} ≤ S and g, h ∈ S such that P_{1} ∈ P, ^{g}P_{1} ≤ Q_{2}, and ^{h}Q_{2} ≤ Q_{3}. Set P_{2} = ^{g}P_{1} and P_{3} = ^{h}P_{2} = ^{hg}P_{1}. If P_{2} = Q_{2} ∈ P, then γ(P_{2}, h) ∘ γ(P_{1}, g) = γ(P_{1}, hg) by (3) and (8), and so

δ̂_{P_{2},Q_{3}}(h) ∘ δ̂_{P_{1},P_{2}}(g) = γ(P_{2}, h)_{P_{2}}^{P_{3},Q_{3}} ∘ γ(P_{1}, g)_{P_{1}}^{P_{2}}
= γ(P_{1}, hg)_{P_{1}}^{P_{3},Q_{3}} = δ̂_{P_{1},Q_{3}}(hg)

by (5). If P_{2} < Q_{2} ∈ H, set N_{i} = N_{Q_{i}}(P_{i}) (i = 2, 3) and N_{i}^{∗} = N_{S}(P_{i}) (i = 1, 2, 3); then

δ̂_{Q_{2},Q_{3}}(h) ∘ δ̂_{P_{1},Q_{2}}(g) = δ_{Q_{2},Q_{3}}(h) ∘ γ(P_{1}, g)_{P_{1}}^{P_{2},Q_{2}}

= (ρτ_{P_{2},P_{3}}(N_{2}, δ_{N_{2},N_{3}}(h), N_{3}) ∘ ρτ_{P_{1},P_{2}}(N_{1}^{∗}, δ_{N_{1}^{∗},N_{2}^{∗}}(g), N_{2}^{∗}))_{P_{1}}^{P_{3},Q_{3}} (by (6) and (8))

= (ρτ_{P_{1},P_{3}}(N_{1}^{∗}, δ_{N_{1}^{∗},N_{3}^{∗}}(hg), N_{3}^{∗}))_{P_{1}}^{P_{3},Q_{3}}  (by (3))
= γ(P_{1}, hg)_{P_{1}}^{P_{3},Q_{3}} = δ̂_{P_{1},Q_{3}}(hg) .

\underline{Proof of axiom (B):} Fix P, Q ∈ Ĥ and g ∈ G such that ^{g}P ≤ Q. If P ∈ H, then π̂(δ̂_{P,Q}(g)) = c_{g}|_{P,Q} by axiom (B) for L. If P ∈ P, set P_{2} = ^{g}P ≤ Q, and then

π̂(δ̂_{P,Q}(g)) = π̂(γ(P, g)_{P}^{P_{2},Q}) = incl_{P_{2}}^{Q} ∘ φ_{P_{2}}^{−1} ∘ π(γ(P, g))|_{P_{∗},P_{∗}} ∘ φ_{P}

= incl_{P_{2}}^{Q} ∘ φ_{P_{2}}^{−1} ∘ (φ_{P_{2}} ∘ π(δ_{N_{S}(P),N_{S}(P_{2})}(g))|_{P,P_{2}} ∘ φ_{P}^{−1}) ∘ φ_{P} = c_{g}|_{P,Q} .

\underline{Proof of axiom (C):} It suffices to consider morphisms ψ in L̂ of the following three types (since each morphism in L̂ not in L is a composite of such morphisms).

• ψ = γ_{P_{∗}}^{P_{∗}} ∈ AutL̂(P_{∗}) for γ ∈ Γ. For g ∈ P_{∗},

ψ ∘ δ̂_{P_{∗}}(g) = (γ ∘ δP̂(g))_{P_{∗}}^{P_{∗}} = (δP̂(π(γ)(g)) ∘ γ)_{P_{∗}}^{P_{∗}} = δ̂_{P_{∗}}(π̂(ψ)(g)) ∘ ψ ,

where the second equality holds by axiom (C) for L, and the first and third each holds by (9) and (5).

• ψ = 1_{P}^{P_{∗}} ∈ IsoL̂(P, P_{∗}) for P ∈ P. For g ∈ P, δ̂_{P} (g) = (δP̂(φ_{P} (g)))_{P}^{P} by (9), and so

ψ ∘ δ̂_{P} (g) = (δP̂(φ_{P} (g)))_{P}^{P_{∗}} = δ̂_{P_{∗}}(φ_{P} (g)) ∘ ψ

by (5). (Note that π̂(ψ) = φ_{P} in this case.)

• ψ = 1_{P}^{P,Q} ∈ MorL̂(P, Q) for Q > P ∈ P (the inclusion of P in Q). For g ∈ P,

ψ ∘ δ̂_{P} (g) = (δP̂(φ_{P} (g)))_{P}^{P,Q} = (ρτ_{P,P} (Q, δ_{Q}(g), Q))_{P}^{P,Q} = δ_{Q}(g) ∘ ψ ,

where the first equality holds by (9) and (5), the second by (4), and the third by (6).

\underline{Proof of (A1), (A2):} Axiom (A1) follows from the assumptions on Ĥ, and the first statement in axiom (A2) holds by construction. Since the rest of (A2) holds for L, it suffices to prove it for morphism sets MorL̂(P, Q) for P ∈ P and Q ∈ Ĥ, where P is fully centralized in F. In this case, π̂_{P,Q} is onto by (1) (and (7)). Also, δ̂_{P} (C_{S}(P)) acts freely on MorL̂(P, Q) by construction and by (9), and π̂(δ̂_{P} (C_{S}(P))) = 1 by (B).

Now assume β_{P}^{P_{2},Q} and γ_{P}^{P_{3},Q} are morphisms in MorL̂(P, Q) such that π̂(β_{P}^{P_{2},Q}) = π̂(γ_{P}^{P_{3},Q}), where P_{2}, P_{3} ∈ P. Since P is fully centralized in F, φ_{P} (C_{S}(P)) = C_{S}(P_{∗}). Then P_{2} = Im(π̂(β_{P}^{P_{2},Q})) = Im(π̂(γ_{P}^{P_{3},Q})) = P_{3}, and π(β)|_{P_{∗}} = π(γ)|_{P_{∗}} ∈ Aut_{F}(P_{∗}) by (7). By Lemma 4.6(c), there is x ∈ C_{S}(P_{∗}) ≤ P̂ such that π(β) = π(γ) ∘c_{x} in Aut_{F}(P̂). By axiom (A2) for the linking system L, there is y ∈ C_{S}(P̂) = Z(P̂) such that β = γ ∘ δP̂(xy) in Γ ≤ AutL̂(P̂) (P̂ is fully centralized in F by Lemma 4.6(d)). Then xy ∈ C_{S}(P_{∗}), xy = φ_{P} (z) for some z ∈ C_{S}(P), and

β_{P}^{P_{2},Q} = γ_{P}^{P_{3},Q} ∘ (δP̂(xy))_{P}^{P} = γ_{P}^{P_{3},Q} ∘ δ̂_{P} (z)

by (5) and (9). This proves that π̂_{P,Q} is the orbit map for the action of C_{S}(P) on MorL̂(P, Q), and finishes the proof of axiom (A2).

Step 4: Let L^{∗} be another linking system associated to F with Ob(L^{∗}) = Ĥ, and assume there is an isomorphism of linking systems χ: L ⟶ L^{∗}|_{H}. Let δ^{∗} and π^{∗} be the structure functors of L^{∗}. We claim χ can be extended to an isomorphism of linking systems χ̂: L̂ ⟶ L^{∗}.

Let χ̂ be the identity on objects, and set χ̂(ψ) = χ(ψ) for ψ ∈ Mor(L). For each P ∈ P, Q ∈ Ĥ, and γ_{P}^{R,Q} ∈ MorL̂(P, Q), set

χ̂(γ_{P}^{R,Q}) = δ_{R,Q}^{∗}(1)∘(χ(ψ_{R})|_{R,P_{∗}})^{−1}∘χ(γ)|_{P_{∗},P_{∗}} ∘χ(ψ_{P})|_{P,P_{∗}} ∈ Mor_{L^{∗}}(P, Q) .

By definition of ρ in Step 2, for each (R, ψ, T) ∈ Ψ, ψ and ρ(R, ψ, T) are both restrictions of some morphism ψ̂ ∈ Mor_{L}(RP̂, T P̂), and thus

χ(ρ(R, ψ, T))|_{P_{∗},P_{∗}} = χ(ψ)|_{P_{∗},P_{∗}} .  (10)

Hence for each P, P_{2} ∈ P, Q_{2} > P_{2}, Q_{3} ∈ H, γ_{P}^{P_{2},Q_{2}} ∈ MorL̂(P, Q_{2}), and ψ ∈ Mor_{L}(Q_{2}, Q_{3}), if we set P_{3} = π(ψ)(P_{2}) < Q_{3} and N_{i} = N_{Q_{i}}(P_{i}) (i = 2, 3), then by (6),

χ̂(ψ ∘ γ_{P}^{P_{2},Q_{2}}) = δ_{P_{3},Q_{3}}^{∗}(1) ∘ (χ(ψ_{P_{3}})|_{P_{3},P_{∗}})^{−1}

∘ χ(ρτ_{P_{2},P_{3}}(N_{2}, ψ|_{N_{2},N_{3}}, N_{3}))|_{P_{∗},P_{∗}} ∘ χ(γ)|_{P_{∗},P_{∗}} ∘ χ(ψ_{P})|_{P,P_{∗}}

= δ_{P_{3},Q_{3}}^{∗}(1) ∘ (χ(ψ_{P_{3}})|_{P_{3},P_{∗}})^{−1}
∘ (χ(ψ_{P_{3}})|_{P_{3},P_{∗}} ∘ χ(ψ)|_{P_{2},P_{3}} ∘ (χ(ψ_{P_{2}})|_{P_{2},P_{∗}})^{−1})
∘ χ(γ)|_{P_{∗},P_{∗}} ∘ χ(ψ_{P})|_{P,P_{∗}}

= δ_{P_{3},Q_{3}}^{∗}(1) ∘ χ(ψ)|_{P_{2},P_{3}} ∘ (χ(ψ_{P_{2}})|_{P_{2},P_{∗}})^{−1} ∘ χ(γ)|_{P_{∗},P_{∗}} ∘ χ(ψ_{P})|_{P,P_{∗}}

= χ(ψ) ∘ δ_{P_{2},Q_{2}}^{∗}(1) ∘ (χ(ψ_{P_{2}})|_{P_{2},P_{∗}})^{−1} ∘ χ(γ)|_{P_{∗},P_{∗}} ∘ χ(ψ_{P})|_{P,P_{∗}}
= χ̂(ψ) ∘ χ̂(γ_{P}^{P_{2},Q_{2}}) .

Composition in the situation of (5) is easily handled, and thus χ̂ is a functor.

Now, π^{∗} ∘χ̂ = π̂ and δ^{∗} = χ̂∘δ̂ by the definitions of π̂ and δ̂ in Step 3, and by (10) in the latter case. This finishes the proof that χ̂ is an isomorphism of linking systems. □

4.3. Automorphisms of fusion and linking systems.

We next look at certain groups of self equivalences of fusion and linking systems. We first consider automorphisms of fusion systems. In this case, rather than look at self equivalences or automorphisms of the category, it turns out to be enough (and much easier) to look at automorphisms of the underlying p-group.

Fusion preserving isomorphisms between Sylow subgroups of finite groups were defined in Definition 1.14(a). When F is an (abstract) saturated fusion system over a p-group S, an automorphism α ∈ Aut(S) is fusion preserving if ^{α}F = F (Definition I.3.4). Set

Aut(S, F) = {α ∈ Aut(S) |^{α}F = F}
Out(S, F) = Aut(S, F)/Aut_{F}(S) .

In other words, α ∈ Aut(S) is fusion preserving exactly when it induces an automorphism c_{α} of F, by sending P to α(P) and φ to ^{α}φ. An arbitrary automorphism of the category F is realized in this way if and only if it sends inclusions to inclusions, and is “isotypical” in the sense defined by Martino and Priddy [MP2]: it commutes up to natural isomorphism of functors with the forgetful functor from F to groups. Note, however, that there can be nontrivial elements of Out(S, F) which act trivially on F; for example, when F is the fusion system of an abelian p-group (and p > 2).

Before discussing automorphisms of linking systems, we first recall some basic facts about equivalences of categories. A functor Φ: C ⟶ D is an equivalence if it induces a bijection between the isomorphism classes of objects, and bijections on all morphism sets. This is equivalent to the condition that there be a functor Ψ in the other direction such that Φ ∘ Ψ and Ψ∘Φ are both naturally isomorphic to the identity on C or D. Hence for any small category C, the set of all self equivalences C ⟶ C up to natural isomorphism is a group under composition, which we denote Out(C).

Now fix a linking system L associated to a saturated fusion system F over a p-group S. Recall that as part of the structure of the linking system, there is a functor δ from T_{Ob(L)}(S) to L. Thus for each P ∈ Ob(L), δ_{P} sends N_{G}(P) to Aut_{L}(P). An equivalence of categories α: L ⟶ L is isotypical if for each P ∈ Ob(L), α(δ_{P} (P)) = δ_{α(P)}(α(P)). Let Out_{typ}(L) ≤ Out(L) be the subgroup of all natural isomorphism classes of isotypical self equivalences of L. (A submonoid of the finite group Out(L) must be a subgroup.)

Finally, define

Aut_{typ}^{I}(L) = {α ∈ Aut_{typ}(L) | α sends inclusions to inclusions} .

By analogy with the definition by Martino and Priddy of an isotypical equivalence of fusion systems, an equivalence α: L_{1} ⟶ L_{2} between centric linking systems was defined in [BLO1] to be isotypical whenever F_{1} ≅ F_{2} ∘ α (naturally isomorphic as functors), where F_{i}: L_{i} ⟶ Gps denotes the forgetful functor. For centric linking systems, this definition of an isotypical equivalence was shown in [BLO2, Lemma 8.2] to be equivalent to the one given above. That proof extends easily to the case of isotypical equivalences of linking systems all of whose objects are centric, but the result is definitely not true in general for equivalences of linking systems which contain noncentric objects. Also, Proposition 4.11 below is false if the earlier definition of “isotypical” (in terms of the forgetful functor to groups) is used. This is why the definition in terms of the “distinguished subgroups” δ_{P} (P) seems more appropriate.

In general, it is much easier to work with isotypical equivalences of a linking system L which sends inclusions to inclusions, than to work with arbitrary isotypical equivalences. Some of the reasons for this will be seen shortly in Proposition 4.11. The next lemma explains how Out_{typ}(L) is described in terms of Aut_{typ}^{I}(L).

For L and F as above, and any γ ∈ Aut_{L}(S), we let c_{γ} ∈ Aut_{typ}^{I}(L) be

the automorphism which sends P ∈ Ob(L) to c_{γ}(P) =^{def} π(γ)(P), and sends ψ ∈ Mor_{L}(P, Q) to

c_{γ}(ψ) =^{def} (γ|_{Q,c_{γ}(Q)}) ∘ ψ ∘ (γ|_{P,c_{γ}(P)})^{−1} ∈ Mor_{L}(c_{γ}(P), c_{γ}(Q))

for P, Q ∈ Ob(L). This is clearly isotypical, since for g ∈ P ∈ Ob(L), c_{γ}(δ_{P} (g)) = δ_{c_{γ}(P)}(π(γ)(g)) by axiom (C). It sends inclusions to inclusions since the restrictions of γ are defined with respect to the inclusions.

Lemma 4.9 ([AOV, Lemma 2.7]). For any saturated fusion system F over a p-group S, and any linking system L associated to F, the sequence

Aut_{L}(S) ⟶^{γ↦c_{γ}} Aut_{typ}^{I}(L) ⟶ Out_{typ}(L) ⟶ 1

is exact. Each α ∈ Aut_{typ}^{I}(L) is an automorphism of the category L, and hence Aut_{typ}^{I}(L) (as well as Out_{typ}(L)) is a group.

The sequence of Lemma 4.9 is analogous to the following exact sequence involving automorphsms of groups. For any finite group G and any S ∈ Syl_{p}(G), the sequence

N_{G}(S) ⟶^{g↦c_{g}} Aut(G, S) ⟶ Out(G) ⟶ 1

is exact, where Aut(G, S) = {α ∈ Aut(G) | α(S) = S}.

Since one sometimes has to work with two or more linking systems associated to the same fusion system but having different objects, it is useful to know that they have the same outer automorphism groups.

Lemma 4.10 ([AOV, Lemma 1.17]). Fix a saturated fusion system F over a p-group S. Let L_{0} ⊆ L be a pair of linking systems associated to F, such that Ob(L_{0}) and Ob(L) are both Aut(S, F)-invariant. Then restriction defines an isomorphism

Out_{typ}(L) ⟶_{≅}^{R} Out_{typ}(L_{0}).

The injectivity of the above map R is an easy consequence of the uniqueness of extensions in a linking system (Proposition 4.3(b)). The surjectivity of R is basically a consequence of the uniqueness statement in Proposition 4.8: each automorphism of the subsystem extends to an automorphism of the larger linking system.

The next proposition describes how an isotypical equivalence of a linking system L induces a fusion preserving automorphism of the associated fusion system F.

Proposition 4.11 ([O4, Proposition 6]). Let L be a linking system associated to a saturated fusion system F over a p-group S. Assume α ∈ Aut_{typ}^{I}(L), and let β ∈ Aut(S) be the automorphism such that α(δ_{S}(g)) = δ_{S}(β(g)) for each g ∈ S. Then β ∈ Aut(S, F), and α sends each P ∈ Ob(L) to β(P). Let c_{β} ∈ Aut(F) be the functor which sends P to β(P) and φ ∈ Mor(F) to ^{β}φ. Then the following diagram of categories and functors commutes

\xymatrix{T_{H}(S) \ar[r]^{δ} \ar[d]^{β_{∗}} & L \ar[r]^{π} \ar[d]^{α} & F \ar[d]^{c_{β}} \\ T_{H}(S) \ar[r]^{δ} & L \ar[r]^{π} & F}

where H = Ob(L), and where β_{∗} sends P ∈ H to β(P) and sends g ∈ T_{S}(P, Q) to β(g) ∈ T_{S}(β(P), β(Q)).

Proof. This is all shown in [O4], except for the claim that the left square in the diagram commutes. It commutes on elements of Aut_{T_{H}(S)}(S) = S by definition of β. It commutes on other objects and morphisms since α sends inclusions to inclusions and hence commutes with restrictions. □

By Proposition 4.11, for each linking system L associated to a saturated fusion system F, there is a well defined homomorphism

μ̃: Aut_{typ}^{I}(L) ⟶ Aut(S, F) ,

which sends α ∈ Aut_{typ}^{I}(L) to the restriction of α_{S} to δ_{S}(S) ≅ S. By the exact sequence of Lemma 4.9, this factors through a homomorphism

μ_{L}: Out_{typ}(L) ⟶ Out(S, F) .

The kernel and cokernel of μ_{L} will be described later, in Proposition 5.12. The automorphism group Out_{typ}(L) plays an important role when describing self equivalences of the space |L_{p}^{∧}| (Theorem 4.22), and also in Section 6.1 (Definition 6.3).

4.4. Normal fusion and linking subsystems.

Recall the definition of a weakly normal fusion subsystem in Section I.6. If F_{0} ⊆ F is a pair of saturated fusion systems over p-groups S_{0} ≤ S, then F_{0} is weakly normal in F (denoted F_{0} ˙⊴ F) if

(i) S_{0} is strongly closed in F;

(ii) (Frattini condition) for each P ≤ S_{0} and each φ ∈ Hom_{F}(P, S_{0}), there are morphisms α ∈ Aut_{F}(S_{0}) and φ_{0} ∈ Hom_{F_{0}}(P, S_{0}) such that φ = α ∘ φ_{0}; and

(iii) (Invariance condition) for each P, Q ≤ S_{0}, each φ ∈ Hom_{F_{0}}(P, Q), and each β ∈ Aut_{F}(S_{0}), ^{β}φ ∈ Hom_{F_{0}}(β(P), β(Q)).

Given this definition, it is natural to impose the analogous conditions when defining a normal linking subsystem. The only problem is choosing exactly which conditions to put on the sets of objects in the two linking systems. The following definition was used in [O4] and [AOV], where normal linking subsystems played an important role.

Definition 4.12. Fix a pair of saturated fusion systems F_{0} ⊴ F over pgroups S_{0} ⊴ S such that F_{0} is weakly normal in F, and let L_{0} ⊆ L be associated linking systems. Then L_{0} is normal in L (L_{0} ⊴ L) if

(i) Ob(L) = {P ≤ S | P ∩ S_{0} ∈ Ob(L_{0})};

(ii) for all P ∈ Ob(L_{0}) and ψ ∈ Mor_{L}(P, S_{0}), there are morphisms γ ∈ Aut_{L}(S_{0}) and ψ_{0} ∈ Mor_{L_{0}}(P, S_{0}) such that ψ = γ ∘ ψ_{0}; and

(iii) for all γ ∈ Aut_{L}(S_{0}), P, Q ∈ Ob(L_{0}), and ψ ∈ Mor_{L_{0}}(P, Q),

γ|_{Q,γ(Q)} ∘ ψ ∘ γ|_{P,γ(P)}^{−1} ∈ Mor_{L_{0}}(γ(P), γ(Q)).

Here, we write γ(P) = π(γ)(P) and γ(Q) = π(γ)(Q) for short. In this situation, we define

L/L_{0} = Aut_{L}(S_{0})/Aut_{L_{0}}(S_{0}) .

In fact, condition (ii) in Definition 4.12 follows from other conditions in that definition, together with the Frattini condition in Definition I.6.1. It is included here only to make the analogy between the two definitions clearer.

The following lemma helps to motivate the notation L/L_{0}: it describes a functor from L to B(L/L_{0}) with L_{0} as a “kernel”. By Proposition 2.8, this in turn induces a surjection of π_{1}(|L|) onto L/L_{0}.

Lemma 4.13. For each linking system L associated to a fusion system F over S, and each normal subsystem L_{0} ⊴ L associated to F_{0} over S_{0}, there is a functor

F: L ⟶ B(L/L_{0})

which sends Aut_{L}(S_{0}) to L/L_{0} = Aut_{L}(S_{0})/Aut_{L_{0}}(S_{0}) via the projection and sends inclusion morphisms to the identity. Also, S_{0} = Ker(Δ_{S} ∘ δ_{S}); and for P, Q ∈ Ob(L_{0}) and φ ∈ Mor_{L}(P, Q), φ ∈ Mor_{L_{0}}(P, Q) if and only if F (φ) = 1.

Proof. In general, for P ∈ Ob(L), we write P_{0} = P ∩S_{0} for short. Likewise, for φ ∈ Mor_{L}(P, Q), set φ_{0} = φ|_{P_{0},Q_{0}} ∈ Mor_{L}(P_{0}, Q_{0}). By condition (ii), for any such φ, we can write φ_{0} = φ^{∗} ∘ α|_{P_{0},α(P_{0})} for some α ∈ Aut_{L}(S_{0}) and some φ^{∗} ∈ Mor_{L_{0}}(α(P_{0}), Q_{0}). Set F (φ) = [α] ∈ L/L_{0}: the class of α modulo Aut_{L_{0}}(S_{0}).

If in addition, φ_{0} = ψ^{∗} ∘ β|_{P_{0},β(P_{0})}, where β ∈ Aut_{L}(S_{0}) and ψ^{∗} ∈ Mor_{L_{0}}(β(P_{0}), Q_{0}), then

(αβ^{−1})|_{β(P_{0}),α(P_{0})} = (φ^{∗})^{−1} ∘ ψ^{∗} ∈ Mor_{L_{0}}(β(P_{0}), α(P_{0})) .

Since (φ^{∗})^{−1} ∘ ψ^{∗} extends to an automorphism of S_{0} in L, it satisfies (by axiom (C)) the hypotheses in Proposition 4.3(b) for it to have an extension in the category L_{0} with domain N_{S_{0}}(β(P_{0})). By the same proposition, the extension is unique in L, and hence is again the restriction of αβ^{−1}. Upon repeating this procedure, we see that (φ^{∗})^{−1} ∘ψ^{∗} extends to a unique automorphism of S_{0} in L_{0}, and hence that αβ^{−1} ∈ Aut_{L_{0}}(S_{0}). Thus [α] = [β] in L/L_{0}, and this proves that F (φ) is well defined. The other properties of F follow immediately from this construction. □

A converse to Lemma 4.13 is given in [OV1, Proposition 4.1]. If L is a linking system associated to F, Γ is a group, and F: L ⟶ B(Γ) is a functor, then under certain conditions (especially conditions on the objects in L), a kernel category L_{0} can be defined which is a normal linking subsystem of L associated to a saturated fusion subsystem F_{0} ˙⊴ F. This is stated in [OV1] in terms of transporter systems, but restricts to a result about linking systems as a special case.

The following lemma describes how, under certain conditions, one constructs a normal linking subsystem associated to a given weakly normal fusion subsystem.

Lemma 4.14 ([AOV, Lemma 1.30]). Fix a weakly normal pair of fusion systems F_{0} ˙⊴ F over p-groups S_{0} ⊴ S. Assume there are sets H_{0} and H of subgroups of S_{0} and S, respectively, both closed under F_{0}- or F-conjugacy and overgroups, such that H = {P ≤ S | P ∩ S_{0} ∈ H_{0}}, H ⊆ F^{c}, and H_{0} ⊇ F_{0}^{cr}. Assume F has an associated centric linking system L^{c}, and let L ⊆ L^{c} be the full subcategory with Ob(L) = H. Let L_{0} ⊆ L be the subcategory with Ob(L_{0}) = H_{0}, where for P, Q ∈ H_{0},

Mor_{L_{0}}(P, Q) = {ψ ∈ Mor_{L}(P, Q) | π(ψ) ∈ Hom_{F_{0}}(P, Q)} . (11)

Then L_{0} ⊴ L is a normal pair of linking systems associated to F_{0} ˙⊴ F.

Once we have defined normal linking subsystems, and the “quotient group” of a normal pair of linking systems, it is natural to ask what conditions are needed to specify an extension of a linking system by a finite group. This is what is described in the following theorem.

Theorem 4.15 ([O4, Theorem 9]). Fix a saturated fusion system F_{0} over a p-group S_{0}, and let L_{0} be a linking system associated to F_{0}. Set H_{0} = Ob(L_{0}), and assume it is closed under overgroups. Set Γ_{0} = Aut_{L_{0}}(S_{0}), and regard S_{0} as a subgroup of Γ_{0} via the inclusion of T_{H_{0}}(S_{0}) into L_{0}. Fix a finite group Γ such that Γ_{0} ⊴ Γ, and a homomorphism τ: Γ → Aut_{typ}^{I}(L_{0}) which makes both triangles in the following diagram commute:

\xymatrix{Aut_{L_{0}}(S_{0}) = Γ_{0} \ar[rr]^{conj} \ar[dr]|{incl} &  & Aut_{typ}^{I}(L_{0}) \ar[d]^{(α↦α_{S_{0}} )} \\  & Γ \ar[r]^{conj} \ar[ur]^{τ} & Aut(Γ_{0}) ,}

Let F_{1} be the smallest fusion system over S_{0} (not necessarily saturated) which contains F_{0} and Aut_{Γ}(S_{0}), where Γ acts on S_{0} via conjugation on S_{0} = O_{p}(Γ_{0}) ⊴ Γ. Fix S ∈ Syl_{p}(Γ). Then there is a saturated fusion system F over S which contains F_{1} as full subcategory, and such that F_{0} ˙⊴ F.

Assume, in addition, that C_{Γ}(S_{0}) is a p-group; and also that either Γ/Γ_{0} is a p-group or H_{0} ⊆ F_{0}^{c}. Then F can be chosen so as to have an associated linking system L for which L_{0} ⊴ L,

Ob(L) = H =^{def} {P ≤ S | P ∩ S_{0} ∈ H_{0}} ,

and Aut_{L}(S_{0}) = Γ with the given action on L_{0}. If L′ is another linking system, associated to a saturated fusion system F′ over S, such that L_{0} ⊴ L′, F_{0} ˙⊴ F′, and Aut_{L′}(S_{0}) = Γ with the given action on L_{0} and the given inclusion S ≤ Γ, then F′ = F and L′ ≅ L.

An early version of Theorem 4.15 was proven in [5a2, Theorem 4.6], and applied to construct extensions containing a given p-local finite group with p-power index. Later, a different version was proven by Castellana and Libman [CL, Theorem 5.2], and applied to define wreath products of p-local finite groups by permutation groups. The result in the above form was needed to prove [AOV, Theorem B] (see Theorem 6.5 below).

4.5. Fundamental groups and covering spaces.

Certain constructions made with fusion and linking systems can be explained in terms of the fundamental group and covering spaces of their geometric realizations. By Proposition 2.8, for any fusion system F, π_{1}(|F^{c}|) is isomorphic to the free group on elements [φ] for φ ∈ Mor(F^{c}), modulo the relations [ψ ∘ φ] = [ψ][φ] and [incl_{P}^{S}] = 1. Similarly, for any linking system L, π_{1}(|L|) is isomorphic to the free group on elements [φ] for φ ∈ Mor(L), modulo the relations [ψ ∘ φ] = [ψ][φ] and [ι_{P,S}] = 1.

In contrast, π_{1}(|F|) = 1 for any fusion system F. In fact, |F| is contractible since the trivial subgroup is an initial object in F.

The following proposition is an application of Proposition 2.9, and gives a first example of the usefulness of this viewpoint. It is essentially shown in [5a2, Corollary 3.10], although it is stated there with different hypotheses.

Proposition 4.16. Fix a linking system L associated to a fusion system F over the p-group S. Let L_{0} ⊴ L be a normal subsystem, associated to F_{0} ˙⊴ F over S_{0}. Then the inclusion of the geometric realization |L_{0}| into |L| is homotopy equivalent to a covering space over |L| with covering group L/L_{0}.

Proof. Let L^{∗} ⊆ L be the full subcategory with Ob(L^{∗}) = Ob(L_{0}) = {P ∩ S_{0} | P ∈ Ob(L)}, and let i: L^{∗} ⟶ L be the inclusion. Let r: L ⟶ L^{∗} be the functor which sends P to P ∩ S_{0} and sends a morphism to its restriction (Proposition 4.3(a)). Thus r ∘ i = Id_{L^{∗}}. There is a natural morphism of functors from i ∘ r to Id_{L} which sends an object P to the inclusion ι_{P_{0},P} ∈ Mor_{L}(P_{0}, P) (where P_{0} = P ∩ S_{0}), so |i ∘ r| ≃ Id_{|L|}, and r is a deformation retraction. The inclusion |L^{∗}| ⊆ |L| is a thus a homotopy equivalence.

Let F: L^{∗} ⟶ B(L/L_{0}) be the functor defined in Lemma 4.13. Then L_{0} is the “kernel” of F in the sense of Proposition 2.9: it has the same objects as L^{∗}, and φ ∈ Mor(L^{∗}) is in Mor(L_{0}) if and only if F (φ) = 1 by Lemma 4.13 again. For each P ∈ Ob(L^{∗}) and each α ∈ Aut_{L}(S_{0}), there is some P^{∗} such that α restricts to an isomorphism β = α|_{P,P^{∗}} ∈ Iso_{L}(P, P^{∗}).

The hypotheses of Proposition 2.9 are thus satisfied, and so |L_{0}| has the homotopy type of a covering space over |L^{∗}| (hence over |L|) with covering group L/L_{0}. □

We next look at the properties of π_{1}(|L|) and π_{1}(|L|_{p}^{∧}). For example, it is important that the space |L| is p-good (otherwise we don’t expect any reasonable behavior from its p-completion). Recall the definition of the hyperfocal subgroup in a fusion system (Definition I.7.1):

𝔥𝔶𝔭(F) = ⟨g^{−1}α(g) | g ∈ P ≤ S, α ∈ O^{p}(Aut_{F}(P))⟩ ⊴ S .

Theorem 4.17 ([BLO2] & [5a2]). Let (S, F, L) be any p-local finite group. Then |L| is p-good. Let ∗ ∈ |L| be the vertex corresponding to S ∈ Ob(L). Consider the composite

Ψ: S ⟶^{δ_{S}} Aut_{L}(S) ⟶^{θ_{S}} π_{1}(|L|, ∗) ⟶ π_{1}(|L|_{p}^{∧}, ∗) ,

where θ_{S} sends an automorphism of S to the corresponding loop in |L|. Then Ψ is surjective, Ker(Ψ) = 𝔥𝔶𝔭(F), and thus Ψ induces an isomorphism π_{1}(|L|_{p}^{∧}, ∗) ≅ S/𝔥𝔶𝔭(F).

Proof. By [BLO2, Proposition 1.11], |L| is p-good, and S surjects onto π_{1}(|L|_{p}^{∧}). We sketch here the idea of the proof. Let H ⊴ π_{1}(|L|, ∗) be the subgroup generated by all elements of finite order prime to p. In particular, H is p-perfect. One first shows, using the presentation in Proposition 2.8, that π_{1}(|L|, ∗) is generated by H together with θ_{S}(δ_{S}(S)). Thus H has finite index in π_{1}(|L|, ∗). It then follows by Proposition 1.11 that |L| is p-good, and that π_{1}(|L|_{p}^{∧}, ∗) ≅ π_{1}(|L|, ∗)/H.

The kernel of Ψ was shown in [5a2, Theorem B] to be equal to 𝔥𝔶𝔭(F). The hard part, the inclusion Ker(Ψ) ≤ 𝔥𝔶𝔭(F), was shown by explicitly (and laboriously) constructing a functor from L to B(S/𝔥𝔶𝔭(F)) which sends δ_{S}(g) (for g ∈ S) to the class of g modulo 𝔥𝔶𝔭(F). □

Thus the fundamental group |L|_{p}^{∧} for a linking system L depends only on the fusion system to which L is associated. We will see later that many of the other homotopy theoretic properties of |L|_{p}^{∧} also depend only on the fusion system.

In [5a2], we showed that for each p-local finite group (S, F, L), each covering space over |L|_{p}^{∧} is the classifying space of another p-local finite group, and each covering space over |F^{c}| is the geometric realization of the centric fusion system in another p-local finite group. This came out of the study of fusion subsystems of p-power index and of index prime to p, as defined in Part I (Definition I.7.3).

We saw in Theorem I.7.4 that for each saturated fusion system F over a p-group S, and each T ≤ S which contains 𝔥𝔶𝔭(F), there is a unique saturated fusion subsystem F_{T} ⊆ F over T of p-power index. We now look at associated linking systems in this context.

Theorem 4.18 ([5a2, Theorem 4.4]). Fix a saturated fusion system F over a p-group S, and a quasicentric linking system L associated to F. Fix a subgroup T ≤ S which contains 𝔥𝔶𝔭(F), and let F_{T} ⊆ F be the unique saturated fusion subsystem of p-power index over T. Then there is a quasicentric linking subsystem L_{T} ⊆ L associated to F_{T}, such that |L_{T} | is homotopy equivalent to a covering space over |L| of degree [S:T], and such that |L_{T} |_{p}^{∧} is homotopy equivalent to the covering space over |L|_{p}^{∧} with fundamental group T/𝔥𝔶𝔭(F). Also, if T ⊴ S, then L_{T} ⊴ L.

Proof. Except for the last statement (L_{T} ⊴ L), this was stated explicitly in [5a2, Theorem 4.4]. The normality of L_{T} is shown in [AOV, Propositions 1.31] for T = 𝔥𝔶𝔭(F) (and for a slightly different choice of objects in the linking systems). The key point is that L_{T} can be described as the “kernel” of a map λ: Mor(L) ⟶ S/T constructed in [5a2, Proposition 2.4]. The general case follows by a similar argument. □

Since π_{1}(|L|_{p}^{∧}) ≅ S/𝔥𝔶𝔭(F) by Theorem 4.17, Theorem 4.18 says that every connected covering space of |L|_{p}^{∧} is realized (up to homotopy type) as the classifying space of a p-local finite subgroup of p-power index in (S, F, L). We thus have a bijective correspondence between subgroups of π_{1}(|L|_{p}^{∧}) (equivalently, connected covering spaces of |L|_{p}^{∧}), and the fusion subsystems of p-power index in F. This is analogous to the situation for the classifying space of a finite group G: since π_{1}(BG_{p}^{∧}) ≅ G/O^{p}(G), there is a bijective correspondence between connected covering spaces of BG_{p}^{∧} and subgroups of G containing O^{p}(G).

We now look at subsystems of index prime to p, and their connection with π_{1}(|F^{c}|) which was first shown by Aschbacher.

Theorem 4.19 ([5a2, Theorem 5.4]). Fix a saturated fusion system F over a p-group S. Let ∗ ∈ |F^{c}| be the vertex corresponding to S ∈ Ob(F), and let

θ: Mor(F^{c}) ⟶ π_{1}(|F^{c}|, ∗)

be the function of Proposition 2.8 which sends φ ∈ Hom_{F}(P, Q) to the class of the loop formed by the edges in |F^{c}| corresponding to φ, incl_{P}^{S}, and incl_{Q}^{S} in F^{c}.

(a) For each H ≤ π_{1}(|F^{c}|, ∗), there is a unique saturated fusion subsystem F_{H} ⊆ F of index prime to p such that Mor(F_{H}^{c}) = θ^{−1}(H) and Ob(F_{H}^{c}) = Ob(F^{c}). Conversely, for each saturated fusion subsystem F_{∗} ⊆ F of index prime to p, F_{∗} = F_{H} for some H.

(b) For each H ≤ π_{1}(|F^{c}|, ∗), |F_{H}^{c}| has the homotopy type of the connected covering space of |F^{c}| with fundamental group H.

(c) The restriction of θ to Aut_{F}(S) induces an isomorphism

π_{1}(|F^{c}|, ∗) ≅ Aut_{F}(S)/Aut_{O^{p′} (F)}(S) ,

where O^{p′}(F) = F_{1}. In particular, π_{1}(|F^{c}|, ∗) is finite of order prime to p.

Proof. Let θ′: Mor(F^{c}) ⟶ Γ_{F} be the map and group defined in Theorem I.7.7. Thus θ′ sends composites of morphisms to products in Γ_{F}, θ′(O^{p′}(Aut_{F}(P))) = 1 for each P ∈ F^{c}, and θ′ is universal among all maps from Mor(F^{c}) to groups having these properties. By Theorem I.7.7 and Lemma I.7.6, points (a) and (c) hold if π_{1}(|F^{c}|, ∗) and θ are replaced by Γ_{F} and θ′. Also, (b) follows from (a) and Proposition 2.9.

By Proposition 2.8, θ is universal among all maps from Mor(F^{c}) to a group which send composites of morphisms to products, and such that θ(incl_{P}^{S}) = 1 for each P ∈ F^{c}. Hence there is a surjective homomorphism λ: π_{1}(|F^{c}|, ∗) ⟶^{≅} Γ_{F} such that λ ∘ θ = θ′, and we must show that λ is an isomorphism. Set Γ = π_{1}(|F^{c}|, ∗) for short.

For g ∈ S, we write θ(c_{g}) ∈ Γ to denote the image of c_{g} ∈ Inn(S). For each P ∈ F^{c} and c_{g}|_{P} ∈ Hom_{F}(P,^{g}P), θ(c_{g}|_{P}) = θ(c_{g}) since θ(incl_{P}^{S}) = θ(incl_{^{g}P}^{S}) = 1. If P, Q ∈ F^{c}, φ ∈ Iso_{F}(P, Q), and g ∈ P, then θ(c_{g}|_{P}) and θ(c_{φ(g)}|_{Q}) are conjugate in Γ by θ(φ), and thus θ(c_{g}) is Γ-conjugate to θ(c_{φ(g)}). Hence by Alperin’s fusion theorem (Theorem I.3.6), if g, h ∈ S are F-conjugate, then θ(c_{g}) is Γ-conjugate to θ(c_{h}).

For each g ∈ S such that ⟨g⟩ is fully centralized, C_{S}(g) ∈ F^{c}, and c_{g}|_{C_{S}(g)} = Id ∈ Aut_{F}(C_{S}(g)). Hence θ(c_{g}|_{C_{S}(g)}) = 1, and so θ(c_{g}) = 1 in this case. Since every element of S is F-conjugate to such an element g, this proves that θ(Inn(S)) = 1. So by restriction, θ(Aut_{S}(P)) = 1 for all P. Thus θ(O^{p′}(Aut_{F}(P))) = 1 when P is fully normalized in F, since O^{p′}(Aut_{F}(P)) is the normal closure of Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) in this case; and θ(O^{p′}(Aut_{F}(P))) = 1 for all P since this clearly depends only on the F-isomorphism class of P. This finishes the proof that λ sends Γ isomorphically to Γ_{F}. □

When L is a centric linking system associated to F, and F_{∗} ⊆ F is a saturated fusion subsystem of index prime to p, then by Lemma 4.14, a centric linking subsystem L_{∗} ⊆ L associated to F_{∗} can be defined simply as the pullback of L and F_{∗} over F. See [5a2, Theorem 5.5] for more details.

4.6. Homotopy properties of classifying spaces.

We now list some of the other basic results on classifying spaces of plocal finite groups, which show that they have many of the nice homotopy theoretic properties of the p-completed classifying spaces BG_{p}^{∧}. We begin with the following result, which shows that the homotopy type of |L| is independent of the choice of objects.

Theorem 4.20 ([5a1, Theorem 3.5]). Fix a saturated fusion system F over a p-group S. Let L_{0} ⊆ L be two linking systems associated to F with different sets of objects. Then the inclusion induces a homotopy equivalence of spaces |L_{0}| ≃ |L|.

Proof. This is stated in [5a1] under the assumption that L is a quasicentric linking system, and L_{0} ⊆ L is any full subcategory which contains all subgroups which are F-centric and F-radical. Axiom (A1) in Definition 4.1 says that Ob(L_{0}) ⊇ F^{cr}, and axiom (A2) implies that L_{0} is a full subcategory. Thus the only thing assumed in [5a2] and not assumed here is that Ob(L) = F^{q}. But in fact, the proof of [5a1, Proposition 3.11] (which is the main induction step when proving the theorem and is based on Quillen’s Theorem A) does not use this assumption at all, and so the result holds as stated above. □

Theorem 4.20 helps to motivate using the homotopy type of |L| and |L|_{p}^{∧}, and their homotopy properties, as important invariants of the p-local finite group. It also allows a certain flexibility when working with geometric realizations of linking systems which is crucial, for example, when comparing linking systems associated to a pair F_{0} ⊆ F of fusion systems.

We next look at mapping spaces. The following result is the version of Theorem 1.16 for classifying spaces of p-local finite groups. When Q is a p-group and F is a saturated fusion system over a p-group S, we set

Rep(Q, F) = Hom(Q, S)/∼ ,

where for ρ, σ ∈ Hom(Q, S), ρ ∼ σ if there is α ∈ Iso_{F}(ρ(Q), σ(Q)) such that α ∘ ρ = σ.

Theorem 4.21 ([BLO2, Corollary 4.5]). For any p-group Q and any plocal finite group (S, F, L), the map

Rep(Q, F) ⟶^{≅} [BQ, |L|_{p}^{∧}],

defined by sending the class of ρ: Q ⟶ S to the class of the composite

BQ ⟶^{Bρ} BS ⟶^{incl} |L| ⟶^{ϕ_{|L|}} |L|_{p}^{∧}

is a bijection.

The components of the mapping space map(BQ, |L|_{p}^{∧}) can also be described. By [BLO2, Theorem 6.3], for ρ ∈ Hom(Q, S) such that ρ(Q) is fully centralized in F, the component of the above composite induced by ρ has the homotopy type of |C_{L}(ρ(Q))|_{p}^{∧}, where C_{L}(ρ(Q)) is a certain linking system associated to the centralizer fusion system C_{F}(ρ(Q)).

For any topological space X, we define

Out(X) = {homotopy equivalences f: X ⟶^{≃} X}/(homotopy) :

the group of homotopy classes of self homotopy equivalences. One other property which the classifying spaces |L|_{p}^{∧} have is a very nice combinatorial description of Out(|L|_{p}^{∧}) in terms of self equivalences of the category L. Recall the definitions of isotypical equivalences of a linking system, and of the group Out_{typ}(L), in Section 4.3. The next theorem helps to explain the importance of Out_{typ}(L) as an automorphism group of the p-local finite group (S, F, L).

Theorem 4.22 ([BLO2, Theorem 8.1]). For any p-local finite group (S, F, L), there is an isomorphism of groups

Out_{typ}(L) ⟶^{≅} Out(|L|_{p}^{∧})

which sends the class of α ∈ Aut_{typ}^{I}(L) to the class of |α|_{p}^{∧}: |L|_{p}^{∧} ⟶ |L|_{p}^{∧}.

This isomorphism Out_{typ}(L) ≅ Out(|L|_{p}^{∧}) was proven in [BLO2] only for centric linking systems. However, with the help of Lemma 4.10 and Theorem 4.20 (and Proposition 4.8), it can be shown to hold for an arbitrary linking system whose set of objects is Aut(S, F)-invariant.

For any fusion system F over a p-group S, let H^{∗}(F; 𝔽_{p}) be the subring of H^{∗}(BS; 𝔽_{p}) consisting of those elements which are stable under all fusion in F; i.e.,

H^{∗}(F; 𝔽_{p}) = {x ∈ H^{∗}(BS; 𝔽_{p}) | α^{∗}(x) = x|_{BP} , all α ∈ Hom_{F}(P, S)}

≅ \underleftarrow{lim}_{F} H^{∗}(−; 𝔽_{p}) .

Theorem 4.23 ([BLO2, Theorem B]). For any p-local finite group (S, F, L), the natural homomorphism

H^{∗}(|L|_{p}^{∧}; 𝔽_{p}) ⟶^{≅} H^{∗}(F; 𝔽_{p}),

induced by the inclusion of BS in |L|, is an isomorphism. Furthermore, the ring H^{∗}(|L|_{p}^{∧}; 𝔽_{p}) is noetherian.

The proof of Theorem 4.23 depends on the existence of a characteristic biset for F (Definition I.8.3).

An isomorphism (S_{1}, F_{1}, L_{1}) ⟶ (S_{2}, F_{2}, L_{2}) of p-local finite groups consists of a triple

S_{1} ⟶_{≅}^{α} S_{2},  F_{1} ⟶_{≅}^{α_{F}} F_{2},  and  L_{1} ⟶_{≅}^{α_{L}} L_{2}

of isomorphisms of groups and categories, such that α_{F}(P) = α(P) for all P ≤ S_{1}, α_{L}(P) = α(P) for all P ∈ Ob(L_{1}), and such that they commute in the obvious way with the structural functors L_{i} ⟶^{π_{i}} F_{i} and

T_{Ob(L_{i})}(S_{i}) ⟶^{δ_{i}} L_{i}.

This is slightly stronger than the definition in [BLO2, § 7] in that in [BLO2], α and α_{L} were not required to commute with δ, but only with the restricted homomorphisms δ_{P} |_{P} : P ⟶ Aut_{L}(P). So we check here that the two definitions are equivalent. In fact, we can show the following:

Lemma 4.24. For i = 1, 2, let F_{i} be a saturated fusion system over the p-group S_{i}, and let L_{i} be a linking system associated to F_{i}. Let

S_{1} ⟶_{≅}^{α} S_{2},  F_{1} ⟶_{≅}^{α_{F}} F_{2},  and  L_{1} ⟶_{≅}^{α_{L}} L_{2}

be isomorphisms of groups and categories such that π_{2} ∘ α_{L} = α_{F} ∘ π_{1}, α_{F}(P) = α(P) for each P ≤ S_{1} and α_{L}(δ_{1S_{1}}(g)) = δ_{2S_{2}}(α(g)) for each g ∈ S_{1}. Then there is α′_{L} ∈ Iso(L_{1}, L_{2}) such that (α, α_{F}, α′_{L}) is an isomorphism of p-local finite groups.

Proof. For each P ∈ Ob(L_{1}), let

ι_{P} ∈ Mor_{L_{1}}(P, S_{1})  and  ι_{α(P)} ∈ Mor_{L_{2}}(α(P), S_{2})

be the two inclusions. Since π_{2}(α_{L}(ι_{P})) = π_{2}(ι_{α(P)}), there is by Proposition 4.2(a) a unique element ζ_{P} ∈ Aut_{L_{1}}(P) such that π_{1}(ζ_{P}) = Id_{P} and α_{L}(ι_{P}) = ι_{α(P)} ∘ ζ_{P}. In particular, ζ_{S_{1}} = Id. Define α′_{L} by setting α′_{L}(P) = α(P) for each P ∈ Ob(L_{1}); and for each φ ∈ Mor_{L_{1}}(P, Q),

α′_{L}(φ) = ζ_{Q} ∘ α_{L}(φ) ∘ ζ_{P}^{−1} .

Then α′_{L} sends inclusions to inclusions, π_{2} ∘ α′_{L} = π_{2} ∘ α_{L} = α_{F} ∘ π_{1}, and α′_{L}(φ) = α_{L}(φ) for all φ ∈ Aut_{L_{1}}(S_{1}). Since the δ_{i} are determined uniquely by the inclusions and δ_{iS_{i}}, α′_{L} ∘ δ_{1} = δ_{2} ∘ T (α). Thus (α, α_{F}, α′_{L}) is an isomorphism of p-local finite groups. □

The next theorem says that the isomorphism type of (S_{i}, F_{i}, L_{i}) is completely determined by the homotopy type of the space |L_{i}|_{p}^{∧}.

Theorem 4.25. If (S_{1}, F_{1}, L_{1}) and (S_{2}, F_{2}, L_{2}) are p-local finite groups, then any homotopy equivalence |L_{1}|_{p}^{∧} ⟶^{≃} |L_{2}|_{p}^{∧} induces an isomorphism

(S_{1}, F_{1}, L_{1}) ⟶^{≅} (S_{2}, F_{2}, L_{2}) of p-local finite groups.

Proof. See [BLO2, Theorem 7.4]. The basic idea of the proof is to show that F_{i} and L_{i} are isomorphic to the fusion and linking categories of the space |L_{i}|_{p}^{∧} in a natural way which commutes with the structural functors.

More precisely, by [BLO2, Proposition 7.3], if (S, F, L) is a p-local finite group, and f: BS ⟶ |L|_{p}^{∧} is the natural inclusion, then there are equivalences of categories

F ≅ F_{S,f}(|L|_{p}^{∧})  and  L ≅ L_{S,f}^{c}(|L|_{p}^{∧}).

This implies the theorem, since by Theorem 4.21, for any homotopy equivalence |L_{1}|_{p}^{∧} ⟶_{≃}^{ψ} |L_{2}|_{p}^{∧}, there is an isomorphism S_{1} ⟶_{≅}^{α} S_{2} such that the following square commutes up to homotopy:

\xymatrix{BS_{1} \ar[r]^{f_{1}} \ar[d]_{Bα} & |L_{1}|_{p}^{∧} \ar[d]_{ψ} \\ BS_{2} \ar[r]^{f_{2}} & |L_{2}|_{p}^{∧}}

where f_{1} and f_{2} are the natural inclusions. □

4.7. Classifying spectra of fusion systems.

This subsection is mainly intended for algebraic topologists who already know what a spectrum is. But to keep it from being completely inaccessible to others, we begin with a quick summary of some basic definitions.

For any space X, the suspension of X is the space

ΣX = (X × I)/∼  where  (x, s) ∼ (y, t) if s = t ∈ {0, 1}.

In other words, each of the subspaces X × {0} and X × {1} in X × [0, 1] is identified to a point. One important property of the suspension is that it increases the degree of all (reduced) homology groups: H̃_{q}(ΣX) ≅ H̃_{q−1}(X) for all q, where H̃_{q}(X) = H_{q}(X)/H_{q}(pt). A spectrum consists of a sequence X = {X_{n}} of spaces, together with maps ΣX_{n} ⟶ X_{n+1} for each n. These maps induce homomorphisms H_{q}(X_{n}) ⟶ H_{q+1}(X_{n+1}) for each q and n; and the spectrum homology of X = {X_{n}} is defined by setting H_{q}(X) = colim_{n} H̃_{q+n}(X_{n}) and similarly for cohomology. The suspension spectrum of a space X is the sequence Σ^{∞}X = {Σ^{n}X}_{n≥0} of iterated suspensions of X, and H_{q}(Σ^{∞}X) ≅ H̃_{q}(X).

To avoid technical details, we omit here the general definition of maps (and homotopy classes of maps) between spectra, and refer to [Ad, § III.2] and [Sw, 8.12–15]. In general, for any pair of spectra X and Y, [X, Y] has the structure of an abelian group (defined roughly by adding along the suspension coordinates), and the structure of a ring (with multiplication defined by composition) when X = Y. When X and Y are CW complexes, one writes {X, Y } = [Σ^{∞}X, Σ^{∞}Y] for short; this is equal to colim_{n}[Σ^{n}X, Σ^{n}Y] when X is a finite complex.

In [BLO2, p. 815], a procedure was very loosely described for constructing a classifying spectrum for a saturated fusion system F, whether or not a classifying space (equivalently, a linking system) exists. This construction, which takes as starting point a characteristic biset for F as described in Lemma I.8.3, was made more precise by Kári Ragnarsson [Rg]. He showed how to assign, to each saturated fusion system F over a p-group S, a canonical idempotent e_{F} ∈ {BS, BS}, a “classifying spectrum” 𝔹F, together with a structure map σ_{F} ∈ [Σ^{∞}BS, 𝔹F] (projection onto a direct factor), such that

H^{∗}(𝔹F; 𝔽_{p}) ≅ H^{∗}(e_{F})·H^{∗}(BS; 𝔽_{p}) ≅ \underleftarrow{lim}_{F} H^{∗}(−; 𝔽_{p}) .

Hence by Theorem 4.23, for any centric linking system L associated to F, H^{∗}(𝔹F; 𝔽_{p}) ≅ H^{∗}(|L|_{p}^{∧}; 𝔽_{p}). Using this, he then showed that 𝔹F is the suspension spectrum of |L|_{p}^{∧}. In particular, if F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G), then 𝔹F is the suspension spectrum of BG_{p}^{∧}, and σ_{F} is the suspension of the canonical map BS ⟶ BG_{p}^{∧} induced by the inclusion S ≤ G.

Thus whether or not classifying spaces exist or are unique, a unique classifying spectrum can be associated to every saturated fusion system, a spectrum which does have some of the properties (such as homology) which the classifying spaces would have if they existed.

In [Rg, Theorem A], Ragnarsson proves that for any F and S, and any pair of subgroups P, Q ≤ S, Hom_{F}(P, Q) is precisely the set of all homomorphisms φ ∈ Hom(P, Q) such that σ_{F}|_{Σ^{∞}BP} and σ_{F}|_{Σ^{∞}BQ} ∘ Σ^{∞}(Bφ) are equal in [Σ^{∞}BP, 𝔹F]. In other words, the fusion system F can be recovered from the classifying spectrum 𝔹F together with the structure map σ_{F}, in a way very similar to that in which F can be recovered from |L|_{p}^{∧} when there is an associated linking system L (see Theorem 4.25 and its proof). It is, however, important here that σ_{F} is part of the structure: Martino and Priddy [MP1, Example 5.2] gave examples of pairs of finite groups G_{1} and G_{2} whose fusion systems at p are not isomorphic, but such that Σ^{∞}(BG_{1}_{p}^{∧}) ≃ Σ^{∞}(BG_{2}_{p}^{∧}).

In addition, Ragnarsson described the group of homotopy classes of stable maps [𝔹F_{1}, 𝔹F_{2}] between the classifying spectra of two saturated fusion systems by giving an explicit basis [Rg, Theorem B]. This was then used in [Rg, Theorem C] to show that these classifying spectra are functorial: that 𝔹 defines a functor from the category of saturated fusion systems (for fixed p), with fusion preserving homomorphisms as its morphisms, to the category of p-local spectra and homotopy classes of stable maps between them.

More recently, Ragnarsson and Stancu [RSt] have shown that saturated fusion systems over a p-group S are in bijective correspondence with a certain family of idempotents in the ring {BS, BS} (which can be described purely algebraically as the “p-adic double Burnside ring” A(S, S)_{p}^{∧} for S). The conditions on the idempotents [RSt, Theorem D] are that they not be in the Nishida ideal (a certain ideal in A(S, S)_{p}^{∧}), and that they satisfy a Frobenius reciprocity condition. This result provides a completely new way to characterize saturated fusion systems, this time using stable homotopy theory.

4.8. An infinite version: p-local compact groups.

A discrete p-toral group is a group S which contains a normal subgroup S_{0} ⊴ S such that S/S_{0} is a (finite) p-group, and S_{0} ≅ (ℤ/p^{∞})^{r} for some r. Here, ℤ/p^{∞} ≅ ℤ[\frac{1}{p}]/ℤ is the increasing union of the ℤ/p^{n} for all n.

In [BLO3], a saturated fusion system over a discrete p-toral group S is defined to be a category F whose objects are the subgroups of S, which satisfies the same conditions as those used to define a fusion system over a p-group (Definitions I.2.1 and I.2.2), and which in addition satisfies a “continuity” axiom: if P ≤ S is the increasing union of subgroups P_{n} for n ≥ 1, and φ ∈ Hom(P, S) is such that φ|_{P_{n}} ∈ Hom_{F}(P_{n}, S) for all n, then φ ∈ Hom_{F}(P, S). A centric linking system associated to F is then defined exactly as in the finite case (Definition 4.1).

A p-local compact group is a triple (S, F, L), where S is a discrete p-toral group, F is a saturated fusion system over S, and L is a linking system associated to F. As in the finite case, the classifying space of the triple (S, F, L) is defined to be the p-completed space |L|_{p}^{∧}. By results in [BLO3, Sections 6–7], classifying spaces of p-local compact groups satisfy homotopy theoretic properties similar to those just described in Section 4.6.

By [BLO3, Theorem 9.10], for every compact Lie group G, there is a maximal discrete p-toral subgroup S ≤ G which is unique up to conjugacy, a saturated fusion system F_{S}(G) over S, and an associated linking system L_{S}^{c}(G) with the property that |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧}. The definition of F_{S}(G) is essentially identical to that in the finite case, but the definition of L_{S}^{c}(G) turns out to be quite a bit trickier.

Other examples of p-local compact groups studied in [BLO3] include those coming from p-compact groups, as defined by Dwyer and Wilkerson, and those coming from torsion linear groups (subgroups of GL_{n}(K) for char(K) ≠ p, all of whose elements have finite order). We do not know any examples for which we can prove that they do not come from one of these three sources, but we have constructed many for which this is unknown. So while p-local compact groups were originally defined in order to construct a “compact Lie group version” of p-local finite groups, it seems to be a much more varied class of objects, and we are far from understanding what the class really contains.

5. The orbit category and its applications

The orbit category O(G) of a group G is usually defined to be the category with one object G/H = {gH | g ∈ G} for each subgroup H ≤ G (the “orbits” of G), and with morphism sets defined by

Mor_{O(G)}(G/H, G/K) = map_{G}(G/H, G/K) .

However, by analogy with the notation for fusion and linking systems, it will be more convenient for our purposes to regard the objects of O(G) as subgroups rather than as orbits of G, and to define morphisms from H to K in terms of the transporter set T_{G}(H, K) (defined in Section 3.1).

Definition 5.1. For any group G, the orbit category of G is the category O(G) whose objects are the subgroups of G, and where for each H, K ≤ G,

Mor_{O(G)}(H, K) = K\T_{G}(H, K) .

When G is finite and S ∈ Syl_{p}(G), then O_{S}(G) ⊆ O(G) denotes the full subcategory whose objects are the subgroups of S. More generally, for any set H of subgroups of G, O_{H}(G) ⊆ O(G) is the full subcategory with object set H.

The equivalence of these two definitions comes from identifying f ∈ map_{G}(G/H, G/K) with Kx^{−1} ∈ K\T_{G}(H, K), when x ∈ G is such that f(H) = xK. Note that for h ∈ H, xK = f(hH) = hxK, so H^{x} ≤ K, and thus x^{−1} ∈ T_{G}(H, K). Conversely, for y ∈ T_{G}(H, K), the coset Ky corresponds to the map of orbits (gH ↦ gy^{−1}K).

Thus O(G) can be thought of as a quotient category of the transporter category T (G), where we divide out by the action of the target group on the morphism sets. By analogy, the “orbit category” of a fusion system is defined by replacing the homomorphisms in the fusion system by conjugacy classes of homomorphisms.

Definition 5.2. Let F be a saturated fusion system over a p-group S. The orbit category of F is the category O(F) where Ob(O(F)) = Ob(F), and where

Mor_{O(F)}(P, Q) = Rep_{F}(P, Q) =^{def} Inn(Q)\Hom_{F}(P, Q).

For any full subcategory F_{0} ⊆ F, O(F_{0}) ⊆ O(F) denotes the full subcategory with Ob(O(F_{0})) = Ob(F_{0}). In particular, the centric orbit category O(F^{c}) of F is the full subcategory of O(F) with Ob(O(F^{c})) = F^{c}.

When F = F_{S}(G) for a finite group G with S ∈ Syl_{p}(G), then for arbitrary subgroups P, Q ≤ S,

Mor_{O(G)}(P, Q) ≅ Q\T_{G}(P, Q)

while

Mor_{O(F)}(P, Q) ≅ Inn(Q)\Hom_{G}(P, Q) ≅ Q\T_{G}(P, Q)/C_{G}(P).

Thus there is a natural functor

O_{S}(G) ⟶ O(F_{S}(G)) ,

which is the identity on objects and surjective on morphism sets, but where the morphism sets can be very different. For example, the trivial subgroup 1 ≤ G is an initial object in O(F_{S}(G)), while Aut_{O(G)}(1) ≅ G.

However, when P, Q ≤ S are both p-centric in G (equivalently, F_{S}(G)- centric), then since C_{G}(P) = Z(P) × C′_{G}(P),

Mor_{O(F^{c})}(P, Q) ≅ Q\T_{G}(P, Q)/C_{G}(P) ≅ Q\T_{G}(P, Q)/C′_{G}(P).

In other words, by comparison with Mor_{O(G)}(P, Q), we are dividing out by the group C′_{G}(P) of order prime to p. The restricted functor

O_{S}^{c}(G) ⟶ O(F_{S}^{c}(G))

thus satisfies the hypotheses of Lemma 2.3, and hence induces a p equivalence from |O_{S}^{c}(G)| to |O(F_{S}^{c}(G))| and a homotopy equivalence |O_{S}^{c}(G)|_{p}^{∧} ≃ |O(F_{S}^{c}(G))|_{p}^{∧}. This helps to explain why the centric orbit category O(F^{c}) of a fusion system F in practice seems to be more useful than the full orbit category O(F).

These orbit categories play a central role when proving many of the results listed in earlier sections. For example, when (S, F, L) is a p-local finite group, then O(F^{c}) is the indexing category of one of the important decompositions of |L| as a “homotopy colimit” of the classifying spaces of its p-subgroups (Section 5.6). Orbit categories are also used when defining many of the obstruction groups which arise in connection with fusion and linking systems, such as those to the existence and uniqueness of linking systems (Section 5.3).

The original motivation in [BLO2] for defining abstract linking systems associated to fusion systems was as a tool for constructing a classifying space associated to a given fusion system F. We now sketch briefly a different, and equivalent, way of doing this (details will be given in Section 5.6). For given F, one would like to define a functor from O(F^{c}) to spaces by sending P ≤ S to the classifying space BP, but on morphisms this is defined only up to homotopy. Proposition 5.31 describes a bijective correspondence between centric linking systems L associated to F and “rigidifications” (actual functors) B̃: O(F^{c}) ⟶ Top of the “homotopy functor” P ↦ BP. Under this correspondence, the homotopy type of the space |L| can be recovered as a direct limit (or homotopy version of a direct limit) of the functor B̃. This correspondence provides a justification (at least to a topologist!) for why centric linking systems are natural objects to choose as a means of “linking” fusion systems to associated classifying spaces.

We begin this section by describing the bar resolution for higher derived functors of inverse limits. This does not seem to be very useful for making computations, at least not directly, but it is the key to showing how these groups appear as obstruction groups for certain lifting problems. Next, as a first application of the bar resolution for functors defined on a group, we prove that constrained fusion systems (see Definition I.4.8) are always (uniquely) realizable (as announced in Theorem I.4.9). Afterwards, we describe concretely the obstructions to the existence and uniqueness of linking systems, and also the obstructions to existence and uniqueness when lifting automorphisms of a fusion system to automorphisms of an associated linking system.

We then explain what a homotopy decomposition is in general, and describe in particular the subgroup decomposition of |L| for a p-local finite group (S, F, L). On the way, we also describe some of the techniques which can be used when computing in these obstruction groups. We end the section with an outline of the proofs of Theorems 4.21 and 4.22 (two results describing certain sets of maps between classifying spaces), to illustrate the role which homotopy decompositions and the orbit category play in those proofs.

5.1. Higher limits of functors and the bar resolution.

Let C be a small category. For any functor F: C^{op} ⟶ Ab, the inverse limit of F is defined by setting

\underleftarrow{lim}_{C}(F) = \{(x_{c})_{c∈Ob(C)} | x_{c} ∈ F (c) for all c ∈ Ob(C),

x_{c} = F (α)(x_{d}) for all α ∈ Mor_{C}(c, d)} .

This is characterized by the universal property: there are homomorphisms ϕ_{c}: \underleftarrow{lim}(F) ⟶ F (c) which make the obvious triangles (one for each morphism in C) commute. Of course, all of this applies equally well to covariant functors C ⟶ Ab, but to simplify the discussion (and since the applications we will describe all involve contravariant functors), we restrict to that case.

Inverse limits are left exact, and hence have (right) derived functors \underleftarrow{lim}^{i}(F) for i ≥ 0. When describing these, it is useful to work in the category C-mod whose objects are the functors C^{op} ⟶ Ab (contravariant functors from C to abelian groups), and whose morphisms are the natural transformations of functors. A sequence of functors F′ → F → F′′ in C-mod is exact if F′(c) → F (c) → F′′(c) is exact for each c ∈ Ob(C).

Injective and surjective morphisms in C-mod are defined similarly. A functor I: C^{op} ⟶ Ab is injective if for each injective morphism F ⟶ F′, every morphism φ: F ⟶ I can be extended to some ψ: F′ ⟶ I.

An injective resolution of F: C^{op} ⟶ Ab is an exact sequence

0 ⟶ F ⟶^{d_{0}} I_{0} ⟶^{d_{1}} I_{1} ⟶^{d_{2}} ⋯

in C-mod such that each I_{n} is injective. Such a resolution exists for any functor F; see, e.g., [Wei, Example 2.3.13, p. 43] or [BK, p.305] for a proof. For any such injective resolution, the (right) derived functors of \underleftarrow{lim}(F) are defined by setting, for each i ≥ 0,

\underleftarrow{lim}_{C}^{i}(F) = H^{i}(\underleftarrow{lim}_{C}(I_{∗}), d_{∗}) .

Since \underleftarrow{lim}(−) is left exact, \underleftarrow{lim}(F) ≅ \underleftarrow{lim}^{0}(F). For a proof that these groups are independent (up to isomorphism) of the choice of resolution, we refer to [McL, Theorem IX.4.3] or [Wei, Lemma 2.4.1]. (Those result are stated and proven for projective resolutions and left derived functors, but our situation can be reduced to that case upon replacing Ab by its opposite category Ab^{op}.)

We now want to present a more concrete formula for describing these higher limit functors. For any functor F: C^{op} ⟶ Ab, let C^{∗}(C; F) denote the chain complex induced by the “bar resolution” for C:

C^{n}(C; F) = ∏_{c_{0}→···→c_{n}} F (c_{0}),

where the product is taken over all composable sequences of n morphisms in C. We regard an element ξ ∈ C^{n}(C; F) as a function which sends each (c_{0} → ⋯ → c_{n}) ∈ N (C)_{n} to an element of F (c_{0}). Coboundary maps

d: C^{n}(C; F) ⟶ C^{n+1}(C; F),

are defined by setting

dξ(c_{0} ⟶^{α} c_{1} → ⋯ → c_{n+1}) = F (α)(ξ(c_{1} → ⋯ → c_{n+1}))

+ ∑_{i=1}^{n+1}(−1)^{i}ξ(c_{0} → ⋯ → c_{i−1} → c_{i+1} → ⋯) ∈ F (c_{0}).

This is easily checked to define a chain complex (C^{∗}(C; F), d). There is also a subcomplex (C^{∗}(C; F), d) of normalized chains, where C^{n}(C, F) is the group of those ξ ∈ C^{n}(C; F) such that ξ(η) = 0 for each η ∈ N (C)_{n} which contains an identity morphism.

When C = B(G) for a group G, this is just the chain complex induced by the classical bar resolution for G (cf. [McL, § IV.5]). So we think of the above complexes as being induced by a “bar resolution” for functors over categories. This bar resolution will be constructed explicitly in the proof of the following proposition.

Proposition 5.3. For any small category C and any functor F: C^{op} → Ab,

\underleftarrow{lim}_{C}^{i}(F) ≅ H^{i}(C^{∗}(C; F), d) ≅ H^{i}(C^{∗}(C; F), d).

Proof. The first isomorphism is shown in [GZ, Appendix II, Proposition 3.3], and the second follows upon modifying the proof of the first. They also follow from [BK, Proposition XI.6.2]. A different, more explicit proof is given in [O1, Lemma 2]. We sketch that proof here, for the normalized chain complex (C^{∗}(C; F), d).

Let \underline{ℤ}: C^{op} ⟶ Ab be the constant functor which sends each object to ℤ and each morphism to the identity. From the above definition of inverse limits, it follows immediately that for any F,

\underleftarrow{lim}_{C}(F) ≅ Hom_{C-mod} (\underline{ℤ}, F) .

We can thus identify the derived functors of \underleftarrow{lim}(F) with the derived functors of Hom_{C-mod} (\underline{ℤ}, F); i.e., with the groups Ext_{C-mod}^{i} (\underline{ℤ}, F). As usual, these groups can be defined not only via a C-mod-injective resolution of F, but also using a C-mod-projective resolution of \underline{ℤ} (cf. [Wei, Theorem 2.7.6]).

To construct a projective resolution of \underline{ℤ}, define functors

C_{n}(C): C^{op} ⟶ Ab

(for each n ≥ 0) as follows. For c ∈ Ob(C), let ch_{n}(C, c) be the set of all chains c → c_{0} → ⋯ → c_{n} of morphisms in C. Let ch_{n}^{0}(C, c) ⊆ ch_{n}(C, c) be the subset of those chains which are “degenerate”: those where at least one of the morphisms c_{i} → c_{i+1} (for 0 ≤ i ≤ n − 1) is an identity morphism. Let C_{n}(C)(c) be the free abelian group with basis ch_{n}(C, c), and set C_{n}(C)(c) = C_{n}(C)(c)/⟨ch_{n}^{0}(C, c)⟩. A morphism c → d in C induces a map from ch_{n}(d) to ch_{n}(c) by composition with the first map in the chain, and hence homomorphisms from C_{n}(C)(d) to C_{n}(C)(c) and from C_{n}(C)(d) to C_{n}(C)(c).

For each n and each F: C^{op} ⟶ Ab, there is a natural isomorphism

Hom_{C-mod} (C_{n}(C), F) ≅ C^{n}(C; F) .

Since C^{n}(C; −) sends surjections in C-mod to surjections of groups, this shows that C_{n}(C) is projective in C-mod.

For each n and c, define ∂: C_{n}(C)(c) ⟶ C_{n−1}(C)(c) by setting

∂(c → c_{0} → ⋯ c_{n}) = ∑_{k=0}^{n}(−1)^{k}(c → c_{0} → ⋯ ĉ_{k} ⋯ → c_{n}) .

Here, ĉ_{k} means that that term is removed from the chain. It is straightforward to check that these are well defined, and define a chain complex

⋯ ⟶^{∂} C_{2}(C) ⟶^{∂} C_{1}(C) ⟶^{∂} C_{0}(C) ⟶^{ε} \underline{ℤ} ⟶ 0 (1)

of functors on C, where \underline{ℤ} denotes the constant functor with value ℤ, and ε(c → c_{0}) = 1 for each basis element (c → c_{0}) in C_{0}(C)(c) = C_{0}(C)(c). Furthermore, for each c ∈ Ob(C), the functions T_{n}: C_{n}(C)(c) ⟶ C_{n+1}(C)(c)

which send a chain (c → c_{0} → ⋯ → c_{n}) to (c ⟶^{Id} c → c_{0} → ⋯ → c_{n})

(or send 1 ∈ ℤ to (c ⟶^{Id} c)) define a splitting of the chain complex (C_{∗}(C)(c), ∂) for each c ∈ Ob(C). Thus (1) is a projective resolution of \underline{ℤ}, (Hom_{C-mod} (C_{∗}(C), F), ∂^{∗}) ≅ (C^{∗}(C; F), d), and its homology groups are the higher derived functors of Hom_{C-mod} (\underline{ℤ}, F) ≅ \underleftarrow{lim}_{C}(F). □

The following is an immediate consequence of Proposition 5.3, together with the description in Section 2.1 of the cohomology of the realization of a simplicial set.

Corollary 5.4. Fix a small category C and an abelian group A, and let \underline{A}: C^{op} ⟶ Ab be the constant functor: \underline{A} sends each object to A and sends each morphism to Id_{A}. Then H^{∗}(|C|; A) ≅ \underleftarrow{lim}_{C}^{∗}(\underline{A}).

When C is a finite category, the bar resolution provides an easy way to prove finite generation of higher limits of certain functors on C.

Proposition 5.5. Assume C has finitely many objects and finitely many morphisms. Then the following hold for any commutative noetherian ring R and any functor F: C^{op} → R-mod.

(a) If F (c) is finitely generated as an R-module for each c ∈ Ob(C), then \underleftarrow{lim}^{i}(F) is finitely generated as an R-module for each i.

(b) In general, for each i, \underleftarrow{lim}^{i}(F) is the direct limit of the groups \underleftarrow{lim}^{i}(F_{0}), taken over all subfunctors F_{0} ⊆ F for which F_{0}(c) is finitely generated for each c.

Proof. For convenience, we say F is finitely generated if F (c) is finitely generated for each c. When F is finitely generated, then C^{i}(C; F) is finitely generated for each i since the set N (C)_{i} is finite. Since R is noetherian, this implies the cohomology groups \underleftarrow{lim}^{i}(F) are all finitely generated.

Now let F be arbitrary. For each c ∈ Ob(C) and each x ∈ F (c), let F_{x} ⊆ F be the subfunctor where F_{x}(d) = ⟨φ^{∗}(x) | φ ∈ Mor_{C}(d, c)⟩ for each d ∈ Ob(C). This is a finitely generated subfunctor, and x ∈ F_{x}(c). Thus F is the direct limit (or union) of its finitely generated subfunctors, so C^{∗}(C; F) is the direct limit of the subcomplexes C^{∗}(C; F_{0}) for F_{0} ⊆ F finitely generated. Since (filtered) direct limits preserve exact sequences, we conclude that for each i, \underleftarrow{lim}^{i}(F) is the direct limit of the groups \underleftarrow{lim}^{i}(F_{0}) for F_{0} ⊆ F finitely generated. □

When C and D are small categories, and Φ: C → D and F: D^{op} → Ab are functors, composition with Φ defines a homomorphism from C^{∗}(D; F) to C^{∗}(C; F ∘ Φ^{op}), and hence a homomorphism

Φ^{#} : \underleftarrow{lim}_{D}^{∗}(F) ⟶ \underleftarrow{lim}_{C}^{∗}(F ∘ Φ^{op}) .

Of course, this can also be defined directly using injective resolutions of these functors. By the analogy with cohomology of spaces, it is natural to expect this to be invariant under natural isomorphisms of functors, interpreted correctly. This is true, but we will not prove it here. Instead, we just note the following special case.

Proposition 5.6. Let Φ: C ⟶ D be an equivalence of small categories. Then for any functor F: D^{op} ⟶ Ab, the induced homomorphism

Φ^{#} : \underleftarrow{lim}_{D}^{∗}(F) ⟶^{≅} \underleftarrow{lim}_{C}^{∗}(F ∘ Φ^{op})

is an isomorphism. In particular, restricting a functor to an equivalent subcategory doesn’t change its higher limits.

Proof. Since Φ is an equivalence of categories, one easily checks that for any injective functor I: C^{op} ⟶ Ab, the composite I ∘ Φ is also injective. Hence for any injective resolution

0 ⟶ F ⟶ I_{0} ⟶ I_{1} ⟶ I_{2} ⟶ ⋯

of F, composition with Φ defines an injective resolution

0 ⟶ F ∘ Φ ⟶ I_{0} ∘ Φ ⟶ I_{1} ∘ Φ ⟶ I_{2} ∘ Φ ⟶ ⋯

of F ∘ Φ. Also,

\underleftarrow{lim}_{D}(F) ≅ \underleftarrow{lim}_{C}(F ∘ Φ)  and  \underleftarrow{lim}_{D}(I_{k}) ≅ \underleftarrow{lim}_{C}(I_{k} ∘ Φ) ,

and the proposition follows immediately. □

As usual when working with cohomology, we write Z^{n}(C; F) to denote the group of normalized n-cocycles; i.e., the kernel of the coboundary map from C^{n}(C; F) to C^{n+1}(C; F). Similarly, B^{n}(C; F) denotes the group of normalized n-coboundaries; i.e., the image of the coboundary map from C^{n−1}(C; F) to C^{n}(C; F). Thus

\underleftarrow{lim}_{C}^{n}(F) ≅ Z^{n}(C; F)/B^{n}(C; F) .

5.2. Constrained fusion systems.

Recall that a saturated fusion system F over a p-group S is constrained if it contains a normal centric p-subgroup; i.e., if O_{p}(F) ∈ F^{c}. In this section, we prove the “model theorem” for constrained fusion systems (Theorem I.4.9): if F is a constrained fusion system over S, and Q ⊴ F is normal and centric, then there is a finite group G, unique up to isomorphism, such that S ∈ Syl_{p}(G), Q ⊴ G, C_{G}(Q) ≤ Q, and F ≅ F_{S}(G). The existence of such G, and a weak form of the uniqueness statement, were proven in [5a1, Proposition 4.3], by first constructing a linking system L associated to F, using the obstruction theory described in Section 5.3, and then showing that we can set G = Aut_{L}(Q). We give a more direct proof of this result here; one which does not involve linking systems.

The first lemma describes the connection between certain group automorphisms and cohomology. In its proof, and also in the proof of Proposition 5.8 which follows, we describe elements in group cohomology in terms of the classical bar resolution.

When G is a group and M is a ℤ[G]-module, we can regard M as a functor from B(G) to abelian groups, and H^{n}(G; M) is just the n-th derived functor of the inverse limit of M. So we can describe elements in the cohomology using the bar resolution of Section 5.1. Thus H^{∗}(G; M) is the homology of the cochain complex (C^{∗}(G; M), d), where C^{n}(G; M) is the group of all θ: G^{n} ⟶ M such that θ(g_{1}, …, g_{n}) = 0 if g_{i} = 1 for some i. However, we use here the more traditional definition of the boundary map (cf. [Br, p. 59] or [McL, p. 116]): for ξ ∈ C^{n}(G; M),

dξ(g_{1}, …, g_{n+1}) = g_{1}(ξ(g_{2}, …, g_{n+1}))

+ ∑_{i=1}^{n}(−1)^{i}ξ(g_{1}, …, g_{i}g_{i+1}, …, g_{n+1}) + (−1)^{n+1}ξ(g_{1}, …, g_{n}).

Lemma 5.7. Fix a group G and a normal subgroup H ⊴ G such that C_{G}(H) ≤ H. Set A(G) = {α ∈ Aut(G) | α|_{H} = Id}. Then there is an isomorphism

ψ: A(G) ⟶^{≅} Z^{1}(G/H; Z(H)),

where ψ(α)(gH) = α(g)g^{−1} ∈ Z(H) for all g ∈ G and α ∈ A(G). Also, B^{1}(G/H; Z(H)) = ψ(Aut_{Z(H)}(G)), and thus

A/Aut_{Z(H)}(G) ≅ H^{1}(G/H; Z(H)).

Proof. Fix α ∈ A(G). For each g ∈ G and each x ∈ H, ^{α(g)}x = α(^{g}x) = ^{g}x, so α(g) ≡ g (mod C_{G}(H) = Z(H)), and α(g)g^{−1} ∈ Z(H). Also, α(e) = e, and α(g)g^{−1} = α(g′)(g′)^{−1} if g ≡ g′ (mod H). Thus ψ(α) as defined above is a normalized 1-cochain in C^{1}(G/H; Z(H)). For all g, h ∈ G,

ψ(α)(gh) = α(gh)(gh)^{−1} = α(g)ψ(α)(h)g^{−1} = ψ(α)(g)·gψ(α)(h)g^{−1} ,

and so ψ(α) ∈ Z^{1}(G/H; Z(H)).

Conversely, for any normalized cochain ξ ∈ Z^{1}(G/H; Z(H)), one easily checks that the coboundary condition implies that the map α: G ⟶ G, defined by setting α(g) = ξ(gH)·g, is an automorphism. This proves that ψ is a bijection; and this bijection is easily checked to be an isomorphism of groups.

The last statement follows since for each a ∈ Z(H), regarded as a 0chain (as an element in C^{0}(G/H; Z(H))), ψ^{−1}(da) ∈ A(G) sends g ∈ G to (^{g}a·a^{−1})g = (a^{−1}·^{g}a)g = ^{a^{−1}}g, and thus ψ^{−1}(da) = c_{a}^{−1} ∈ Aut_{Z(H)}(G). □

In the following proposition, we collect those results involving models for constrained fusion systems which can be proven using group cohomology. We are not yet claiming that the group G constructed there is a model for the fusion system F.

Recall, for any fusion system F over a p-group S, that Aut(S, F) denotes the group of fusion preserving automorphisms of S: the group of all α ∈ Aut(S) such that ^{α}F = F.

Proposition 5.8. Let F be a constrained saturated fusion system over a p-group S. Fix Q ∈ F^{c} such that Q ⊴ F.

(a) There is a finite group G containing S as a Sylow p-subgroup such that Q ⊴ G, C_{G}(Q) ≤ Q, and Aut_{G}(Q) = Aut_{F}(Q).

(b) If G_{1} and G_{2} are two groups as in (a), then there is an isomorphism ψ: G_{1} ⟶^{≅} G_{2} such that ψ|_{Q} = Id_{Q}.

(c) If G is as in (a), then for any α ∈ Aut(S, F_{S}(G)) such that α|_{Q} = Id_{Q}, there is β ∈ Aut(G) such that β|_{S} = α. If β, β′ ∈ Aut(G) are two extensions of α, then there is z ∈ Z(S) such that β′ = β ∘ c_{z}.

Proof. Set G = Out_{F}(Q). In general, for subgroups P ≤ S containing Q, we write P = Out_{P} (Q) ≤ G; thus P ≅ P/Q. Also, S ∈ Syl_{p}(G). We regard Z(Q) as a ℤ[G]-module.

Consider, for n > 0, the homomorphism

Rs_{G,S}: H^{n}(G; Z(Q)) ⟶ H^{n}(S; Z(Q))

induced by restriction. By [CE, Theorem XII.10.1] or [Br, Theorem III.10.3], (i) Rs_{G,S} is injective; and

(ii) Im(Rs_{G,S}) is the subgroup of “stable elements” in H^{n}(S; Z(Q)): the

group of all ξ ∈ H^{n}(S; Z(Q)) such that for each P , R ≤ S and each x ∈ G such that ^{x}P = R, c_{x}(Rs_{S,P} (ξ)) = Rs_{S,R}(ξ).

Here, when ^{x}P = R, c_{x}: H^{n}(P; Z(Q)) ⟶ H^{n}(R; Z(Q)) sends the class of a cocycle θ: P^{n} ⟶ Z(Q) to the class of the cocycle c_{x}(θ), where

c_{x}(θ)(g_{1}, …, g_{n}) = ^{x}(θ(g_{1}^{x}, …, g_{n}^{x}))

(see [CE, § XII.8] or [Br, § III.8]).

(a) Let 𝔛 be the set of all isomorphism classes of extensions

1 ⟶ Q ⟶ G ⟶ G ⟶ 1

where the conjugation action of G = Out_{F}(Q) on Q is the canonical one. For each T ≤ S containing Q, let 𝔛_{T} be the set of isomorphism classes of extensions of Q by T = Out_{T} (Q), and let [T] ∈ 𝔛_{T} be the class of the extension T. There are obvious maps R_{T} : 𝔛 ⟶ 𝔛_{T} induced by restriction. We must show there is [G] ∈ 𝔛 such that R_{S}([G]) = [S].

By [McL, Theorem IV.8.7], the obstruction to 𝔛 being nonempty lies in H^{3}(G; Z(Q)). This obstruction vanishes by (i), since its restriction to H^{3}(S; Z(Q)) vanishes ([S] ∈ 𝔛_{S} ≠ ∅). Fix an extension G^{∗} of Q by G such that [G^{∗}] ∈ 𝔛. For each Q ≤ T ≤ S, let T^{∗} ≤ G^{∗} be the subextension of Q by T. Thus [T^{∗}] = R_{T} ([G^{∗}]) ∈ 𝔛_{T}.

For P, R ≤ S containing Q, each γ ∈ T_{G}(P , R) (the transporter set) induces a map γ^{∗}: 𝔛_{R} ⟶ 𝔛_{P} by taking a pullback over c_{γ}: P ⟶ R and twisting by γ ∈ G = Out_{F}(Q). This is uniquely defined, since twisting by an element of Inn(Q) (and the identity on P) induces the identity on 𝔛_{P}. If γ = c_{g} ∈ Out_{G^{∗}}(Q) for g ∈ G^{∗}, then c_{g} ∈ Hom(P^{∗}, R^{∗}) induces a monomorphism of extensions which implies φ^{∗}([R^{∗}]) = [P^{∗}]. Since Q is receptive in F, any γ̃ ∈ Aut_{F}(Q) in the class of γ ∈ Out_{F}(Q) extends to some γ ∈ Hom_{F}(P, R), and this monomorphism of extensions shows that γ^{∗}([R]) = [P].

By [McL, Theorem IV.8.8], there is a natural action of H^{2}(G; Z(Q)) on 𝔛 which is free and transitive. Similarly, for each Q ≤ P ≤ S, the group H^{2}(P; Z(Q)) acts freely and transitively on 𝔛_{P}. Let ξ_{S} ∈ H^{2}(S; Z(Q)) be such that ξ_{S}([S^{∗}]) = [S]. Since the elements [P^{∗}] ∈ 𝔛_{P} are stable under fusion in F_{S}(G), and similarly for the elements [P] ∈ 𝔛_{P}, ξ_{S} is also stable under such fusion. Hence by (ii) above, ξ_{S} is the restriction of an element ξ ∈ H^{2}(G; Z(Q)). Set [G] = ξ([G^{∗}]) ∈ 𝔛; then R_{S}([G]) = [S].

(b) Now let G_{1} and G_{2} be two groups such that S ∈ Syl_{p}(G_{i}), Q ⊴ G_{i}, C_{G_{i}}(Q) ≤ Q, and Aut_{G_{i}}(Q) = Aut_{F}(Q) for i = 1, 2. Then [G_{1}], [G_{2}] ∈ 𝔛, and R_{S}([G_{i}]) = [S]. Since the restriction map from H^{2}(G; Z(Q)) to H^{2}(S; Z(Q)) is injective by (i) again, R_{S} is also injective, and so [G_{1}] = [G_{2}]. Thus there is an isomorphism ψ: G_{1} ⟶^{≅} G_{2} such that ψ|_{Q} = Id_{Q}.

(c) Fix G as in (a), and identify G = G/Q. Set g = gQ ∈ G/Q for all g ∈ G.

Fix α ∈ Aut(S, F_{S}(G)) such that α|_{Q} = Id_{Q}. Let θ ∈ Z^{1}(S; Z(Q)) be the 1-cocycle of Lemma 5.7: θ(g) = α(g)g^{−1} for all g ∈ S. We must show that θ extends to a 1-cocycle on G. By (ii), it suffices to show that [θ] ∈ H^{1}(S; Z(Q)) is a stable element in the sense defined above.

Fix x ∈ G and P = P/Q ≤ S such that ^{x}P ≤ S. Thus x = c_{x}|_{Q} for some x ∈ G such that ^{x}P ≤ S and hence c_{x} ∈ Hom_{F_{S}(G)}(P, S). Since α(P) = P, α(S) = S, and α ∈ Aut(S, F_{S}(G)) (α is fusion preserving), there is y ∈ G such that α ∘ c_{x} ∘ (α|_{P})^{−1} = c_{y} as homomorphisms from P to S. Since α|_{Q} = Id_{Q}, this means that c_{x}|_{Q} = c_{y}|_{Q}, and thus that x^{−1}y ∈ C_{G}(Q) = Z(Q). Set z = x^{−1}y, so y = xz.

For all g ∈ P,

α(xgx^{−1}) = yα(g)y^{−1} ⟹ θ(xgx^{−1})·xgx^{−1} = xz(θ(g)g)z^{−1}x^{−1}; and hence (since [z, θ(g)] ∈ [Z(Q), Z(Q)] = 1)

x^{−1}θ(^{x}g) = zθ(g)gz^{−1}g^{−1} = θ(g)·zgz^{−1}g^{−1} = θ(g)·(dz(g))^{−1} .

Here, we regard z ∈ Z(Q) as a 0-cochain, so that dz ∈ B^{1}(P; Z(Q)) (and dz(g) = ^{g}z·z^{−1}). Hence if we define θ_{x} ∈ Z^{1}(P; Z(Q)) by setting θ_{x}(g) = x^{−1}θ(^{x}g), then [θ_{x}] = [θ|_{P}] in H^{1}(P; Z(Q)).

By definition, the conjugation homomorphism

c_{x}: H^{1}(P; Z(Q)) ⟶ H^{1}(^{x}P; Z(Q))

sends the class of θ_{x} as defined above to the class of θ|_{^{x}P}. So what we have shown is that for each P ≤ S containing Q and each x ∈ G with ^{x}P ≤ S, c_{x}([θ|_{P}]) = [θ|_{^{x}P}]. In other words, [θ] ∈ H^{1}(S; Z(Q)) is a stable element with respect to the inclusion S ≤ G, and so by (ii), it is the restriction of an element [η] ∈ H^{1}(G; Z(Q)). Since C^{0}(G; Z(Q)) = C^{0}(S; Z(Q)) = Z(Q), this means that there is η ∈ Z^{1}(G; Z(Q)) such that η|_{S} = θ.

Define β: G ⟶ G by setting β(g) = η(g)g. Then β ∈ Aut(G) by Lemma 5.7, and β|_{S} = α.

Assume β′ ∈ Aut(G) is such that β′|_{S} = β|_{S}. Set η′ = ψ(β′) in the notation of Lemma 5.7; thus [η], [η′] ∈ H^{1}(G; Z(Q)) and η|_{S} = η′|_{S}. So [η] = [η′] by (i) again, and η′ = η·dz for some z ∈ Z(Q). In other words, for each g ∈ G, η′(g) = η(g)gzg^{−1}z^{−1}, so

β′(g) = η′(g)g = η(g)·gzg^{−1}·z^{−1}·g = η(g)·z^{−1}·gzg^{−1}·g = β(z^{−1}gz).

Thus β′ = β ∘ c_{z}^{−1} for some z ∈ Z(Q), and z ∈ Z(S) since β′|_{S} = β|_{S}. □

The proof of the model theorem given here, like the proof in [5a1], depends on the vanishing of a certain group of higher limits over the orbit category of Out_{F}(Q). This vanishing result follows as a special case of a much more general theorem of Jackowski and McClure.

Lemma 5.9. Fix a finite group G, a Sylow subgroup S ∈ Syl_{p}(G), and a ℤ[G]-module M. For i ≥ 0, let H^{i}M: O_{S}(G)^{op} ⟶ Ab be the functor (H^{i}M)(P) = H^{i}(P; M), where a morphism [g] ∈ Q\T_{G}(P, Q) in O_{S}(G) induces a homomorphism between the cohomology groups via composition with c_{g} ∈ Hom(P, Q) and the action of g^{−1} on M. Then \underleftarrow{lim}^{k}_{O_{S}(G)}(H^{i}M) = 0

for all k ≥ 1.

Proof. Via the transfer homomorphism for group cohomology (see [CE, § XII.8] or [Br, § III.9]), H^{i}M becomes a Mackey functor in the sense of Dress [Dr] and Jackowski and McClure [JM]. Hence \underleftarrow{lim}^{k}(H^{i}M) = 0 for k > 0 by [JM, Proposition 5.14], which says that higher limits vanish for all Mackey functors over O_{S}(G). □

We are now ready to prove Theorem I.4.9: all constrained fusion systems have models, which are unique in a very strong sense. What makes this difficult is that when F, S, Q, and G are as in Proposition 5.8(a), then F need not be equal to F_{S}(G) as fusion systems over S. Instead, as we will see, there is always β ∈ Aut(S) such that β|_{Q} = Id_{Q} and F_{S}(G) = ^{β}F. In other words, F and F_{S}(G) are always isomorphic, but need not be equal. When choosing the automorphism β as just described, we must take all morphisms in the fusion system into account, and this is why Lemma 5.9 is needed.

Theorem 5.10 (Model theorem for constrained fusion systems). Let F be a constrained, saturated fusion system over a p-group S. Fix Q ∈ F^{c} such that Q ⊴ F. Then the following hold.

(a) There is a model for F: a finite group G with S ∈ Syl_{p}(G) such that Q ⊴ G, C_{G}(Q) ≤ Q, and F_{S}(G) = F.

(b) For any finite group G such that S ∈ Syl_{p}(G), Q ⊴ G, C_{G}(Q) ≤ Q, and Aut_{G}(Q) = Aut_{F}(Q), there is β ∈ Aut(S) such that β|_{Q} = Id_{Q} and F_{S}(G) = ^{β}F.

(c) The model G is unique in the following strong sense: if G_{1}, G_{2} are two finite groups such that S ∈ Syl_{p}(G_{i}), Q ⊴ G_{i}, F_{S}(G_{i}) = F, and

C_{G_{i}}(Q) ≤ Q for i = 1, 2, then there is an isomorphism ψ: G_{1} ⟶^{≅} G_{2} such that ψ|_{S} = Id_{S}. If ψ and ψ′ are two such isomorphisms, then ψ′ = ψ ∘ c_{z} for some z ∈ Z(S).

Proof. Points (a) and (b), and a weaker version of (c), were shown in [5a1, Proposition C]. We give a different proof here.

(a,b) Let G be as in Proposition 5.8(a): S ∈ Syl_{p}(G), Q ⊴ G, C_{G}(Q) ≤ Q, and Aut_{G}(Q) = Aut_{F}(Q). We must show that F_{S}(G) = ^{β}F for some β ∈ Aut(S) such that β|_{Q} = Id_{Q}. Set F_{1} = F and F_{2} = F_{S}(G).

Let O = O_{S}(G) be the orbit category of the group G with objects the subgroups of S. Thus Mor_{O}(P , R) = R\T_{G}(P , R). Consider the functor

F: O^{op} ⟶ p-Gps

which sends P to H^{1}(P; Z(Q)). A morphism φ = [γ] ∈ Mor_{O}(P , R) is sent to the homomorphism γ^{∗} from H^{1}(R; Z(Q)) to H^{1}(P; Z(Q)) induced by c_{γ} ∈ Hom(P , R) and by (γ|_{Z(Q)})^{−1}.

For each P ≤ S containing Q, set A(P) = {α ∈ Aut(P) | α|_{Q} = Id}. For each P ≤ S, consider the isomorphism

ψ_{P} : A(P)/Aut_{Z(Q)}(P) ⟶^{≅} H^{1}(P; Z(Q)) = F (P)

of Lemma 5.7, which sends the class of α ∈ A(P) to the class of the 1cocycle ψ̂_{P} (α) defined by ψ̂_{P} (α)(g) = α(g)·g^{−1}. Here, for g ∈ P, g = [c_{g}|_{Q}] ∈ P = Out_{P} (Q).

Fix P, R ≤ S containing Q and γ ∈ T_{G}(P , R), and choose γ̃ ∈ Aut_{F}(Q) in the class of γ ∈ G = Out_{F}(Q). Since Q is receptive in F_{1} and F_{2}, γ̃ extends to morphisms γ_{i} ∈ Hom_{F_{i}}(P, R) (i = 1, 2). We claim that

γ^{∗} ∘ ψ_{R} = ψ_{P} ∘ c_{γ_{i}}^{−1} : A(R)/Aut_{Z(Q)}(R) ⟶ H^{1}(P; Z(Q)) (2)

for i = 1, 2. Here, c_{γ_{i}}^{−1} sends the class of α ∈ A(R) to the class of γ_{i}^{−1}αγ_{i}|_{P} ∈ A(P), which is defined since α ≡ Id (mod Q). For g ∈ P, since g = [c_{g}|_{Q}] ∈ P, we have γ_{i}(g) = [c_{γ_{i}(g)}|_{Q}] = γ[c_{g}|_{Q}]γ^{−1} = c_{γ}(g) ∈ R. Thus for each α ∈ A(R) and each g ∈ P,

(γ^{∗}(ψ̂_{R}(α)))(g) = γ^{−1}(ψ̂_{R}(α)(c_{γ}(g))) = γ^{−1}(ψ̂_{R}(α)(γ_{i}(g)))
= γ^{−1}(α(γ_{i}(g))·γ_{i}(g)^{−1}) = γ_{i}^{−1}(α(γ_{i}(g)))·g^{−1}
= c_{γ_{i}}^{−1}(α)(g)·g^{−1} = ψ̂_{P} (c_{γ_{i}}^{−1}(α))(g) ,

and this proves (2).

Again fix φ = [γ] ∈ Mor_{O}(P , R), choose γ̃ ∈ Aut_{F}(Q) in the class of γ ∈ G = Out_{F}(Q), and let γ_{i} ∈ Hom_{F_{i}}(P, R) (i = 1, 2) be extensions of γ̃. Then Im(γ_{1}) = Im(γ_{2}), so there is some θ_{γ} ∈ A(P) such that γ_{2} = γ_{1} ∘ θ_{γ}. By Lemma I.5.6 or 4.6(c), for a given choice of γ̃, the extensions γ_{1} and γ_{2} (and hence θ_{γ}) are unique modulo Aut_{Z(Q)}(P). Also, if γ̃ is replaced by c_{h} ∘ γ̃ for some h ∈ R, then each γ_{i} can be replaced by c_{h} ∘ γ_{i} without changing θ_{γ}. Thus [θ_{γ}] ∈ A(P)/Aut_{Z(Q)}(P) depends only on the morphism φ in the orbit category, and not on γ or γ̃ itself.

Now set t(φ) = ψ_{P} ([θ_{γ}]) ∈ F (P) for φ = [γ] ∈ Mor_{O}(P , R) as above. This defines an element t ∈ C^{1}(O; F). We claim t is a cocycle. To see this, fix a pair of morphisms

P ⟶^{φ=[γ]} R ⟶^{ψ=[ν]} T

in O, choose representatives γ̃, ν̃ ∈ Aut_{F}(Q) of γ, ν ∈ G = Out_{F}(Q), and choose extensions γ_{i} ∈ Hom_{F_{i}}(P, R) of γ̃ and ν_{i} ∈ Hom_{F_{i}}(R, T) of ν̃. Let θ_{γ}, θ_{νγ} ∈ A(P) and θ_{ν} ∈ A(Q) be such that γ_{2} = γ_{1} ∘ θ_{γ}, ν_{2} = ν_{1} ∘ θ_{ν}, and ν_{2}γ_{2} = ν_{1}γ_{1}θ_{νγ}. Thus θ_{ν} ∘ γ_{1} ∘ θ_{γ} = γ_{1} ∘ θ_{νγ}, since all of these are injective group homomorphisms, and hence

t(ψφ) = ψ_{P} ([θ_{νγ}]) = ψ_{P} ([c_{γ_{1}}^{−1}(θ_{ν}) ∘ θ_{γ}])
= γ^{∗}(ψ_{R}([θ_{ν}]))·ψ_{P} ([θ_{γ}]) = γ^{∗}(t(ψ))·t(φ) .

Thus dt(φ, ψ) = 1 for each pair (φ, ψ), and so t ∈ Z^{1}(O; F).

Now, \underleftarrow{lim}^{1}(F) = 0 by Lemma 5.9. Hence there is u ∈ C^{0}(O; F) such that t = du. In other words, for each φ = [γ] ∈ Mor_{O}(P , R),

t(φ) = γ^{∗}(u(R))·u(P)^{−1} ∈ H^{1}(P; Z(Q)) .  (3)

Since all inclusions into S are in both categories F and F_{S}(G), t(incl_{P}^{S}) = 0 for each P ≤ S, and so u(P) is the restriction of u(S). Thus u is determined by u(S) ∈ H^{1}(S; Z(Q)).

Choose any β ∈ A(S) such that ψ_{S}([β]) = u(S). For each morphism φ = [γ] ∈ Mor_{O}(P , R), (3) and (2) imply

ψ_{P} ([θ_{γ}]) = t(φ) = γ^{∗}(u(R))·u(P)^{−1}

= γ^{∗}(ψ_{R}([β|_{R}]))·(ψ_{P} ([β|_{P}]))^{−1} = ψ_{P} ([c_{γ_{1}}^{−1}(β|_{R}) ∘ (β|_{P})^{−1}])

and hence θ_{γ} ≡ (γ_{1}^{−1}βγ_{1})|_{P} ∘ β^{−1}|_{P} (mod Aut_{Z(Q)}(P)). Thus

γ_{2} = γ_{1}θ_{γ} ≡ (β|_{R})γ_{1}(β|_{P})^{−1} (mod Aut_{Z(Q)}(R)),

where γ_{i} ∈ Hom_{F_{i}}(P, R) are extensions defined as before.

This proves that F_{2} and ^{β}F_{1} have the same morphisms between subgroups of S which contain Q. By Proposition I.4.5, for each i = 1, 2, all subgroups which are F_{i}-essential contain Q. Hence F_{2} = ^{β}F_{1} by Alperin’s fusion theorem (Theorem I.3.5).

(c) Assume G_{1} and G_{2} are two models for F. By Proposition 5.8(b), there is an isomorphism ψ_{0} : G_{1} ⟶^{≅} G_{2} such that ψ_{0}|_{Q} = Id_{Q}. We must find an isomorphism which is the identity on S.

Set α = ψ_{0}|_{S} ∈ Aut(S). Thus α|_{Q} = Id_{Q}. Also, αc_{x}α^{−1} = c_{α(x)} for x ∈ G_{1}, and so ^{α}F = ^{α}(F_{S}(G_{1})) = F_{S}(G_{2}) = F. Thus α ∈ Aut(S, F), and by Proposition 5.8(c), α extends to an automorphism β ∈ Aut(G_{1}). Set ψ = ψ_{0} ∘ β^{−1}; then ψ ∈ Iso(G_{1}, G_{2}) and ψ|_{S} = Id_{S}.

If ψ′ ∈ Iso(G_{1}, G_{2}) is another isomorphism such that ψ′|_{S} = Id_{S}, then ψ^{−1}ψ′ ∈ Aut_{Z(S)}(G_{1}) by the last statement in Proposition 5.8(c), and so ψ′ = ψ ∘ c_{z} for some z ∈ Z(S). □

5.3. Existence, uniqueness, and automorphisms of linking systems.

Consider the functor

Z_{F}: O(F^{c})^{op} ⟶ Ab,

defined for any fusion system F by setting Z_{F}(P) = Z(P) and

Z_{F}(P ⟶^{φ} Q) = (Z(Q) ⟶^{incl} Z(φ(P)) ⟶^{φ^{−1}} Z(P)).

(Note that Z(Q) ≤ Z(φ(P)) since Q ∈ F^{c}.) The higher derived functors of the inverse limit of Z_{F} turn out to be obstruction groups to various problems involving fusion and linking systems.

Two linking systems L_{1} and L_{2} associated to the same fusion system F over S, with Ob(L_{1}) = Ob(L_{2}) = H, are isomorphic if there is an isomorphism of categories from L_{1} to L_{2} which commutes with the structural functors to F and from T_{H}(S). By Lemma 4.24, applied with α = Id_{S} and α_{F} = Id_{F}, this is equivalent to the definition used in [BLO2] for an isomorphism of centric linking systems associated to F.

Proposition 5.11 ([BLO2, Proposition 3.1]). Fix a saturated fusion system F over the p-group S. The obstruction to the existence of a centric linking system associated to F lies in \underleftarrow{lim}^{3}_{O(F^{c})}(Z_{F}). The group \underleftarrow{lim}^{2}_{O(F^{c})}(Z_{F})

acts freely and transitively on the set of isomorphism classes of all centric linking systems if the set is nonempty.

Brief sketch of proof. Assume first L_{1} and L_{2} are two centric linking systems associated to F, with structural functors δ_{i} and π_{i} (i = 1, 2). Choose maps

Φ_{P,Q}: Mor_{L_{1}}(P, Q) ⟶ Mor_{L_{2}}(P, Q) ,

for each pair of objects P, Q, which commute with the projections π_{1} and π_{2} to Hom_{F}(P, Q). By first defining them on orbit representatives for the free actions of Q (defined via δ_{i,Q} and left composition), this can be done so that Φ_{P,Q} is Q-equivariant for each P and Q. Thus Φ_{P,Q}(δ_{1,Q}(g) ∘ φ) = δ_{2,Q}(g) ∘ Φ_{P,Q}(φ) for all g ∈ Q and φ ∈ Mor_{L_{1}}(P, Q).

For each composable pair of morphisms P ⟶^{φ} Q ⟶^{ψ} R in L_{1}, Φ_{P,R}(ψ ∘φ) differs from Φ_{Q,R}(ψ) ∘ Φ_{P,Q}(φ) by δ_{2,P} (t(φ, ψ)) for some unique t(φ, ψ) ∈ Z(P) (axiom (A2)). This depends only on the classes of φ and ψ in O(F^{c}); and together the elements t(φ, ψ) define a normalized 2-cocycle t ∈ Z^{2}(O(F^{c}); Z_{F}). If t = du for some 1-cochain u ∈ C^{1}(O(F^{c}); Z_{F}), then if we set Φ′_{P,Q}(φ) = Φ_{P,Q}(φ) ∘ δ_{P} (u([φ])) for each φ ∈ Mor_{L_{1}}(P, Q), we obtain a new set of maps {Φ′_{P,Q}} which commute with composition, and hence define an isomorphism of categories Φ′ from L_{1} to L_{2}.

Thus the choice of maps Φ_{P,Q} determines an element [t] ∈ \underleftarrow{lim}^{2}(Z_{F}), and L_{1} ≅ L_{2} if [t] = 0. If {Ψ_{P,Q}}_{P,Q∈Ob(L)} is another choice of maps, defining a 2-cocycle u, one can show that t − u ∈ B^{2}(O(F^{c}); Z_{F}), and hence that [u] = [t]. In other words, there is a unique element of \underleftarrow{lim}^{2}(Z_{F}) which measures the “difference” between any pair of linking systems associated to F. More precisely, the group \underleftarrow{lim}^{2}(Z_{F}) acts freely and transitively on the set of isomorphism classes of associated centric linking systems, and the above procedure describes how to construct the unique element that sends L_{1} to L_{2}.

The existence obstruction is constructed in a similar way. One defines a “precategory” L by setting Ob(L) = Ob(F^{c}), and Mor_{L}(P, Q) = Q × Mor_{O(F^{c})}(P, Q) for each pair of objects P and Q. One then defines a projection of each Mor_{L}(P, Q) to Hom_{F}(P, Q), and composition maps for each triple P, Q, R of objects. If this is done with sufficient care (see the proof of [BLO2, Proposition 3.1] for details), then the error in associativity defines a function which sends each composable triple of morphisms with source P to Z(P), and which depends only on the classes of those morphisms in O(F^{c}). In this way, we get a 3-cocycle, whose class in \underleftarrow{lim}^{3}(Z_{F}) vanishes if and only if there exists some linking system associated to F. □

These obstructions are analogous to the obstructions to the existence and uniqueness of group extensions. For any pair of groups G and K, with an outer action χ: G ⟶ Out(K) of G on K, the obstruction to the existence of an extension

1 ⟶ K ⟶ Γ ⟶ G ⟶ 1

for which the conjugation action of G on K equals χ lies in H^{3}(G; Z(K)); while the obstruction to its uniqueness lies in H^{2}(G; Z(K)). More precisely, the group H^{2}(G; Z(K)) acts freely and transitively on the set of isomorphism classes of all extensions if it is nonempty. See, e.g., [McL, Theorems IV.8.7–8] for a detailed description of this theory. By comparison, we think of a linking system L associated to F as an extension

1 ⟶ \{P \}_{P ∈Ob(F^{c})} ⟶ L \underrightarrow{π̃} O(F^{c}) ⟶ 1.

We know of no examples of saturated fusion systems for which either of these obstruction groups is nonvanishing, and it is quite possible that there is a unique linking system associated to each saturated fusion system. This question is discussed in more detail later in the section. But we first look at the (very closely related) obstruction groups which appear when comparing automorphisms of fusion and linking systems.

Fix a saturated fusion system F over a p-group S, and an associated (centric) linking system L. We refer to Section 4.3 for the definitions of outer automorphism groups Out(S, F) and Out_{typ}(L). By Theorem 4.22, Out(|L|_{p}^{∧}) ≅ Out_{typ}(L) for any p-local finite group (S, F, L), and the group Out(|L|_{p}^{∧}) of homotopy classes of self equivalences of the classifying space is thus described combinatorially in terms of automorphisms of the finite category L. For this and other reasons (see, e.g., Section 6.1), Out_{typ}(L) seems to be the most important automorphism group of (S, F, L). However, in practice, the group Out(S, F) of fusion preserving (outer) automorphisms of F is much easier to compute. Hence the importance of being able to compare these two groups.

Recall the formula of Lemma 4.9: for any linking system L associated to F over S,

Out_{typ}(L) ≅ Aut_{typ}^{I}(L)/{c_{γ} | γ ∈ Aut_{L}(S)} .

Here, Aut_{typ}^{I}(L) is the group of isotypical automorphisms of L which send inclusions to inclusions.

Proposition 5.12. For each p-local finite group (S, F, L), there is an exact sequence

1 ⟶ \underleftarrow{lim}^{1}_{O(F^{c})}(Z_{F}) ⟶^{λ_{L}} Out_{typ}(L) ⟶^{μ_{L}} Out(S, F) ⟶^{ω_{L}} \underleftarrow{lim}^{2}_{O(F^{c})}(Z_{F}).

Here, λ_{L} and μ_{L} are homomorphisms, and μ_{L}([α]) = [β] for α ∈ Aut_{typ}^{I}(L) and β ∈ Aut(S, F) such that δ_{S}(β(g)) = α_{S}(δ_{S}(g)) for g ∈ S.

Proof. When (S, F, L) is realized by a finite group G with S ∈ Syl_{p}(G), this was shown in [BL, Theorem 1.6] with Out(BG_{p}^{∧}) instead of Out_{typ}(L_{S}(G)), and in [BLO1, Theorem 6.2] in the above form. The general case (for arbitrary linking and fusion systems) follows by essentially the same proof as that of [BLO1]. Here, we just give part of the proof, enough to illustrate how the bar resolution is used to identify these obstruction groups.

For each φ ∈ Mor(L), we write [φ] = π̃(φ) ∈ Mor(O(F^{c})) for short. A (normalized) 1-cochain t ∈ C^{1}(O(F^{c}); Z_{F}) is a map from Mor(O(F^{c})) to S which sends Mor_{L}(P, Q) to Z(P) for each P, Q ∈ Ob(L), and such that t(Id_{P}) = 1 for each P. Such a 1-cochain is a 1-cocycle (and hence represents an element in \underleftarrow{lim}^{1}(Z_{F})) if for each (P ⟶^{φ} Q ⟶^{ψ} R) in L,

dt(P ⟶^{φ} Q ⟶^{ψ} R) = Z_{F}(φ)(t([ψ]))·t([ψφ])^{−1}·t([φ]) = 1 ∈ Z(P). (4)

Note that t([δ_{P} (g)]) = t(Id_{P}) = 1 for all g ∈ P, since [δ_{P} (g)] = Id_{P} in O(F^{c}).

For each such 1-cocycle t, set λ̂(t) = Ψ_{t}, where Ψ_{t} ∈ Aut_{typ}(L) is the identity on objects, and

Ψ_{t}(φ) = φ ∘ δ_{P} (t([φ])) for all φ ∈ Mor_{L}(P, Q).

Then Ψ_{t} is a functor, since for each sequence (P ⟶^{φ} Q ⟶^{ψ} R) of morphisms in L,

Ψ_{t}(ψ ∘ φ) = ψ ∘ φ ∘ δ_{P} (t([ψφ]))
= ψ ∘ φ ∘ δ_{P} (π(φ)^{−1}(t([ψ]))) ∘ δ_{P} (t([φ]))  (by (4))
= ψ ∘ δ_{Q}(t([ψ])) ∘ φ ∘ δ_{P} (t([φ]))  (by axiom (C))
= Ψ_{t}(ψ) ∘ Ψ_{t}(φ).

In fact, this computation shows that for all t ∈ C^{1}(O(F^{c}); Z_{F}), Ψ_{t} is a functor if and only if dt = 0. Clearly, Ψ_{t} sends δ_{P} (P) to itself for each P ∈ Ob(L), and hence is isotypical.

Let Ẑ^{1}(O(F^{c}); Z_{F}) be the subgroup of all t ∈ Z^{1}(O(F^{c}); Z_{F}) such that t([ι_{P,S}]) = 1 for each P ∈ Ob(L). By definition, Ψ_{t} ∈ Aut_{typ}^{I}(L) if and only if t ∈ Ẑ^{1}(O(F^{c}); Z_{F}). Fix t ∈ Z^{1}(O(F^{c}); Z_{F}), and let u ∈ C^{0}(O(F^{c}); Z_{F}) be the 0-cochain where u(P) = t([ι_{P,S}]) for all P ∈ Ob(L). Then

du([ι_{P,S}]) = u(S)·u(P)^{−1} = t([ι_{P,S}])^{−1},

and hence t + du ∈ Ẑ^{1}(O(F^{c}); Z_{F}). Thus every element in \underleftarrow{lim}^{1}(Z_{F}) is represented by a cocycle in Ẑ^{1}(O(F^{c}); Z_{F}). Also, the same computation shows that for u ∈ C^{0}(O(F^{c}); Z_{F}), du ∈ Ẑ^{1}(O(F^{c}); Z_{F}) if and only if u is a constant cochain; i.e., there is some g ∈ Z(S) such that u(P) = g for all P.

Consider the following sequence

1 ⟶ Ẑ^{1}(O(F^{c}); Z_{F}) \underrightarrow{λ̂} Aut_{typ}^{I}(L) \underrightarrow{μ̂} Aut(S, F) . (5)

Here, μ̂ sends α ∈ Aut_{typ}^{I}(L) to δ_{S}^{−1} ∘ α_{S} ∘ δ_{S} ∈ Aut(S), and μ̂(α) is fusion preserving by Proposition 4.11(b). For each t ∈ Ẑ^{1}(O(F^{c}); Z_{F}) and each g ∈ =S, [δ_{S}(g)] is the identity in O(F^{c}), so t([δ_{S}(g)]) = 1, and Ψ_{t}(δ_{S}(g)) = δ_{S}(g) by definition of Ψ_{t}. In other words, μ̂ ∘ λ̂ = 0. If Ψ ∈ Ker(μ̂), then by Proposition 4.11(b) again, π ∘ Ψ = π where π: L ⟶ F is the canonical functor. By axiom (A2), for each morphism φ ∈ Mor_{L}(P, Q) in L, Ψ(φ) = φ ∘ δ_{P} (t(φ)) for some unique t(φ) ∈ Z(P), this depends only on the class of φ in Mor_{O(F^{c})}(P, Q) since Ψ is the identity on δ_{S}(S) and on inclusions, and thus Ψ = Ψ_{t}, and [Ψ] = λ̂([t]), for t ∈ Ẑ^{1}(O(F^{c}); Z_{F}). Since λ̂ is clearly injective, this proves the exactness of (5).

Now consider the following diagram:

\xymatrix{ & 1 \ar[d] & 1 \ar[d] & 1 \ar[d] \\  & Z(S) \ar[r]^{d} \ar[d]^{δ_{S}} & Ẑ^{1}(O(F^{c}); Z_{F}) \ar[r] \ar[d]^{λ̂} & \underleftarrow{lim}^{1}(Z_{F}) \ar[r] \ar[d]^{λ_{L}} & 1 \\  & Aut_{L}(S) \ar[r]^{γ↦c_{γ}} \ar@{->>}[d]^{π_{S}} & Aut_{typ}^{I}(L) \ar[r] \ar[d]^{μ̂} & Out_{typ}(L) \ar[r] \ar[d]^{μ_{L}} & 1 \\ 1 \ar[r] & Aut_{F}(S) \ar[r]^{incl} & Aut(S, F) \ar[r] & Out(S, F) \ar[r] & 1 .}

Here, d sends g ∈ Z(S) to d(u_{g}), where u_{g} is the constant 0-cochain with value g. The diagram commutes by construction. We have just shown the top row and the second column are exact. The second row is exact by Lemma 4.9, and the third by definition of Out(S, F). The first column is exact by axiom (A2) for a linking system. By an easy diagram chase, one now sees that the third column in the diagram is exact.

The definition of the map ω_{L} and the exactness of the sequence at Out(S, F) are really a consequence of Proposition 5.11. Very roughly, for each β ∈ Aut(S, F), let L_{β} be the linking system whose underlying category is L, and with structural functors

T_{Ob(L)}(S) ⟶^{δ∘T (β^{−1})} L_{β} ⟶^{c_{β} ∘π} F

(and where the identification of objects in L with subgroups of S is changed accordingly). Then ω_{L}([β]) ∈ \underleftarrow{lim}^{2}(Z_{F}) is defined to be the obstruction to the existence of an isomorphism L ≅ L_{β}. In particular, [β] ∈ Im(μ_{L}) if and only if this obstruction vanishes. □

Proposition 5.12 thus gives a very precise description of the kernel of the natural map μ_{L} from Out_{typ}(L) to Out(S, F). A very different method for doing this, which in practice seems to give an easy way to make this computation in many cases without using the machinery for computing higher limits, is given in [AOV, Proposition 4.2].

We now sketch what is known about these obstruction groups \underleftarrow{lim}^{i}(C_{F}), beginning with the case where F is realizable.

Theorem 5.13. If F = F_{S}(G) for some finite group G and some S ∈ Syl_{p}(G), then

\underleftarrow{lim}^{i}_{O(F^{c})}(Z_{F}) = 0

for all i ≥ 2, and for all i ≥ 1 if p is odd.

Proof. This is shown as [O2, Theorems A & B] when p is odd, and in [O3, Theorems A & B] when p = 2. In both cases, the proof consisted first of a reduction to a question involving simple groups, and then a case-by-case check using the classification theorem. □

When p = 2, examples do occur of (realizable) fusion systems F for which \underleftarrow{lim}^{1}(Z_{F}) ≠ 0. For example, when F = F_{S}(G) and G = P SL_{2}(q) for an odd prime power q ≡ ±1 (mod 8), then \underleftarrow{lim}^{1}(Z_{F}) ≅ ℤ/2 by [BL, Theorem 1.7(3)] when q = 3^{2^{s}} for some s ≥ 1, and the same argument applies in the general case. The proof in [BL] is based on a comparison of the automorphisms for P SL_{2}(q) and SL_{2}(q); a more direct algebraic proof (using some of the techniques described in Section 5.4) is given in [O3, Proposition 1.6]. Similarly, \underleftarrow{lim}^{1}(Z_{F}) ≅ ℤ/2 when F = F_{S}(G) and G ≅ A_{n} (the alternating group) for n ≡ 2, 3 (mod 4) and n ≥ 6: this is shown indirectly (via the exact sequence of Proposition 5.12) in [AOV, proof of Proposition 4.8], and a more direct proof is sketched in [O3, Chapter 10].

Upon combining Theorem 5.13 with Propositions 5.11 and 5.12 and Theorem 4.22, we get the following result:

Corollary 5.14. For any finite group G, and any Sylow p-subgroup S ≤ G, L_{S}^{c}(G) is the unique centric linking system associated to F_{S}(G). If p is odd, then

Out(BG_{p}^{∧}) ≅ Out_{typ}(L_{S}^{c}(G)) ≅ Out(S, F_{S}(G)).

If G_{1} and G_{2} are two finite groups, and S_{i} ∈ Syl_{p}(G_{i}) are such that there

is a fusion preserving isomorphism S_{1} ⟶^{≅} S_{2}, then L_{S_{1}}^{c}(G_{1}) ≅ L_{S_{2}}^{c}(G_{2}) by Corollary 5.14, and hence BG_{1}_{p}^{∧} ≃ BG_{2}_{p}^{∧} by Theorem 3.7. In other words, the Martino-Priddy conjecture (the “if” part of Theorem 1.17) follows as a special case of Theorem 5.13.

Computations of higher limits over these functors will be discussed in more detail in Section 5.4. For now, we just note a few additional cases where we know the existence and/or uniqueness of linking systems.

Proposition 5.15 ([BLO2, Corollary 3.5]). Let F be a saturated fusion system over a p-group S. If rk(S) < p^{3} (if (C_{p})^{p^{3}} ≰ S), then there is a centric linking system associated to F. If rk(S) < p^{2} (if (C_{p})^{p^{2}} ≰ S), then there is a unique centric linking system associated to F.

Proof. This follows from Proposition 5.11, together with Proposition 5.26 below. □

When p is odd, the general question of whether \underleftarrow{lim}^{i}(Z_{F}) = 0 for all F and all i > 0 was reduced in [O2] to a purely group-theoretic question about p-groups. Another definition is needed to describe this. For any group G and any H, K ≤ G, we write [H, K; n] for the n-fold iterated commutator subgroup: thus [H, K; 1] = [H, K], [H, K; 2] = [[H, K], K], and [H, K; n] = [[H, K; n − 1], K].

Definition 5.16 ([O2, Definition 3.1]). Fix a prime p and a p-group S. For any p-group S, let 𝔛(S) denote the unique largest subgroup of S for which there is a sequence

1 = Q_{0} ≤ Q_{1} ≤ ⋯ ≤ Q_{n} = 𝔛(S) ≤ S

of subgroups, all normal in S, such that

[Ω_{1}(C_{S}(Q_{i−1})), Q_{i}; p−1] = 1  (6)

for each i = 1, …, n.

To see that 𝔛(S) is well defined (that there is a unique such largest subgroup), note that any two chains which satisfy (6) and terminate in Q_{n} and Q′_{m} can be combined to give a chain ending in Q_{n}Q′_{m}. It follows immediately from the definition that 𝔛(S) = S if [S, S; p − 1] = 1; i.e., if S has nilpotence class at most p − 1.

Recall (Definition A.15) that the Thompson subgroup J(S) of a p-group S is be the subgroup generated by all elementary abelian subgroups of maximal rank in S. The following is one of the main results in [O2].

Proposition 5.17 ([O2, Proposition 3.5 & Corollary 3.8]). Assume p is odd, and let F be a saturated fusion system over a p-group S.

(a) Assume there is a subgroup P ≤ 𝔛(S) which is F-centric and weakly closed in F. Then \underleftarrow{lim}^{i}(Z_{F}) = 0 for all i > 0.

(b) Let 1 = T_{0} ≤ T_{1} ≤ ⋯ ≤ T_{k} = S be any sequence of subgroups which are all strongly F-closed in S. Assume, for all 1 ≤ i ≤ k, that 𝔛(T_{i}/T_{i−1}) ≥ J(T_{i}/T_{i−1}). Then \underleftarrow{lim}^{i}(Z_{F}) = 0 for all i > 0.

We have stated these two versions of the proposition for simplicity, but there is also a more complicated version, a consequence of [O2, Proposition 3.5], which includes points (a) and (b) as special cases. Note that in point (a), for P to be weakly closed in F, it suffices that it be characteristic in each subgroup of S which contains it. In particular, we can take P = J(S) if it is contained in 𝔛(S).

Proposition 5.17 motivates the following conjecture, which if true would imply the existence and uniqueness of centric linking systems associated to any saturated fusion system over a p-group for p odd.

Conjecture 5.18 ([O2, Conjecture 3.9]). For each odd prime p and each p-group S, 𝔛(S) ≥ J(S).

In fact, by Proposition 5.17, in order to prove the existence and uniqueness of centric linking systems associated to saturated fusion systems over p-groups (p-odd), it suffices to prove Conjecture 5.18 for all p-groups S for which there is a saturated fusion system over S with no proper strongly closed subgroups of S. In principle, this restricts greatly the list of p-groups which must be considered, but it is difficult to see how to use this restriction in practice.

This conjecture has recently been reformulated in terms of representation theory by Green, Héthelyi, and Lilienthal [GHL]. Using that reformulation, they show among other things that Conjecture 5.18 holds whenever S/𝔛(S) has nilpotency class at most two [GHL, Theorem 1.1]. Building on this, Green, Héthelyi, and Mazza [GHM] then showed that Conjecture 5.18 holds whenever S/𝔛(S) has nilpotency class ≤ 4, or is metabelian, or is of maximal class, or has rank ≤ p.

5.4. Some computational techniques for higher limits over orbit categories.

The bar resolution is very useful for identifying certain obstruction groups as higher derived functors of limits over orbit categories. However, it does not seem to be very useful in practice for making actual computations. For that, other techniques are needed. We describe some of those here, with examples to illustrate how they are applied.

One technique, which works very well in practice when computing higher limits of a functor F over an orbit category, is to filter F by a sequence of subfunctors, 0 = F_{0} ⊆ F_{1} ⊆ ⋯ ⊆ F_{n} = F, in such a way that each quotient functor F_{i}/F_{i−1} vanishes except on one isomorphism class of objects. Surprisingly, when a functor F′ vanishes except on the isomorphism class of one object P, then \underleftarrow{lim}^{∗}(F′) depends only on the automorphism group of P in the orbit category and its action on F′(P). The following definition helps us to formulate this precisely.

Definition 5.19. Fix a finite group Γ and a ℤ[Γ]-module M. Let O_{p}(Γ) ⊆ O(Γ) be the full subcategory whose objects are the p-subgroups of Γ. Define F_{M}: O_{p}(Γ)^{op} ⟶ Ab by setting

F_{M}(P) = \begin{cases}M & if P = 1 \\ 0 & if P ≠ 1 . \end{cases}

Here, F_{M}(1) = M has the given action of Aut_{O_{p}(Γ)}(1) = Γ. Set

Λ^{∗}(Γ; M) = \underleftarrow{lim}^{∗}_{O_{p}(Γ)}(F_{M}).

Proposition 5.20. (a) Fix a finite group G. Let

F: O_{p}(G)^{op} ⟶ ℤ_{(p)}-mod

be any functor which vanishes except on the isomorphism class of some p-subgroup P ≤ G. Then

\underleftarrow{lim}^{∗}_{O_{p}(G)}(F) ≅ Λ^{∗}(N_{G}(P)/P; F (P)).

(b) Let F be a saturated fusion system over a p-group S. Let

F: O(F^{c})^{op} ⟶ ℤ_{(p)}-mod

be any functor which vanishes except on the isomorphism class of some subgroup Q ∈ F^{c}. Then

\underleftarrow{lim}^{∗}_{O(F^{c})}(F) ≅ Λ^{∗}(Out_{F}(Q); F (Q)).

Proof. Part (a) is shown in [JMO, Proposition 5.4], and part (b) in [BLO2, Proposition 3.2]. A more general versiom of both of these results is shown in [BLO3, Proposition 5.3].

The proofs in [BLO2] and [BLO3] and the first proof in [JMO] are purely algebraic, based on a series of comparisons of higher limits of different functors over different categories. There is also a geometric proof of (a) in [JMO], based on interpreting higher limits of functors over O_{p}(G) as equivariant cohomology over a certain G-space associated to this category.

□

The idea now, for an arbitrary functor F: O(F^{c})^{op} ⟶ ℤ_{(p)}-mod, is to reduce the computation of \underleftarrow{lim}^{∗}(F) to that of the graded groups Λ^{∗}(G; M) for appropriate G and M. The way of doing this is illustrated by the following corollary to Proposition 5.20.

Corollary 5.21. Let C be one of the following categories: either C = O_{p}(G) for some finite group G, or C = O(F^{c}) for some saturated fusion system F over a p-group S. Fix a functor F: C^{op} ⟶ ℤ_{(p)}-mod.

(a) Assume, for some i ≥ 0, that Λ^{i}(Aut_{C}(P); F (P)) = 0 for each P ∈ Ob(C). Then \underleftarrow{lim}^{i}(F) = 0.

(b) Assume, for some Q ∈ Ob(C), that Λ^{i}(Aut_{C}(P); F (P)) = 0 for each P ∈ Ob(C) not C-isomorphic to Q. Then

\underleftarrow{lim}^{∗}(F) ≅ Λ^{∗}(Aut_{C}(Q); F (Q)).

Proof. Let P_{1}, …, P_{m} be the set of C-isomorphism classes of objects in C, arranged from smallest to largest. Thus P ∈ P_{i}, Q ∈ P_{j}, and i ≤ j imply |P | ≤ |Q|. For each i = 0, …, m, let F_{i} ⊆ F be the subfunctor

F_{i}(P) = \begin{cases}F (P) & if P ∈ P_{1} ∪ ⋯ ∪ P_{i} \\ 0 & otherwise. \end{cases}

This defines a sequence of subfunctors

0 = F_{0} ⊆ F_{1} ⊆ F_{2} ⊆ ⋯ ⊆ F_{m} = F,

such that for each 1 ≤ i ≤ m,

(F_{i}/F_{i−1})(P) = \begin{cases}F (P) & for P ∈ P_{i} \\ 0 & for P ∉ P_{i} . \end{cases}

By Proposition 5.20, for each j = 1, …, m,

\underleftarrow{lim}^{∗}(F_{j}/F_{j−1}) ≅ Λ^{∗}(Aut_{C}(P_{j}); F (P_{j}))

for all P_{j} ∈ P_{j}. Hence under the assumption of (a), \underleftarrow{lim}^{i}(F_{j}/F_{j−1}) = 0 for each j. For each triple of indices 0 ≤ j < k < ℓ ≤ m, there is an exact sequence

\underleftarrow{lim}^{i}(F_{k}/F_{j}) ⟶ \underleftarrow{lim}^{i}(F_{ℓ}/F_{j}) ⟶ \underleftarrow{lim}^{i}(F_{ℓ}/F_{k}) ,

and so \underleftarrow{lim}^{i}(F) = \underleftarrow{lim}^{i}(F_{m}/F_{0}) = 0 by induction.

Now assume we are in the situation of (b), and let j be such that Q ∈ P_{j}. Then \underleftarrow{lim}^{∗}(F_{i}/F_{i−1}) = 0 for all i ≠ j. Using the exact sequences for an extension of functors, we now see that \underleftarrow{lim}^{∗}(F) ≅ \underleftarrow{lim}^{∗}(F_{j}/F_{j−1}) ≅ Λ^{∗}(Aut_{C}(Q); F (Q)). □

A similar technique can also be used to compute \underleftarrow{lim}^{∗}(F) in many other cases. As long as most of the groups \underleftarrow{lim}^{∗}(F_{j}/F_{j−1}) vanish, the exact sequences for higher limits of an extension of functors can be used to compute \underleftarrow{lim}^{∗}(F) once one knows the higher limits of all subquotient functors. This procedure works surprisingly well in practice, mostly because the functors Λ^{∗} vanish in many cases, as will be seen in Proposition 5.24.

The computation of \underleftarrow{lim}^{∗}(F) using such a filtration can also be organized as a spectral sequence, as was done explicitly by Grodal [Gr, Theorem 1.3]. This spectral sequence was used by Ziemiański [Zi] when making the computations needed to construct maps between 2-completed classifying spaces of certain compact Lie groups.

We now need some tools which allow us to compute Λ^{∗}(G; M) in favorable cases. There are two very different ways of doing this, based on the next two propositions. The first is due to Grodal.

Proposition 5.22 ([Gr]). Fix a prime p, a finite group G such that p||G|, and a ℤ_{(p)}[G]-module M. Let S_{p}(G) be the poset of p-subgroups 1 ≠ P ≤ G, and let |S_{p}(G)| be its geometric realization when regarded as a category. Let C_{∗}(|S_{p}(G)|) be the reduced cellular chain complex for |S_{p}(G)|; i.e., the chain complex which has ℤ in degree −1, and which in degree k is the free abelian group with basis the set of all chains P_{0} < ⋯ < P_{k} of length k in S_{p}(G). Then for all i > 0,

Λ^{i}(G; M) ≅ H^{i−1}(Hom_{ℤ[G]}(C_{∗}(|S_{p}(G)|), M)),

where G acts on C_{∗}(|S_{p}(G)|) via its conjugation action on S_{p}(G).

Proof. This is a special case of [Gr, Theorem 1.2]. The proof is based on a comparison of the orbit and poset categories O_{p}(G) and S_{p}(G). A similar result, which also involves comparing higher limits over a poset and an associated category but in a more abstract setting, was shown by Słomińska ([JS, Theorem 6.6]). □

We refer to [Gr, § 5] for several examples of how Proposition 5.22 can be applied. Many of these are based on a theorem of Webb [Wb1], which says that the p-adic “Steinberg complex” C_{∗}(|S_{p}(G)|; ℤ_{p}) splits as a sum of an acyclic complex and a complex of projective modules, together with a very careful analysis of which indecomposable projective ℤ_{p}[G]-modules can occur in the complex.

The next proposition provides a very different tool for making such computations. Here, for any ℤ[G]-module M and any H ≤ G, we write M^{H} for the subgroup of elements of M fixed by H. Note that part of this proposition is a special case of Lemma 5.9.

Proposition 5.23 ([JMO]). Fix a finite group and a ℤ_{(p)}[G]-module M. For each H ≤ G, set 𝔑_{H} = ∑_{h∈H} h ∈ ℤ[G]. Let

H^{0}M, 𝔑M: O_{p}(G) ⟶ ℤ_{(p)}-mod

be the functors H^{0}M(P) = H^{0}(P; M) = M^{P} = C_{M}(P) and 𝔑M(P) = 𝔑_{P} ·M. Then

\underleftarrow{lim}^{i}_{O_{p}(G)}(H^{0}M) = \begin{cases}M^{G} = C_{M}(G) & if i = 0 \\ 0 & if i > 0 \end{cases}

and

\underleftarrow{lim}^{i}_{O_{p}(G)}(𝔑M) = \begin{cases}𝔑_{G}·M & if i = 0 \\ 0 & if i > 0 . \end{cases}

Proof. This is shown in [JMO, Proposition 5.2] for the functors H^{0}M and H^{0}M/𝔑M (note that (H^{0}M/𝔑M)(P) = Ĥ^{0}(P; M)). This, in turn, is a special case of a general result of Jackowski and McClure [JM, Proposition 5.14]. □

In practice, Proposition 5.22 seems to be most useful for getting information about the groups Λ^{∗}(−; −) in general situations, while a procedure using Proposition 5.23 often works better for specific computations. Proposition 5.24 was shown in [JMO] mostly using repeated applications of Proposition 5.23 (together with exact sequences for higher limits of a pair of functors), but most or all of its points can also be proven using Proposition 5.22.

Proposition 5.24. The following hold for each finite group G and each ℤ_{(p)}[G]-module M.

(a) If (p, |G|) = 1, then

Λ^{i}(G; M) = \begin{cases}M^{G} & if i = 0 \\ 0 & if i > 0. \end{cases}

If H ⊴ G is a normal subgroup which acts trivially on M, and (p, |H|) = 1, then Λ^{∗}(G; M) ≅ Λ^{∗}(G/H; M).

(b) If p||G|, then Λ^{0}(G; M) = 0. If H ⊴ G is a normal subgroup which acts trivially on M, and p||H|, then Λ^{∗}(G; M) = 0. If O_{p}(G) ≠ 1, then Λ^{∗}(G; M) = 0.

(c) Assume p||G|, fix S ∈ Syl_{p}(G), and let ∼ be the equivalence relation among Sylow p-subgroups of Γ generated by nontrivial intersection. Set

H = {g ∈ G |^{g}S ∼ S}.

Thus H < G is the minimal strongly p-embedded subgroup in G containing S if there is one (see Definition A.6 and Proposition A.7), and H = G otherwise. Then Λ^{1}(G; M) ≅ M^{H}/M^{G}, and Λ^{i}(G; M) ≅ Λ^{i}(H; M) for i ≥ 2.

(d) A short exact sequence 0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0 of ℤ_{(p)}[G]- modules induces a long exact sequence

⟶ Λ^{i}(G; M′) ⟶ Λ^{i}(G; M) ⟶ Λ^{i}(G; M′′) ⟶ Λ^{i+1}(G; M′) ⟶ .

Proof. Except for the last statement in (c) (Λ^{i}(G; M) ≅ Λ^{i}(H; M) for i ≥ 2), this is shown in [JMO, Propositions 6.1 and 6.2]. So we prove that here, using Proposition 5.23. We refer to Definition A.6 and Proposition A.7 for the definition and properties of strongly p-embedded subgroups.

Let F_{M} ⊆ H^{0}M: O_{p}(G) ⟶ ℤ_{(p)}-mod be the functors F_{M}(1) = M, F_{M}(P) = 0 for P ≠ 1, and H^{0}(M)(P) = M^{P} = C_{M}(P) for all p-subgroups P ≤ G. We also use the same names for the restrictions of these functors to O_{p}(H). We claim that for all i ≥ 2,

Λ^{i}(G; M) ≅ \underleftarrow{lim}^{i−1}_{O_{p}(G)}(H^{0}M/F_{M}) ≅ \underleftarrow{lim}^{i−1}_{O_{S}(G)}(H^{0}M/F_{M})

≅ \underleftarrow{lim}^{i−1}_{O_{S}(H)}(H^{0}M/F_{M}) ≅ \underleftarrow{lim}^{i−1}_{O_{p}(H)}(H^{0}M/F_{M}) ≅ Λ^{i}(H; M) .

The first and last isomorphisms follow from the exact sequence of higher limits for the pair F_{M} ⊆ H^{0}M, and since \underleftarrow{lim}^{i}(H^{0}M) = 0 for i > 0 by Proposition 5.23. The second and fourth isomorphisms follow from Corollary 5.6. The third isomorphism follows from Proposition 5.3 (\underleftarrow{lim}^{∗}(−) is the cohomology of the bar resolution): the two chain complexes are isomorphic since Mor_{O_{S}(G)}(P, Q) = Mor_{O_{S}(H)}(P, Q) for all 1 ≠ P, Q ≤ S, and since (H^{0}M/F_{M})(1) = 0. This proves the last statement in (c). □

The next proposition provides a simple example of how Propositions 5.22 and 5.23 can both be applied in practice.

Proposition 5.25 ([BLO2, Corollary 3.4]). Let F be a saturated fusion system over a p-group S of order p^{m}. Then for each F: O(F^{c})^{op} ⟶ ℤ_{(p)}-mod, \underleftarrow{lim}^{i}(F) = 0 for all i > m.

Proof. By Lemma 5.21(a), it suffices to prove, for each finite group G, each ℤ_{(p)}[G]-module M, and each k ≥ 0,

G = p^{k}r where p∤r  implies  Λ^{i}(G; M) = 0 for all i > k. (7)

This follows immediately from Proposition 5.22, since the geometric realization of the poset S_{p}(G) has dimension at most k − 1.

We now give an inductive proof of (7) using Proposition 5.23. When k = 0, it follows from Proposition 5.23 (F_{M} = H^{0}M as functors on O_{p}(G)), so assume k > 0. Let F_{M}: O_{p}(G) ⟶ ℤ_{(p)}-mod be the functor F_{M}(1) = M and F_{M}(P) = 0 for p-subgroups 1 ≠ P ≤ G, regarded as a subfunctor of H^{0}M. Thus (H^{0}M/F_{M})(P) = M^{P} for 1 ≠ P ≤ G, and (H^{0}M/F_{M})(1) = 0. By the induction hypothesis, Λ^{i}(N_{G}(P)/P; M^{P}) = 0 for each 1 ≠ P ≤ G and each i ≥ k; and hence \underleftarrow{lim}^{i}(H^{0}M/F_{M}) = 0 for all i ≥ k by Lemma 5.21(a). Since \underleftarrow{lim}^{i}(H^{0}M) = 0 for all i > 0 by Proposition 5.23, (7) now follows from the long exact sequence of the pair F_{M} ⊆ H^{0}M. □

Here is another vanishing result. In fact, it includes Proposition 5.25 as a special case, but its proof is less elementary. As usual, for a p-group P, Fr(P) denotes its Fratini subgroup: the subgroup generated by all commutators and p-th powers in P. Thus Fr(P) is the smallest normal subgroup such that P/Fr(P) is elementary abelian.

Proposition 5.26 ([BLO2, Corollary 3.5]). Let F be a saturated fusion system over a p-group S, and let F: O(F^{c})^{op} ⟶ ℤ_{(p)}-mod be a functor. Assume, for some k > 0, that either rk(P/Fr(P)) < k for each P ≤ S, or that rk_{𝔽_{p}}(M/pM) < p^{k} for each P ∈ Ob(F^{c}) and each M ⊆ F (P). Then \underleftarrow{lim}^{i}(F) = 0 for all i ≥ k.

Proof. By Lemma 5.21(a), it suffices to prove that Λ^{∗}(G; M) = 0 for each finite group G and each ℤ_{(p)}[G]-module M such that rk_{p}(G) < k (i.e., G contains no subgroup ≅ C_{p}^{k}), or rk_{𝔽_{p}}(M_{0}/pM_{0}) < p^{k} for all M_{0} ⊆ M.

By [BLO1, Proposition 6.3], for each finite group G, each 𝔽_{p}[G]-module M, and each k ≥ 1, Λ^{k}(G; M) = 0 if rk(M) < p^{k} or if rk_{p}(G) < k (i.e., G contains no subgroup C_{p}^{k}). This is proven in [BLO1] using Grodal’s theorem (Theorem 5.22), together with a decomposition of the Steinberg complex due to Peter Webb [Wb1, Theorem 2.7.1]. A second argument, using Proposition 5.23, is given in the proof of Lemma 5.27 below.

The general case (M not necessarily of exponent p) now follows by standard manipulations of the groups Λ^{∗}(G; −), using Propositions 5.5 and 5.24(f). If M is a finitely generated ℤ_{(p)}[G]-module, then Λ^{k}(G; M) is finitely generated by Proposition 5.5(a) (and since O_{p}(G) is a finite category). If, in addition, rk(M/pM) < p^{k} or rk_{p}(G) < k, then rk(M_{0}/pM_{0}) < p^{k} for each ℤ_{(p)}[G]-submodule M_{0} ⊆ M, and hence Λ^{k}(G; M_{0}/pM_{0}) = 0 for each such M_{0} by the above remarks. Via the exact sequences of Proposition 5.24(f), this implies that each inclusion p^{r+1}M ⊆ p^{r}M induces a surjection from Λ^{k}(G; p^{r+1}M) onto Λ^{k}(G; p^{r}M). Since M is finitely generated, p^{r}M is torsion free for r large enough, hence Λ^{k}(G; p^{r}M) is p-divisible, and Λ^{k}(G; p^{r}M) = 0 since it is also finitely generated. We just saw that Λ^{k}(G; p^{i}M/p^{i+1}M) = 0 for each i, and hence Λ^{k}(G; M) = 0 by the exact sequences of Proposition 5.24(f) again.

If M is infinitely generated, then by Proposition 5.5(b), Λ^{k}(G; M) is the direct limit of the groups Λ^{k}(G; M_{0}) for M_{0} ⊆ M finitely generated. Thus Λ^{k}(G; M) = 0 if rk_{p}(G) < k or rk(M_{0}/pM_{0}) < p^{k} for all M_{0} ⊆ M. □

We now state a more precise version of Proposition 5.26, and show how it can be proven using Proposition 5.23. It is a special case of the more technical result [O3, Proposition 3.5], and the proof given here is a much shortened version of that given in [O3].

For use in the following lemma only, for any set P_{1}, …, P_{m} ≤ G of subgroups, we set N_{G}(P_{1}, …, P_{m}) = ⋂_{i=1}^{m} N_{G}(P_{i}). A radical p-chain of length n in G is a sequence

O_{p}(G) = P_{0} < P_{1} < ⋯ < P_{n}

of distinct p-subgroups of G such that P_{i} = O_{p}(N_{G}(P_{0}, …, P_{i})) for all i, and such that P_{n} ∈ Syl_{p}(N_{G}(P_{0}, …, P_{n−1})). Note that the first condition implies that P_{i} ⊴ P_{j} for i < j, and that P_{i}/P_{i−1} is a radical p-subgroup of N_{G}(P_{0}, …, P_{i−1})/P_{i−1} for all i.

Lemma 5.27. Fix a finite group G, and a ℤ_{(p)}[G]-module M. Assume, for some n ≥ 1, that Λ^{n}(G; M) ≠ 0. Then there is a radical p-chain

1 = P_{0} < P_{1} < P_{2} < ⋯ < P_{n}

of length n in G such that 𝔑_{N_{G}(P_{1},...,P_{n})}·M ≠ 0. If M is finitely generated, then M/pM (when regarded as an 𝔽_{p}[P_{n}]-module) contains a copy of the free module 𝔽_{p}[P_{n}]. In particular,

rk(M/pM) ≥ |P_{n}| ≥ p^{n}.

Proof. We prove this by induction on n ≥ 1. By assumption, Λ^{n}(G; M) = \underleftarrow{lim}^{n}(F_{M}) ≠ 0, where F_{M}(1) = M, and F_{M}(P) = 0 for all p-subgroups 1 ≠ P ≤ G. Regard F_{M} as a subfunctor of the acyclic functor 𝔑M of Proposition 5.23: the functor on O_{p}(G) which sends P to 𝔑_{P} ·M. Thus \underleftarrow{lim}^{n}(𝔑M) = 0, and hence \underleftarrow{lim}^{n−1}(𝔑M/F_{M}) ≠ 0.

By Corollary 5.21(a), there is a p-subgroup 1 ≠ P_{1} ≤ G such that

Λ^{n−1}(N_{G}(P_{1})/P_{1}; 𝔑_{P_{1}}·M) ≅ Λ^{n−1}(N_{G}(P_{1})/P_{1}; (𝔑M/F_{M})(P_{1})) ≠ 0 .

By Proposition 5.24(b), O_{p}(N_{G}(P_{1})) = P_{1}. If n = 1, then N_{G}(P_{1})/P_{1} has order prime to p by Proposition 5.24(b); so P_{1} ∈ Syl_{p}(G) (Lemma A.1). Thus (1 < P_{1}) is a radical p-chain of length one in this case, and 𝔑_{N_{G}(P_{1})}·M = (𝔑_{P_{1}}·M)^{N_{G}(P_{1})/P_{1}} ≠ 0.

If n > 1, then set G_{1} = N_{G}(P_{1})/P_{1} and M_{1} = 𝔑_{P_{1}}·M. By the induction hypothesis applied to Λ^{n−1}(G_{1}; M_{1}) ≠ 0, there is a radical p-chain (1 < P_{2}/P_{1} < ⋯ < P_{n}/P_{1}) in G_{1} such that

𝔑_{N_{G_{1}} (P_{2}/P_{1},...,P_{n}/P_{1})}·M_{1} = 𝔑_{N_{G}(P_{1},...,P_{n})}·M ≠ 0 .

This proves the first statement.

Assume pM = 0, so that M is an 𝔽_{p}[G]-module. Fix x ∈ M such that 𝔑_{P_{n}}·x ≠ 0, and consider the 𝔽_{p}[P_{n}]-linear homomorphism φ: 𝔽_{p}[P_{n}] → M which sends 1 to x. Every nonzero submodule of 𝔽_{p}[P_{n}] contains fixed elements and hence contains 𝔑_{P_{n}}, so Ker(φ) = 0, and φ embeds 𝔽_{p}[P_{n}] as a submodule of M. The general case (pM ≠ 0) follows by the same arguments as those used in the proof of Proposition 5.26. □

All of these vanishing results suggest that if there is a fusion system F for which \underleftarrow{lim}^{i}(Z_{F}) ≠ 0 for some i > 1, then this must be due to some very complicated combination of conditions being fulfilled. On the other hand, most of the general techniques we have for proving vanishing of higher limits are designed to prove that \underleftarrow{lim}^{i}(−) = 0 for all i > 0, and hence cannot be applied to this situation since we know examples where \underleftarrow{lim}^{1}(Z_{F}) ≠ 0. This might help to explain why the question of the existence and uniqueness of linking systems seems to be so difficult.

5.5. Homotopy colimits and homotopy decompositions.

By a homotopy decomposition of a space X is meant a homotopy equivalence of X with the homotopy colimit of a functor defined on some small category. So before discussing homotopy decompositions of |L| and |L|_{p}^{∧}, we first explain what a homotopy colimit is.

Let C be a small category, and let F: C ⟶ Topbe a (covariant) functor to the category of topological spaces. The ordinary colimit (or direct limit) of F is defined by setting

colim_{C}(F) = ( ∐_{c∈Ob(C)} F (c))/∼ ,

where ∼ is the equivalence relation generated by identifying x ∈ F (c) with f_{∗}(x) ∈ F (d) for each morphism f ∈ Mor_{C}(c, d) in C. The problem with this construction is that the homotopy types of the spaces F (c) and the maps between them do not, in general, in any way determine the homotopy type of the colimit. This will be explained more concretely in some of the examples given below.

The homotopy colimit of F: C ⟶ Top is defined by setting

hocolim_{C}(F) = (∐_{n≥0} ∐_{c_{0}→···→c_{n}} F (c_{0}) × Δ^{n})/∼ .

Here, each face or degeneracy map between the sequences c_{0} → ⋯ → c_{n} gives rise to an obvious identification between the corresponding spaces. In other words, we begin by taking the disjoint union of the spaces F (c), for all c ∈ Ob(C). Then, for each nonidentity morphism φ: c → d in C, we attach a copy of F (c) × I, by identifying F (c) × 0 with F (c) and attaching F (c)×1 to F (d) via F (φ): F (c) ⟶ F (d). This can be thought of as the “1-skeleton” of the homotopy colimit. Afterwards, one attaches higher dimensional simplices corresponding to each sequence of two or more composable morphisms in C.

More precisely, let C_{F} be the simplicial space Δ^{op} ⟶ Top defined by sending the object [n] to the disjoint union of the F (c_{0}), taken over all sequences c_{0} → ⋯ → c_{n} in N (C)_{n} (see Section 2.2). A morphism φ ∈ Mor_{Δ}([n], [m]) sends F (c_{0})_{ξ}, where ξ = (c_{0} → ⋯ → c_{m}) ∈ N (C)_{m}, to F (c_{φ(0)})_{φ^{∗}(ξ)} via the map F (c_{0} → c_{φ(0)}). We then set

hocolim_{C}(F) = |C_{F} | = (∐_{n=0}^{∞}(C_{F} ([n]) × Δ^{n}))/∼

with the quotient topology, where just as for the geometric realization of a simplicial set, (x, φ_{∗}(y)) ∼ (φ^{∗}(x), y) for all x ∈ C_{F} ([m]), y ∈ Δ^{n}, and φ ∈ Mor_{Δ}([n], [m]).

If F_{1}, F_{2} : C ⟶ Top are two functors, and φ: F_{1} ⟶ F_{2} is a natural transformation of functors, then φ induces a map of spaces hocolim(φ) from hocolim(F_{1}) to hocolim(F_{2}). If φ induces a homotopy equivalence φ(c): F_{1}(c) ⟶^{≃} F_{2}(c) for each c ∈ Ob(C), then hocolim(φ) is a homotopy equivalence: at least if F_{1}, F_{2}, and φ take values in the category of simplicial sets and maps (cf. [GJ, Proposition IV.1.7]). Similarly, if φ(c) induces an isomorphism in cohomology (with any given group of coefficients) for each c ∈ Ob(C), then so does hocolim(φ). Thus, if we think of a functor F: C ⟶ Top as a diagram of spaces and maps between them, then the the homotopy type or homology of the homotopy colimit depends only on the “homotopy type” or “homology” of the diagram in a very weak sense. This is definitely not true for induced maps between ordinary colimits of diagrams of spaces, and explains why this construction is called a “homotopy” colimit.

Among some simple examples of homotopy colimits, we note the following:

• Let C be the “pushout category”: C = (c_{1} ⟵^{f_{1}} c_{0} ⟶^{f_{2}} c_{2}). The homotopy colimit of a functor F: C ⟶ Top can be identified with

the double mapping cylinder of the pair of maps F (c_{1}) ⟵^{F (f_{1})}

F (c_{0}) ⟶^{F (f_{2})} F (c_{2}); i.e., with the space

hocolim_{C}(F) = (F (c_{1}) ⨿ (F (c_{0}) × I) ⨿ F (c_{2}))/∼ ,

where for each x ∈ F (c_{0}), (x, 0) ∼ f_{1}(x) and (x, 1) ∼ f_{2}(x). In contrast,

colim_{C}(F) ≅ (F (c_{1}) ⨿ F (c_{2}))/∼ ,

where f_{1}(x) ∼ f_{2}(x) for each x ∈ F (c_{0}). Thus, for example, if F (c_{1}) and F (c_{2}) are both points, then colim(F) is a point, and the structure of F (c_{0}) is lost. However, hocolim(F) is the suspension of F (c_{0}) in this case; and the cohomology (at least) of F (c_{0}) can be recovered from the cohomology of the homotopy colimit.

• Fix a discrete group G, and a space X upon which G acts. We can regard X as a functor from B(G) to spaces: the functor which sends the unique object to X and which sends a morphism g ∈ G to the action of g on X (see Definition 2.4). The colimit of this functor is the orbit space X/G, while the homotopy colimit of this functor is the Borel construction EG ×_{G} X. Any equivariant map of G-spaces f: X ⟶ Y which is also an (ordinary) homotopy equivalence induces a homotopy equivalence between the Borel constructions, but not in general between the orbit spaces.

• Let C be any category, and let ∗: C ⟶ Top be the functor which sends each object to a point. Then hocolim_{C}(∗) ≅ |C|. Thus for arbitrary F: C ⟶ Top, the natural morphism of functors from F to ∗ induces a map of spaces from hocolim_{C}(F) to |C| whose fibers (point inverses) all have the form F (−).

There is a spectral sequence which links the cohomology of a homotopy colimit to that of its pieces.

Proposition 5.28 ([BK, XII.4.5]). For any small category C, any functor F: C ⟶ Top, and any coefficient ring A, there is a spectral sequence

E_{2}^{nk} = \underleftarrow{lim}_{C}^{n}(H^{k}(F (−); A)) ⟹ H^{n+k}(hocolim_{C}(F); A) .

Proof. This is seen most simply by filtering the homotopy colimit by its “skeleta”. For each n ≥ 0, let hocolim^{(n)}(F) be the union of the cells F (c_{0}) × Δ^{m}, taken over all c_{0} → ⋯ → c_{m} and all m ≤ n. We can identify

E_{1}^{nk} = H^{n+k}(hocolim^{(n)}(F), hocolim^{(n−1)}(F); A)

≅ ∏_{c_{0}→···→c_{n}} H^{n+k}(F (c_{0}) × Δ^{n}, F (c_{0}) × ∂Δ^{n}; A)

≅ ∏_{c_{0}→···→c_{n}} H^{k}(F (c_{0}); A) ≅ C^{n}(H^{k}(F (−); A)) ,

where both products are taken over nondegenerate simplices only. Hence the spectral sequence of this filtration has E_{2}-term

E_{2}^{nk} ≅ H^{n}(C^{∗}(H^{k}(F (−); A)), d) ≅ \underleftarrow{lim}^{n}(H^{k}(F (−); A)) ,

and converges to H^{∗}(hocolim(F); A). A different argument is given in [BK, XII.4.5]. □

Note that in the first example (over the pushout category), this spectral sequence reduces to the usual Meyer-Vietoris exact sequence for a union of two spaces. In the second example, it is just the Serre spectral sequence of the fibration X ⟶ EG ×_{G} X ⟶ BG.

5.6. The subgroup decomposition of |L|.

Fix a p-local finite group (S, F, L), and let L \underrightarrow{π̃} O(F^{c}) be the projection. We define a functor

B̃: O(F^{c}) ⟶ Top

as follows. For P ∈ Ob(O(F^{c})) = Ob(L), let π̃↓P be the overcategory whose objects are the pairs (Q, α) for Q ∈ Ob(L) and α ∈ Rep_{F}(Q, P) = Mor_{O(F^{c})}(Q, P), and where

Mor_{π̃↓P} ((Q, α), (R, β)) = {ψ ∈ Mor_{L}(Q, R) | β ∘ π̃(ψ) = α}.

Define B̃ on objects by setting

B̃(P) = hocolim_{π̃↓P}(∗) = |π̃↓P | .

For each morphism φ ∈ Rep_{F}(P_{1}, P_{2}), set

B̃(φ) = |π̃↓φ|: |π̃↓P_{1}| ⟶ |π̃↓P_{2}|,

where π̃↓φ sends (Q, α) in π̃↓P_{1} to (Q, φ ∘ α) in π̃↓P_{2}. Thus B̃ is the left homotopy Kan extension over π̃ of the constant functor ∗: L ⟶ Top.

Proposition 5.29. Fix a saturated fusion system F and an associated centric linking system L. Let π̃: L ⟶ O(F^{c}) be the projection functor, and let

B̃: O(F^{c}) ⟶ Top

be as defined above. Then the following hold:

(a) |L| ≃ hocolim_{O(F^{c})}(B̃).

(b) For each P ∈ Ob(L), the functor e_{P} : B(P) ⟶ π̃↓P, which sends the unique object in B(P) to (P, Id) and sends g ∈ P to δ_{P} (g), induces a homotopy equivalence ε_{P} : BP ⟶ B̃(P). For each φ ∈ Hom_{F}(P, Q),

ε_{Q} ∘ Bφ ≃ B̃([φ]) ∘ ε_{P} ,

where [φ] ∈ Rep_{F}(P, Q) denotes the class of φ modulo Inn(Q).

Proof. See [BLO2, Proposition 2.2]. Point (a) is a very general fact: if Ψ: C ⟶ D and F: C ⟶ Top are two functors, then hocolim_{C}(F) is homotopy equivalent to the homotopy colimit over D of the left homotopy Kan extension over Ψ of F [HV, Theorem 5.5]. In this case, that reduces to the equivalence

hocolim_{O(F^{c})}(B̃) ≃ hocolim_{L}(∗) = |L|.  □

To help motivate this property of Kan extensions, we note here as an exercise the analogous (but easier) algebraic result. Let Ψ: C ⟶ D and T: C ⟶ Ab be a pair of functors. Let T̂ be the left Kan extension over Ψ of T: T̂(d) = colim_{Ψ↓d}(T) for d ∈ Ob(D), with induced morphisms defined in the obvious way. Then it is not hard to see that

colim_{D}(T̂) ≅ colim_{C}(T).

Point (b) in Proposition 5.29 is motivated by regarding L as an extension of the form

1 ⟶ \{P \}_{P ∈Ob(F^{c})} ⟶ L \underrightarrow{π̃} O(F^{c}) ⟶ 1.

The actual proof is by straightforward manupulation of categories.

The decomposition of Proposition 5.29 is known as the “subgroup decomposition” of |L|, because it identifies this space with a homotopy colimit of spaces having the homotopy type of classifying spaces of p-subgroups of S. We will see in Section 5.7 the important role played by this decomposition when studying certain mapping spaces involving |L|_{p}^{∧}.

Now fix a finite group G and S ∈ Syl_{p}(G). If we apply the above procedure to the projection functor π̃: T_{S}(G) ⟶ O_{S}(G), we obtain a functor B̃: O_{S}(G) ⟶ Top, where B̃(P) = |π̃↓P | for each P ≤ S, and B̃(P) ≃ BP. Hence by [HV, Theorem 5.5] again, for each set H of subgroups of S,

|T_{H}(G)| = hocolim_{T_{H}(G)}(∗) ≃ hocolim_{O_{H}(G)}(B̃|_{H}) .

Using this, we prove the following lemma, which is essentially due to Dwyer [Dw, Theorem 8.3], and which was used in Section 3.1 when showing that |L_{S}^{c}(G)|_{p}^{∧} ≃ |T_{S}(G)|_{p}^{∧} (and hence that |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧}).

Lemma 5.30. Fix a finite group G and S ∈ Syl_{p}(G). Then the inclusion of |T_{S}^{c}(G)| into |T_{S}(G)| induces an isomorphism of mod p cohomology.

Proof. We just saw that

|T_{S}(G)| ≃ hocolim_{O_{S}(G)}(B̃) and |T_{S}^{c}(G)| ≃ hocolim_{O_{S}^{c}(G)}(B̃^{c}) ,

where B̃^{c} is the restriction of B̃ to O_{S}^{c}(G). By Proposition 5.28, there is a spectral sequence of the form

E_{2}^{ij} = \underleftarrow{lim}^{i}_{O_{S}(G)}(Φ^{j}) ⟹ H^{∗}(|T_{S}(G)|, |T_{S}^{c}(G)|; A)

where for each P ≤ S,

Φ^{j}(P) = H^{j}(B̃(P), B̃^{c}(P); A) ≅ \begin{cases}0 & if P is p-centric in G \\ H^{j}(BP; A) & otherwise. \end{cases} When P is not p-centric, P C_{G}(P)/P acts trivially on H^{j}(BP; A) under the action of N_{G}(P)/P, and p||P C_{G}(P)/P | since Z(P) ∉ Syl_{p}(C_{G}(P)).

Thus by Proposition 5.24(b), Λ^{∗}(N_{G}(P)/P; Φ^{j}(P)) = 0 for each P ≤ S and each j. Hence by Corollary 5.21(a), \underleftarrow{lim}^{i}(Φ^{j}) = 0 for each i and j, and so H^{∗}(|T_{S}(G)|, |T_{S}^{c}(G)|; A) = 0 by the above spectral sequence. □

The functor B̃ defined in Proposition 5.29 is an example of a rigidification of a homotopy functor. To make precise what this means, let hoTop be the category whose objects are topological spaces and whose morphisms are homotopy classes of continuous maps (the “homotopy category”), and let h: Top⟶ hoTop be the quotient functor. Thus h is the identity on objects, and sends a map to its homotopy class. A homotopy functor is a functor from any small category C to hoTop. For any homotopy functor F: C ⟶ hoTop, a rigidification of F is a functor F̃: C ⟶ Top, together with a natural isomorphism of functors from h ∘ F̃ to F.

In general, the homotopy colimit of a homotopy functor is not defined, not even up to homotopy equivalence. Hence one must first replace it by a rigidification. The rigidification of homotopy functors was studied by Dwyer and Kan [DK1, DK2], who developed an obstruction theory for the existence and uniqueness of rigidifications.

When F is a saturated fusion system over a p-group S, there is a natural homotopy functor B: O(F^{c}) ⟶ hoTop which sends P ≤ S to BP and sends the class of φ ∈ Mor(F) to Bφ. This is not a functor to Top, since inner automorphisms of P do not induce the identity on BP. Proposition 5.29 says that any linking system L associated to F determines a rigidification of B, and also that the homotopy colimit of that realization has the homotopy type of |L|. The next proposition says that this in fact defines a bijective correspondence between linking systems associated to F and rigidifications of B up to homotopy.

The obstructions of Dwyer and Kan to the existence and uniqueness of rigidifications of B: O(F^{c}) ⟶ hoTop, in the form defined in [DK2], are in fact exactly the same groups as the obstructions of Proposition 5.11 to the existence and uniqueness of centric linking systems associated to F. This was the observation which first motivated us to prove the next proposition. We also note that it was in [DK2] that Dwyer and Kan introduced the word “centric”, but as a condition on a continuous map rather than on a subgroup.

Proposition 5.31. Fix a saturated fusion system F over a p-group S. Then there are mutually inverse bijections

\{ \begin{matrix}centric linking systems \\ associated to F \\ up to isomorphism \end{matrix} \} \overrightarrow{⟵}_{ls}^{ke} \{ \begin{matrix}rigidifications O(F^{c}) → Top \\ of (P ↦ BP) up to \\ natural homotopy equiv. \end{matrix} \}

where ke sends the class of a centric linking system L to the class of the rigidification

ke(L): O(F^{c}) ⟶ Top,

defined to be the left homotopy Kan extension of the constant functor L ⟶^{∗} Top along the projection π̃: L ⟶ O(F^{c}). Furthermore, for each linking system L,

|L| ≃ hocolim_{O(F^{c})}(ke(L)) .

Proof. A weaker version of this is stated in [BLO2, Proposition 2.3]. The full proposition (in a more general context) is shown in [BLO3, Proposition 4.6]. Part of it, of course, follows from Proposition 5.29.

One way to construct the linking system L = ls(B̃) associated to a rigidification B̃ is as the linking category of the space hocolim_{O(F^{c})}(B̃) in the sense of Definition 3.5 (taken with respect to S and the obvious inclusion of BS). The difficulty when doing it this way is to show that the linking category of the homotopy colimit actually is a linking system associated to F.

We sketch here another, more direct construction of ls(B̃). It is carried out in detail in the proof of [BLO3, Proposition 4.6]. Fix B̃: O(F^{c}) → Top. We are given, as part of the rigidification data, homotopy equivalences BP ⟶^{ε_{P}} B̃(P), for all P ∈ Ob(F^{c}), which make {[ε_{P}]}_{P ∈Ob(F^{c})} into a natural isomorphism from B to h ∘ B̃. For each P ∈ Ob(F^{c}), let ∗_{P} ∈ B̃(P) be the image under ε_{P} of the base point of BP, and let

γ_{P} : P ⟶^{≅} π_{1}(B̃(P), ∗_{P})

be the isomorphism induced by ε_{P} on fundamental groups.

The category L = ls(B̃) is defined by setting Ob(L) = Ob(F^{c}), and

Mor_{L}(P, Q) = {(φ, u) | φ ∈ Rep_{F}(P, Q), u ∈ π_{1}(B̃(Q); B̃φ(∗_{P}), ∗_{Q})}.

Here, π_{1}(X; x_{0}, x_{1}) means the set of homotopy classes of paths in X from x_{0} to x_{1}. Composition in L is defined by setting

(ψ, v) ∘ (φ, u) = (ψφ, v·(B̃ψ ∘ u)) ,

where paths are composed from right to left. Also, π: L ⟶ F^{c} is the functor which is the identity on objects, and which sends (φ, u) ∈ Mor_{L}(P, Q) to the composite

P ⟶_{≅}^{γ_{P}} π_{1}(B̃(P), ∗_{P}) ⟶^{B̃φ_{#}} π_{1}(B̃(Q), B̃φ(∗_{P}))

⟶^{c_{u}} π_{1}(B̃(Q), ∗_{Q}) ⟶_{≅}^{γ_{Q}^{−1}} Q.

It remains only to check that L and π, together with a functor δ constructed in an appropriate way, form a centric linking system associated to F. □

Recall that by Theorem 4.25, a p-local finite group (S, F, L) is determined up to isomorphism by the homotopy type of its classifying space |L|_{p}^{∧}. So Proposition 5.31 also says that there is a bijective correspondence between classifying spaces of F (up to homotopy type) and rigidifications of the homotopy functor B, where a rigidification corresponds to its homotopy colimit.

5.7. An outline of the proofs of Theorems 4.21 and 4.22.

To illustrate the role played by the subgroup decomposition, we now sketch the proofs of two of the results stated earlier about maps to the space |L|_{p}^{∧}, with emphasis on those parts which most directly involve the decomposition. To be carried out completely, both proofs require more information on the actual components of certain mapping spaces; but we skip over most of those details in this presentation. In fact, both of these theorems in their original form in [BLO2] contain results about the homotopy types of the full mapping spaces — the space of all maps from BQ to |L|_{p}^{∧} and the space of self homotopy equivalences of |L|_{p}^{∧} — but we restrict attention here as far as possible to the set (or group) of homotopy classes of maps.

Theorem 4.21 says that for any p-group Q and any p-local finite group (S, F, L), the natural map from Rep(Q, F) to [BQ, |L|_{p}^{∧}] is a bijection. The starting point for proving this is the following proposition, due originally to Broto and Kitchloo. We say that a category C has bounded limits at p if there is some N such that for any functor Φ: C^{op} ⟶ ℤ_{(p)}-mod, \underleftarrow{lim}^{i}(Φ) = 0 for all i > N.

Proposition 5.32 ([BLO2, Proposition 4.2]). Fix a prime p and a pgroup Q. Let C be a finite category with bounded limits at p, and let F: C ⟶ Top be a functor such that for each c ∈ Ob(C) and each Q_{0} ≤ Q, map(BQ_{0}, F (c)) is p-complete and has finite mod p cohomology in each degree. Then the natural map

(hocolim_{C}(map(BQ, F (−))))_{p}^{∧} ⟶ map(BQ, (hocolim_{C}(F))_{p}^{∧})

is a homotopy equivalence.

When Q is elementary abelian, Proposition 5.32 follows as a consequence of the description by Lannes [La] of mapping spaces map(BQ, −) and their cohomology. The general case then follows by induction on |Q|.

By Proposition 5.25, the orbit category O(F^{c}) of a saturated fusion system over a p-group S has bounded limits at p: if |S| = p^{k}, then \underleftarrow{lim}^{i}(F) = 0 for every i > k and every contravariant functor F from O(F^{c}) to ℤ_{(p)}- modules. When Q and P are two p-groups, [BQ, BP] ≅ Rep(Q, P) by Corollary 1.6. The component of the mapping space containing Bρ (for ρ ∈ Hom(Q, P)) is homotopy equivalent to BC_{P} (ρ(Q)) by Proposition 2.10(b), and hence map(BQ, BP) is p-complete (Proposition 1.10) and has finite mod p cohomology in each degree. So by Proposition 5.32, for any p-group Q and any p-local finite group (S, F, L), the space of maps from BQ to |L|_{p}^{∧} has the homotopy type of

(hocolim_{O(F^{c})}(map(BQ, B̃(−))))_{p}^{∧},

where B̃: O(F^{c}) ⟶ Top is as in Proposition 5.29. Using the above description of the homotopy type of map(BQ, B̃(P)) ≃ map(BQ, BP), and some straightforward manipulations of the homotopy colimit, we are led to the following result:

Proposition 5.33 ([BLO2, Proposition 4.3]). Fix a p-local finite group (S, F, L) and a p-group Q. Let L_{Q} be the category whose objects are the pairs (P, α) for P ∈ Ob(L) and α ∈ Hom(Q, P), and where

Mor_{L_{Q}}((P, α), (R, β)) = {φ ∈ Mor_{L}(P, R) | β = π(φ) ∘ α ∈ Hom(Q, R)}.

Let Φ: L_{Q} × B(Q) ⟶ L be the functor defined by setting

Φ((P, α), o_{Q}) = P  and  Φ((P, α) ⟶^{φ} (R, β) , x) = φ ∘ δ_{P} (α(x))

Then the map

|Φ|′: |L_{Q}|_{p}^{∧} ⟶ map(BQ, |L|_{p}^{∧})

adjoint to |Φ| is a homotopy equivalence.

Thus the set [BQ, |L|_{p}^{∧}] is in bijective correspondence with the set of connected components of |L_{Q}|_{p}^{∧} and hence of |L_{Q}|, and this is in bijective correspondence with Rep(Q, F). This finishes the proof of Theorem 4.21.

The proof of Theorem 4.22 — the description of Out(|L|_{p}^{∧}) in terms of automorphisms of the category L — is based on a very different use of the subgroup decomposition of |L|. One wants to understand maps from |L|_{p}^{∧} to itself using the subgroup decomposition of |L|, and this is a special case of the following theorem of Wojtkowiak.

Proposition 5.34 ([Wo]). Fix a small category C, a functor F: C ⟶ Top, and a space Y. Consider the map

R: [hocolim_{C}(F), Y ] ⟶ \underleftarrow{lim}_{C}([F (−), Y])

which sends the homotopy class [f] of a map f to the family {[f|_{F (c)}]}_{c∈Ob(C)} of homotopy classes of its restrictions. Fix maps f_{c}: F (c) ⟶ Y for each object c, such that f =^{def} \{[f_{c}]\}_{c∈Ob(C)} lies in \underleftarrow{lim}([F (−), Y]). Then the obstructions to the existence of an element in R^{−1}(f) lie in the groups \underleftarrow{lim}^{i+1}(π_{i}(map(F (−), Y), f_{c})) for i ≥ 1, and the obstructions to the uniqueness of such an element lie in \underleftarrow{lim}^{i}(π_{i}(map(F (−), Y), f_{c})) for i ≥ 1.

More precisely, there is a sequence of obstructions to the existence of an element in R^{−1}({[f_{c}]}), where one obstruction is defined only if the preceeding one vanishes. So it is more precise (but also more restrictive) to say that if all of the groups \underleftarrow{lim}^{i+1}(π_{i}(map(F (−), Y), f_{c})) vanish, then there is a map f such that f|_{F (c)} ≃ f_{c} for each c ∈ Ob(C) (and similarly for uniqueness). This is what will be used here.

Consider the following diagram:

[figure: 1; \underleftarrow{lim}^{1}(Z); λ_{L}; Out_{typ}(L); μ_{L}; |−|_{p}^{∧}; Out(S, F); ω_{L}; r̂; ≅; \underleftarrow{lim}^{2}(Z); 1; \underleftarrow{lim}^{1}(Z); λ̂; Out(|L|_{p}^{∧}); μ̂; \underleftarrow{lim} IRep(−, F); ω̂; \underleftarrow{lim}^{2}(Z),] (8)

where all limits are taken over O(F^{c}). Here, Z = Z_{F}, as defined in Section 5.3. The top row is defined and exact by Proposition 5.12. To prove that | − |_{p}^{∧} is an isomorphism, it suffices to construct the rest of the diagram and prove that it commutes, show the bottom row is exact, and that r̂ is an isomorphism.

Let IRep(P, F) ⊆ Rep(P, F) (for P ≤ S) denote the set of F-conjugacy classes of injective homomorphisms from P to S. Thus the set IRep(S, F) = Aut(S)/Aut_{F}(S) contains Out(S, F) = Aut(S, F)/Aut_{F}(S) as a subgroup. We can also consider \underleftarrow{lim} IRep(−, F) as a subset of IRep(S, F), and this is equal to Out(S, F) by definition of fusion preserving. In other words, there is a bijection r̂ from Out(S, F) to \underleftarrow{lim} IRep(−, F), defined by setting r̂([α]) = {[α|_{P}]}_{P ∈Ob(F^{c})} for α ∈ Aut(S, F).

Let μ̂ be the homomorphism defined by restriction:

Out(|L|_{p}^{∧}) ⊆ [|L|_{p}^{∧}, |L|_{p}^{∧}] ⟶ \underleftarrow{lim}_{O(F^{c})}[B(−), |L|_{p}^{∧}] ≅ \underleftarrow{lim}_{O(F^{c})} Rep(−, F).

Each self equivalence of |L|_{p}^{∧} restricts to an automorphism of S (well defined up to F-conjugacy), and hence Im(μ̂) is contained in \underleftarrow{lim} IRep(−, F). This is exactly the type of problem — describing the point inverses of this map — which Wojtkowiak’s theorem is designed to solve.

Let map(BP, |L|_{p}^{∧})_{incl} be the space of maps homotopic to the inclusion. By [BLO2, Theorem 4.4(c)], when P ∈ F^{c}, map(BP, |L|_{p}^{∧})_{incl} ≃ BZ(P); and thus its only nonvanishing homotopy group is π_{1}(map(BP, |L|_{p}^{∧})_{incl}) ≅ Z(P). Furthermore, as a functor on O(F^{c}), π_{1}(map(B(−), |L|_{p}^{∧})) ≅ Z_{F}. The only obstruction to lifting an element in \underleftarrow{lim} IRep(−, F) to Out(|L|_{p}^{∧}) thus lies in \underleftarrow{lim}^{2}(Z_{F}), and the only obstruction to the uniqueness of such a lifting lies in \underleftarrow{lim}^{1}(Z_{F}). The definition and exactness of the bottom row of (8) are thus special cases of Proposition 5.34.

The second square in (8) clearly commutes. The proofs that the other two squares commute are based on a closer examination of the obstruction maps, and we refer to Step 2 in the proof of [BLO3, Theorem 7.1] for details.

5.8. The centralizer and normalizer decompositions of |L|.

In [Dw], Dwyer studied homotopy decompositions of BG and BG_{p}^{∧}, when G is a finite group, and described three general families of such decompositions. A subgroup decomposition identifies BG_{p}^{∧} as the homotopy colimit of a functor defined on some full subcategory of O(G), which sends each object P to a space with the homotopy type of BP. A centralizer decomposition describes BG_{p}^{∧} as the homotopy colimit of a functor on a category of p-subgroups of G, which sends each object P ≤ G to a space with the homotopy type of BC_{G}(P)_{p}^{∧}. A normalizer decomposition of BG_{p}^{∧} is the homotopy colimit of a functor defined on a certain category of conjugacy classes of chains of subgroups, and sends each chain to a space with the homotopy type of the classifying space of a certain automorphism group of the chain.

Each of these families of decompositions can be realized as a decomposition of the classifying space of a p-local finite group (S, F, L). In fact, in some cases, this gives us a decomposition of |L| without p-completion. The subgroup decomposition of L, and some of its applications, was discussed in Sections 5.6 and 5.7.

We now describe centralizer and normalizer decompositions of |L|, beginning with the centralizer decomposition. For any p-local finite group (S, F, L), let F^{e} denote the full subcategory of F whose objects are those nontrivial elementary abelian p-subgroups of S which are fully centralized in F. For each E ∈ Ob(F^{e}), let C_{L}(E) be the category whose objects are the pairs (P, α) for P in L and α ∈ Hom_{F}(E, Z(P)), and where

Mor_{C_{L}(E)}((P, α), (Q, β)) = {φ ∈ Mor_{L}(P, Q) | π(φ) ∘ α = β} .

The following decomposition of |L| played an important role in the proof in [BLO2] of Theorem 4.23 (the computation of H^{∗}(|L|; 𝔽_{p})).

Theorem 5.35. For any p-local finite group (S, F, L), the natural map

hocolim_{E∈(F^{e})^{op}} |C_{L}(E)| ⟶ |L| ,

induced by the forgetful functors (P, α) ↦ P, is a homotopy equivalence. Also, for each E ∈ Ob(F^{e}), the functor P ↦ (P, incl) induces a homotopy equivalence |C_{L}(E)| ⟶ |C_{L}(E)|.

Normalizer decompositions of |L| have been constructed by Assaf Libman [Lb]. We describe here just one version of his decomposition: that based on F-centric subgroups. Fix a p-local finite group (S, F, L). Let sd(F^{c}) be the poset of all chains P_{0} < P_{1} < ⋯ < P_{n} of elements of F^{c}, ordered by inclusions of subchains. Let \overline{s}d(F^{c}) be the quotient poset of F-conjugacy classes of such chains. We regard \overline{s}d(F^{c}) as a category in the usual way: there is a unique morphism from one conjugacy class of chains to another if some representative of the first class is a subchain of a representative of the second.

For each P = (P_{0} < ⋯ < P_{n}) ∈ sd(F^{c}), let Aut_{L}(P) ≤ ∏_{i=0}^{n} Aut_{L}(P_{i}) be the subgroup of all n-tuples (α_{0}, …, α_{n}) such that for each 1 ≤ i ≤ n, α_{i}(P_{i−1}) = P_{i−1} and α_{i}|_{P_{i−1},P_{i−1}} = α_{i−1}. This is identified with a subgroup of Aut_{L}(P_{0}) in the obvious way (using the uniqueness of extensions of morphisms in L), and thus B(Aut_{L}(P)) is identified as a subcategory of L.

Theorem 5.36 ([Lb, Theorem A]). Fix a p-local finite group (S, F, L), and let \overline{s}d(F^{c}) be defined as above. Then there is a functor δ: \overline{s}d(F^{c}) ⟶ Top such that the following hold.

(a) There is a homotopy equivalence

hocolim_{\overline{s}d(F^{c})}(δ) ⟶^{≃} |L| .

(b) For each P ∈ sd(F^{c}), there is a natural homotopy equivalence

BAut_{L}(P) ⟶^{≃} δ([P]) .

(c) For each P ∈ sd(F^{c}), the map from BAut_{L}(P) to |L| induced by the equivalences in (a) and (b) is that induced by the inclusion of categories from B(Aut_{L}(P)) to L.

This decomposition has been applied by Libman and Viruel [LV], to give precise descriptions of the homotopy type of |L| in many cases.

6. Examples of exotic fusion systems

We say that a saturated fusion system is realizable if it is the fusion system of a finite group, and is exotic otherwise. There are several reasons for looking for exotic saturated fusion systems. From the homotopy theory point of view, their classifying spaces provide new spaces which are not p-completed classifying spaces of finite groups (Theorem 4.25), but which have many of the very nice homotopy theoretic properties of such spaces. From the group theory point of view, some exotic fusion systems have properties which are similar to those of certain sporadic simple groups. Also, one can hope that the search for exotic fusion systems, at least at the prime 2, could give some more insight into the proof of the classification theorem for finite simple groups. More generally, we look for exotic fusion system simply because we want to understand better how they arise, and how frequently.

We emphasise that when we say that a saturated fusion system over a p-group S is “exotic”, we mean it is not isomorphic to F_{S}(G) for any finite group G which contains S as a Sylow p-subgroup. Ian Leary and Radu Stancu [LS] and Geoff Robinson [Ro3] have given two very different constructions which show that every saturated fusion system over a p-group S is at least realized as the fusion system of some infinite group G which contains S as Sylow subgroup, in the sense that every finite p-subgroup of G is contained in a subgroup conjugate to S. Also, Sejong Park [Pa2] has shown that every saturated fusion system over a p-group S is isomorphic to F_{S}(G) for some finite G which contains S — but not necessarily as Sylow subgroup.

In almost all cases, fusion systems are shown to be exotic using the classification of finite simple groups, together with the following lemma. As usual, a finite group G is almost simple if for some nonabelian simple group H, G is isomorphic to a subgroup of Aut(H) which contains Inn(H) ≅ H. Equivalently, G contains a nonabelian simple subgroup H ⊴ G such that C_{G}(H) = 1.

Lemma 6.1 ([DRV]). Let F be a saturated fusion system over a p-group S. Assume, for each strongly closed subgroup 1 ≠ P ⊴ S in F, that P ∈ F^{c}, P is not elementary abelian, and does not factor as a product of two or more subgroups which are permuted transitively by Aut_{F}(P). Then if F is realizable, it is the fusion system of a finite almost simple group.

Proof. This is shown in [DRV, Proposition 2.19], and we outline their argument here. Assume F is realized by G, where |G| is the smallest possible. Then O_{p′}(G) = 1, since G/O_{p′}(G) also realizes F. Let 1 ≠ H ⊴ G be a minimal normal subgroup, and set P = S ∩ H. By Lemma A.5, either H is elementary abelian, or it is a product of simple groups which are permuted transitively under conjugation in G. Since P is strongly closed in F, the hypotheses of the lemma imply H must be nonabelian and simple. Since C_{G}(H) ⊴ G, this implies that C_{G}(H) ∩ H = 1. Also, C_{G}(H) has order prime to p since P ∈ Syl_{p}(H) is F-centric, and C_{G}(H) = 1 since O_{p′}(G) = 1. Thus G is almost simple. □

Throughout the rest of this section, we attempt to at least list all known examples of exotic fusion systems, giving more discussion in cases where it seems appropriate.

6.1. Reduced fusion systems and tame fusion systems.

When searching systematically for exotic fusion systems, it is convenient to have a way of restricting the class of saturated fusion systems one has to examine, without risking to overlook some exotic cases. Also, one would like to focus on exotic fusion systems which are minimal in some sense, without including those which can easily be derived from smaller exotic fusion systems. The class of simple fusion systems doesn’t seem appropriate for doing this, since saturated fusion systems in general do not seem to be built up via extensions of simple fusion systems.

This is the question addressed in [AOV], where we settled on the following definition of “reduced fusion systems”. Recall the definition of a normal subgroup in a fusion system (Definition I.4.1), and those of fusion subsystems of p-power index or of index prime to p (Definition I.7.3).

Definition 6.2 ([AOV]). A reduced fusion system is a saturated fusion system F such that

• F has no nontrivial normal p-subgroups,

• F has no proper subsystem of p-power index, and

• F has no proper subsystem of index prime to p.

For any fusion system F, the reduction of F is the fusion system 𝔯𝔢𝔡(F) obtained by first setting F_{0} = C_{F}(O_{p}(F))/Z(O_{p}(F)) (see Definition I.5.3 for the definition of the centralizer fusion system), and then alternately taking O^{p}(−) and O^{p′}(−). More explicitly, let

F_{0} ⊇ F_{1} ⊇ F_{2} ⊇ ⋯ ⊇ F_{m}

be such that F_{i} = O^{p}(F_{i−1}) if i is odd, F_{i} = O^{p′}(F_{i−1}) if i is even, and O^{p}(F_{m}) = O^{p′}(F_{m}) = F_{m}. Then 𝔯𝔢𝔡(F) = F_{m}.

By [AOV, Proposition 2.2], the reduction of any saturated fusion system is reduced. (The only condition which is not obvious is that O_{p}(𝔯𝔢𝔡(F)) = 1.) By [AOV, Proposition 2.4], 𝔯𝔢𝔡(F) = 1 (the fusion system over the trivial group) if and only if F is constrained (i.e., O_{p}(F) ∈ F^{c}).

We do not know an example of an exotic fusion system whose reduction is realizable, but it seems likely that this could happen.

The following definition was formulated to provide a criterion for testing whether or not a given reduced fusion system can be the reduction of an exotic one. Recall the outer automorphism group Out_{typ}(L) of a centric linking system L which was defined in Section 4.3, and its homotopy theoretic interpretation Out_{typ}(L) ≅ Out(|L|_{p}^{∧}) in Theorem 4.22.

Definition 6.3. A saturated fusion system F over S is tame if there is a finite group G which satisfies:

• S ∈ Syl_{p}(G) and F ≅ F_{S}(G); and

• the natural map

κ_{G}: Out(G) ⟶ Out_{typ}(L_{S}^{c}(G)) ≅ Out(BG_{p}^{∧})

is split surjective.

The following theorem describes how the problem of finding exotic fusion systems is reduced to one involving only reduced fusion systems.

Theorem 6.4 ([AOV, Theorem A]). For any saturated fusion system F over a p-group S, if 𝔯𝔢𝔡(F) is tame, then F is also tame, and hence realizable.

So far, we know of no examples of realizable fusion systems which are not tame, and hence no examples of exotic fusion systems whose reduction is realizable. By contrast, Ruiz [Rz] gave examples of realizable (in fact, tame) fusion systems F = F_{S}(G) for which 𝔯𝔢𝔡(F) is exotic (see Section 6.5). However, the following theorem provides a different type of converse to Theorem 6.4.

Theorem 6.5 ([AOV, Theorem B]). Let F be a reduced fusion system which is not tame. Then there is an exotic fusion system F̃ such that 𝔯𝔢𝔡(F̃) ≅ F.

It is Theorem 6.5 which motivates the split surjectivity condition in the definition of tameness. Assume F is a saturated fusion system over the p-group S which is realizable but not tame. In particular, it has an associated linking system L. Let A be any elementary abelian p-group upon which Out_{typ}(L) acts faithfully. Consider the p-local finite group (S × A, F × A, L × A), defined so that if (S, F, L) is realized by G, then (S × A, F × A, L × A) is realized by G × A. Using Theorem 4.15, one can construct a fusion system F̂, with an associated linking system L̂ which is extension of L × A by Out_{typ}(L), where Out_{typ}(L) acts on L in the canonical way and on A via the given faithful action. One now checks that 𝔯𝔢𝔡(F̂) ≅ F. If F̂ is realizable, then it is realized by a group Ĝ such that A ⊴ Ĝ and Ĝ/CĜ(A) ≅ Out_{typ}(L), and also such that G^{∗} =^{def} CĜ(A)/A realizes F. Then conjugation induces a homomorphism from Out_{typ}(L) to Out(G^{∗}) which splits κ_{G^{∗}}, contradicting the original assumption that F is not tame.

Thus Theorems 6.4 and 6.5 show that a reduced fusion system is tame if and only if it is not the reduction of any exotic fusion system.

As mentioned above, reduced fusion systems are very far from being simple in any sense. For example, any product of reduced fusion systems is again reduced. (See Definition I.3.4(c) for the definition of a product of fusion systems.) However, the following theorem says that when searching for reduced fusion systems which are not tame, it suffices to look for those which are indecomposable: those which do not factor as products of proper fusion subsystems.

Theorem 6.6 ([AOV, Theorem C]). Each reduced fusion system F has a unique factorisation F = F_{1} × ⋯ × F_{m} as a product of indecomposable fusion subsystems. If F_{i} is tame for each i, then F is tame.

6.2. The Ruiz-Viruel examples.

Probably the simplest examples of exotic fusion systems are those constructed by Ruiz and Viruel [RV]. They classified all saturated fusion systems over extraspecial p-groups of order p^{3} and exponent p (for odd p), of which three turned out to be exotic (all when p = 7). We give a quick sketch here of their results.

Let S be extraspecial of order p^{3} and exponent p. Then S contains p + 1 subgroups V_{0}, …, V_{p} of order p^{2}, each of which is isomorphic to C_{p}^{2}. The V_{i} are the only proper subgroups which are centric in S. So by Theorem I.3.5, each saturated fusion system F over S is generated by (hence determined by) Out_{F}(S), together with the groups Aut_{F}(V_{i}) for those V_{i} which are F-radical. If V_{i} is F-radical, then Aut_{F}(V_{i}) ≤ Aut(V_{i}) ≅ GL_{2}(p) contains at least two subgroups of order p, and hence must contain SL_{2}(p).

This, together with the saturation axioms, lead to the following conditions which relate these different groups:

• Out_{F}(S) has order prime to p, and is a subgroup of Out(S) ≅ GL_{2}(p).

• For each i, and each α ∈ Aut_{F}(S) such that α(V_{i}) = V_{i}, α|_{V_{i}} ∈ Aut_{F}(V_{i}).

• If V_{i} is F-radical, then SL_{2}(p) ≤ Aut_{F}(V_{i}) ≤ GL_{2}(p). If V_{i} is not F-radical, then Aut_{F}(V_{i}) contains only the restrictions of automorphisms in Aut_{F}(S) which leave V_{i} invariant.

• If V_{i} is F-radical, then for each β ∈ Aut_{F}(V_{i}) such that β(Z(S)) = Z(S), β extends to an element of Aut_{F}(S).

Conversely, any fusion system over S which satisfies these conditions, and for which all other morphisms are composites of restrictions of automorphisms of these subgroups, is saturated by Theorem I.3.10.

Using these criteria, Ruiz and Viruel made a complete list of all saturated fusion systems over S (for any odd prime p). To simplify the statement of their result, we list in Table 6.1 only those fusion systems which are reduced. In this situation, F is reduced if and only if there are at least two F-radical subgroups among the V_{i} and F has no fusion subsystems of index prime to p.

p  Out_{F}(S)  nr. V_{i} rad.  Aut_{F}(V_{i})  group
3∤(p−1) C_{p−1} × C_{p−1}  1 + 1  GL_{2}(p)  P SL_{3}(p)
3|(p−1) C_{p−1}×C_{(p−1)/3}  1 + 1  SL_{2}(p) ⋊ C_{(p−1)/3} P SL_{3}(p)
3  D_{8}  2 + 2  GL_{2}(p)  ^{2}F_{4}(2)′
5  4S_{4}  6  GL_{2}(p)  F_{3} = Th
7  S_{3} × C_{3}  3  SL_{2}(p)  He
7  S_{3} × C_{6}  3 + 3  SL_{2}(p) ⋊ C_{2}  Fi′_{24}
7  (C_{6} × C_{6}) ⋊ C_{2} 6 + 2 SL_{2}(p) ⋊ C_{2}, GL_{2}(p)  —
7  D_{8} × C_{3}  2 + 2  SL_{2}(p) ⋊ C_{2}  O’N
7  D_{16} × C_{3}  4 + 4  SL_{2}(p) ⋊ C_{2}  —
13  C_{3} × 4S_{4}  6  SL_{2}(p) ⋊ C_{4}  F_{1} = M

Table 6.1

The third column in the table gives the numbers of subgroups V_{i} in each F-conjugacy class of radical subgroups. Thus “6” means there are six F-radical subgroups among the V_{i}, all of which are F-conjugate, while “1 + 1” means there are two F-conjugacy classes containing one subgroup each. The fusion systems listed in the seventh and ninth lines of the table were shown to be exotic using a version of Lemma 6.1. (The third exotic fusion system found by Ruiz and Viruel is not reduced.)

6.3. Saturated fusion systems over 2-groups.

The following was proven in [LO] (with a correction in [LO2]).

Theorem 6.7. Let q be an odd prime power, and fix S ∈ Syl_{2}(Spin_{7}(q)). Let z ∈ Z(Spin_{7}(q)) be the central element of order 2. Then there is a saturated fusion system F = F_{Sol}(q) which satisfies the following conditions:

(a) C_{F}(z) = F_{S}(Spin_{7}(q)) as fusion systems over S.

(b) All involutions of S are F-conjugate.

Furthermore, there is a unique centric linking system L = L_{Sol}^{c}(q) associated to F.

The existence of these fusion systems, and certain homotopical properties of their classifying spaces, were predicted by Benson [Be3]. This, in turn, was motivated by a theorem of Solomon [So, Theorem 3.2], which said that when q ≡ ±3 (mod 8), there is no finite group G whose 2-fusion system satisfies the above condtions. Thus F_{Sol}(q) is exotic in these cases by Solomon’s theorem. In fact, Solomon showed that this is true for each prime power q, and the remaining details of his argument were written down in the proof of [LO, Proposition 3.4].

We sketch here the construction of F_{Sol}(q) given in [LO, §2]. Set G = Spin_{7}(q) for short, and fix S ∈ Syl_{2}(G). Let z ∈ Z(G) = Z(S) be the central involution, and choose U ⊴ S such that z ∈ U ≅ C_{2}^{2}. The centralizer of U in Spin_{7}(𝔽_{q}) is isomorphic to SL_{2}(𝔽_{q})^{3}/⟨(−I, −I, −I)⟩, and C_{G}(U) contains SL_{2}(q)^{3}/⟨(−I, −I, −I)⟩ with index two. Thus S_{3} acts on C_{G}(U) and on C_{S}(U) in a natural way. (Note, however, that the action depends on the choice of identification of C_{G}(U) as a subgroup of SL_{2}(𝔽_{q})^{3}/⟨(−I, −I, −I)⟩; this was the source of the error which made a correction necessary in [LO2].) The fusion system F_{Sol}(q) was defined to be the fusion system over S generated by F_{S}(G), together with this action of S_{3} on C_{S}(U) (and its restrictions).

A different construction of these fusion and linking systems was given by Aschbacher and Chermak in [AC]. Again set G = Spin_{7}(q) and fix U ⊴ S ∈ Syl_{2}(G). Set K = N_{G}(U) and K_{0} = C_{G}(U). There is a group H ≥ K such that K_{0} ⊴ H and H/K_{0} ≅ S_{3} acts on K_{0} = C_{G}(U) via the same action as that described in the last paragraph. Aschbacher and Chermak showed that F_{Sol}(q) is the fusion system of the amalgamated free product G ∗_{K} H, and gave a new proof that this fusion system is saturated.

They also constructed an associated linking system L_{Sol}^{c}(q) as the quotient of the transporter system of G ∗_{K} H by a certain “signaliser functor”.

In contrast, the linking system L_{Sol}^{c}(q) was not constructed explicitly in [LO], but rather shown to exist because the obstruction groups of Proposition 5.11 to its existence and uniqueness vanish. In fact, the existence of a linking system follows from Proposition 5.15, since the Sylow 2-subgroups of Spin_{7}(q) have rank four, but its uniquenss requires some more calculations.

The smallest of the Solomon fusion systems is over a group of order 2^{10}. Work in progress by Andersen, Oliver, and Ventura seems to show that there are no exotic fusion systems over 2-groups of smaller order. Using a computer search for 2-groups which meet certain necessary conditions, we explicitly listed all indecomposable, reduced fusion systems over 2-groups of order ≤ 2^{9}, and showed that they are all tame.

6.4. Mixing related fusion systems.

Fix a prime p ≥ 5. Let q be a prime power such that p|(q − 1) but p^{2}∤(q − 1). Set G = P SL_{p}(q), let T ≅ C_{q−1}^{p−2} × C_{(q−1)/p} be the subgroup of classes of diagonal matrices, and set N = N_{G}(T) ≅ T ⋊ S_{p}. Fix S ∈ Syl_{p}(N); then S ≅ C_{p}^{p−2} ⋊ C_{p} is also a Sylow p-subgroup of G.

Let V_{1}, …, V_{p} be conjugacy class representatives for those subgroups V ≤ S such that V ≅ C_{p}^{2} and V ≰ T. (All of these lift to extraspecial subgroups in SL_{p}(q) of order p^{3} and exponent p, and they are conjugate to each other in P GL_{p}(q).) The fusion systems F_{S}(N) ⊆ F_{S}(G) differ only in that all of the V_{i} are radical in F_{S}(G) (with Aut_{G}(V_{i}) ≅ SL_{2}(p)), while none of them are radical in F_{S}(N). For each subset I ⊆ {1, …, p}, let F_{I} be the fusion system such that F_{S}(N) ⊆ F_{I} ⊆ F_{S}(G), and such that Aut_{F_{I}} (V_{i}) = Aut_{G}(V_{i}) when i ∈ I, and Aut_{F_{I}} (V_{i}) = Aut_{N}(V_{i}) when i ∉ I. Thus, for example, F_{all} = F_{S}(G) and F_{∅} = F_{S}(N).

Using the fact that F_{S}(G) and F_{S}(N) are both saturated, it is not hard to see that these “mixed” fusion systems F_{I} are also saturated. Using Lemma 6.1, they are all shown to be exotic when 1 ≤ |I| < p (see [BLO2, Example 9.3] for details).

6.5. Other examples.

We end with a brief survey of all other examples we know of exotic fusion systems.

We first describe some examples constructed by Ruiz [Rz]. Fix a prime p ≥ 5, and a prime power q such that p∤q. Let e be the multiplicative order of q in 𝔽_{p}^{×}, and assume e > 1 (p∤(q − 1)). Fix any n ≥ ep, set G_{n,q} = GL_{n}(q) (a simple group), fix S_{n,q} ∈ Syl_{p}(G), and consider the fusion system F_{n,q} = F_{S_{n,q}}(G). By [Rz, Theorem B], O^{p′}(F) has index e in F_{n,q}, and thus F_{n,q} is not reduced. Furthermore, for each divisor d|e with d > 2, the normal fusion subsystem of index d in F_{n,q} is exotic.

In [DRV, Theorem 1.1], Díaz, Ruiz, and Viruel classify all saturated fusion systems over p-groups of rank two when p is odd. In addition to the exotic fusion systems over extraspecial groups of order 7^{3} found in [RV], they showed in [DRV, Theorem 5.10] that there are several infinite families of exotic fusion systems over 3-groups of rank two.

Several families of exotic saturated fusion systems were constructed by Clelland and Parker [CP], and by Broto, Levi, and Oliver [BLO4], as the fusion systems of the amalgamated free products associated to certain trees of finite groups.

Another source of exotic fusion systems comes from nonabelian p-groups with abelian subgroup of index p. When p = 2, each reduced fusion system over a 2-group S of this type is the fusion system of one of the simple groups P SL_{2}(q) or P SL_{3}(q) for q odd, and S is dihedral, semidihedral, or a wreath product C_{2^{n}} ≀C_{2} (to appear in a future paper by Andersen, Oliver, and Ventura). But when p is odd, this author has shown in unpublished work that there are many families of exotic (reduced) fusion systems over groups of this form. For example, one can give a complete list of all reduced fusion systems F over p-groups S (p odd) which contain a unique abelian subgroup A ⊴ S of index p, and such that A is not F-essential — and almost all of these (with a few exceptions when p = 3) are exotic.

In addition, a few other examples of exotic fusion systems are given in [BLO2, Example 9.4].

7. Open problems

We finish this survey with a list of important problems in the subject which as far as we know are still open.

1. Prove the existence and uniqueness of centric linking systems (equivalently, of classifying spaces) associated to any given saturated fusion system, or find a counterexample. This was discussed in detail in Section 5.3.

2. Define morphisms between p-local finite groups. Once p-local finite groups have been defined, it is natural to want to construct a category with them as objects. But it is not clear how morphisms should be defined in such a category. The simplest way to define a morphism from (S_{1}, F_{1}, L_{1}) to (S_{2}, F_{2}, L_{2}) is as a basepoint preserving map (or homotopy class of maps) from |L_{1}|_{p}^{∧} to |L_{2}|_{p}^{∧}. However, one would like to find a more combinatorial definition; for example, one induced by a functor between finite categories. A map between the classifying spaces as above does induce a functor Φ: F_{1} ⟶ F_{2}. But it need not induce a functor between the linking systems (at least not in an obvious way) since Φ need not send F_{1}^{c} into F_{2}^{c} (not even when it is induced by a homomorphism of groups). This problem has been studied in detail by Chermak (in work as yet unpublished).

A related problem was studied by Castellana and Libman [CL]: that of extending a homomorphism between Sylow subgroups to a map between classifying spaces. First, they construct wreath products in this setting: for each p-local finite group (S, F, L), each n > 1, and each K ≤ S_{n} (the symmetric group on n letters), they define a p-local finite group (S ≀ K, F ≀ K, L ≀ K) such that |L ≀ K| has the homotopy type of the wreath product of spaces |L| ≀ K = EK ×_{K} (|L|^{n}). They then show [CL, Theorem B] that for any pair of plocal finite groups (S_{i}, F_{i}, L_{i}) (i = 1, 2), and any fusion preserving homomorphism φ: S_{1} ⟶ S_{2} (whether or not it defines a functor between the fusion systems), there is a map of spaces from |L_{1}|_{p}^{∧} to |L_{2} ≀ S_{p^{k}}|_{p}^{∧}, for k large enough, which realizes φ in a certain explicit sense.

3. Find more exotic fusion systems at the prime 2, or prove there are none. As noted in Section 6.3, there is only one known family of exotic fusion systems over 2-groups. Motivated by the proof of the classification of finite simple groups, Ron Solomon has conjectured that these are the only ones which are simple. Whether or not the fusion systems F_{Sol}(q) are the only exotic fusion systems (aside from the obvious constructions taking them as starting point), we currently have little idea how to search for other examples.

Here is a slightly more general question. Does there exist an exotic, reduced fusion system over a finite 2-group which does not contain a normal subsystem isomorphic to a product of Solomon fusion systems F_{Sol}(q) (Section 6.3)? Note that this includes as a special case the question of whether there exist any realizable fusion systems over finite 2-groups which are not tame.

One project which could lead to such examples is that of Aschbacher, to repeat some of the steps in the proof of the classification theorem in the context of fusion systems. This is discussed in much more detail in Part II (Sections II.14–15), and also in [A5, A6, A7]. Another (less ambitious) project is the attempt by Andersen, Oliver, and Ventura to carry out a systematic computer search for exotic examples.

4. Try to better understand how exotic fusion systems arise at odd primes; or (more realistically) look for patterns which explain how certain large families of them arise. There seem to be many different examples of exotic fusion systems at odd primes, and they are constructed using a large range of techniques. Is there any way to categorise certain types? Is there an algorithm for systematically constructing large families of exotic examples, other than under certain very restrictive conditions?

5. Find criteria, especially at odd primes, which at least in some cases allow us to prove that a saturated fusion system is exotic without using the classification theorem. Currently, there is no known easily checked property of fusion systems which holds for all realizable fusion systems, but which fails for at least some exotic ones. In fact, there is no fusion system over a p-group for p odd which we can prove is exotic without using the classification of finite simple groups. This is a surprising and unsatisfactory situation. Any property which allows us to distinguish (even in some cases) between realizable and exotic fusion systems would be of great interest.

6. Find an example of a saturated fusion system which is realizable but not tame, or prove that there are none. As described in Theorem 6.5, finding such an example would provide a new way of constructing exotic fusion system as extensions of realizable fusion systems.

7. Prove that each p-local finite group (S, F, L) can be realized by some (possibly infinite) group, and some choice of “signaliser functor” in the sense of Aschbacher and Chermak. As noted earlier, Leary and Stancu [LS], and Robinson [Ro3] have shown via separate constructions that one can always find an infinite group G with Sylow p-subgroup S such that F ≅ F_{S}(G). Can G always be chosen so that L can be constructed as a quotient category of the centric transporter category T_{S}^{c}(G)? (See Definition 3.1.) This would mean choosing subgroups C_{G}^{∗}(P) ≤ C_{G}(P) for each P ∈ F^{c} (the “signaliser functor”), such that Mor_{L}(P, Q) ≅ T_{G}(P, Q)/C_{G}^{∗}(P) for each P and Q.

8. Fundamental groups of linking systems. What, in general, can be said about π_{1}(|L|), when L is a linking system, especially the linking system of a finite group G? For example, in [COS], it was shown that π_{1}(|L_{Sol}^{c}(q)|) = 1 when L_{Sol}^{c}(q) is the centric linking system associated to the Solomon fusion system F_{Sol}(q) (Section 6.3). In unpublished work by Grodal and Oliver, they construct some examples where π_{1}(|L_{S}^{c}(G)|) is isomorphic to G, others where it is a group of Lie type over an infinite ring, and yet others where it is infinite without any obvious group to identify it with. Also, together with Shpectorov, they show that when G is the sporadic simple group of Lyons, π_{1}(|L_{S}^{c}(G)|) = 1. So far, these are relatively isolated examples, and the computations in most cases depend on earlier results about geometries for G. Is there anything more general one can say about these groups? What, if any, significance might they have for the properties of G, its fusion system, and its p-completed classifying space?

9. Get a more precise understanding of H^{∗}(|L|; 𝔽_{p}) for a p-local finite group (S, F, L), and find ways to compute H^{∗}(|L|; M) when M is a finite abelian p-group with action of π_{1}(|L|). For example, for a p-local finite group (S, F, L), it would be useful to know whether the functors H^{i}(−; 𝔽_{p}) (as functors on O(F^{c})) all have vanishing higher limits. If so, this would give a new proof of Theorem 4.23, by showing that the cohomology spectral sequence for the subgroup decomposition of L (see Proposition 5.28) collapses (thus this would show that the decomposition is “sharp” in the terminology of Dwyer).

More generally, when M is any finite abelian p-group with action of π_{1}(|L|), we know examples where H^{∗}(|L|; M) is not the inverse limit over O(F^{c}) of the cohomology groups H^{∗}(−; M), and we know examples where it is the inverse limit (including all cases where the action is trivial). But we do not yet have a good understanding of what properties of the action distinguish these two cases. Some results along these lines have been obtained by Levi and Ragnarsson [LR].

10. Find other classes of fusion systems, or fusion systems over classes of groups other than p-groups or discrete p-toral groups, which could be of interest in group theory or homotopy theory. The generalization to fusion systems over discrete p-toral groups is described in Section 4.8. In work which has not yet appeared, Stancu and Symonds develop a theory of fusion systems over pro-p-groups. In [ABC, § X.4.1], the authors discuss the possibility of defining fusion systems over p-unipotent groups of finite Morley rank.

Are there any other generalizations which arise naturally?

Part IV. Fusion and Representation theory

Radha Kessar

Let G be a finite group, p a prime number dividing the order of G and k a field of characteristic p. In the 1930’s Richard Brauer, partially in joint work with Nesbitt, initiated the systematic study of the representations of G over k, nearly four decades after the first paper on representations of G over ℂ by Frobenius. In contrast to the complex group algebra ℂG of G, the modular group algebra kG is not a direct product of simple algebras; the indecomposable factors of kG, called blocks, have a very rich representation theory.

Between 1942 and 1946, Brauer introduced two fundamental notions. The first, which is now known as the Brauer homomorphism, is a k-algebra homomorphism from the center Z(kG) of kG to Z(kH), for H a local subgroup of G, that is H is a subgroup of G such that C_{G}(Q) ≤ H ≤ N_{G}(Q), for some p-subgroup Q of G. To each block B of kG the Brauer homomorphism associates a set of blocks of kH called the Brauer correspondents of B. The second notion is that of the defect groups of a block algebra- a certain conjugacy class of p-subgroups of G. The two notions are intimately connected, for instance Brauer’s First Main Theorem gives a bijection, via the Brauer homomorphism, between the set of blocks of kG with a given defect group P and the set of blocks of kN_{G}(P) with defect group P. The representation theory of B is deeply influenced by the nature of the defect groups of B, the local subgroups of G and the represention theory of Brauer correspondents of B.

Building on the ideas behind Brauer’s First Main Theorem, in the late seventies Alperin and Broué introduced the G-poset of Brauer pairs associated to a block B of kG. The elements of this poset are pairs (Q, e) where Q is a p-subgroup of G and e is a block of kC_{G}(Q) in Brauer correspondence with B; the inclusion relationship is defined through the Brauer homomorphism and the first component of any maximal pair is a defect group of B. A certain sub-poset of these pairs (which in the language of this part of the book corresponds to the poset of centric Brauer pairs) had been studied earlier by Brauer. Alperin and Broué showed that the G-conjugation pattern of Brauer pairs associated to B has very similar properties to the G-poset of p-subgroups of G. It was these similarities that led Puig to distill the essence of ”fusion” through abstractly defined categories and Alperin and Broué’s results may be reformulated as the statement that for any maximal Brauer pair (P, e) associated to B, the category F whose objects are subgroups of P and whose morphism sets Hom_{F}(Q, R) consist of morphisms induced by conjugations in the G-sub poset of Brauer pairs contained in (P, e) is a saturated fusion system on P.

The first 3 sections of this part of the book discuss Brauer pairs and associated fusion systems. The theory of Brauer pairs as developed by Alperin and Broué and later by Broué and Puig applies more generally to p-permutation G-algebras and we work in this setting. In particular, to every p-permutation G-algebra A over k and to every primitive idempotent b of the subalgebra of A of G-fixed points is associated an isomorphism class of fusion system through the fusion of Brauer pairs. These fusion systems are not always saturated. Section 3 gives a sufficient condition for saturation. The special case of block fusion systems is recovered in subsections 3.4 and 3.5.

Let B be a block of kG and let F be the saturated fusion system arising from some maximal Brauer pair (P, e) associated to B. Many results and conjectures in modular representation theory are statements relating the representation theory of B with F. Section 4 contains some background results and terminology from group representation theory. Section 5 is a selection of topics which highlight the connections between the representation theory of B and F. This section is not a comprehensive survey, but are rather intended to give a flavour of the subject. For instance, the nilpotent block theorem of Puig states that if F = F_{P} (P) then the module category of B is equivalent to that of the algebra kP. As another example, by a result of Rickard, building on the Brauer-Thompson-Dade-Green-Janush theory of cyclic blocks, if P is cyclic then the derived module category B is equivalent to the derived module category of the algebra k(P ⋊ Out_{F}(P)). Broué’s abelian defect group conjecture asserts that an analogue of the above result holds whenever P is abelian, with k(P ⋊ Out_{F}(P)) replaced by a twisted group algebra k_{α}(P ⋊ Out_{F}(P)), where α is a certain element in the second cohomogy group H^{2}(Out_{F}(P), k^{∗}). Alperin’s weight conjecture is a formula which expresses the number of isomorphism classes of simple B-modules in terms of the corresponding saturated fusion system F and certain elements in H^{2}(Out_{F}(Q), k^{∗}), where Q runs over the centric-radical objects of F.

A saturated fusion system is called block realisable if it is the fusion system of a block of a finite group, and is called block-exotic otherwise. By Brauer’s Third Main Theorem, any realisable saturated fusion system is block realisable, but the converse is open. A possible approach is to reduce the problem to blocks of finite simple or quasi-simple groups. For this it is important to understand the relationship between the fusion sytem of the blocks of kG and those of kN for N a normal subgroup of G. This is discussed in Section 6.

Sections 2 and 3 contain detailed proofs-the intention is that a reader reasonably familiar with the background material of Section 1 will be able to get a detailed picture of how fusion systems arise in p-permutation Galgebras and in particular, in block algebras. By contrast, Sections 4 and 6 should be read as a ”guided tour” and contain hardly any proofs, but on occasion an outline of a proof has been provided.

I would like to thank Kasper Andersen, Fei Xu, Ellen Henke and Matthew Gelvin for their careful reading of this part of the manuscript, and their many corrections and suggestions.

Notation. Group actions and maps are written on the left. If a group X acts on a set B, denote by Orb_{X}(B) the set of orbits of X on B, by B^{X} the subset of X-fixed points in B and by C_{X}(a) the stabilizer in X of an element a of B. If B is a X-poset, Y is a subgroup of X and C is a subset of B such that ^{y}C ⊆ C, for all y ∈ Y, then C is considered as a Y -poset through the structure inherited from B, and C is said to be a Y -subposet of B. If (A, +) is an abelian group, then for any finite subset C of A, denote by C^{+} the sum of the elements of C in A.

1. Algebras and G-algebras

This section is intended as a quick introduction to some basic notions and terminology for finite dimensional G-algebras over fields. Throughout this section, k will denote an algebraically closed field, and A will denote a ring with identity 1_{A} ≠ 0. We will denote by A^{∗} the group of invertible elements of A. If a ∈ A and u ∈ A^{∗}, we will denote by ^{u}a the conjugate uau^{−1} of a by u. By an ideal of A, we will always mean a two-sided ideal. If A is a k-algebra, we will always assume that A is finite dimensional as vector space over k.

All A-modules will be left A-modules, and will be assumed to be finitely generated over A, so in particular, if A is a k-algebra, then all A-modules considered will be finite dimensional as k-vector spaces. An A-module M is simple if M ≠ 0 and M does not properly contain any non-zero A-module, M is indecomposable if M ≅ M̸_{1} ⊕ M_{2} for any non-zero A-modules M_{1}, M_{2}, and M is projective if M ≠ 0 and there exists a positive integer n such that for some A-module N, A^{n} ≅ M ⊕N, where A^{n} denotes the direct sum of n copies of A.

1.1. Ideals and Idempotents.

This subsection recalls some background results from ring theory. This is standard material and can be found in many representation theory texts, for example, in the books of Benson [Be1], Curtis and Reiner [CuR1], [CuR2], [CuR3], Külshammer [Ku] and Nagao and Tsushima [NT]. The treatment below is not comprehensive and only results which are used in the sequel are mentioned here.

An element a of A is called nilpotent if a^{n} = 0 for some natural number n. Similarly, an ideal I of A is nilpotent if I^{n} = 0 for some natural number n. Clearly, every element of a nilpotent ideal of A is nilpotent. The Jacobson radical J(A) of A is the intersection of the maximal left ideals of A.

Proposition 1.1. Suppose that A is a k-algebra.

(a) A has finitely many maximal two-sided ideals and J(A) is the intersection of these. In particular, J(A) is an ideal of A.

(b) J(A) is nilpotent and J(A) contains every nilpotent ideal of A.

(c) A/J(A) is isomorphic to a direct product

A/J(A) = ∏_{i=1}^{r} Mat_{n_{i}}(k)

of matrix algebras over k. Further, J(A) is contained in any ideal I of A such that A/I is a direct product of matrix algebras over k.

(d) An element a of A is in J(A) if and only if aV = 0 for every simple A-module V.

An idempotent of A is an element e of A such that 0 ≠ e = e^{2}. In particular, 1_{A} is an idempotent of A. Two idempotents e, f of A are orthogonal if ef = fe = 0. An idempotent e of A is primitive in A if e cannot be decomposed as a direct sum e = e_{1} + e_{2}, with e_{1}, e_{2} idempotent and orthogonal. For an idempotent e of A, a (primitive) decomposition of e in A is a finite set of (primitive) idempotents of A, which are pairwise orthogonal and which sum to e. If X is a decomposition of e, we often abusively say that e = ∑_{f∈X} f is an idempotent decomposition of e.

Proposition 1.2. Suppose that A is a k-algebra and let e ∈ A be an idempotent. Then e has a primitive decomposition. Further, this primitive decomposition is unique up to conjugation in A. Precisely, if I and I′ are two primitive decompositions of e, then there exists a bijection i → i′ from I to I′ and u ∈ A^{∗} such that i′ = ^{u}i for all i ∈ I.

Proposition 1.3. Suppose that A is a k-algebra and let e and f be primitive idempotents of A. Then e and f are conjugate in A if and only if eAf is not contained in J(A).

Proposition 1.4. (Idempotent Lifting) Let π: A → B be a surjective homomorphism of k-algebras.

(a) Let e ∈ A be an idempotent. If e is primitive, then π(e) is either 0 or a primitive idempotent of B. If π(e) is primitive, then there exists an orthogonal idempotent decomposition e = e′ + e′′, where e′ is primitive in A, and π(e′′) = 0, so that π(e′) = π(e).

(b) If Ker(π) is nilpotent, then π(e) is a primitive idempotent of B for any primitive idempotent e of A.

(c) Let {j_{1}, ⋯, j_{t}} be a primitive decomposition of an idempotent f in B. Then there exists an idempotent e ∈ A and a primitive decomposition {i_{1}, ⋯, i_{t}} of e in A such that π(e) = f and π(i_{s}) = j_{s} for all s, 1 ≤ s ≤ t. In particular, any primitive idempotent f of B lifts to a primitive idempotent of A via π.

Lemma 1.5. (Rosenberg’s Lemma) Suppose that A is a k-algebra. Let e be a primitive idempotent of A, and let X be a family of ideals of A. If e ∈ ∑_{I∈X} I, then e ∈ I for some I ∈ X.

Definition 1.6. Let e and f be idempotents of A. We say that f is contained in e and write f ≤ e if

fe = f = ef.

It is immediate from the definition that the relation ≤ is transitive on the set of idempotents of A. Also, it is easy to see that f ≤ e if and only if f = efe. If f ≤ e and f ≠ e, then {f, e−f} is an idempotent decomposition of e. In particular, if A is a k-algebra and if f is primitive in A, then f ≤ e if and only if there is a primitive decomposition of e containing f.

If e is an idempotent of A, then eAe is a subring of A with identity element 1_{eAe} = e. An element a ∈ A belongs to eAe if and only if ea = a = ae. In particular, e is a primitive idempotent of A if and only if e is the only idempotent of eAe.

The ring A is local if the set of its non-invertible elements forms an ideal of A.

Proposition 1.7. Suppose that A is a k-algebra and let e be an idempotent of A. Then J(eAe) = eJ(A)e. Further, the following are equivalent.

(a) eAe is local.

(b) e is a primitive idempotent of A.

(c) e is a primitive idempotent of eAe.

(d) eJ(A)e is the unique maximal ideal of eAe.

(e) Every proper ideal of eAe is nilpotent.

(f) eAe/eJ(A)e ≅ k as k-algebras.

If e is an idempotent of A, then Ae and A(1_{A} − e) are A-modules via left multiplication by elements of A, and A = Ae ⊕ A(1_{A} − e) is a direct sum of A-modules. In particular, Ae is a projective A-module.

Proposition 1.8. Suppose that A is a k-algebra and let I be a set of representatives of the A-conjugacy classes of primitive idempotents of A.

(a) The set {Ai: i ∈ I} is a set of representatives of the projective indecomposable A-modules.

(b) The set {Ai/J(A)i: i ∈ I} is a set of representatives of the simple A-modules.

A central idempotent of A is an idempotent of the center Z(A) of A. A primitive idempotent of Z(A) is called a block of A. If e is a central idempotent of A, then for any a ∈ A, ae = ea = eae. By applying Proposition 1.2 to the commutative k-algebra Z(A), we get the following.

Proposition 1.9. Suppose that A is a k-algebra.

(a) Any central idempotent of A has a unique primitive decomposition in Z(A) and this decomposition is a subset of the unique primitive decomposition of 1_{A} in Z(A). In particular, the blocks of A all belong to the unique primitive decomposition of 1_{A} in Z(A) and any two distinct blocks of A are orthogonal.

(b) If b is a block of A and d is a central idempotent of A, then b ≤ d if and only if bd ≠ 0.

(c) If e is a primitive idempotent of A, then there is a unique block, b of A such that be ≠ 0 and for this b, e ≤ b.

(d) Let

1_{A} = b_{1} + ⋯ + b_{r}

be the primitive decomposition of 1_{A} in Z(A). Then for each i, 1 ≤ i ≤ r, Ab_{i} = b_{i}Ab_{i} is a k-algebra, with identity b_{i}; Ab_{i} is an indecomposable ring and

A = Ab_{1} × ⋯ × Ab_{r}

is the unique decomposition of A into a product of indecomposable factors. Further, for each i, 1 ≤ i ≤ r, J(Ab_{i}) = J(A)b_{i} and Ab_{i}/J(A)b_{i} is isomorphic to Mat_{n_{i}}(k) for some n_{i} ≥ 1.

The decomposition of 1_{A} (respectively A) in (d) is called the block decomposition of 1_{A} (respectively A).

1.2. G-algebras.

Thevenaz’s book [Th] is a good source for the material in this and the following subsections. The material here is less standard than in the previous subsection, and proofs are provided for many of the statements. Throughout, G will denote a finite group.

Definition 1.10. A G-algebra (over k) is a pair (A, ψ) where A is a kalgebra and ψ: G → Aut(A) is a group homomorphism.

Thus, a G-algebra is a k-algebra A endowed with an action of G by algebra automorphisms. If (A, ψ) is a G-algebra, we suppress the notation ψ by writing ^{g}a for ψ(g)(a) for g ∈ G and a ∈ A.

Definition 1.11. An interior G-algebra over k is a pair (A, φ), such that that A is a k-algebra and φ: G → A^{∗} is a group homomorphism.

An interior G-algebra (A, φ) is a G-algebra via ^{g}a = ^{φ(g)}a, for g ∈ G, a ∈ A. The G-algebra structure on an interior G-algebra will always be the inherited one unless stated otherwise. The map φ will usually be clear from context and will be suppressed. Note that if (A, φ) is an interior Galgebra over k then A is an k[G × G]-module through the map (g, h).a = φ(g)aφ(h)^{−1}, for a ∈ A and (g, h) ∈ G × H.

We list some examples of G-algebras and interior G-algebras - the main example is the first one.

Example 1.12. (a) As a k-vector space the group algebra kG has a basis consisting of the elements of G, multiplication in kG is the unique k-linear extension to kG of the group multiplication in G and the k-algebra structure is through the map

λ → λ1_{G}, λ ∈ k.

The natural inclusion of G in kG gives kG an interior G-algebra structure; the corresponding G-algebra structure on kG is through conjugation, that is

ga = gag^{−1}, g ∈ G, a ∈ kG.

(b) If N is a normal subgroup of G, then kN is a G-algebra via the restriction of the G-action on kG to the subalgebra kN. The G-algebra structure on kN may not be the restriction of an interior G-algebra structure on kN.

(c) Let V be a left kG-module. For each g ∈ G let φ_{g} ∈ End_{k}(V) be defined by φ_{g}(v) = gv, v ∈ V. Clearly, φ_{g} is invertible and the map g → φ_{g}, g ∈ G endows End_{k}(V) with an interior G-algebra structure. The inherited G-algebra structure satisfies ^{g}(ϕ)(v) = gϕ(g^{−1}v) for g ∈ G, ϕ ∈ End_{k}(V) and v ∈ V.

(d) Let A be an (interior) G-algebra and let H ≤ G. Then, A is an (interior) H-algebra via restriction.

(e) Let A be a G-algebra and let e be an idempotent of A such that ^{g}e = e for all g ∈ G. Then eAe is a G-algebra through the restriction of the G action on A to eAe; if in addition, A is an interior G-algebra, then eAe is an interior G-algebra through the map which sends an element g of G to the element ege of eAe.

1.3. Relative trace maps and Brauer homomorphisms.

If A is a G-algebra, and H is a subgroup of G, then the H-fixed point set A^{H} = {a ∈ A:^{h}a = a ∀h ∈ H} of A is a k-subalgebra of A and if L ≤ H, then A^{H} ⊆ A^{L}. Further, if g ∈ G, then ^{g}(A^{H}) = A^{^{g}H}.

Definition 1.13. Let A be a G-algebra and let L ≤ H ≤ G.

(a) Denote by Res_{L}^{H} : A^{H} → A^{L} the inclusion map.

(b) The relative trace map from H to L is the map Tr_{L}^{H} : A^{L} → A^{H}, defined by Tr_{L}^{H}(a): = ∑_{[h∈H/L]} ^{h}a, for a ∈ A^{L}. We denote by A_{L}^{H} the image of Tr_{L}^{H}.

(c) For g ∈ G, denote by c_{g,H} : A^{H} → A^{^{g}H} the conjugation map defined by a → ^{g}a for a ∈ A^{H}.

The maps Res_{{−}}^{{−}} are inclusions. So, for subgroups L ≤ H of G, a map f with domain A^{L} and a subset X of A^{H}, we will simply write f(X) instead of f(Res_{L}^{H}(X)).

We record some of the basic properties of the above maps-the proofs are an easy exercise and can be found in [Th, Proposition 11.4].

Proposition 1.14. Let A be a G-algebra.

(a) Tr_{L}^{H}, Res_{L}^{H} and c_{g,H} are k-linear and Res_{L}^{H} and c_{g,H} are unitary kalgebra homomorphisms for all L ≤ H ≤ G and all g ∈ G.

(b) c_{xg,H} = c_{x,^{g}H}c_{g,H} for all g, x ∈ G and all H ≤ G.

(c) Tr_{H}^{H}, Res_{H}^{H} and c_{h,H} are identity maps for all H ≤ G and all h ∈ H.

(d) Tr_{L}^{H}Tr_{K}^{L} = Tr_{K}^{H} and Res_{K}^{L}Res_{L}^{H} = Res_{K}^{H} for all K ≤ L ≤ H ≤ G.

(e) Res _{^{g}L}^{^{g}H}c_{g,H} = c_{g,L}Res_{L}^{H} and Tr _{^{g}L}^{^{g}H}c_{g,L} = c_{g,H}Tr_{L}^{H} for all L ≤ H ≤ G and all g ∈ G.

(f) (Mackey formula) Res_{L}^{H}Tr_{K}^{H} = ∑_{x∈[L\H/K]} Tr_{L∩^{x}K}^{L}c_{x,^{x^{−1}} L∩K}Res_{^{x^{−1}} L∩K}^{K} for all L, K ≤ H ≤ G. In other words, Tr_{K}^{H}(a) = ∑_{x∈[L\H/K]} Tr_{L∩^{x}K}^{L}(^{x}a) for all L, K ≤ H ≤ G and all a ∈ A^{K}.

(g) (Frobenius relations) Tr_{L}^{H}(ea) = eTr_{L}^{H}(a) and Tr_{L}^{H}(ae) = Tr_{L}^{H}(a)e for all L ≤ H ≤ G and all a ∈ A^{L}, e ∈ A^{H}. In particular, if I is an ideal of A^{L}, then Tr_{L}^{H}(I) is an ideal of A^{H}.

(h) Tr_{L}^{H}(a) = |H:L|·a for all L ≤ H ≤ G and all a ∈ A^{H}.

The above properties can be summed up by saying that the datum

(A^{H}, Res_{L}^{H}, Tr_{L}^{H}, c_{g,H})_{L≤H≤G,g∈G}

defines a cohomological Green functor for G over k (see [Th, Examples 53.1, 53.2]).

Definition 1.15. Let A be a G-algebra. For H ≤ G, let A_{<H}^{H} be the sum of all relative traces A_{L}^{H} of proper subgroups L of H, and let A(H) denote the quotient A^{H}/A_{<H}^{H} . The Brauer homomorphism Br_{H}^{A} is the canonical surjection

Br_{H}^{A} : A^{H} → A(H).

By Proposition 1.14 (g), for H ≤ G, A_{<H}^{H} is an ideal of A^{H}, hence Br_{H}^{A} is a homomorphism of k-algebras. If g ∈ G, then ^{g}(A^{H}) = A^{^{g}H} and by Proposition 1.14(e) ^{g}Tr_{L}^{H}(a) = Tr _{^{g}L}^{^{g}H}(^{g}a) for L ≤ H ≤ G, and a ∈ A^{L}. So, ^{g}Ker(Br_{H}^{A}) = Ker(Br_{^{g}H}^{A}) and conjugation by g induces an algebra isomorphism between A(H) and A(^{g}H). In particular, if A(H) ≠ 0, then A(H) inherits a natural structure of N_{G}(H)-algebra. Further, if K ≤ N_{G}(H), then the image of A(H)^{K} under the isomorphism between A(H) and A(^{g}H) induced by g is A(^{g}H)^{^{g}K} and the image of Ker(Br_{K}^{A(H)}) is Ker(Br _{^{g}K}^{^{g}A(H)}), hence g induces an isomorphism between A(H)(K) and A(^{g}H)(^{g}K).

Definition 1.16. For H ≤ G, a ∈ A^{H} and g ∈ G, define ^{g}(Br_{H}^{A}(a)): = Br_{^{g}H}^{A}(^{g}a) and set ^{g}A(H) = A(^{g}H) ; for K ≤ N_{G}(H) and α ∈ A(H)^{K}, denote by ^{g}Br_{K}^{A(H)}(α) the element Br_{^{g}H}^{A(^{g}H)}(^{g}α) of A(^{g}H)(^{g}K) and set gA(H)(K) = A(^{g}H)(^{g}K).

If k has characteristic p and S is a Sylow p-subgroup of H, then by Proposition 1.14 (h), the k-linearity of trace maps, and the fact that the index of S in H is invertible in k,

1_{A} = \frac{1}{|H:S|}Tr_{S}^{H}(1_{A}) = Tr_{S}^{H}(\frac{1}{|H:S|}1_{A}).

So, if S is proper in H, then 1_{A} ∈ A_{<H}^{H} and A(H) = 0. Thus we get the following.

Proposition 1.17. Suppose that char(k) = p and let A be a G-algebra. Then A(H) is non-zero only if H is a p-group.

The next few results sum up the interaction between relative trace maps, Brauer homomorphisms and primitive idempotents when k is of characteristic p. They will be crucial in the next sections. The first is a consequence of the Mackey formula.

Lemma 1.18. Suppose that char(k) = p. Let A be a G-algebra and let P, Q be subgroups of G. Then,

(a) For all a ∈ A^{P},

Br_{Q}^{A}(Tr_{P}^{G}(a)) = ∑_{\begin{matrix}x∈[Q\G/P], \\ Q≤^{x}P \end{matrix}} Br_{Q}^{A}(^{x}a).

(b) Br_{Q}^{A}(Tr_{Q}^{G}(A^{Q})) = Tr_{Q}^{N_{G}(Q)}(A(Q)).

Proof. The statement of (b) makes sense as A(Q) is an N_{G}(Q)-algebra (see discussion before Definition 1.16). Let a ∈ A^{P}. By the Mackey formula (Lemma 1.14 (f)), Tr_{P}^{G}(a) = ∑_{x∈[Q\G/P]} Tr_{Q∩^{x}P}^{Q} (^{x}a), hence Br_{Q}^{A}(Tr_{P}^{G}(a)) = ∑_{x∈[Q\G/P]} Br_{Q}^{A}(Tr_{Q∩^{x}P}^{Q} (^{x}a)). If Q ∩ ^{x}P is a proper subgroup of Q, then Br_{Q}^{A}(Tr_{Q∩^{x}P}^{Q} (^{x}a)) = 0. On the other hand, if Q ≤ ^{x}P, then Tr_{Q∩^{x}P}^{Q} (^{x}a) = xa. This proves (a). If P = Q, the sum in (a) runs over a set of coset representatives of Q in N_{G}(Q). So,

Br_{Q}^{A}(Tr_{Q}^{G}(a)) = ∑_{x∈[N_{G}(Q)/Q]} Br_{Q}^{A}(^{x}a) = ∑_{x∈[N_{G}(Q)/Q]} ^{x}Br_{Q}^{A}(a)
= Tr_{Q}^{N_{G}(Q)}(Br_{Q}^{A}(a)),

for all a ∈ A^{P} and this proves (b). □

In the next sections, we will often be dealing with idempotents in A^{H} for various subgroups H of G. For instance, if A = kG, then A^{G} = Z(kG), and hence the idempotents of A^{G} are the central idempotents of kG and the primitive idempotents of A^{G} are the blocks of kG. Note that if L ≤ H ≤ G, and if i is a primitive idempotent in A^{H}, then since A^{H} ⊆ A^{L}, i is also an idempotent of A^{L}, but since A^{L} may be strictly larger than A^{H}, i may no longer be primitive in A^{L}.

Lemma 1.19. Suppose that char(k) = p. Let A be a G-algebra, b a primitive idempotent of A^{G} and P a p-subgroup of G.

(a) If Br_{P}^{A}(b) ≠ 0, then there exists a primitive idempotent i of A^{P} such that i ≤ b and Br_{P}^{A}(i) ≠ 0.

(b) If b ∈ Tr_{P}^{G}(A^{P}), then there exists a primitive idempotent i of A^{P} such that i ≤ b and b ∈ Tr_{P}^{G}(A^{P} iA^{P}). Here A^{P} iA^{P} denotes the smallest ideal of A^{P} containing i.

Proof. Let X be primitive idempotent decomposition of b in A^{P}. Note that this makes sense by the remark above the statement of the lemma. Suppose that Br_{P}^{A}(b) ≠ 0. Applying Br_{P}^{A} to both sides of the equation b = ∑_{i∈X} i gives

0 ≠ Br_{P}^{A}(b) = ∑_{i∈X} Br_{P}^{A}(i).

Thus, Br_{P}^{A}(i) ≠ 0 for some i ∈ X, proving (a).

Now suppose that b ∈ Tr_{P}^{G}(A^{P}), say b = Tr_{P}^{G}(a), a ∈ A^{P}. So,

b = b^{2} = bTr_{P}^{G}(a) = Tr_{P}^{G}(ba) = Tr_{P}^{G}(∑_{i∈X} ia) = ∑_{i∈X} Tr_{P}^{G}(ia).

Here, the third equality follows from Proposition 1.14 (g) and the fifth by Proposition 1.14 (a). Now Tr_{P}^{G}(ia) ∈ Tr_{P}^{G}(A^{P} iA^{P}), for all i ∈ X. So, the above displayed equation gives that b ∈ ∑_{i∈X} Tr_{P}^{G}(A^{P} iA^{P}). By Proposition 1.14 (g), for each i ∈ X, Tr_{P}^{G}(A^{P} iA^{P}) is an ideal of A^{G}, and b is a primitive idempotent of A^{G}. Thus, by Rosenberg’s Lemma (Lemma 1.5), b ∈ Tr_{P}^{G}(A^{P} iA^{P}) for some i ∈ X, proving (b). □

Lemma 1.20. Suppose that char(k) = p. Let A be a G-algebra, and let P and Q be subgroups of G. Let b be a primitive idempotent of A^{G}, i a primitive idempotent of A^{P} and j a primitive idempotent of A^{Q} such that i ≤ b and j ≤ b. Suppose that Br_{Q}^{A}(j) ≠ 0 and b ∈ Tr_{P}^{G}(A^{P} iA^{P}). Then, there exists x ∈ G and u ∈ A^{Q}^{∗} such that Q ≤ ^{x}P and ^{u}j ≤ ^{x}i.

Proof. Write b = Tr_{P}^{G}(a), a ∈ A^{P} iA^{P}. By Lemma 1.18 (a)

Br_{Q}^{A}(b) = ∑_{\begin{matrix}x∈[Q\G/P], \\ Q≤^{x}P \end{matrix}} Br_{Q}^{A}(^{x}a)

and by hypothesis, j = jbj. So,

Br_{Q}^{A}(j) = ∑_{\begin{matrix}x∈[Q\G/P], \\ Q≤^{x}P \end{matrix}} Br_{Q}^{A}(j^{x}aj).

By hypothesis Br_{Q}^{A}(j) ≠ 0 and Br_{Q}^{A} is a surjective homomorphism of kalgebras. So, by idempotent lifting, Br_{Q}^{A}(j) is a primitive idempotent of A(Q). By Proposition 1.7, it follows that Br_{Q}^{A}(j)A(Q)Br_{Q}^{A}(j) is a local algebra, with identity Br_{Q}^{A}(j). Since by definition, the set of non-invertible elements of a local algebra forms a proper ideal of the algebra, it follows from the above equation that there exists some x ∈ G such that Q ≤ ^{x}P and such that Br_{Q}^{A}(j^{x}aj) = Br_{Q}^{A}(j)Br_{Q}^{A}(^{x}a)Br_{Q}^{A}(j) is an invertible element of Br_{Q}^{A}(j)A(Q)Br_{Q}^{A}(j). Consequently, Br_{Q}^{A}(j^{x}aj) and hence j^{x}aj ∈ A^{Q} is not nilpotent. Since J(A^{Q}) contains only nilpotent elements, this means that j^{x}aj ∉ J(A^{Q}). By hypothesis, a ∈ A^{P} iA^{P}, hence j^{x}aj is a finite sum of elements of the form jd^{x}icj, where d, c ∈ A^{^{x}P} ⊆ A^{Q}. Further, j ∈ A^{Q} and J(A^{Q}) is an ideal of A^{Q}. So, j^{x}aj ∉ J(A^{Q}) implies that there exists d ∈ A^{Q} such that jd^{x}i ∉ J(A^{Q}). Let I be a primitive idempotent decomposition of ^{x}i in A^{Q}, so jd^{x}i = jd ∑_{i′∈I} i′. Again, the fact that J(A^{Q}) is an ideal of A^{Q} implies that jdi′ ∉ J(A^{Q}) for some i′ ∈ I. So, by Proposition 1.3, applied to the primitive idempotents j and i′ of the algebra A^{Q}, i′ = ^{u}j for some u ∈ A^{Q}^{∗}. This proves the result as clearly, i′ ≤ ^{x}i. □

Lemma 1.21. Suppose that char(k) = p. Let A be a G-algebra and b a primitive idempotent of A^{G}. There exists a p-subgroup P of G and a primitive idempotent i of A^{P} such that

i ≤ b, Br_{P}^{A}(i) ≠ 0 and b ∈ Tr_{P}^{G}(A^{P} iA^{P}).

Proof. Let P be minimal amongst subgroups G such that b ∈ Tr_{P}^{G}(A^{P}). By Lemma 1.19, there exists a primitive idempotent i of A^{P} with i ≤ b and b = Tr_{P}^{G}(a) for some a ∈ A^{P} iA^{P}. We will show that Br_{P}^{A}(i) ≠ 0. This will prove the lemma, since by Proposition 1.17, Br_{P}^{A}(A^{P}) being non-zero implies that P is a p-group. Suppose if possible that Br_{P}^{A}(i) = 0, that is i ∈ A_{<P}^{P}. The idempotent i is primitive in A^{P} and A_{<P}^{P} is a sum of ideals of the form A_{Q}^{P}, Q a proper subgroup of G. Hence by Lemma 1.5, i ∈ Tr_{Q}^{P}(A^{Q}) for some proper subgroup Q of P. By Proposition 1.14 (g), Tr_{Q}^{P}(A^{Q}) is an ideal of A^{P}, hence a ∈ A^{P} iA^{P} ⊆ Tr_{Q}^{P}(A^{Q}) . But then by Proposition 1.14(d), b ∈ Tr_{Q}^{P}(A^{Q}), a contradiction to the minimality of P. □

We will also need the following result.

Lemma 1.22. Assume that char(k) = p and let A be a G-algebra. Let R be a subgroup of G and let C be a normal subgroup of G. Suppose that 1_{A} ∈ Tr_{R}^{G}(A^{R}) and 1_{A} is primitive in A^{C}. Then, RC/C contains a Sylow p-subgroup of G/C.

Proof. Let a ∈ A^{R} be such that

1_{A} = Tr_{R}^{G}(a) = Tr_{RC}^{G}(Tr_{R}^{RC}(a)),

and set u: = Tr_{R}^{RC}(a). Then, u ∈ A^{RC} ⊆ A^{C}. By hypothesis, the identity 1_{A} = 1_{A^{C}} of A^{C} is the only idempotent of A^{C}. In other words, A^{C} is a local algebra which means that J(A^{C}) has co-dimension 1 in A^{C} (see Proposition 1.7 (f)). Thus, u = λ1_{A} + v for some λ ∈ k and v ∈ J(A^{C}). This gives

1_{A} = Tr_{RC}^{G}(λ1_{A} + v) = |G:RC|λ1_{A} + Tr_{RC}^{G}(v).

Now, since C is normal in G, A^{C} is invariant under the action of G. In other words, the restriction of the action of G on A to the subalgebra A^{C} gives A^{C} the structure of an G-algebra. Since the Jacobson radical is an invariant of algebra automorphisms, it follows that J(A^{C}) is A-invariant. In particular, Tr_{RC}^{G}(v) ∈ J(A^{C}). But 1_{A} ∉ J(A^{C}). Hence, it follows from the above displayed equation that |G:RC|λ1_{A} ≠ 0, and consequently that |G:RC| is not divisible by p. □

2. p-permutation algebras, Brauer pairs and fusion systems

In [AB], Alperin and Broué introduced the poset of Brauer pairs associated to a block of finite group algebra. A prototype of these pairs had been considered earlier by Brauer [Br4] and were also considered by Olsson[Ols2]. The results of [AB] were reinterpreted and extended to (certain) idempotents of a more general class of G-algebras in [BP2], G a finite group. The aim in this section is to describe the main properties of Brauer pairs and to show how fusion systems arise from them (Proposition 2.22). For the moment, we will work in the general setting of p-permutation G-algebras; however the case of most interest for block theory is A = kN for N a normal subgroup of G, considered as G-algebra through the conjugation action of G; and even more particularly the sub case A = kG. While reading the present section it might be helpful for the reader to occasionally refer ahead to subsection 3.4, where we discuss how the concepts introduced in this section may be understood in the above special case.

Throughout this section k will denote an algebraically closed field of characteristic p > 0, G a finite group and A a G-algebra over k, finite dimensional as k-vector space.

2.1. p-permutation algebras and the Brauer homomorphisms.

Definition 2.1. • A subset B of A is P -stable for a subgroup P of G if xa ∈ B for all a ∈ B and all x ∈ P.

• The G-algebra A is a p-permutation algebra if for any p-subgroup P of G, A has a k-basis which is P -stable.

If B is an S-stable k-basis of A, for S a subgroup of G, then B is P -stable for any subgroup P of S, and ^{g}B is a ^{g}S-stable k-basis of A for any g ∈ G. So A is a p-permutation algebra if and only if some Sylow p-subgroup of G stabilises a k-basis of A. The group algebra kG is a p-permutation algebra since the k-basis consisting of the elements of G is P -stable for any subgroup P of G. Similarly, the algebra kN, for N a normal subgroup of G (see Example 1.12 (b)) is a p-permutation G-algebra.

The existence of a P -stable basis, for p-subgroups P of G yields convenient alternate descriptions of relative trace maps and Brauer homomorphisms. Recall from Proposition 1.17 that if H ≤ G is not a p-group, then A(H) = 0.

Lemma 2.2. Let P be a p-subgroup of G and suppose that B is a P -stable k-basis of A.

(a) The set {C^{+}: C ∈ Orb_{P} (B)} is a k-basis of A^{P}.

(b) For Q ≤ P, {C^{+} : C ∈ Orb_{P} (B) s.t. C_{P} (x) ≤ Q for x ∈ C} is a k-basis of Tr_{Q}^{P}(A^{Q}).

(c) The set {C^{+} : C ∈ Orb_{P} (B) s.t. C ∩ B^{P} = ∅} = {C^{+} : C ∈ Orb_{P} (B) s.t. |C| > 1} is a basis of A_{<P}^{P}.

(d) For Q ≤ P, the set {C^{+} : C ∈ Orb_{P} (B) s.t. C ∩B^{Q} = ∅}, is a k-basis of A_{<Q}^{Q} ∩ A^{P}. In particular, A^{P} ∩ A_{<Q}^{Q} ⊆ A_{<P}^{P}.

(e) The set {x + A_{<P}^{P} : x ∈ B^{P} } is a k-basis of A(P).

(f) For any element a = ∑_{x∈B} α_{x}x of A^{P}, Br_{P}^{A}(a) = ∑_{x∈B^{P}} α_{x}x + A_{<P}^{P} .

Proof. (a) is clear. Before proceeding with the other parts of the proof, we record an elementary fact. If V_{1} and V_{2} are k-vector spaces with basis A_{1} and A_{2} respectively and f : V_{1} → V_{2} is a linear map such that f(A_{1}) ⊆ f(A_{2}), then f(A_{1}) is a basis of the image Im(f) of f and {f(a)+Im(f) : a ∈ A_{2} − f(A_{1})} is a basis of the cokernel V_{2}/Im(f) of f.

Let Q ≤ P, and let C ∈ Orb_{Q}(B), x ∈ C. Clearly, C^{+} = Tr_{C_{Q}(x)}^{Q}(x), hence by Proposition 1.14(c),

Tr_{Q}^{P}(C^{+}) = Tr_{Q}^{P}Tr_{C_{Q}(x)}^{Q}(x)
= Tr_{C_{P} (x)}^{P}Tr_{C_{Q}(x)}^{C_{P} (x)}(x) = |C_{P} (x):C_{Q}(x)|Tr_{C_{P} (x)}^{P}(x).

Since k has characteristic p, the above is 0 if C_{Q}(x) is a proper subgroup of C_{P} (x) and is the P -orbit sum of x if C_{P} (x) ≤ Q. By (a), the Q (respectively P)-orbit sums of B form a basis of A^{Q} (respectively A^{P}). Thus (b) follows by the observation above, applied with V_{1} = A^{Q}, V_{2} = A^{P} and f = Tr_{Q}^{P}. Assertion (c) is immediate from (b). Next, we prove (d). Let Q ≤ P and let a ∈ A_{<Q}^{Q} ∩ A^{P}. By (a), we may write a = ∑_{C∈Orb_{P} (B)} α_{C}C^{+}. Suppose that C ∈ B is such that α_{C} ≠ 0. Since a ∈ A_{<Q}^{Q}, by part (c), applied to Q instead of P, it follows that for any x ∈ C, the Q-orbit of x has size greater than 1, that is x ∉ B^{Q}. Thus, A_{<Q}^{Q} ∩ A^{P} is contained in the k-span of {C^{+} : C ∈ Orb_{P} (B) s.t. C ∩ B^{Q} = ∅}. Conversely, let C be a P -orbit of B with C ∩ B^{Q} = ∅. Then C is a union of Q-orbits each of length greater than 1, hence by (c) applied to Q, C^{+} ∈ A_{<Q}^{Q}. This proves the first assertion of (d). The second follows from the first and (c). Part (e) follows from (c), by applying the observation on linear transformations above to the inclusion of A_{<P}^{P} in A^{P}. Finally, (f) is immediate from (e) and the definition of Brauer homomorphisms. □

Recall from the remarks after Proposition 1.17, that if P ≤ G is a pgroup with A(P) ≠ 0, then A(P) is a N_{G}(P)-algebra. The following is an easy consequence of Lemma 2.2 (e).

Proposition 2.3. If A is a p-permutation G-algebra, then A(P) is a ppermutation N_{G}(P)-algebra for any p-subgroup P of G such that A(P) ≠ 0.

Proof. Let P ≤ G be such that A(P) ≠ 0 and let S be a Sylow p-subgroup of N_{G}(P). By the remark following Definition 2.1, it suffices to prove that A(P) has an S-stable k-basis. So, let B be an S-stable (and hence P - stable) k-basis of A. By Lemma 2.2 (e), {x + A_{<P}^{P} : x ∈ B^{P} } is a k-basis of A(P). This basis is S-stable since P being normal in S implies that B^{P} is S-stable. □

If A is a p-permutation G-algebra, and Q ≤ P are p-subgroups of G, then by Lemma 2.2 (c), Ker(Br_{Q}^{A}) ∩ A^{P} is contained in Ker(Br_{P}^{A}). Thus, Br_{P}^{A} factors through the restriction of Br_{Q}^{A} to A^{P}.

Definition 2.4. Let A be a p-permutation G-algebra and let Q ≤ P of G be p-subgroups of G. Define

Br_{P,Q}^{A}: Br_{Q}^{A}(A^{P}) → A(P)

by Br_{P,Q}^{A}(Br_{Q}^{A}(a)) = Br_{P}^{A}(a) for a ∈ A^{P} .

Clearly, if A(P) ≠ 0, then Br_{P,Q}^{A} is a surjective homomorphism of kalgebras. In particular, if A(P) ≠ 0 then A(Q) ≠ 0. Also, note that while the fact that Br_{P,Q}^{A} is well defined depends on the existence of a P -stable k-basis of A, the definition of Br_{P,Q}^{A} is independent of the choice of such a basis.

Proposition 2.5. Let A be a p-permutation G-algebra and let Q ⊴ P be p-subgroups of G such that A(P) ≠ 0. Then,

A(Q)^{P} = Br_{Q}^{A}(A^{P}) and Ker(Br_{P,Q}^{A}) = Ker(Br_{P}^{A(Q)}).

Proof. The inclusion Br_{Q}^{A}(A^{P}) ⊆ A(Q)^{P} is clear since Br_{Q}^{A} is a homomorphism of N_{G}(Q) algebras. For the reverse inclusion, let B be a P -stable basis of A. By Lemma 2.2 (c) applied to Q, {Br_{Q}^{A}(x) : x ∈ B^{Q}} is a k-basis of A(Q). Since Q is normal in P, this basis is P -stable, and it follows by Lemma 2.2 (a), that the P -orbit sums of {Br_{Q}^{A}(x) : x ∈ B^{Q}} is a k-basis of A(Q)^{P}, that is the set {Br_{Q}^{A}(C^{+})} as C runs through the P -orbits of B^{Q} is a k-basis of A(Q)^{P}. Now since Q is normal in P, any P -orbit C of B^{Q} is also a P -orbit of B and consequently C^{+} is an element of A^{P}. This shows that A(Q)^{P} ⊆ Br_{Q}^{A}(A^{P}). Again by Lemma 2.2 (c) applied to the P -stable basis {Br_{Q}^{A}(x) : x ∈ B^{Q}} of A(Q), Ker(Br_{P}^{A(Q)}) has basis {Br_{Q}^{A}(C^{+})}, where C runs over the P -orbits of B^{Q} of size greater than 1. But by Lemma 2.2 (c) and (f) applied to P and Q respectively, this set is precisely the image of the basis {C^{+} : C ∈ Orb_{P} B, |C| > 1} of Ker(Br_{P}^{A}) under Br_{Q}^{A}. Thus, Ker(Br_{P}^{A(Q)}) = Br_{Q}^{A}(Ker(Br_{P}^{A})) = Ker(Br_{P,Q}^{A}). □

It follows from the above that if A is a p-permutation G-algebra, then for p-subgroups of G, Q ⊴ P of G, Br_{P,Q}^{A} induces a k-algebra isomorphism between A(Q)(P) and A(P).

Definition 2.6. With the notation and hypothesis of the above proposition let b_{P,Q}^{A}: A(Q)(P) → A(P) denote the isomorphism induced by Br_{P,Q}^{A} .

Note that for P a p-subgroup of N_{G}(Q) containing Q, b_{P,Q}^{A} satisfies and is completely determined by the condition

b_{P,Q}^{A}(Br_{P}^{A(Q)}(Br_{Q}^{A}(x))) = Br_{P,Q}^{A}(Br_{P}^{A}(x)) = Br_{P}^{A}(x) for all x ∈ A^{P} .

The map b_{P,Q} is G-equivariant in the sense that for all g ∈ G, w ∈ A(Q)(P), and z ∈ A^{P}

b_{^{g}P,^{g}Q}^{A}(^{g}w) = ^{g}(b_{P,Q}^{A}(w)).

In particular, b_{P,Q}^{A} is an isomorphism of N_{G}(Q) ∩ N_{G}(P)-algebras.

2.2. (A, G)-Brauer pairs and inclusion.

For the rest of this section A will denote a p-permutation G-algebra.

Definition 2.7. An (A, G)-Brauer pair is a pair (P, e), where P is a psubgroup of G such that A(P) ≠ 0 and e is a block of A(P). If (P, e) is an (A, G)-Brauer pair and i is an idempotent of A^{P}, we say that i is associated to (P, e) or more simply that i is associated to e if Br_{P}^{A}(i) ≠ 0 and Br_{P}^{A}(i) ≤ e. In other words, i is associated to e if and only if

eBr_{P}^{A}(i) = Br_{P}^{A}(i)e = Br_{P}^{A}(i) ≠ 0.

Lemma 2.8. Let P be a p-subgroup of G. For a primitive idempotent i of A^{P} and a block e of A(P), i is associated to (P, e) if and only if Br_{P}^{A}(i)e ≠ 0. Moreover, for any block e of A(P), there exists a primitive idempotent of A^{P} associated to (P, e) and for any primitive idempotent i of A^{P} such that Br_{P}^{A}(i) ≠ 0, there is a unique block e of A(P) such that i is associated to (P, e).

Proof. If i is a primitive idempotent of A(P) such that Br_{P}^{A}(i) ≠ 0, then by Proposition 1.4 (a), Br_{P}^{A}(i) is a primitive idempotent of A(P). Hence, if Br_{P}^{A}(i) ≠ 0, then by Proposition 1.9(c), there is a unique block e of of A(P) such that Br_{P}^{A}(i)e ≠ 0 and this e contains Br_{P}^{A}(i). This proves the first and third assertions. For the second, let (P, e) be an (A, G)-Brauer pair and let I be a primitive idempotent decomposition of 1_{A} in A^{P}. Since

0 ≠ e = Br_{P}^{A}(1_{A})e = ∑_{i∈I} Br_{P}^{A}(i)e,

Br_{P}^{A}(i)e ≠ 0 for some i ∈ I, and by the first assertion i is asociated to (P, e). □

Definition 2.9. Let (Q, f) and (P, e) be (A, G)-Brauer pairs. We say that (Q, f) is contained in (P, e) and write (Q, f) ≤ (P, e) if Q ≤ P and if any primitive idempotent i of A^{P} which is associated to e is also associated to f.

Note that if Q ≤ P then A^{P} ≤ A^{Q}, hence if i ∈ A^{P} is a primitive idempotent then i is a (not necessarily primitive) idempotent of A^{Q}. Further since Br_{P}^{A} factors through the restriction of Br_{Q}^{A} to A^{P}, if Br_{P}^{A}(i) ≠ 0, then Br_{Q}^{A}(i) ≠ 0. Thus, (Q, f) ≤ (P, e) if and only if Br_{Q}^{A}(i)f = Br_{Q}^{A}(i) for any primitive idempotent i of A^{P} associated to e.

The next result gives the fundamental properties of inclusion of Brauer pairs. Recall from Definition 2.4 and Proposition 2.5 the map Br_{P,Q}^{A}:A(Q)^{P} → A(P) for p-subgroups Q ⊴ P, of G.

Theorem 2.10. Let (P, e) be an (A, G)-Brauer pair and let Q ≤ P.

(a) There exists a unique block f of A(Q) such that (Q, f) ≤ (P, e).

(b) If Q is normal in P, then the block f of (a) is the unique block of A(Q) which is P -stable and such that Br_{P,Q}^{A}(f)e = e.

(c) Inclusion of (A, G)-pairs is a transitive relation.

The above theorem is due to Alperin-Broué [AB] for the case of group algebras and to Broué-Puig [BP2] in the general case. The proof is essentially in the book [Th, Section 40]. We give details here since [Th] treats only the case A = kG and uses the language of pointed groups which we have not introduced.

Lemma 2.11. Let (P, e) and (Q, f) be (A, G)-Brauer pairs such that (Q, f) ≤ (P, e).

(a) If f′ is a block of A(Q) different from f, then for any primitive idempotent i ∈ A^{P} associated to e, we have

Br_{Q}^{A}(i)f′ = 0.

In particular, f is the unique block of A(Q) with (Q, f) ≤ (P, e).

(b) Let (S, d) and (S, d′) be (A, G)-Brauer pairs such that (S, d′) ≤ (Q, f) and (S, d) ≤ (P, e). Then d = d′.

Proof. Let f′ ≠ f be as in (a) and let i ∈ A^{P} be a primitive idempotent associated to e. By hypothesis, (Q, f) ≤ (P, e), hence

0 ≠ Br_{Q}^{A}(i) = Br_{Q}^{A}(i)f.

On the other hand, f′f = 0, so

Br_{Q}^{A}(i)f′ = Br_{Q}^{A}(i)ff′ = 0.

This proves (a).

Now let S, d and d′ be as in (b) and let i be a primitive idempotent of A^{P} associated to e. We will show that Br_{S}^{A}(i)d′ ≠ 0 and by (a), applied with (Q, f) replaced by (S, d) and (Q, f′) replaced by (S, d′), it will follow that d′ = d, thus proving (b). Let I be a primitive idempotent decomposition of i in A^{Q}. Since (Q, f) ≤ (P, e), i is also associated to f. Hence,

∑_{j∈I} Br_{Q}^{A}(j)f = Br_{Q}^{A}(i)f = Br_{Q}^{A}(i) ≠ 0

and it follows that Br_{Q}^{A}(j)f ≠ 0 for some j ∈ I. By Lemma 2.8, j is associated to (Q, f). Since (S, d′) ≤ (Q, f), j is also associated to d′. So, Br_{S}^{A}(i)d′ ≠ 0 as required. □

Recall from Proposition 1.9 that any idempotent of a commutative kalgebra has a unique primtitive decomposition in the algebra and that this is a subset of the unique primitive decomposition of the identity of the algebra.

Lemma 2.12. Let H be a finite group, B a commutative H-algebra, finite dimensional as a vector space over k and let I be the unique primitive idempotent decomposition of 1_{B} in B. The set {J^{+}} as J runs over the set of H-orbits of I is the unique primitive idempotent decomposition of 1_{B} in B^{H}.

Proof. The statement makes sense since by Proposition 1.9, I is the set of primitive idempotents of B and is therefore H-stable. Let e be an idempotent of B^{H}, and let J ⊆ I be the primitive decomposition of e in B. Since ^{h}e = e, ^{h}J is also a primitive decomposition of e in B. Since J is the unique primitive decomposition of e in B, it follows that J is a union of H-orbits of I and e = J^{+}. On the other hand, if J is an H-orbit of I, then J^{+} is an idempotent of B^{H}. Thus, the set of primitive idempotents of B^{H} is precisely the set of orbit sums of H on I. □

We now prove Theorem 2.10.

Proof. We will first prove (b) and (a) in the case that Q is normal in P. Then, we will prove (a) in the general case. Part (c) will follow easily part (a) and Lemma 2.11 (b). Before proceeding, note that once the existence of a pair (Q, f) such that (Q, f) ≤ (P, e) is proved, uniqueness follows from Lemma 2.11 (a). Suppose that Q is normal in P. Then A(Q) and hence Z(A(Q)) is a P -algebra and P acts on the blocks of A(Q). We will show that there exists a P -stable block f of A(Q) such that Br_{P,Q}^{A}(f)e = e. The map Br_{P,Q}^{A}: A(Q)^{P} → A(P) is surjective hence maps Z(A(Q)^{P}) into Z(A(P)) and since Z(A(Q))^{P} ⊆ Z(A(Q)^{P}), the image of Z(A(Q))^{P} under Br_{P,Q}^{A} also lies in Z(A(P)). Let I be the unique primitive decomposition of 1_{A(Q)} in Z(A(Q))^{P}. The set of non-zero elements of the image of I under Br_{P,Q}^{A} is a decomposition of 1_{A(P)} in Z(A(P)) and e is primitive in Z(A(P)). Thus there is a unique primitive idempotent f of Z(A(Q))^{P} such that Br_{P,Q}^{A}(f)e = e. By Lemma 2.12 applied to the commutative P -algebra Z(A(Q)), f is the sum of a P -orbit of blocks of A(Q), that is f = Tr_{R}^{P}(f′) for some block f′ of A(Q) and some subgroup R of P. Suppose, if possible, that R is a proper subgroup of P. Then f ∈ Ker(Br_{P}^{A(Q)}) and by Proposition 2.5, Ker(Br_{P}^{A(Q)}) = Ker(Br_{P,Q}^{A}). So Br_{P,Q}^{A}(f) = 0, whence 0 = Br_{P,Q}^{A}(f)e = e, a contradiction. So R = P and f = f′ is a P -stable block of A(Q) as required.

Thus, in order to prove part (b) (and part (a) when Q⊴P), it suffices to prove that if f is a P -stable block of A(Q) such that Br_{P,Q}^{A}(f)e = e, then (Q, f) ≤ (P, e). So, let f be a P -stable block of A(Q) with Br_{P,Q}^{A}(f)e = e, and let i be a primitive idempotent of A^{P} associated to e. Then,

Br_{P,Q}^{A}(Br_{Q}^{A}(i)f) = Br_{P}^{A}(i)Br_{P,Q}^{A}(f) = Br_{P}^{A}(i)eBr_{P,Q}^{A}(f) = Br_{P}^{A}(i)e

= Br_{P}^{A}(i) ≠ 0,

and in particular, Br_{Q}^{A}(i)f ≠ 0. By Lemma 2.8, i is associated to f, hence (Q, f) ≤ (P, e).

Next we prove (a) in the general case. We proceed by induction on |P:Q|. Let Q be a proper subgroup of P and let N = N_{P} (Q). By the inductive hypothesis, for every subgroup R of P which properly contains Q, there is a unique block, say e_{R} of A(R) such that (R, e_{R}) ≤ (P, e). Since Q is normal in N, by (b) there exists a block f of Z(A(Q)) such that (Q, f) ≤ (N, e_{N}). Note that e_{N} is defined as Q is proper in P and hence in N_{P} (Q). We claim that for any pair R and T such that Q < R ≤ T ≤ P, (R, e_{R}) ≤ (T, e_{T}). Indeed, since |T:R| < |P:Q|, by the inductive hypothesis, there exists a block e′_{R} of A(R) such that (R, e′_{R}) ≤ (T, e_{T}). By Lemma 2.11 (b), applied to the groups R, T, P, e′_{R} = e_{R} and (R, e_{R}) ≤ (T, e_{T}) as claimed. A similar argument to the one just given shows that for any R such that Q < R ≤ N, (Q, f) ≤ (R, e_{R}).

We will show that (Q, f) ≤ (P, e). So, let i be a primitive idempotent of A^{P} associated to e. We must show that Br_{Q}^{A}(i)f = Br_{Q}^{A}(i) or equivalently that Br_{Q}^{A}(i)(1 − f) = 0. As a first step, we show that

Br_{Q}^{A}(i)(1 − f) ∈ ∩_{Q<R≤N}Ker(Br_{R}^{A(Q)}) ∩ A(Q)^{N}.

Let R be a subgroup of N properly containing Q. Then, (Q, f) ≤ (R, e_{R}) and Q is normal in R. So, by (b), f is R-stable and Br_{R,Q}^{A}(f)e_{R} = e_{R}. Also, (R, e_{R}) ≤ (P, e), so Br_{R}^{A}(i)e_{R} = Br_{R}^{A}(i) and

Br_{R}^{A}(i)Br_{R,Q}^{A}(f) = Br_{R}^{A}(i)e_{R}Br_{R,Q}^{A}(f) = Br_{R}^{A}(i)e_{R} = Br_{R}^{A}(i).

Since Br_{R}^{A}(i) = Br_{R,Q}^{A}(Br_{Q}^{A}(i)), the above gives that Br_{Q}^{A}(i)(1 − f) ∈ Ker(Br_{R,Q}^{A}) and by Proposition 2.5, Ker(Br_{R,Q}^{A}) = Ker(Br_{R}^{A(Q)}). Since i ∈ A^{P} and f ∈ A(Q)^{N}, it follows that

Br_{Q}^{A}(i)(1 − f) ∈ ∩_{Q<R≤N}Ker(Br_{R}^{A(Q)}) ∩ A(Q)^{N}.

Next, we claim that ∩_{Q<R≤N} Ker(Br_{R}^{A(Q)}) ∩ A(Q)^{N} = Br_{Q}^{A}(Tr_{Q}^{P}(A^{Q})). For this, let B be an N-stable basis of the N_{G}(Q)-algebra A(Q) (see Proposition 2.3). Since Q acts trivially on A(Q), Q ≤ C_{N}(x) for all x ∈ B. Thus, by Lemma 2.2 (d), ∩_{Q<R≤N}Ker(Br_{R}^{A(Q)})∩A(Q)^{N} is spanned by the set {C^{+}}, where C runs over the set of those N-orbits of B for which C_{N}(x) = Q for x ∈ C. Consequently, by Lemma 2.2 (b), ∩_{Q<R≤N}Ker(Br_{R}^{A(Q)}) ∩ A(Q)^{N} = Tr_{Q}^{N}(A(Q)). The claim follows as by Lemma 1.18(b), applied with G = P, Tr_{Q}^{N}(A(Q)) = Br_{Q}^{A}(Tr_{Q}^{P}(A^{Q})).

By what we have shown so far, Br_{Q}^{A}(i)(1 − f) ∈ Br_{Q}^{A}(Tr_{Q}^{P}(A^{Q})). Since Br_{Q}^{A}(i) is an idempotent of A(Q) and (1 − f) is a central element of A(Q), we have Br_{Q}^{A}(i)(1 − f) = Br_{Q}^{A}(i)Br_{Q}^{A}(i)(1 − f)Br_{Q}^{A}(i), hence

Br_{Q}^{A}(i)(1 − f) ∈ Br_{Q}^{A}(i)(Tr_{Q}^{P}(A^{Q}))Br_{Q}^{A}(i) = Br_{Q}^{A}(Tr_{Q}^{P}(iA^{Q}i)).

Here, the last equality follows by Lemma 1.14(g). We claim that Br_{Q}^{A}(Tr_{Q}^{P}(iA^{Q}i)) contains only nilpotent elements. Since (Br_{Q}^{A}(i)(1 − f))^{2} = Br_{Q}^{A}(i)(1 − f), this will prove that Br_{Q}^{A}(i)(1−f) = 0 and complete the proof of part (a). In order to prove the claim, note that since iA^{Q}i = (iAi)^{Q}, by 1.14 (g), applied to the P -algebra iAi, Tr_{Q}^{P}(iA^{Q}i) is an ideal of iA^{P} i. Further, Tr_{Q}^{P}(iA^{Q}i) is a proper ideal of iA^{P} i, since Br_{P} (i) ≠ 0 implies that i ∉ Tr_{Q}^{P}(A^{Q}). Since i is primitive in A^{P}, iA^{P} i is a local ring and any proper ideal of iA^{P} i is nilpotent (see Proposition 1.7). In particular, every element of Tr_{Q}^{P}(iA^{Q}i) is nilpotent and hence so is every element of Br_{Q}^{A}(Tr_{Q}^{P}(iA^{Q}i)). This proves the claim and completes the proof of (a).

It remains to prove (c). Let (Q, f) ≤ (P, e) and let (S, d) ≤ (Q, f). By (a), (S, d′) ≤ (P, e) for some block d′ of A(S). By part (b) of the lemma, d = d′, hence (S, d) ≤ (P, e) as required. □

Definition 2.13. Let (Q, f) and (P, e) be (A, G)-Brauer pairs with Q ≤ P. We write that (Q, f) ⊴ (P, e) if Q is a normal subgroup of P, f is P -stable and Br_{P,Q}^{A}(f)e = e .

Thus, by Theorem 2.10 (b), (Q, f) ⊴ (P, e) if Q is normal in P and (Q, f) ≤ (P, e).

Proposition 2.14. Let (Q, f) and (P, e) be (A, G)-Brauer pairs. The following are equivalent.

(a) (Q, f) ≤ (P, e).

(b) There exist primitive idempotents i ∈ A^{P} and j ∈ A^{Q} such that j ≤ i, i is associated to e and j is associated to f.

(c) There exist (A, G)-Brauer pairs (S_{i}, d_{i}), 1 ≤ i ≤ n such that

(Q, f) ⊴ (S_{1}, d_{1}) ⊴ (S_{2}, d_{2}) ⊴ ⋯ ⊴ (S_{n}, d_{n}) ⊴ (P, e).

Proof. (a ⇒ b) First note that for any primitive idempotent j of A^{Q}, either Br_{Q}^{A}(j) = 0 or Br_{Q}^{A}(j) is a primitive idempotent of A(Q). In particuar, j is associated to f if and only if Br_{Q}^{A}(j)f ≠ 0. Suppose that (Q, f) ≤ (P, e). By Lemma 2.8, there exists a primitive idempotent, say i, of A^{P} associated to e. Let I be a primitive decomposition of i in A^{Q}. Since Br_{Q}^{A}(i)f = Br_{Q}^{A}(i), Br_{Q}^{A}(j)f ≠ 0 for some j ∈ I. Thus, as pointed out above, j is associated to f and since j ∈ I, j ≤ i. So, (b) holds.

(b ⇒ a) Let i and j be as in (b). By part (a) of Theorem 2.10, there exists a block d of A(Q) such that (Q, d) ≤ (P, e). Since ij = j = ji and Br_{Q}^{A}(j)f ≠ 0, Br_{Q}^{A}(i)f ≠ 0. So by part (a) of Lemma 2.11, d = f, hence (a) holds.

(a ⇔ c) This is clear from Theorem 2.10 and the fact that any inclusion of finite p-groups can be refined to a chain of normal inclusions. □

If (P, e) is an (A, G)-Brauer pair and x ∈ G, then (^{x}P, ^{x}e) is again an (A, G)-Brauer pair.

Definition 2.15. Let (P, e) be an (A, G)-Brauer pairs and let x ∈ G. The x-conjugate of (P, e) is the (A, G)-pair (^{x}P, ^{x}e); we set ^{x}(P, e) = (^{x}P, ^{x}e).

An easy “transport of structure” argument shows that if (Q, f) ≤ (P, e), then ^{x}(Q, f) ≤ ^{x}(P, e) for any x ∈ G. Thus, by Theorem 2.10 we have the following.

Theorem 2.16. The set of (A, G)-Brauer pairs is a G-poset via the map sending an (A, G)-Brauer pair (P, e) and an element x ∈ G to the (A, G)- Brauer pair ^{x}(P, e).

2.3. (A, b, G)-Brauer pairs and inclusion.

As before A is a p-permutation G-algebra.

Definition 2.17. Let b be a primitive idempotent of A^{G}. An (A, b, G)- Brauer pair is an (A, G)-Brauer pair (P, e) such that Br_{P}^{A}(b) ≠ 0 and Br_{P}^{A}(b)e ≠ 0.

(A, b, G)-Brauer pairs are called (b, G)-Brauer pairs in [BP2]. We have adopted the more cumbersome notation as we will need to simultaneously consider Brauer pairs for different k-algebras. Note that if A = kG then the primitive idempotents of A^{G} are the blocks of kG. However, the above definition does not require b to be central in A in general.

Lemma 2.18. Let (P, e) be an (A, G)-Brauer pair and let b be a primitive idempotent of A^{G}. If (P, e) is an (A, b, G)-Brauer pair, then there exists a primitive idempotent i of A^{P} such that i ≤ b and i is associated to e. Conversely, if i ∈ A^{P} is an idempotent such that i ≤ b and i is associated to e, then (P, e) is an (A, b, G)-Brauer pair.

Proof. The proof of the first part is as in Lemma 2.8 with 1_{A} replaced by b. Now let i ∈ A^{P} be an idempotent such that i ≤ b and i is associated to e. By the first condition, Br_{P}^{A}(i) = Br_{P}^{A}(i)Br_{P}^{A}(b) and by the second, Br_{P}^{A}(i) = Br_{P}^{A}(i)e. Hence,

0 ≠ Br_{P}^{A}(i) = Br_{P}^{A}(i)eBr_{P}^{A}(b),

from which it follows that Br_{P}^{A}(b)e ≠ 0 and therefore that (P, e) is an (A, b, G)-Brauer pair. □

The set of (A, b, G)-Brauer pairs is a downwardly closed sub-G-poset of the set of (A, G)-Brauer pairs. More precisely,

Lemma 2.19. Let (Q, f) ≤ (P, e) be (A, G)-Brauer pairs and let x ∈ G. If (P, e) is an (A, b, G)-Brauer pair then so are (Q, f) and ^{x}(P, e).

Proof. By the previous lemma, there exists an idempotent i ∈ A^{P} such that i ≤ b and i is associated to e. Since (Q, f) ≤ (P, e), i is also associated to f. But then by the previous lemma applied to Q, (Q, f) is an (A, b, G)-Brauer pair as claimed. Since b ∈ A^{G} (see also Defintion 1.16),

0 ≠ ^{x}(Br_{P}^{A}(b)e) = Br_{^{x}P}^{A} (^{x}b)^{x}e = Br_{^{x}P}^{A} (b)^{x}e,

hence ^{x}(P, e) is also an (A, b, G)-Brauer pair. □

The crucial difference between the G-poset of (A, G)-Brauer pairs and the sub-poset of (A, b, G)-Brauer pairs is the nature of maximal objects. We have the following result, proved in [AB, Theorem 3.10] and [BP2, Theorem 1.14].

Theorem 2.20. Let A be a p-permutation G-algebra and let b be a primitive idempotent of A^{G}. Then,

(a) The group G acts transitively on the set of maximal (A, b, G)-Brauer pairs.

(b) Let (P, e) be an (A, b, G)-Brauer pair. The following are equivalent.

(i) (P, e) is a maximal (A, b, G)-Brauer pair.

(ii) Br_{P}^{A}(b) ≠ 0 and P is maximal amongst subgroups H of G with the property that Br_{H}^{A}(b) ≠ 0.

(iii) b ∈ Tr_{P}^{G}(A^{P}) and P is minimal amongst subgroups H of G such that b ∈ Tr_{H}^{G}(A^{H}).

Proof. (a) By Lemma 1.21, there exist a p-subgroup P of G and a primitive idempotent i of A^{P} such that i ≤ b, Br_{P}^{A}(i) ≠ 0 and b ∈ Tr_{P}^{G}(A^{P} iA^{P}). By Lemma 2.8, there is a unique block, say e, of A(Q) to which i is associated, and by Lemma 2.18, (P, e) is an (A, b, G)-Brauer pair. We will show that for any (A, b, G)-Brauer pair (Q, f), (Q, f) ≤ ^{x}(P, e) for some x ∈ G. This will prove part (a) of the theorem since by Theorem 2.19, the set of (A, b, G)-Brauer pairs is a G-poset. So, let (Q, f) be an (A, b, G)-Brauer pair. By Lemma 2.18, there exists a primitive idempotent j of A^{Q} which is associated to f and such that j ≤ b. But j being associated to e means in particular that Br_{Q}^{A}(j) ≠ 0. Hence, by Lemma 1.20, Q ≤ ^{x}P and ^{u}j ≤ ^{x}i for some x ∈ G and some u ∈ (A^{Q})^{∗}. Now, since i is associated to e, ^{x}i is associated to ^{x}e. Further, since u ∈ (A^{Q})^{∗} and f is central in A(Q), the fact that j is associated to f implies that

Br_{Q}^{A}(^{u}j)f = Br_{Q}^{A}(u)Br_{Q}^{A}(j)Br_{Q}^{A}(u)^{−1}f = Br_{Q}^{A}(u)Br_{Q}^{A}(j)fBr_{Q}^{A}(u)^{−1}
= Br_{Q}^{A}(u)Br_{Q}^{A}(j)Br_{Q}^{A}(u)^{−1},

and hence also that Br_{Q}^{A}(^{u}j)f ≠ 0. Thus, ^{u}j is associated to f and ^{x}i is associated to ^{x}e. By Proposition 2.14, (Q, f) ≤ ^{x}(P, e).

We now prove the equivalences of part (b) of the theorem. First of all note by the proof of (a), that there is a maximal (A, b, G)-Brauer pair (P′, e′) with the following property: there is a primitive idempotent of i′ of A^{P′} such that i′ ≤ b, i′ is associated to e′ and b ∈ Tr_{P′}^{G}(A^{P′}i′A^{P′}). Further, if x ∈ G, then ^{x}i′ is a primitive idempotent of A^{^{x}P′}, ^{x}i′ ≤ b, xi′ is associated to ^{x}e′ and b ∈ Tr_{^{x}P′}^{G}(A^{^{x}P′} ^{x}i′A^{^{x}P′}). Since, as has been just proved, all maximal (A, b, G)-Brauer pairs are G-conjugate, it follows that every maximal (A, b, G)-Brauer pair has this property. In particular, Br_{P′}^{A}(b) ≠ 0 and b ∈ Tr_{P′}^{G}(A^{P′}) for any maximal (A, b, G)-Brauer pair (P′, e′) and the implications, ((ii) ⇒ (i)) and ((iii) ⇒ (i)) are immediate.

Assume that (i) holds and let i ∈ A^{P} be a primitive idempotent such that i ≤ b, i is associated to e and b ∈ Tr_{P}^{G}(A^{P} iA^{P}) . Suppose Q ≥ P is such that Br_{Q}^{A}(b) ≠ 0. Then, by Lemma 1.19 (a) there is a primitive idempotent, say j of A^{Q} such that j ≤ b and Br_{Q}^{A}(j) ≠ 0. By Lemma 1.20, Q ≤ ^{x}P for some x ∈ G. But P ≤ Q. So, P = Q and (ii) holds. Now, let R ≤ P be such that b ∈ Tr_{R}^{G}(A^{R}). By Lemma 1.19, there exists a primitive idempotent t in A^{R} such that t ≤ b and b ∈ Tr_{R}^{G}(A^{R}tA^{R}). Thus, Br_{P}^{A}(i) ≠ 0, b ∈ Tr_{R}^{G}(A^{R}tA^{R}) and i ≤ b, t ≤ b. So, by Lemma 1.20, ^{x}P ≤ R for some x ∈ G. But R ≤ P. Hence P = R and (iii) holds. □

2.4. (A, b, G)-Brauer pairs and fusion systems.

For each maximal (A, b, G)-Brauer pair, fusion of subpairs gives rise to a fusion category on the ambient p-group. More precisely, for (A, b, G)- Brauer pairs (Q, f), (R, u) and X a subgroup of G, we define

Hom_{X}((Q, f), (R, u)) = {φ ∈ Hom(Q, R) : φ = c_{g} for some g ∈ X

such that ^{g}(Q, f) ≤ (R, u)}.

Definition 2.21. Let A be a p-permutation G-algebra, b a primitive idempotent of A^{G} and (P, e_{P}) a maximal (A, b, G)-Brauer pair. For each Q ≤ S, let e_{Q} denote the unique block of A(Q) such that (Q, e_{Q}) ≤ (P, e_{P}). The fusion category of (A, b, G) over (P, e_{P}) is the category F_{(P,e_{P})}(A, b, G) whose objects are the subgroups of P and which has morphism sets

Mor_{F_{(P,e_{P} )}(A,b,G)}(Q, R) = Hom_{G}((Q, e_{Q}), (R, e_{R}))

for Q, R ≤ P, and where composition of morphisms is the usual composition of functions.

We will write Hom_{(A,G,e_{P})}(Q, R) to denote Mor_{F_{(P,e_{P} )}(A,b,G)}(Q, R). Recall the definition of fusion systems from Definition I.2.1.

Proposition 2.22. Let A be a p-permutation G-algebra, b a primitive idempotent of A^{G} and (P, e_{P}) a maximal (A, b, G)-Brauer pair. Then

(a) F_{(P,e_{P})}(A, b, G) is a fusion system over P.

(b) If (P′, e_{P′}) is another maximal (A, b, G)-Brauer pair, then

F_{(P′,e_{P})}(A, b, G) = ^{φ}F_{(P,e_{P})}(A, b, G)

for some group isomorphism φ: P′ → P in Hom_{G}(P′, P).

Proof. (a) Let F = F_{(P,e_{P})}(A, b, G) and for each Q ≤ P, let e_{Q} be the unique block of Z(A(Q)) such that (Q, e_{Q}) ≤ (P, e_{P}). If Q, R, S ≤ P and g, h ∈ G are such that ^{g}(Q, e_{Q}) ≤ (R, e_{R}) and ^{h}(R, e_{R}) ≤ (S, e_{S}), then by Proposition 2.16

hg(Q, e_{Q}) ≤ ^{h}(R, e_{R}) ≤ (S, e_{S}),

so the composition of F-morphisms is an F-morphism. Since the identity homomorphism is clearly an F-morphism for any subgroup Q of P, F is a category.

By Theorem 2.10, if Q, R ≤ P and g ∈ G are such that ^{g}Q ≤ R, then ^{g}(Q, e_{Q}) ≤ (R, e_{R}) if and only if ^{g}(Q, e_{Q}) ≤ (P, e_{P}) and ^{g}(Q, e_{Q}) ≤ (P, e_{P}) if and only if ^{g}e_{Q} = e_{^{g}Q}. In particular, if ^{g}(Q, e_{Q}) ≤ (R, e_{R}), then c_{g}: Q → ^{g}Q is a morphism in F. This shows that each F-morphism is the composition of an F-isomorphism followed by an inclusion. Now let g ∈ P be such that ^{g}Q ≤ R. Since g ∈ P, ^{g}(P, e_{P}) = (P, e_{P}). So by Theorem 2.16

g(Q, e_{Q}) ≤ ^{g}(P, e_{P}) = (P, e_{P}).

So, by the remark above ^{g}(Q, e_{Q}) ≤ ^{g}(R, e_{R}) and it follows that c_{g}: Q → R is a morphism in F. This shows that Hom_{P} (Q, R) ≤ Hom_{F}(Q, R). This proves that F is a fusion system on P.

(b) This is immediate from the fact that all maximal (A, b, G)-Brauer pairs are G-conjugate (see Theorem 2.20). □

3. p-permutation algebras and saturated fusion systems

Throughout this section, A will denote a p-permutation G-algebra, and b a primitive idempotent of A^{G}. As seen above the category F_{(P,e_{P})}(A, b, G) is a fusion system on P for any maximal (A, b, G)-Brauer pair (P, e_{P}). This section presents a criterion for F_{(P,e_{P})}(A, b, G) to be saturated.

For an (A, b, G)-Brauer pair, (Q, e), let C_{G}(Q, e) denote the subgroup of C_{G}(Q) stabilising the pair (Q, e) and let N_{G}(Q, e) denote the subgroup of N_{G}(Q) stabilising the pair (Q, e).

3.1. Saturated triples.

By definition, if (Q, e) ia an (A, b, G)-Brauer pair, then e is a block of A(Q). Now A(Q) is an N_{G}(Q)-algebra, hence a C_{G}(Q, e)-algebra and by definition of C_{G}(Q, e_{Q}), e ∈ A(Q)^{C_{G}(Q,e)}.

Definition 3.1. The triple (A, b, G) is a saturated triple if the following two conditions hold.

• b is a central idempotent of A.

• For each (A, b, G)-Brauer pair (Q, e) the idempotent e is primitive in A(Q)^{C_{G}(Q,e)}.

The reason for the above terminology is the following result.

Theorem 3.2. Suppose that (A, b, G) is a saturated triple. Then, F_{(P,e_{P})}(A, b, G) is a saturated fusion system on P for any maximal (A, b, G)-Brauer pair (P, e_{P}).

The proof requires some lemmas.

Lemma 3.3. Let H be a finite group, B a p-permutation H-algebra and e a primitive idempotent of B^{H}. If e ∈ Z(B), then for a p-subgroup Q of H and a block f of B(Q), (Q, f) is an (B, e, H)-Brauer pair if and only if Br_{Q}^{B}(e)f = f.

Proof. Suppose that e ∈ Z(B) and let Q be a p-subgroup of H. Since

Z(B) ∩ B^{H} ⊆ Z(B) ∩ B^{Q} ⊆ Z(B^{Q}),

e is a central idempotet of B^{Q}. Hence, either Br_{Q}^{B}(e) = 0 or Br_{Q}^{B}(e) is a central idempotent of B(Q) and for any block f of B(Q), either Br_{Q}^{B}(e)f = f, or Br_{Q}^{B}(e)f = 0. The result follows. □

Recall that for an (A, b, G)-Brauer pair (Q, e), A(Q) is a N_{G}(Q)-algebra and e is an idempotent of A(Q)^{N_{G}(Q,e)}. Thus, if e is primitive in A(Q)^{C_{G}(Q,e)}, then e is a primitive idempotent of A(Q)^{H} for any H such that C_{G}(Q, e) ≤ H ≤ N_{G}(Q, e) and it makes sense to speak of (A(Q), e, H)-Brauer pairs. The next result compares (A(Q), e, H)-Brauer pairs for different H.

Lemma 3.4. Suppose that (Q, e) is an (A, b, G)-Brauer pair such that e is primitive in A(Q)^{C_{G}(Q,e)} and let H be a subgroup of G with C_{G}(Q, e) ≤ H ≤ N_{G}(Q, e).

(a) The H-poset of (A(Q), e, H)-Brauer pairs is the H-subposet of (A(Q), e, N_{G}(Q, e))-Brauer pairs consisting of those pairs whose first component is contained in H.

(b) The map

(R, α) → (QR, α)

is an H-poset map from the set of (A(Q), e, H)-Brauer pairs to the set of (A(Q), e, QH)-Brauer pairs and induces a bijection between the set of (A(Q), e, H)-Brauer pairs whose first component contains Q∩H and the set of (A(Q), e, QH)-Brauer pairs whose first component contains Q.

(c) If Q ≤ H, then (Q, e) is the unique (A(Q), e, H)-Brauer pair with first component Q and (Q, e) is contained in every maximal (A(Q), e, H)- Brauer pair.

Proof. (a) This is immediate from the definitions.

(b) Since Q acts trivially on A(Q), for any p-subgroup R of H, A(Q)^{R} = A(Q)^{QR} and Br_{R}^{A(Q)} = Br_{QR}^{A(Q)}. This proves the first assertion. The second follows from the first and the fact that R → QR is a bijection between subgroups of H containing Q ∩ H and subgroups of QH containing Q.

(c) By definition, A(Q)^{Q} = A(Q), A(Q)_{<Q}^{Q} = 0, and Br_{Q}^{A(Q)} is the identity map on A(Q). Thus, the set of (A(Q), e, H)-Brauer pairs with first component Q consists precisely of the pairs (Q, α), where α is a block of A(Q) such that eα ≠ 0. Since e itself is a block of A(Q) and any two distinct blocks of A(Q) are orthogonal, it follows that (Q, e) is an (A(Q), e, H)- Braur pair and that it is the unique one with first component Q. Since h(Q, e) = (Q, e) for all h ∈ H and by Theorem 2.20(a) H acts transitively on the set of maximal (A(Q), e, H)-Brauer pairs, (Q, e) is contained in every maximal (A(Q), e, H)-Brauer pair. □

The proof of Thoerem 3.2 follows the lines of the proof of Theorem I.2.3 with Brauer pairs playing the role of p-subgroups and Theorem 2.20 and Lemma 1.22 playing the role of Sylow’s theorem and Lemma A.3. The isomorphisms b_{R,Q}^{A}: A(Q)(R) → A(R) for p-subgroups Q ⊴ R of G introduced in Definition 2.6 allow us to pass back and forth between (A, b, G)- Brauer pairs and (A(Q), e, H)-Brauer pairs for subgroups H of G such that QC_{G}(Q, e) ≤ H ≤ N_{G}(Q, e).

Lemma 3.5. Suppose that (Q, e) is an (A, b, G)-Brauer pair such that e is primitive in (A(Q))^{C_{G}(Q,e)} and let H be a subgroup of G with QC_{G}(Q, e) ≤ H ≤ N_{G}(Q, e).

The map

(R, α) → (R, b_{R,Q}^{A}(α))

is an H-poset equivalence between the subset of (A(Q), e, H)-Brauer pairs consisting of those pairs whose first component contains Q, and the subset of (A, b, G)-Brauer pairs containing (Q, e) and whose first component is contained in H.

In particular, H acts transitively on the subset of (A, b, G)-Brauer pairs which are maximal among those containing (Q, e) and having first component contained in H.

Proof. Let P_{1} be the subset of (A(Q), e, H)-Brauer pairs consisting of those pairs whose first component contains Q, and let P_{2} be the subset of (A, b, G)-Brauer pairs containing (Q, e) and whose first component is contained in H. Since H ≤ N_{G}(Q, e) ≤ N_{G}(Q), P_{1} and P_{2} are H-posets. Now let Q ≤ R ≤ H, and let α be a block of A(Q)(R). By Lemma 3.3, e = Br_{Q}^{A}(b)e, hence

Br_{R,Q}^{A}(e) = b_{R,Q}^{A}(Br_{R}^{A(Q)}(e)) = b_{R,Q}^{A}(Br_{R}^{A(Q)}(Br_{Q}^{A}(b)e)) = Br_{R}^{A}(b)Br_{R,Q}^{A}(e).

Suppose first that (R, α) is an (A(Q), e, H)-Brauer pair. By Lemma 3.3, α = Br_{R}^{A(Q)}(e)α. Applying b_{R,Q}^{A} to both sides of this equation, and using the displayed equation above, we get that

b_{R,Q}^{A}(α) = Br_{R,Q}^{A}(e)b_{R,Q}^{A}(α) = Br_{R}^{A}(b)Br_{R,Q}^{A}(e)b_{R,Q}^{A}(α).

In particular, Br_{R}^{A}(b)b_{R,Q}^{A}(α) ≠ 0, whence (R, b_{R,Q}^{A}(α)) is an (A, b, G)- Brauer pair. By Theorem 2.10(b) and the first equality above, (Q, e) ≤ (R, b_{R,Q}^{A}(α)) as (A, b, G)-Brauer pairs.

Conversely, if (Q, e) ≤ (R, b_{R,Q}^{A}(α)), then by Theorem 2.10(b), b_{R,Q}^{A}(α) = Br_{R,Q}^{A}(e)b_{R,Q}^{A}(α). Applying the inverse of b_{R,Q}^{A} yields that α = Br_{R}^{A(Q)}(e)α, hence that (R, α) is an (A(Q), e, H)-Brauer pair. This shows that (R, α) → (R, b_{R,Q}^{A}(α)) is a bijection between P_{1} and P_{2}. Since Q is normal in H,

b_{^{h}R,Q}^{A}(^{h}α) = b_{^{h}R,^{h}Q}^{A}(^{h}α) = ^{h}b_{R,Q}^{A}(α)

for all h ∈ H, all p-subgroups R of G containing Q as normal subgroup and all α ∈ A(Q)(R), hence the bijection is compatible with the H-action on P_{1} and P_{2}.

We show that the bijection is inclusion preserving. Let (R, α) and (S, β) be (A(Q), e, H)-Brauer pairs with Q ≤ R ≤ S. By Proposition 2.14, it suffices to consider the case that R ⊴ S. By the equation displayed above, α is S-stable if and only if b_{R,Q}^{A}(α) is S-stable. Further, the restrictions of the maps b_{S,Q}^{A} ∘ Br_{S,R}^{A(Q)} ∘ Br_{R}^{A(Q)} ∘ Br_{Q}^{A} and Br_{S,R}^{A} ∘ b_{R,Q}^{A} ∘ Br_{R}^{A(Q)} ∘ Br_{Q}^{A} to

A^{S} both equal Br_{S}^{A}. Since Br_{R}^{A(Q)} ∘ Br_{Q}^{A}(A^{S}) = [A(Q)(R)]^{S}, it follows that b_{S,Q}^{A} ∘Br_{S,R}^{A(Q)} is equal to the restriction of Br_{S,R}^{A} ∘b_{R,Q}^{A} to [A(Q)(R)]^{S}. Since b_{S,Q}^{A} is invertible, it follows that Br_{S,R}^{A}(b_{R,Q}^{A}(α))b_{S,Q}^{A}(β) = b_{S,Q}^{A}(β) if and only if Br_{S,R}^{A(Q)}(α)β = β Thus, by Theorem 2.10 (R, b_{R,Q}^{A}(α)) ≤ (S, b_{S,Q}^{A}(β)) if and only if (R, α) ≤ (S, β). This proves that the bijection is an H-poset equivalence between P_{1} and P_{2}.

The given map induces a bijection between the set of maximal elements of P_{1} and P_{2}. But by Lemma 3.4 (c), the set of maximal elements in P_{1} is precisely the set of maximal (A(Q), e, H)-Brauer pairs. The final assertion follows from this and from the fact that H acts transitively on the set of maximal (A(Q), e, H)-pairs (see 2.20 (a)). □

Lemma 3.6. Suppose that (A, b, G) is a saturated triple and let (P, e_{P}) be a maximal (A, b, G)-Brauer pair. For each Q ≤ P let e_{Q} be the unique block of A(Q) such that (Q, e_{Q}) ≤ (P, e_{P}) and let F = F_{(P,e_{P})}(A, b, G). If Q ≤ P is such that (N_{P} (Q), e_{N_{P} (Q)}) is maximal amongst (A, b, G)-Brauer pairs (R, f) with (Q, e_{Q}) ≤ (R, f) and R ≤ N_{G}(Q, e_{Q}), then Q is fully F-automised and F-receptive.

Proof. Suppose that (N_{P} (Q), e_{N_{P} (Q)}) is maximal amongst (A, b, G)-Brauer pairs (R, f) such that (Q, e_{Q}) ≤ (R, f) and R ≤ N_{G}(Q, e_{Q}). Let α = b_{N_{P} (Q),Q}^{A}(e_{N_{P} (Q)}). By Lemma 3.5, (N_{P} (Q), α) is a maximal (A(Q), e_{Q}, N_{G}(Q, e_{Q}))-

Brauer pair. Thus, by Theorem 2.20 (b), e_{Q} ∈ Tr_{N_{P} (Q)}^{N_{G}(Q,e_{Q})}(A(Q)^{N_{P} (Q)}). Since e_{Q} is central in A(Q), idempotent and an element of N_{G}(Q, e_{Q}), multiplying on both sides by e_{Q} gives that

e_{Q} ∈ Tr_{N_{P} (Q)}^{N_{G}(Q,e_{Q})}((e_{Q}A(Q)e_{Q})^{N_{P} (Q)}).

Now, C_{G}(Q, e_{Q}) is a normal subgroup of N_{G}(Q, e_{Q}) and since (A, b, G) is a saturated triple e_{Q} is a primitive idempotent of (A(Q))^{C_{G}(Q,e_{Q})} and hence of (e_{Q}A(Q)e_{Q})^{C_{G}(Q,e_{Q})}. Thus, by Lemma 1.22 applied to the N_{G}(Q, e_{Q})- algebra e_{Q}A(Q)e_{Q}, and with R = N_{P} (Q) and C = C_{G}(Q, e_{Q}), we have that N_{P} (Q)C_{G}(Q, e_{Q})/C_{G}(Q, e_{Q}) is a Sylow p-subgroup of N_{G}(Q, e_{Q})/C_{G}(Q, e_{Q}). Since N_{P} (Q)C_{G}(Q, e_{Q})/C_{G}(Q, e_{Q}) ≅ N_{P} (Q)/C_{P} (Q) ≅ Aut_{P} (Q) and N_{G}(Q, e_{Q})/C_{G}(Q, e_{Q}) ≅ Aut_{F}(Q), it follows that Q is fully F-automised.

It remains to show that Q is F-receptive. For this, we first observe that the hypothesis on Q implies that (N_{P} (Q), e_{N_{P} (Q)}) is also maximal amongst (A, b, G)-Brauer pairs (R, f) such that (Q, e_{Q}) ≤ (R, f) and R ≤ N_{P} (Q)C_{G}(Q, e_{Q}). Hence, by Lemma 3.5, now applied with H = N_{P} (Q)C_{G}(Q, e_{Q}), (N_{P} (Q), e_{N_{P} (Q)}) contains an N_{P} (Q)C_{G}(Q, e_{Q}) conjugate of any (A, b, G)- Brauer pair which contains (Q, e_{Q}) and whose first component is contained in N_{P} (Q)C_{G}(Q, e_{Q}). Now let φ: R → Q be an isomorphism in F, and let g ∈ G induce φ, that is, ^{g}(R, e_{R}) = (Q, e_{Q}) and φ(x) = gxg^{−1} for all x ∈ R. Then, it is an easy check that N_{φ} = N_{P} (R) ∩ ^{g^{−1}}N_{P} (Q)C_{G}(Q, e_{Q}). Set N′ = ^{g}N_{φ} = ^{g}N_{P} (R) ∩ N_{P} (Q)C_{G}(Q, e_{Q}), set e′_{N′} = ^{g}e_{N_{φ}} and consider the (A, b, G)-Brauer pair (N′, e′_{N′}). Since (R, e_{R}) ≤ (N_{φ}, e_{N_{φ}}), (Q, e_{Q}) ≤ ^{g}(N_{φ}, e_{N_{φ}}) = (N′, e′_{N′}). Also, N′ ≤ N_{P} (Q)C_{G}(Q, e_{Q}). Thus, by what was pointed out above, ^{h}(N′, e′_{N′}) ≤ (N_{P} (Q), e_{N_{P} (Q)}) for some h ∈ N_{P} (Q)C_{G}(Q, e_{Q}). Multiplying by some element of N_{P} (Q) if necessary, we may assume that h ∈ C_{G}(Q, e_{Q}). Since ^{hg}(N_{φ}, e_{N_{φ}}) ≤ (P, e_{P}), φ̄: = c_{hg}: N_{φ} → P is a morphism in F and since h ∈ C_{G}(Q, e_{Q}), φ̄ extends φ. Thus Q is F-receptive. □

We now give the proof of Theorem 3.2.

Proof. Keep the notation of the theorem, set F = F_{(P,e_{P})}(A, b, G) and for each Q ≤ P, let e_{Q} be the unique block of A(Q) such that (Q, e_{Q}) ≤ (P, e_{P}). We have shown in Proposition 2.22 that F is a fusion system on P. Thus, by Lemma 3.6 it suffices to show that each subgroup of P is F-conjugate to a subgroup Q of P such that (N_{P} (Q), e_{N_{P} (Q)}) is maximal amongst (A, b, G)- Brauer pairs (R, f) with (Q, e_{Q}) ≤ (R, f) and R ≤ N_{G}(Q, e_{Q}). So, let Q′ ≤ P, and let (T, α) be a maximal (A(Q′), e_{Q′}, N_{G}(Q′, e_{Q′}))-Brauer pair. By Lemma 3.4 (c), Q′ ≤ T. Set f = b_{R,Q}^{A^{−1}}(α). By Lemma 3.5, (T, f) is an (A, b, G)-Brauer pair with (Q′, e_{Q′}) ≤ (T, f). Since (P, e_{P}) is a maximal (A, b, G)-Brauer pair, we have

g(Q′, e_{Q′}) ≤ ^{g}(T, f) ≤ (P, e_{P})

for some g ∈ G. Set Q = ^{g}Q′. By the above, c_{g}: Q′ → Q is a morphism in F, so Q is F-conjugate to Q′. We will show that (N_{P} (Q), e_{N_{P} (Q)}) has the required maximality property. Note that by Lemma 3.5, (T, f) is maximal amongst (A, b, G)-Brauer pairs which contain (Q′, e_{Q′}) and whose first component is contained in N_{G}(Q′, e_{Q′}). Thus, by transport of structure ^{g}(T, f) is maximal amongst (A, b, G)-Brauer pairs which contain (Q, e_{Q}) and whose first component is contained in N_{G}(Q, e_{Q}). Since ^{g}(T, f) ≤ (P, e_{P}), ^{g}T ≤ N_{P} (Q) and ^{g}f = e_{^{g}T}. Consequently, ^{g}(T, f) ≤ (N_{P} (Q), e_{N_{P} (Q)}). Since (N_{P} (Q), e_{N_{P} (Q)}) contains (Q, e_{Q}) and N_{P} (Q) is contained in N_{G}(Q, e_{Q}), the maximality of ^{g}(T, f) forces ^{g}(T, f) = (N_{P} (Q), e_{N_{P} (Q)}). This completes the proof of the theorem. □

3.2. Normaliser systems and saturated triples.

The notion of saturated triples is compatible with passage to local subgroups and normal subgroups. Just as in the previous subsection, the arguments to prove the results below are essentially the same as for the analogous results for fusion systems of finite groups, with p-subgroups replaced by Brauer pairs. Lemmas 3.4 and 3.5 again form the technical backbone.

Proposition 3.7. If (A, b, G) is a saturated triple, then so is (A(Q), e, H) for any (A, b, G)-Brauer pair (Q, e) and any subgroup H of G with C_{G}(Q, e) ≤ H ≤ N_{G}(Q, e).

Proof. Suppose that (A, b, G) is a saturated triple, let (Q, e) be an (A, b, G)- Brauer pair and let H ≤ G be such that C_{G}(Q, e) ≤ H ≤ N_{G}(Q, e). Note that by the remarks before Lemma 3.4, it makes sense to speak of (A(Q), e, H)-Brauer pairs. Also, the first condition of Definition 3.1 holds as by definition of Brauer pairs, e is central in A(Q). It remains to show that for any (A(Q), e, H)-Brauer pair (R, α), α is primitive in A(Q)(R)^{C_{H}(R,α)}. By Lemma 3.4(b), (QR, α) is an (A(Q), e, QH)-Brauer pair. Further,

C_{H}(R, α) ≥ C_{H}(QR, α) = C_{QH}(QR, α),

the last equation holding since H ≥ C_{QH}(Q). Hence

[A(Q)(R)]^{C_{H}(R,α)} ⊆ [A(Q)(QR)]^{C_{QH}(QR,α)}.

This shows that we may assume that Q ≤ H and that Q ≤ R. Set f = b_{R,Q}^{A}(α). By Lemma 3.5, (R, f) is an (A, b, G)-Brauer pair and by hypothesis (A, b, G) is a saturated triple. So f is primitive in (A(R))^{C_{G}(R,f)}, and since C_{G}(R, f) = C_{H}(R, f), f is primitive in (A(R))^{C_{H}(R,f)}. Since b_{R,Q} is an N_{G}(Q)∩N_{G}(R)-algebra homomorphism and C_{H}(R) ≤ N_{G}(Q)∩N_{G}(R), it follows that α is primitive in [A(Q)(R)]^{C_{H}(R,α)} as required. □

If (A, b, G) is a saturated triple, then by Theorem 3.2 and the above proposition, for any (A, b, G)-Brauer pair (Q, e), any subgroup H of G with C_{G}(Q, e) ≤ H ≤ N_{G}(Q, e), and any maximal (A(Q), e, H)-Brauer pair (N, α), F_{(N,α)}(A(Q), e, H) is a saturated fusion system on N. The following is an analogue of Proposition I.5.4.

Proposition 3.8. Assume that (A, b, G) is a saturated triple. Let (P, e_{P}) be a maximal (A, b, G)-Brauer pair and let F = F_{(P,e_{P})}(A, b, G). For any R ≤ P, let e_{R} be the unique block of A(R) with (R, e_{R}) ≤ (P, e_{P}). Let Q be a subgroup of P, H a subgroup of G with C_{G}(Q, e_{Q}) ≤ H ≤ N_{G}(Q, e_{Q}) and K = Aut_{H}(Q). Let α be the block of A(Q)(QN_{P}^{K}(Q)) which satisfies b_{QN_{P}^{K}(Q),Q}^{A}(α) = e_{QN_{P}^{K}(Q)}. Then Q is fully K-normalised if and only if (N_{P}^{K}(Q), α) is a maximal (A(Q), e_{Q}, H)-Brauer pair. If this is the case, then

N_{F}^{K}(Q) = F_{(N_{P}^{K}(Q),α)}(A(Q), e_{Q}, H).

Proof. By Lemma 3.4 (b), (N_{P}^{K}(Q), α) is a maximal (A(Q), e_{Q}, H)-Brauer pair if and only if (QN_{P}^{K}(Q), α) is a maximal (A(Q), e_{Q}, QH)-Brauer pair. On the other hand, by Proposition I.5.2 (equivalence of (a) and (b)), Q is fully K-normalised if and only if Q is fully Aut_{Q}(Q)K-normalised. Since clearly, Aut_{Q}(Q)K = Aut_{QH}(Q), it follows that in order to prove the proposition, we may (and will) assume that Q ≤ H and therefore also that Q ≤ N_{P}^{K}(Q).

By Lemma 3.5, it suffices to prove that Q is fully K-normalised if and only if (N_{P}^{K}(Q), e_{N_{P}^{K}(Q)}) is maximal amongst (A, b, G)-Brauer pairs which contain (Q, e_{Q}) and whose first component is contained in H.

Amongst the (A, b, G)-Brauer pairs which are maximal with respect to containing (Q, e_{Q}) and having first component contained in H, choose one, say (R, f) containing (N_{P}^{K}(Q), e_{N_{P}^{K}(Q)}). Then, it suffices to prove that Q is fully K-normalised if and only if (N_{P}^{K}(Q), e_{N_{P}^{K}(Q)}) = (R, f). Since

(Q, e_{Q}) ≤ (N_{P}^{K}(Q), e_{N_{P}^{K}(Q)}) ≤ (R, f),

by Theorem 2.20 there exists g ∈ G such that

g(Q, e_{Q}) ≤ ^{g}(N_{P}^{K}(Q), e_{N_{P}^{K}(Q)}) ≤ ^{g}(R, f) ≤ (P, e_{P}).

If we set τ = c_{g}: Q → P, then τ is a morphism in F and ^{g}R ≤ N_{P}^{^{τ} K}(τ(Q)). So, if Q is fully K-normalised then |N_{P}^{K}(Q)| ≥ |N_{P}^{^{τ} K}(τ(Q))| and it follows from the above that (N_{P}^{K}(Q), e_{N_{P}^{K}(Q)}) = (R, f). Conversely, let t ∈ G be such that ^{t}(Q, e_{Q}) ≤ (P, e_{P}) and set φ = c_{t}: Q → P, a morphism in F. Then,

(φ(Q), e_{φ(Q)}) = (^{t}Q, e_{^{t}Q}) = ^{t}(Q, e_{Q}) ≤ ^{t}(N_{P}^{K}(Q), e_{N_{P}^{K}(Q)}) ≤ ^{t}(R, f)

and ^{t}(R, f) is maximal amongst (A, b, G)-Brauer pairs which contain (^{t}Q, e_{^{t}Q}) and whose first component is contained in ^{t}H. Now, φ(Q) ≤ N_{P}^{^{φ}K}(φ(Q)) ≤ P, hence there is an (A, b, G)-Brauer pair which contains (φ(Q), e_{φ(Q)}) and whose first component is N_{P}^{^{φ}K}(φ(Q)). Since

N_{P}^{^{φ}K}(φ(Q)) = N_{P} (φ(Q)) ∩ ^{t}H

and ^{t}(R, f) is maximal amongst (A, b, G)-Brauer pairs which contain (^{t}Q, e_{^{t}Q}) and whose first component is contained in ^{t}H, by Lemma 3.5 |N_{P}^{^{φ}K}(φ(Q))| ≤ |^{t}R| = |R|. Thus, if (N_{P}^{K}(Q), e_{N_{P}^{K}(Q)}) = (R, f), then Q is fully Knormalised.

Now assume that (N_{P}^{K}(Q), α) is a maximal (A(Q), e_{Q}, H)-Brauer pair. For each R ≤ N_{P}^{K}(Q), let α_{R} be the unique block of A(R) such that (R, α_{R}) ≤ (N_{P}^{K}(Q), α), and let N = F_{(N_{P}^{K}(Q),α)}(A(Q), e_{Q}, H) be the corresponding saturated fusion system. Note that α = α_{N_{P}^{K}(Q)}. For subgroups, R, T ≤ N_{P}^{K}(Q), Hom_{N} (R, T) = Hom_{H}((R, α_{R}), (T, α_{T})), and it is an easy check that also Hom_{N_{F}^{K}(Q)}(R, T) = Hom_{H}((R, e_{R}), (T, e_{T})). Thus, by Lemma 3.4 (a) in order to prove the second assertion of the lemma it suffices to show that if R, T ≤ N_{P}^{K}(Q) and h ∈ N_{G}(Q, e_{Q}) are such that hR ≤ T, then ^{h}(R, e_{R}) ≤ (T, e_{T}) if and only ^{h}(R, α_{R}) ≤ (T, α_{T}).

So, let R, T and h be as above. Then, ^{h}(R, e_{R}) ≤ (T, e_{T}) if and only if ^{h}(QR, e_{QR}) ≤ (RT, e_{RT}) and similarly ^{h}(R, α_{R}) ≤ (T, α_{T}) if and only if ^{h}(QR, α_{QR}) ≤ (RT, α_{RT}). So we may assume that Q ≤ R, T. By definition e_{N_{P}^{K}(P)} = Br_{N_{P}^{K}(P),Q}^{A}(α). Hence by Lemma 3.5, e_{X} = b_{X,Q}^{A}(α_{X}) for X = R, T and ^{h}(R, e_{R}) ≤ (T, e_{T}) if and only ^{h}(R, α_{R}) ≤ (T, α_{T}) as required. □

3.3. Saturated triples and normal subgroups.

Suppose that N is a subgroup of G such that b is also primitive in A^{N}. Then, it makes sense to speak of (A, b, N)-Brauer pairs. Further, the Nposet of (A, b, N)-Brauer pairs is just the N sub-poset of (A, b, G)-Brauer pairs consisting of those pairs whose first component is contained in N, and this leads to an obvious inclusion of the corresponding fusion systems. The next result shows that If (A, b, G) and (A, b, N) are both saturated triples, and if N is normal in G, then this inclusion is normal.

Proposition 3.9. Suppose that N is a normal subgroup of G, b is primitive in A^{N} and (A, b, G) and (A, b, N) are both saturated triples. Let (P, e_{P}) be a maximal (A, b, G)-Brauer pair and for each Q ≤ P, let e_{Q} be the unique block of A(Q) such that (Q, e_{Q}) ≤ (P, e_{P}). Then (P ∩ N, e_{P ∩N}) is a maximal (A, b, N)-Brauer pair. Further, if F = F_{(P,e_{P})}(A, b, G) and E = F_{(P ∩N,e_{P ∩N})}(A, b, N), then E is a normal subsystem of F.

Proof. Set T = P ∩ N. Clearly, (T, e_{T}) is a (A, b, N)-Brauer pair. Suppose if possible that (T, e_{T}) ≤ (R, f) for some (A, b, N)-Brauer pair such that R contains T properly. By Theorem 2.20 (a), there exists g ∈ G such that g(R, f) ≤ (P, e_{P}). Since N is normal in G and R ≤ N, ^{g}R ≤ P ∩ N = T, a contradiction since |R| is strictly larger than |T |. Thus, (T, e_{T}) is a maximal (A, b, G)-brauer pair and this proves the first assertion.

Clearly, T is strongly F-closed and ^{α}E = E for any α ∈ Aut_{F}(T). Also, G acts on the set of maximal (A, b, N)-Brauer pairs and by Theorem 2.20 (a), N acts transitively on the set of maximal (A, b, N)-Brauer pairs. Thus, by the Frattini argument, G = N_{G}(T, e_{T})·N. Let g ∈ G be such that g(Q, e_{Q}) ≤ (R, e_{R}) for some Q, R ≤ T. Then g = xn, for some n ∈ N and some x ∈ N_{G}(T, e_{T}). Since ^{x}(Q, e_{Q}) ≤ ^{x}(T, e_{T}) = (T, e_{T}), it follows that n(Q, e_{Q}) ≤ (T, e_{T}). Thus, c_{n}: Q → T is a morphism in E and c_{g}: Q → R factors as a morphism in E followed by one in Aut_{F}(T). This proves the Frattini condition in Definition I.6.1.

Since T is strongly F-closed in P, T is fully F-centralised. Hence by Proposition 3.8, (C_{P} (T), α) is a maximal (A(T), e_{T} , C_{G}(T, e_{T}))-Brauer pair, where α is defined by e_{T C_{P} (T)} = b_{T C_{P} (T),T}^{A} (α). Consequently, (C_{T} (P), α) is also a maximal (A(T), e_{T} , C_{N}(T, e_{T})C_{P} (T))-Brauer pair. Now by the argument used above for the Frattini condition applied to C_{N}(T, e_{T})C_{P} (T)⊴ N_{N}(T, e_{T})C_{P} (T) and the set of maximal (A(T), e_{T} , C_{N}(T, e_{T})C_{P} (T))-Brauer pairs,

N_{N}(T, e_{T})C_{P} (T) = C_{N}(T, e_{T})C_{P} (T)·N_{N_{N} (T,e_{T})}(C_{P} (T), α).

Thus for each g ∈ N_{N}(T, e_{T}), g = an, where a ∈ C_{NP} (T, e_{T}) and n ∈ N_{N_{N} (T,e_{T})}(C_{P} (T), α). But then n is also clearly in N_{N_{N} (T,e_{T})}(T C_{P} (T), α) and it follows by Lemma 3.5 that n ∈ N_{N_{N} (T,e_{T})}(T C_{P} (T), e_{T C_{P} (T)}). Thus, c_{g} ∈ Aut_{E}(T) extends to c_{n} ∈ Aut_{F}(T C_{P} (T)) where [c_{n}, C_{P} (T)] ∈ C_{P} (T)∩ N = Z(T) as T = P ∩N. This proves the extension condition of Definition I.6.1, and finishes the proof. □

3.4. Block fusion systems.

Let N be a normal subgroup of G. Consider kN as a G-algebra as in Example 1.12. The action of G on kN induces an action of G on Z(kN) and since N ≤ G, (kN)^{G} = Z(kN)^{G}. The following is an immediate consequence of Lemma 2.12, applied with H = G and B = Z(kN).

Proposition 3.10. The set of primitive idempotents of (kN)^{G} is equal to the set {J^{+}} as J runs over the orbits of G on the set of blocks of kN. In particular, the primitive idempotents of (kG)^{G} are precisely the blocks of kG.

Since the k-basis of kN consisting of the elements of N is G-stable, kN is a p-permutation G-algebra. Applying Lemma 2.2 with B = N allows us to identify the homomorphism Br_{P}^{kN}, for P a p-subgroup of G, with the Brauer homomorphism as orginally defined by Brauer.

Definition 3.11. Let P be a p-subgroup of G. Let br_{P}: (kG)^{P} → kC_{G}(P) be the map which sends an element ∑_{x∈G} α_{x}x of kG to the element ∑_{x∈C_{G}(P)} α_{x}x of kC_{G}(P).

The map br_{P} is clearly k-linear and surjective and should be thought of as the truncation map which cuts off the support of an element of kG outside of C_{G}(P). The next result identifies Br_{P}^{kN} with br_{P}. Note that for any subgroup P of G, C_{N}(P) ⊆ (kN)^{P}, and since C_{N}(P) is normalized by N_{G}(P), kC_{N}(P) is a kN_{G}(P)-subalgebra of (kN)^{P}.

Proposition 3.12. Let P be a p-subgroup of G.

(a) The composition of the inclusion kC_{N}(P) → (kN)^{P} with Br_{P}^{kN}: (kN)^{P} → kN(P) induces an isomorphism of kN_{G}(P)-algebras between kC_{N}(P) and kN(P).

(b) If kN(P) is identified with kC_{N}(P) via the isomorphism of (a), then the Brauer homomorphism Br_{P}^{kN} is identified with the restriction of br_{P} to (kN)^{P}.

Proof. Consider the G-invariant basis N of kN. The orbits of P on N are the P -conjugacy classes of N. A P -conjugacy class of N is a singleton {x} if and only if x ∈ C_{N}(P). In other words, the set of fixed points of P on N is C_{N}(P) and all P -class sums of N outside of C_{N}(P) are in the kernel of Br_{P}^{A}. The result follows from Lemma 2.2. □

From now on, for any p-subgroup P of G, we shall identify kN(P) with kC_{N}(P) and Br_{P}^{kN} with br_{P}. Further, if Q is a normal subgroup P, then applying the above discussion to the normal subgroup C_{N}(Q) of N_{G}(Q), we may, and will identify kC_{N}(Q)(P) with kC_{N}(P), and consequently Br_{P}^{kC_{N} (Q)} with the restriction of br_{P} to (kC_{N}(Q))^{P} and Br_{P,Q}^{kN} with the identity map. These identifications allow for an alternate, easier description of the poset of Brauer pairs, which is given in the next proposition. Note that since by Proposition 2.14, inclusion of Brauer pairs is just the transitive extension of the “subnormality” relation ⊴, part (b) below describes inclusion of Brauer pairs completely.

Proposition 3.13. Let b be a primitive idempotent of (kN)^{G}.

(a) A (kN, G)-Brauer pair is a pair (P, e), where P is a p-subgroup of G and e is a block of kC_{N}(P).

(b) If (Q, f) and (P, e) are (kN, G)-Brauer pairs with Q⊴P, then (Q, f) ≤ (P, e) if and only if f ∈ (kC_{N}(Q))^{P} and br_{P} (f)e = e.

(c) A (kN, G)-Brauer pair (P, e) is a (kN, b, G)-Brauer pair if and only if br_{P} (b)e ≠ 0, and this in turn is equivalent to br_{P} (b)e = e.

(d) Any (kN, G)-Brauer pair is a (kN, b, G)-Brauer pair for a unique primitive idempotent of (kN)^{G}.

(e) The pair ({1}, b) is a (kG, G)-Brauer pair and a (kG, b)-Brauer pair (P, e) is a (kG, b, G)-Brauer pair if and only if ({1}, b) ≤ (P, e) as (kG, G)-Brauer pairs.

Proof. (a) and (b) and the first part of (c) are immediate given the identifications specified above. Since (kN)^{G} ⊆ Z(kN), the second assertion of (c) follows from Lemma 3.3. By Proposition 3.10 any two distinct primitive idempotents of (kN)^{G} are orthogonal whence (d). Clearly, b is a block of kC_{G}({1}) = kG, so ({1}, b) is a (kG, G)-Brauer pair. Further by part (b), applied with N = G, ({1}, b) ≤ (P, e) if and only if br_{P} (b)e = e and by part (c),(P, e) is a (kG, G)-Brauer pair if and only if br_{P} (b)e = e. This proves (e). □

Proposition 3.14. Let b be a primitive idempotent of (kN)^{G}. The triple (kN, b, G) is of saturated type.

Proof. By Proposition 3.10, b is a central idempotent of kN, hence the first condition of Definition 3.1 holds. Let (Q, e) be a (kN, G)-Brauer pair. By definition, e is a primitive idempotent of Z(kC_{N}(Q)). Therefore e is primitive in any subalgebra of Z(kC_{N}(Q)) which contains e. Since clearly C_{N}(Q) ≤ C_{G}(Q, e),

Z(kC_{N}(Q)) = kC_{N}(Q)^{C_{N} (Q)} ⊃ (kC_{N}(Q))^{C_{G}(Q,e)}

and it follows that e is primitive in (kC_{N}(Q))^{C_{G}(Q,e)}. Thus the second condition of Definition 3.1 holds. □

By the above proposition and Theorem 3.2, for a primitive idempotent b of (kN)^{G} and a maximal (kN, b, G)-Brauer pair (P, e_{P}), the category F_{(P,e_{P})}(kN, b, G) is a saturated fusion system on P. In the special case that N = G, the primitive idempotents of (kG)^{G} are just the blocks of kG.

Definition 3.15. Let b be a block of kG.

• A p-subgroup P of G is a defect group of kGb (or more simply of b) if there exists a maximal (kG, b, G)-Brauer pair with first component P.

• Fix a maximal (kG, b, G)-Brauer pair (P, e_{P}). The fusion system of kGb (or of b) over (P, e_{P}) is the category F_{(P,e_{P})}(kG, b, G).

By Theorem 2.20, any two maximal (kG, b, G)-Brauer pairs are G-conjugate, hence P is a defect group of kGb if and only if br_{P} (b) ≠ 0 and every (kG, b, G)-Brauer pair with first component P is maximal. By Theorem 2.22 (b), the fusion systems corresponding to any two maximal (kG, b, G)- Brauer pairs are isomorphic. For this reason, we often refer to F_{(P,e_{P})}(G, b) in the definition above as the fusion system of the block b.

3.5. Fusion systems of blocks of local subgroups.

For the rest of the section, we concentrate on block fusion systems, that is the case A = kG. The aim of this subsection is to show that if F is the fusion system of a block b of kG with respect to some maximal (kG, b, G)- Brauer pair (P, e_{P}), then for a fully-F normalised subgroup Q of P, N_{F}(Q) is the fusion system of a block of kN_{G}(Q) and if Q is fully-F centralised, then C_{F}(Q) is the fusion system of a block of kC_{G}(Q). The results of Subsection 3.2 come very close to what we will prove here but the point of view is different.

Lemma 3.16. Suppose that Q is a normal p-subgroup of G. Then x − 1 ∈ J(kG) for all x ∈ Q and Tr_{R}^{Q}((kG)^{R}) ⊆ J(kG) for any proper subgroup R of Q.

Proof. The first statement is equivalent to showing that for any simple kG-module V. (x − 1)V = 0 for all x ∈ Q. Proceed by induction on |Q|. Suppose first that |Q| = p and let W ⊆ V be defined by W = {v ∈ V : xv = v, ∀x ∈ Q}. Since Q is normal in G, W is a kG-submodule of V. Hence either W = 0 or W = V and it suffices to show that W ≠ 0. Let z be a generator of Q. Since z has order p, z^{p}v = v for all v ∈ V which means that

(z − 1)^{p}V = (z^{p} − 1)V = 0.

In particular, for some n, 0 ≤ n < p and for some v ∈ V, we have that (z − 1)^{n}v ≠ 0 and (z − 1)^{n+1}v = 0. Set w = (z − 1)^{n}v. By definiton zw = w, hence z^{i}w = w for any positive integer i, that is w ∈ W and W ≠ 0 as required. Now suppose that |Q| > p, and let U be a proper non-trivial normal subgroup of Q. By induction U acts as the identity on V. Thus, V is a (simple) kG/U-module and Q/U is a normal subgroup of G/U. By induction again, Q/U acts as the identity on V. Thus, Q acts as the identity on V, proving the first statement.

Now, let R < Q and let a ∈ (kG)^{R}. Then by the first part, for any x ∈ Q, xax^{−1} − a ∈ J(kG), hence

Tr_{R}^{Q}(a) = ∑_{x∈Q/R} xax^{−1} − |Q:R|a ∈ J(kG),

thus proving the second assertion. □

Lemma 3.17. Suppose that Q is a normal p-subgroup of G. The set of blocks of kG is the set of sums of G-orbits of blocks of kC_{G}(Q) under the conjugation action of G.

Proof. Since the orbit sum of any G-orbit of blocks of kC_{G}(Q) is a central idempotent of kG, it suffices to show that any central idempotent of kG is an element of kC_{G}(Q).

So, let d be an idempotent of Z(kG). The class sums of G form a basis of Z(kG). Further, since Q is normal in G, any conjugacy class of G is either contained in C_{G}(Q) or intersects C_{G}(Q) trivially. Thus, we may write d = d_{1} + d_{2}, where d_{1} ∈ Z(kG) ∩ kC_{G}(Q) and d_{2} ∈ (kG)_{<Q}^{Q}. By Lemma 3.16, d_{2} ∈ J(kG), hence d_{2}^{p^{n}} = 0 for some natural number n. Since d_{1} and d_{2} commute, d^{p^{n}} = d_{1}^{p^{n}} + d_{2}^{p^{n}} = d_{1}^{p^{n}}. Since d is idempotent, this means that d = d_{1}^{p^{n}} ∈ kC_{G}(Q), as required. □

Lemma 3.18. Let Q be a p-subgroup of G and H a subgroup of G with QC_{G}(Q) ≤ H. Let e be a block of kC_{G}(Q) and b and c be blocks of kG and kH respectively such that br_{Q}(b)e ≠ 0 ≠ br_{Q}(c)e.

(a) The H-poset of (kH, c, H)-Brauer pairs containing (Q, e) is equal to the H-poset of (kG, b, G)-Brauer pairs containing (Q, e) and whose first component is contained in H.

(b) Suppose that Q is normal in G and H = N_{G}(Q, e). Then e = c and the set of maximal (kH, e, H)-Brauer pairs is the set of maximal (kG, b, G)-Brauer pairs containing (Q, e). Further, for any maximal (kH, e, H)-Brauer pair (R, f)

F_{(R,f)}(kH, e, H) = F_{(R,f)}(kG, b, G).

Proof. (a) Since C_{G}(Q) ≤ H, C_{G}(R) = C_{H}(R) for any R such that Q ≤ R ≤ H. Thus, by Proposition 3.13, the H-poset of (kH, H)-Brauer pairs whose first component contains Q equals the the H-poset of (kG, G)-Brauer pairs whose first component contains Q and is contained in H. Note that by Proposition 3.13(b), and the remark just before Proposition 3.13, for such pairs the inclusion as (kH, H)-Brauer pairs is the same as inclusion as (kG, G)-Brauer pairs. Also by Proposition 3.13, any (kG, G)-Brauer pair which contains (Q, e) is a (kG, b, G)-Brauer pair and similarly any (kH, H)-Brauer pair which contains (Q, e) is a (kH, c, H)-Brauer pair.

(b) The assertion that e = c is immediate from Lemma 3.17. Let (R, f) be a (kG, b, G)-Brauer pair containing (Q, e) and let x ∈ R. Then ^{x}(Q, e) ≤ (R, f) and ^{x}Q = Q. Hence, by the uniqueness of inclusion of Brauer pairs, x(Q, e) = (Q, e), that is x ∈ H. So, the first component of any (kG, b, G)- Brauer pair which contains (Q, e) is contained in H. By part (a), it follows that the H-poset of (kH, e, H)-Brauer pairs containing (Q, e) is equal to the H-poset of (kG, b, G)-Brauer pairs containing (Q, e). Since H = N_{H}(Q, e), (Q, e) is contained in any maximal (kH, e, H)-Brauer pair, hence the set of maximal (kH, e, H)-Brauer pairs equals the set of maximal (kG, b, G)- Brauer pairs containing (Q, e).

Now let (R, f) be a maximal (kH, e, H)-Brauer pair and set E = F_{(R,f)}(kH, e, H), F = F_{(R,f)}(kG, b, G). Since H = N_{G}(Q, e), E = N_{F}(Q). Thus, in order to show that E = F, it suffices to show that Q is normal in F. Let U be a F-centric-radical subgroup of R. Since Q is normal in G, Aut_{QU}(U) is a normal p-subgroup of Aut_{F}(U), hence

Aut_{QU}(U) ≤ O_{p}(Aut_{F}(U)) = Aut_{U}(U).

So,

N_{QU}(U) ≤ UC_{G}(U) ∩ R = UC_{R}(U) = U.

Thus, Q ≤ U. Since Q is clearly stongly F-closed, the result follows by Proposition I.4.5. □

Theorem 3.19. Let b a block of kG and (P, e_{P}) a maximal (kG, b, G)- Brauer pair. For each Q ≤ P, let e_{Q} be the unique block of kC_{G}(Q) such that (Q, e_{Q}) ≤ (P, e_{P}) and set F = F_{(P,e_{P})}(kG, b, G).

(a) Let Q ≤ P, let N be a subgroup of G such that N_{G}(Q, e_{Q}) ≤ N ≤ N_{G}(Q) and let c be the unique block of kN with br_{Q}(c)e_{Q} = e_{Q}. Then Q is fully F-normalized if and only if (N_{P} (Q), e_{N_{P} (Q)}) is a maximal (kN, c, N)-Brauer pair. If this is the case, then

N_{F}(Q) = F_{(N_{P} (Q),e_{N_{P} (Q)})}(kN, c, N).

(b) A subgroup Q ≤ P is fully centralised if and only if (C_{P} (Q), e_{C_{P} (Q)}) is a maximal (kC_{G}(Q), e_{Q}, C_{G}(Q))-Brauer pair. If this is the case, then

C_{F}(Q) = F_{(C_{P} (Q),e_{C_{P} (Q)})}(kC_{G}(Q), e_{Q}, C_{G}(Q)).

Proof. (a) By Lemma 3.17 applied with the group G replaced by N and H replaced by N_{G}(Q, e_{Q}), we may assume that N = N_{G}(Q, e_{Q}) and c = e_{Q}. By Lemma 3.5 applied to the Brauer pair (Q, e_{Q}) of the saturated triple (kN, e_{Q}, N) and noting that C_{N}(Q) = C_{G}(Q), the N-poset of (kN, e_{Q}, N)-Brauer pairs which contain (Q, e_{Q}) is equal to the N-poset of (kC_{G}(Q), e_{Q}, N)-Brauer pairs which contain (Q, e_{Q}). In particular, if (R, f) is a maximal (kC_{G}(Q), e_{Q}, N)-Brauer pair containing (Q, e_{Q}), then (R, f) is a maximal (kN, e_{Q}, N)-Brauer pair containing (Q, e_{Q}) and F_{(R,f)}(kN, e_{Q}, N) = F_{(R,f)}(kC_{G}(Q), e_{Q}, N). Now the result is immediate from Proposition 3.8.

(b) Since C_{G}(Q, e_{Q}) = C_{G}(Q), this is immediate from Proposition 3.8. □

As pointed out in the remarks after Definition 3.15, the condition that (N_{P} (Q), e_{N_{P} (Q)}) is a maximal (kN, c, N)-Brauer pair in the above theorem is equivalent to N_{P} (Q) being a defect group of the block c of kN, and similarly the condition that (C_{P} (Q), e_{C_{P} (Q)}) is a maximal (kC_{G}(Q), e_{Q}, C_{G}(Q))- Brauer pair is equivalent to C_{P} (Q) being a defect group of the block e_{Q} of kC_{G}(Q). The next result is a characterization of centric subgroups.

Theorem 3.20. Keep the notation of Theorem 3.19. A subgroup Q of P is F-centric if and only if Z(Q) is a defect group of the block e_{Q} of kC_{G}(Q).

Proof. If Q is F-centric, then Q is fully F-centralized and by Theorem 3.19, Z(Q) = C_{P} (Q) is a defect group of the block e_{Q}. Conversely, suppose that Z(Q) is a defect group of the block e_{Q} of kC_{G}(Q). Let R ≤ P be F-conjugate to Q and let g ∈ G be such that ^{g}(Q, e_{Q}) = (R, e_{R}). By Theorem 2.20, Z(Q) is maximal amongst p-subgroups of C_{G}(Q) such that br_{Z(Q)}(e_{Q}) ≠ 0. So, Z(R) = ^{g}Z(Q) is maximal amongst p-subgroups of C_{G}(R) = ^{g}C_{G}(Q) such that br_{Z(R)}(e_{R}) = br_{Z(R)}(^{g}e_{Q}) ≠ 0. By theorem 2.20, Z(R) is a defect group of the block e_{R} of kC_{G}(R). On the other hand, there is a (kC_{G}(R), e_{R}, C_{G}(R))-Brauer pair with first component C_{P} (R), namely (C_{P} (R), e_{C_{P} (R)}) and Z(R) ≤ C_{P} (R). Thus, Z(R) = C_{P} (R) for all R ≤ P which are F-conjugate to Q. □

4. Background on finite group representations

We have seen in the previous section that every block of a finite group algebra gives rise to an (isomorphism class of) saturated fusion system on a defect group of the block. The central problem of modular representation theory is the connection between a block fusion system and the representation theory of the corresponding block algebra. In order to be able to discuss this connection, we need to invoke some of the standard language and notions of modular group representation theory, and this is what we will do in this section. As in Section 1, we will be very brief- many books on group representation theory give an extensive treatement of the material presented here. Some of these are [CuR1], [CuR2], [NT] and [Fe].

4.1. Ordinary and modular representations.

Let F be a field and let G be a finite group. In order to avoid complications of rationality we will assume that F is a splitting field for G, that is End_{F G}(V) ≅ F for any simple F G-module V. Recall that by the convention of Section 1, all modules are finitely generated left modules.

Definition 4.1. Let H be a subgroup of G. An F G-module V is relatively H-projective if the following holds:

For any surjective homomorphism

φ : W → V

of F G-modules, if φ splits as a map of F H-modules, then φ splits as a map of F G-modules.

It is an easy exercise that a F G-module V is relatively {1}-projective if and only if V is a projective F G-module. Recall from Example 1.12 that if V is a F G-module, then End_{F} (V) is naturally a G-algebra. For a subgroup H of G, the H-fixed point subalgebra (End_{F} (V))^{H} is equal to the subalgebra of F H-module endomorphisms End_{F H}(V).

Theorem 4.2. (Higman’s criterion). Let G be a finite group, H ≤ G and let V be a F G-module. Then V is relatively H-projective if and only if the identity Id_{V} : V → V of End_{k}(V) is a relative trace Tr_{H}^{G}(π) for some π ∈ End_{k}(V)^{H}.

Theorem 4.3. Let H ≤ G. The following are equivalent.

(a) 1_{F G} = Tr_{H}^{G}(a) for some a ∈ (F G)^{H}.

(b) Any finitely generated F G-module is relatively H-projective.

The above is a standard fact, but the statement does not appear often in this form in the literature-we provide a proof for the convenience of the reader.

Proof. (a ⇒ b) Suppose that 1_{F G} = Tr_{H}^{G}(a), a ∈ (F G)^{H} and let V be an F G-module. Let π : V → V be the map given by π(v) = av for v ∈ V. Since a ∈ (F G)^{H}, we have

hπ(v) = hπ(h^{−1}v) = hah^{−1}v = av = π(v), for all h ∈ H, v ∈ V,

so π ∈ End_{F H}(V). Also, we have

Tr_{H}^{G}(π(v)) = ∑_{g∈[G/H]} gπ(g^{−1}v) = ∑_{g[∈G/H]} gag^{−1}v = v for all v ∈ V.

So, by Higman’s criterion, V is relatively H-projective and (b) holds.

(b ⇒ a) Suppose that every F G-module is relatively H-projective. Consider F G as an F G-module via the conjugation action of G on itself. We will prove that (a) holds by applying (b) to F G. Note that since F G is being considered as a F G-module via conjugation, for π ∈ End_{F} (F G), g ∈ G, the map ^{g}π : F G → F G satisfies ^{g}π(a) = gπ(g^{−1}ag)g^{−1} for a ∈ F Gb. In particular, π ∈ End_{F H}(F G) if and only if hπ(a)h^{−1} = π(hah^{−1}) for all a ∈ F G and all h ∈ H.

By hypothesis, F G is relatively H-projective. Hence by Higman’s criterion, the identity map Id_{F G} : F G → F G is a relative trace Tr_{H}^{G}(π) for some π ∈ End_{F H}(F G). Let a = π(1_{F G}). Since π ∈ End_{F H}(F G),

hah^{−1} = hπ(1_{F G})h^{−1} = π(1_{F G}) = a for all h ∈ H,

so a ∈ (F G)^{H}. We have,

Tr_{H}^{G}(a) = ∑_{g∈[G/H]} gag^{−1} = ∑_{g∈[G/H]} gπ(1_{F G})g^{−1} = ∑_{g∈[G/H]} ^{g}π(1_{F G})
= Id_{F G}(1_{F G}) = 1_{F G}.

Hence (a) holds. □

Proposition 4.4. The identity element 1_{F G} belongs to Tr_{1}^{G}(F G) if and only if char(F) does not divide |G|.

Proof. Let g ∈ G and let C be the G-conjugacy class of g. Then Tr_{1}^{G}(g) = |C_{G}(g)|C^{+}, where C^{+} denotes the sum of elements of C in F G. So, the elements C^{+}, where C runs over those conjugacy classes of G for which |C_{G}(x)| is non-zero in F form a basis of Tr_{1}^{G}(F G). The result is immediate from this. □

The results above show the essential difference between representation theory of finite groups over fields of characteristic 0 (ordinary representation theory) and fields of positive characteristic (modular representation theory). For, if char(F) does not divide |G|, then by the above every finitely generated every F G-module is projective (in case char(F) = 0, this fact is known as Maschke’s theorem). Thus the Artin-Wedderburn theory gives that if char(F) does not divide |G|, then F G is a product of matrix algebras. In particular, every block of F G is a matrix algebra. On the other hand, if char(F) does divide |G|, then there exist simple kG-modules which are not projective, hence F G has blocks which are not matrix algebras:

Theorem 4.5. Suppose that char(F) does not divide |G|. Then, J(F G) = 0 and

F G = ∏_{i=1}^{r} Λ_{i},

where each Λ_{i} ≅ Mat_{n_{i}}(F) for some natural number n_{i}. Further, the following holds.

(a) For each i, 1 ≤ i ≤ r, let e_{i} be a primitive idempotent of Λ_{i}, and regard Λ_{i}e_{i} as a F G-module through the projection of F G onto Λ_{i}. Then {e_{i}:1 ≤ i ≤ r} is a set of representatives of conjugacy classes of primitive idempotents of F G and {Λ_{i}e_{i} : 1 ≤ i ≤ r} is a set of representatives of isomorphism classes of simple F G-modules.

(b) The set {1_{Λ_{i}} : 1 ≤ i ≤ r} is the block decomposition of 1_{F G}, and each block of F G is a matrix algebra over F.

If char(F) divides |G|, then there exist blocks b of of F G such that J(F Gb) ≠ 0 (and hence F Gb is not a matrix algebra).

4.2. p-modular systems.

Definition 4.6. A p-modular system is a triple (K, O, k) where O is a local principal ideal domain, K is the field of quotients of O and k = O/𝔭, where 𝔭 is the unique maximal ideal of O such that the following hold:

• O is complete with respect to the natural topology induced by 𝔭,

• K has characteristic 0; and

• k has characteristic p.

Example 4.7. Let L be a number field, R the ring of algebraic integers in L, 𝔭 a prime ideal of R lying over p. Let R_{𝔭} be the localization of A with respect to 𝔭 and let O = \underleftarrow{lim} R_{𝔭}/𝔭^{n}, the completion of A with respect to the 𝔭-adic topology. Then O is a local principal ideal domain, and the corresponding triple (K, O, k) is a p-modular system.

For the rest of the article, we will fix a p-modular system (K, O, k) which satisfies the following additional properties:

• k is algebraically closed and perfect; and

• K is a splitting field for any finite group appearing in the sequel.

The existence of p-modular systems satisfying the above properties relies on two results. The first is that given any natural number n there exists a p-modular system (K, O, k) such that k is algebraically closed and perfect and such that K contains a primitive n-th root of unity (see [Se1, Chapter 2, Section 3, Theorem 1 and Chapter 2, Section 5, Theorem 3]). The second is Brauer’s theorem that any field of characteristic 0 containing a primitive n-th root of unity is a splitting field for any finite group of order n, which in turn is a consequence of Brauer’s characterisation of characters (see [CuR1, Theorem 41.1]).

We record the following fact. Note that the natural surjective ring homomorphism O → k induces a group homomorphism O^{∗} → k^{∗}.

Proposition 4.8. Let 𝔭 denote the maximal ideal of O. The group homomorphism O^{∗} → k^{∗} is surjective with kernel 1 + 𝔭. Further, the short exact sequence

1 → 1 + 𝔭 → O^{∗} → k^{∗} → 1

splits uniquely. In particular, the surjection O^{∗} → k^{∗} restricts to an isomorphism between the torsion subgroup of O^{∗} and k^{∗}.

4.3. Cartan and decomposition maps.

The extension of the natural surjection of O onto k, and of the inclusion of O in K to the corresponding group rings, yields the following diagram of ring homomorphisms

KG ↩ OG ↠ kG,

where the map on the left is inclusion and the map on the right sends an element a: = ∑_{g∈G} α_{g}g of OG to the element ā: = ∑_{g∈G}(α_{g} + 𝔭)g of kG.

Since conjugacy class sums of elements of G form a basis for the center of the group ring RG of G over any commutaitive ring R with 1, restricting the above maps to Z(OG) gives corresponding homomorphisms

Z(KG) ↩ Z(OG) ↠ Z(kG).

The above maps lead to a two way traffic between the characteristic 0 representations and characteristic p representations of G. Note that if M is an OG-module, then by extension of scalars K ⊗_{O} M is a KG-module and similarly k ⊗_{O} M ≅ M/𝔭M is a kG-module.

The completeness of O with respect to the 𝔭-adic topology has as consequence that the idempotent lifting results of Subsection 1.1 carry over to O-algebras. More precisely, Propositions 1.2, 1.3, 1.4, 1.8, 1.9 and Lemma 1.5 hold for O-algebras which are finitely generated as O-modules. In particular, we have the following.

Proposition 4.9. The map a → ā induces a bijection between OG-conjugacy classes of primitive idempotents of OG and kG-conjugacy classes of primitive idempotents of kG and also a bijection between the blocks of OG and the blocks of kG.

There is an analogous result for modules.

Proposition 4.10. Let I be a set of representatives of OG-conjugacy classes of primitive idempotents of OG.

(a) The set {OGi: i ∈ I} is a set of representatives of the isomorphism classes of projective indecomposable OG-modules.

(b) The set {kGī ≅ k ⊗_{O} OG: i ∈ I} is a set of representatives of the isomorphism classes of projective indecomposable kG-modules.

(c) The set {kGī/J(kG)ī: i ∈ I} is a set of representatives of the isomorphism classes of simple kG-modules.

The following definition introduces notation for various Grothendieck groups and subgroups associated to the categories of KG, OG and kGmodules.

Definition 4.11. Let F denote a field. Denote,

• by R_{F} (G) the Grothendieck group (with respect to short exact sequences) of finitely generated F G-modules. For a finitely generated F G-module V, denote by [V] the image of V in R_{F} (G),

• by Irr_{F} (G) the set of images of the simple F G-modules in R_{F} (G),

• by IPr_{F} (G) the set of the projective indecomposable F G-modules in R_{F} (G) and by P r_{F} (G) the subgroup of R_{F} (G) generated by IPr_{F} (G); and

• by IPr_{O}(G) the set of images of modules of the form K ⊗_{O} U, where U is a finitely generated projective indecomposable OG-module and by P r_{O}(G) the subgroup of R_{K}(G) generated by IPr_{O}(G).

Note that for F = K or k, R_{F} (G) is a finitely generated free abelian group with basis Irr_{F} (G). Contrary to what might be expected, the above list does not include the subgroup of R_{K}(G) generated by the scalar extensions of simple OG-modules. The reason for this is the following. If M is a non-zero OG-module, finitely generated as O-module, then 𝔭M is a proper OG-submodule of M (this is a special case of what is known as Nakyama’s lemma). So, if M is a simple OG-module, then 𝔭M = 0 and consequently K ⊗_{O} M = 0. Also, note that by Theorem 4.3 and Proposition 4.4, every finitely generated KG-module is projective and hence IPr_{K}(G) = Irr_{K}(G) and P r_{K}(G) = R_{K}(G). On the other hand, if p divides the order of G, then P r_{k}(G) is a proper subgroup of R_{k}(G).

Definition 4.12. The Cartan map of F G is the inclusion map c: P_{F} (G) → R_{F} (G). For Φ ∈ IPr_{F} (G) and φ ∈ Irr_{F} (G) the Cartan number c_{Φ,φ} of Φ with respect to φ is the number defined by the equation

c(Φ) = ∑_{φ∈Irr_{F} (G)} c_{Φ,φ}φ.

The matrix (c_{Φ,φ})_{Φ∈IPr_{F} (G),φ∈Irr_{F} (G)} is the Cartan matrix of F G.

The Cartan numbers of kG have the following alternative descriptions. For a projective indecomposable kG-module P and a simple kG-module S, the Cartan number c_{[P],[V]} is the multiplicity of V as a composition factor of P and this in turn is equal to dim_{k}(jkGi), for any primitive idempotents i and j of kG such that P ≅ kGi and V ≅ kGj/J(kG)j.

Since O is a principal ideal domain, if V is a finitely generated KGmodule. V has an O-form, i.e., there exists an OG-module V_{0} which is O-free and such that V = K ⊗_{O} V_{0}. Then, V_{0}/𝔭V_{0} is a kG-module of the same k-dimension as the K-dimension of V. Neither the isomorphism class of V_{0} nor that of V_{0}/𝔭V_{0} is determined by the isomorphism class of V in general. However, the image [V_{0}/𝔭V_{0}] of V_{0}/𝔭V_{0} in R_{k}(G) is uniquely determined by the image [V] of V in R_{K}(G). More precisely,

Proposition 4.13. The map d: R_{K}(G) → R_{k}(G) which sends the image [V] of a finitely generated OG-module to the element [V_{0}/𝔭V_{0}] ∈ R_{k}(G), where V_{0} is an O-free OG-module such that V = K ⊗_{O} V_{0} is a well defined group homomorphism.

Definition 4.14. With the notation above,

• The map d is called the decomposition map of OG.

• For each χ ∈ Irr_{K}(G), write

d(χ) = ∑_{φ∈Irr_{k}(G)} d_{χ,φ}φ.

The number d_{χ,φ} is the decomposition number of χ with respect to φ and the matrix (d_{χ,φ})_{χ∈Irr_{K}(G),φ∈Irr_{K}(G)} is the decomposition matrix of G.

It is easy to see that for V a simple KG-module and S a simple kGmodule, the decomposition number d_{[V],[S]} is the multiplicity of S as a composition factor of V_{0}/𝔭V_{0}, where V_{0} is an O-free OG-module such that V = K ⊗_{O} V_{0} .

For the next theorem note that by Proposition 4.10, if I is a set of representatives of conjugacy classes of primitive idempotents of OG, then IPr_{k}(G) = {[kGī] : i ∈ I} and Irr_{k}(G) = {[kGī/J(kG)ī] : i ∈ I}.

Theorem 4.15. Let d: R_{K}(G) → R_{k}(G) be the decomposition map of OG and let c: P r_{K}(G) → R_{k}(G) be the Cartan map of kG. Let I be a set of representatives of conjugacy classes of primitive idempotents of OG and for each i ∈ I, let Φ_{i} = [kGī] and φ_{i} = [kGī/J(kG)ī].

(a) The map d induces an isomorphism between the subgroups P r_{O}(G) and P r_{k}(G).

(b) Let e: P r_{k}(G) → R_{K}(G) be the group homomorphism which for each i in I sends the element [kGī] of P r_{k}(G) to the element [K ⊗_{O} OGi] of R_{K}(G). Then de = c, and

e(Φ_{i}) = ∑_{χ∈Irr_{K}(G)} d_{χ,φ_{i}}χ,

for all i ∈ I. In particular, if D is the decomposition matrix of OG and C is the Cartan matrix of kG, then

C = D^{t}D.

(c) The map d is surjective.

The above notions are compatible with block decompositions. Let R denote one of the rings K, O, or k and let b be a central idempotent of the group ring RG. We say that an RG-module M belongs to b, if bM = M. If M is an RG-module which belongs to b, and if m ∈ M, then writing m = bn for n ∈ M, we have that

bm = bbn = bn = m.

So, if M belongs to b, then b acts as the identity on M and M is therefore an RGb-module. Conversely, if N is an RGb-module, then N is an RGmodule via a.n = abn, for a ∈ RG and m ∈ M. Thus RGb-modules are precisely RG-modules which belong to b. Further, if {b_{i}: 1 ≤ i ≤ r} is an orthogonal decomposition of 1_{RG} in Z(RG), and M is an RG-module, then since b_{i} is central in RG, b_{i}M is an RG-module belonging to b_{i} for 1 ≤ i ≤ r and we have a direct sum decomposition

M = b_{1}M ⊕ ⋯ ⊕ b_{r}M,

into RG-modules. In particular, if M is indecomposable, then M belongs to b_{i} for a unique i, 1 ≤ i ≤ r.

For the next definition, note that by Proposition 4.10, for any central idempotent b of kG, there is a unique central idempotent b̂ of OG such that ¯ b̂ = b and if b is a block of kGb, then b̂ is a block of OGb. The central idempotent b̂ of OG is called the lift of b. Also, since Z(OG) is a subring of Z(KG), any central idempotent of OG is a central idempotent of KG.

Definition 4.16. Let b be a central idempotent of kG and let b̂ ∈ OG be the lift of b. Denote by R_{k}(G, b) (respectively R_{K}(G, b)) the subgroup of R_{k}(G, b) (respectively R_{K}(G)) generated by images of finitely generated kG (respectively KG-)modules which belong to b (respectively b̂); by Irr_{k}(G, b) (respectvely Irr_{K}(G, b)), the images of simple kG-(respectively KG-)modules which belong to b (respectively b̂); by P r_{k}(G, b) (respectively P r_{O}(G, b)) the group P r_{k}(G) ∩ R_{k}(G, b) (respectively P r_{O}(G) ∩ R_{K}(G, b) ); and by IPr_{k}(G, b) (respectively IPr_{O}(G, b)) the set IPr_{k}(G) ∩ R_{k}(G, b) (respectively IPr_{O}(G, b) ∩ R_{K}(G, b)).

For a central idempotent b of kG, |Irr_{k}(G, b)| is the number of isomorphism classes of simple kGb-modules and similarly |Irr_{K}(G, b)| is the number of ismorphism classes of KGb-modules. Also, if b and c are orthogonal idempotents of Z(kGb), then for any φ ∈ Irr_{k}(G, b), χ ∈ Irr_{K}(G, c) and Φ ∈ IPr_{k}(G, c), the corresponding Cartan and decomposition numbers satisfy

d_{χ,φ} = 0 = c_{Φ,φ}.

Definition 4.17. Let b be a central idempotent of kG and let b̂ be the central idempotent of OG with ¯ b̂ = b. The Cartan matrix of kGb is the submatrix of the Cartan matrix of kG whose entries correspond to modules which belong to b and the decomposition matrix of OGb is the submatrix of the decomposition matrix of OG whose entries correspond to modules which belong to b.

4.4. Ordinary and Brauer characters.

Let R be an integral domain and let M be a finitely generated RGmodule such that M is free as R-module. Choose an R-basis of M and identify End_{R}(M) with Mat_{n}(R) through the chosen basis, where n is the cardinality of the basis.

Definition 4.18. With the above notation, the character χ_{M} : RG → R of M is the function defined by χ_{M}(a) = trace(ϕ(a)), a ∈ RG, where ϕ(a) ∈ End_{R}(M) is defined by

ϕ(a)(m) = am, m ∈ M.

The character χ_{M} is independent of the choice of R-basis of M nad is completely determined by its values on the elements of the group G. So it is customary to identify χ_{M} with its restriction to G. As such, χ_{M} : G → R is a class function on G, i.e., χ_{M} has constant value on G-conjugacy classes.

Over fields, characters separate simple modules:

Proposition 4.19. Let F = K or k and let M_{1}, M_{2} be simple F G-modules. Then χ_{M_{1}} = χ_{M_{2}} if and only if M_{1} ≅ M_{2} as F G-modules.

The character of a KG-module is called an an ordinary character, and the character of a simple KG-module is an ordinary irreducible character. The character of a kG-module is a modular character and the character of a simple kG-module is a modular irreducible character. By the above result, we may identify Irr_{K}(G) with the set of ordinary irreducible characters of G and it is customary to do so. This can of course also be done for Irr_{k}(G) but there is another subtle and very useful interpretation of Irr_{k}(G) due to Brauer which we now describe.

Let M be a kG-module, and let X be an abelian subgroup of G. Since k is algebraically closed, by the standard ”simultaneous eigen-vector” argument in linear algebra, there exists a k-basis of M such that identifying End_{k}(M) with Mat_{n}(k) through this basis, the matrix representing the action of x on M is upper triangular, for all x ∈ X. For each x ∈ X, the diagonal entries, say λ_{1}, ⋯, λ_{n} of the corresponding matrix are the eigenvalues of x on M and in particular are t-th roots of unity, where t is the order of x in G. Since char(k) = p, the only p-power root of unity in k is 1. Thus, if x and y are a pair of commuting elements of G with x a p-element, then χ_{M}(xy) = χ_{M}(y). On the other hand, every element x of G is a product x = x_{p}x_{p′}, for uniquely determined elements x_{p}, x_{p′} of G such that x_{p} and x_{p′} commute, x_{p} is a p-element and x_{p′} is a p′-element. So, χ_{M}(x) = χ_{M}(x_{p′}) for all x ∈ G and χ_{M} is completely determined by its values on the p′-elements of G. Now, let x ∈ G be a p′-element and let λ_{1}, ⋯, λ_{n} be the eigenvalues of x on M. For each i, 1 ≤ i ≤ n, λ_{i} is a p′-root of unity, so by Proposition 4.8, there exists a unique root of unity, say λ̂_{i} in O^{∗} such that \overline{λ̂}_{i} = λ_{i}. Let G_{p′} denote the set of p′-elements of G and define χ̂_{M} : G_{p′} → K by χ̂_{M}(x) = λ̂_{1} + ⋯ + λ̂_{n} for x ∈ G_{p′}. So, \overline{χ̂_{M}(x)} = χ_{M}(x), for all x ∈ G_{p′}.

Definition 4.20. With the notation above the map χ̂_{M} : G_{p′} → K is called the Brauer character of M. The Brauer character of a simple kG-module is called an irreducible Brauer character.

By the discussion preceding the above definition, for any kG-module M the modular character χ_{M} of M is completeley determined by the Brauer character χ̂_{M} of M. So, by Proposition 4.19, two simple kG-modules M_{1} and M_{2} are isomorphic if and only they have the same Brauer characters. Thus Irr_{k}(G) may be identified with the set of irreducible Brauer characters of G, and it is customary to do so.

The functional, i.e., character theoretic approach is a big part of the representation theory of finite groups. We describe below a few properties of characters-the choice of statements is dictated by the material of the next sections. The first assertion of the next theorem forms part of the “orthogonality relations” of characters.

Theorem 4.21. With the above identifications, Irr_{K}(G) is a basis of the K-vector space of K-valued class functions on G and Irr_{k}(G) is a basis of the K-vector space of K-valued class functions on G_{p′}.

By the above, the Grothendieck group R_{K}(G) becomes identified with the subgroup of the K-vector space of K-valued class functions on G generated by Irr_{K}(G) and R_{k}(G) ≅ ℤIrr_{k}(G) with the subgroup of the K-vector space of K-valued class functions on G_{p′} generated by Irr_{k}(G).

If χ: G → K is a K-valued class function on G, then the restriction Res_{G_{p′}}^{G} χ of χ to G_{p′} is a K-valued class function on G_{p′}. The map χ → Res_{G_{p′}}^{G} χ is a surjective K-linear map from the space of K-valued class functions on G to the space of K-valued class functions on G_{p′}; restriction of this map to the group R_{K}(G) is just the character theoretic analogue of the decomposition map. This map has two obvious sections: either one can extend a class function Φ on G_{p′} to the class function Φ̃ on G by setting Φ̃(x) = Φ(x_{p′}) for x ∈ G, or to the class function Φ^{∗} on G by setting Φ^{∗}(x) = 0 if x ∈ G − G_{p′}. The next result, which is related to the surjectivity of the decomposition map, shows that these sections are meaningful on restriction to Grothendieck groups. These proofs (see [NT, Chapter 3, Lemma 6.13, Lemma 6.31]) depend on Brauer’s characterisation of characters. Recall that Pr_{O}(G) is a subgroup of R_{K}(G).

Theorem 4.22. Keep the notation above and let p^{a} be the p-part of the order of G.

(a) Pr_{O}(G) consists of the elements of R_{K}(G) which vanish on G − G_{p′}.

(b) For any Φ ∈ R_{k}(G), Φ̃ ∈ R_{K}(G).

(c) For any Φ ∈ R_{k}(G), p^{a}Φ^{∗} ∈ Pr_{O}(G). Consequently, p^{a}(Res_{G_{p′}}^{G} χ)^{∗} ∈ Pr_{O}(G) for all χ ∈ R_{K}(G).

By Theorem 4.5, if F is a splitting field for G such that char(F) does not divide |G|, then the map which sends a simple F G-module to the block of F G containing it is a bijection between Irr_{F} (G) and the set of blocks of F G. The next is a formula which expresses a block of F G as an element of F G in terms of the values of the character of the unique simple F Gmodule contained in the block-this formula also encodes to some extent the orthogonality relations of characters.

Theorem 4.23 (Fourier inversion formula). Let F be a splitting field for G whose characteristic does not divide |G| and identify Irr_{F} (G) with the set of characters of the simple F G-modules. For each χ ∈ Irr_{F} (G), let e_{χ} be the block of F G containing χ. Then,

e_{χ} = \frac{χ(1)}{|G|} ∑_{g∈G} χ(g)g^{−1}.

If b is a block of kG, and b̂ the lift of b in OG, then b̂ is a central idempotent of KG since Z(OG) ⊆ Z(KG), but b̂ is not in general a block of KG.

Theorem 4.24. With the notation above,

(a) For any block b of kG,

b̂ = ∑_{χ∈Irr_{K}(G,b)} e_{χ}.

(b) For any χ ∈ Irr_{K}(G),

\frac{|G|χ(g)}{|C_{G}(g)|χ(1)} ∈ O for all g ∈ G.

Further, if χ, χ′ ∈ Irr_{K}(G), then χ and χ′ are in Irr_{K}(G, b) for the same block b of kG if and only if for any p′-element g ∈ G,

\frac{|G|χ(g)}{|C_{G}(g)|χ(1)} − \frac{|G|χ′(g)}{|C_{G}(g)|χ′(1)} ∈ 𝔭.

The above two results give a recipe for calculating block idempotents for kG from the values of the ordinary irreducible characters of G. We illustrate this with a small example. Examples 5.43 and 6.5 illustrate how this recipe can be extended to also calculate Brauer pairs and the corresponding fusion systems.

Example 4.25. Let G = S_{3}, the symmetric group on 3 letters. By Theorem 4.24, |Irr_{K}(G)| = dim_{K} Z(KG). Also, dim_{K} Z(KG) equals the number of conjugacy classes of G. Thus, up to isomorphism KG has three simple modules. There are two simple modules of dimension 1, corresponding to the trivial and sign representation respectively, and one of dimension 2 -the quotient of the natural permutation module for G by the submodule of G-fixpoints. So,

KG = K × K × Mat_{2}(K).

Denoting by χ_{1} the character of the trivial kG-module, by χ_{2} the character of the sign representation and by χ_{3} the character of the two dimensional simple KG-module, by the Fourier Inversion formula, the primitive central idempotents of KG are

e_{χ_{1}} = \frac{1}{6} ∑_{g∈G} g, e_{χ_{2}} = \frac{1}{6} ∑_{g∈G} sgn(g)g, e_{χ_{3}} = \frac{2}{3}1_{G} − \frac{1}{3} ∑_{g∈A_{3}−1_{G}} g.

Theorem 4.24 gives the kG and hence the OG-block distribution of Irr_{K}(G) for different values of p.

• If p = 2, then χ_{1} and χ_{2} are in the same block of kG, but not in the same block as χ_{3}. Thus OG has two blocks:

b_{1}: = e_{χ_{1}} + e_{χ_{2}} = \frac{1}{3} ∑_{g∈A_{3}} g

and

b_{2} = e_{χ_{3}} = \frac{2}{3}1_{G} − \frac{1}{3} ∑_{g∈A_{3}−1_{G}} g,

where A_{3} is the alternating subgroup of S_{3}.

• If p = 3, then all three irreducible characters of KG are in the same block of kG. Thus, OG has a unique block, i.e. 1_{OG} is primitive in Z(OG), and OG is indecomposable.

• If p ≥ 5, then e_{χ_{i}} is a block idempotent of OG, for each i, 1 ≤ i ≤ 3.

Another useful fact about blocks of OG and of kG is that their support is contained in the p′-part of G (see [NT, Chapter 3, Theorem 6.22]):

Proposition 4.26. Let R = O or k. Any block of RG is an R-linear combination of p′-elements of G.

5. Fusion and structure

Let (K, O, k) be a p-modular system satisfying the conditions of Subsection 4.2. For a p-subgroup P of G, let br_{P} : (kG)^{P} → kC_{G}(P) be the Brauer homomorphism with respect to P, as in Subsection 3.4.

5.1. The three main theorems of Brauer.

The three main theorems are the bedrock of block theory. All involve the Brauer homomorphism, Brauer pairs and hence fusion systems. Of the three the second makes an explicit link between representation theory and Brauer pairs. The ideas of the proof of the first main theorem are also the main ideas behind the existence of saturated fusion systems associated to blocks. The third main theorem describes the fusion system of a particular block (the principal block) of kG.

Theorem 5.1 (Brauer’s First Main Theorem). Let P be a p-subgroup of G. The map br_{P} induces a bijection between the set of blocks of G with defect group P and the set of blocks of N_{G}(P) with defect group P.

Proof. The main ingredients of the proof are Theorem 2.20, Lemma 3.17 and Lemma 3.18. We indicate how the pieces are put together to yield the theorem. Let H = N_{G}(P), let e be a block for kC_{G}(P) and let b and c be blocks of kG and kH respectively such that br_{P} (b)e ≠ 0 ≠ br_{P} (c)e. Then (P, e) is a (kG, b, G)-Brauer pair as well as a (kH, c, H)-Brauer pair. By Lemma 3.18, (P, e) is a maximal (kG, b, G)-Brauer pair if and only if (P, e) is a maximal (kH, c, H)-Brauer pair since if (P, e) is not maximal as (kG, b, G)-Brauer pair, then there exists a (kG, b, G)-Brauer pair containing (P, e) and whose first component normalises P. Since for any block e of kC_{G}(P), the blocks b and c of kG and kH are uniquely determined by the condition br_{P} (b)e ≠ 0 and br_{P} (c)e ≠ 0, it suffices to prove that if b is a block of kG with defect group P, then br_{P} (b) is a block of kH. So, let b be a block of kG with defect group P, and let (P, e) be a maximal (kG, b, G)- Brauer pair. Let c be the unique block of kH such that br_{P} (c)e ≠ 0. By Lemma 3.17, c is the sum of H-conjugates of e. But by Theorem 2.20, H acts transitively on the set of (kG, b, G)-Brauer pairs with first component P. In other words, H acts transitively on the set of blocks of kC_{G}(P) which appear in the block decomposition of br_{P} (b) in kC_{G}(P). Thus c = br_{P} (b). □

Definition 5.2. Let P be a p-subgroup of G and let b be a block of kG with P as a defect group. The Brauer correspondent of b in N_{G}(P) is the block br_{P} (b) of kN_{G}(P).

An immediate consequence of Theorem 3.19 and Brauer’s First Main Theorem is that the fusion system of the Brauer correspondant of a block is the normaliser subsystem (on the underlying p-group) of the fusion system of the block. More precisely:

Corollary 5.3. Suppose that b is a block of kG, P ≤ G is a defect group of kGb and c is the Brauer correspondent of b in kN_{G}(P). Then for any maximal (kG, b, G)-Brauer pair of the form (P, e), (P, e) is a maximal (kN_{G}(P), c, N_{G}(P))-Brauer pair and setting F = F_{(P,e)}(kG, b, G), we have that F_{(P,e)}(kN_{G}(P), c, N_{G}(P)) = N_{F}(P).

Theorem 5.4 (Brauer’s Second Main Theorem). Let b be a block of kG, x a p-element of G and e a block of kC_{G}(x). Let b̂ (respectively ê) be the block of OG (respectively OC_{G}(x) ) lifting b (respectively e). If (⟨x⟩, e) is not an (kG, b, G)-Brauer pair, then χ_{V} (êxy) = 0 for any OGb-module V which is free and of finite rank as O-module and any p′-element y of C_{G}(x).

Proof. We sketch the ideas behind the proof given in [NT, Chapter 5, Theorem 4.1]. Let V be as in the theorem, an O-free finitely generated OGb-module and set H = C_{G}(x). Let

V = W_{1} ⊕ ⋯ ⊕ W_{t},

be a decomposition of Res_{OH}^{OG}V into indecomposable OH-modules and for each i, 1 ≤ i ≤ t, let e_{i} be the block of kH such that W_{i} belongs to the lift , ê_{i}, of e_{i} to a block of OH. Then for 1 ≤ i ≤ t one proves:

• If br_{P} (b)e_{i} = 0, then W_{i} is relatively Q_{i}-projective for some proper subgroup Q_{i} of ⟨x⟩, where the definition of relative projectivity for OHmodules is as in Definition 4.1 with F replaced by O. The proof uses Green’s theory of vertices of indecomposable modules (see [NT, Chapter 4]).

• If W_{i} is relatively Q_{i}-projective for some proper subgroup Q_{i} of ⟨x⟩, then χ_{W_{i}}(xy) = 0 for any p′-element y of H. This is a consequence of a deep result known as Green’s indecomposability theorem (see [NT, Chapter 4, Theorem 7.4]).

Now if e is a block of kH, then multiplying the above decomposition of V with e gives

eV = ⊕_{i∈I}W_{i},

where I ⊂ {1, ⋯, t} consists of those i for which e_{i} = e. This is because for 1 ≤ i ≤ t, W_{i} = e_{i}W_{i} and ee_{i} = 0 if e ≠ e_{i}. Thus, if br_{P} (b)e = 0,

χ_{V} (exy) = χ_{eV} (xy) = ∑_{i∈I} χ_{W_{i}}(xy) = 0.  □

The second main theorem is often stated in terms of generalized decomposition numbers. We indicate briefly the connection with the version we have given above. keeping in mind the identification of modules and characters laid out in Section 4.4, let x be a p-element of G and let χ ∈ Irr_{K}(G). The restriction Res_{C_{G}(x)}^{G}χ of χ to KC_{G}(x) is a sum of ordinary irreducible C_{G}(x) characters, say

Res_{C_{G}(x)}^{G}χ = ∑_{τ∈Irr_{K}(C_{G}(x))} n_{χ,ζ}τ.

Since x is central in C_{G}(x), and K is a splitting field of C_{G}(x), by Schur’s Lemma (see [NT, Chapter 3, Lemma 3.5]), x acts as a scalar linear transformation on any simple KC_{G}(x)-module. Since x is a p-element, this means that for any ζ ∈ Irr_{K}(C_{G}(x)), there exists an element λ_{x,ζ} of K of p-power order such that ζ(xy) = λ_{x,ζ}ζ(y) for all y ∈ C_{G}(x). Let d^{x}:R_{K}(C_{G}(x)) → R_{k}(C_{G}(x)) be the decomposition map of OC_{G}(x), and for each ζ ∈ Irr_{K}(C_{G}(x)) and each τ ∈ Irr_{k}(C_{G}(x)), let d_{ζ,τ}^{(x)} be the corresponding decomposition number. For each χ ∈ Irr_{K}(G) and each τ ∈ Irr_{k}(C_{G}(x)), the generalised decomposition number is defined by

d_{χ,τ}^{x} = ∑_{ζ∈Irr_{K}(C_{G}(x))} n_{χ,ζ}λ_{x,ζ}d_{ζ,τ}^{(x)}.

Then, it is easy to see that for all χ ∈ Irr_{K}(G), τ ∈ Irr_{k}(C_{G}(x)) and any p′-element y of C_{G}(x),

χ(xy) = ∑_{τ∈Irr_{k}(C_{G}(x))} d_{χ,τ}^{x} τ(y),

where each τ ∈ Irr_{k}(C_{G}(x)) is identified to its Brauer character. From this and using the fact that the set of irreducible Brauer characters on C_{G}(x) is a linearly independent subset of the set of K-valued functions on the set of p′-elements of C_{G}(x)(see Theorem 4.21), it can be shown that Theorem 5.4 is equivalent to the following statement.

Theorem 5.5. Let x be a p-element of G, b a block of kG, e a block of kC_{G}(x). If (⟨x⟩, e) is not a (kG, b, G)-Brauer pair, then for any χ ∈ Irr_{K}(G, b) and any τ ∈ Irr_{k}(C_{G}(x), e), d_{χ,τ}^{x} = 0.

A consequence of Theorem 5.4 is that the number of ordinary irreducible characters in a block is “locally” determined, in a sense that is made precise below.

Definition 5.6. Let P be a finite p-group and let F be a saturated fusion system on P. Two elements x, y ∈ P are F-isomorphic if there is a morphism φ:⟨x⟩ → ⟨y⟩ in F such that φ(x) = y. Denote by P/F a set of representatives of the F-isomorphism classes of elements of P.

Theorem 5.7. Let b be a block of kG and let F = F_{(P,e_{P})}(kG, b, G) for some maximal (kG, b, G)-Brauer pair (P, e_{P}). For each Q ≤ P, let e_{Q} be the unique (kG, b, G)-Brauer pair such that (Q, e_{Q}) ≤ (P, e_{P}). Then,

|Irr_{K}(G, b)| = ∑_{x∈P/F} |Irr_{k}(C_{G}(x), e_{⟨x⟩})|.

Proof. See [NT, Chapter 5, Theorem 4.13, Theorem 9.4]. □

Examples 4.25 and 5.43 show how to calcuate the poset of Brauer pairs using character theory. Brauer’s Second Main Theorem can also used for this. In fact, the second main theorem is one of the tools for understanding the fusion systems of blocks of families of groups such as the finite groups of Lie type and their Weyl groups. To explain this properly would require an exposition of the representation theory of these classes of groups, but a rough idea of how the theorem is used may be gleaned from the following situation. Suppose that χ ∈ Irr_{K}(G, b), and x is a p-element of G such that χ(xy) ≠ 0 for some p′-element of C_{G}(x) and such that Res_{G}^{C_{G}(x)}χ ∈ Irr_{K}(C_{G}(x)). Then letting e be the block of kC_{G}(x) to which Res_{G}^{C_{G}(x)}χ belongs (with the obvious abuse of notation), by Theorem 5.4, (⟨x⟩, e) is a (kG, b, G)-Brauer pair.

The above applies to the the character of the trivial KG-module, and can be used to give a proof Brauer’s Third Main Theorem which is stated below.

Definition 5.8. Let G be a finite group. The principal block of kG is the block b of kG containing the trivial kG-module.

Theorem 5.9 (Brauer’s Third Main Theorem). Suppose that b is the principal block of kG. Then, for any p-subgroup Q of G, br_{Q}(b) is the principal block of kC_{G}(Q). Consequently, for any maximal (kG, b, G)-Brauer pair (P, e), P is a Sylow p-subgroup of G and F_{(P,e)}(kG, b, G) = F_{P} (G).

Proof. We sketch a proof of the above given in [AB, Theorem 3.13]. For any finite group H, let p_{H} denote the principal block of kH. Applying the discussion before Definition 5.8 repeatedly (or by using the fact that the principal block is the unique block of kG not contained in the augmentation ideal of kG) one proves that

• For any p-element x of G, ({1}, p_{G}) ≤ (⟨x⟩, p_{C_{G}(x)}); and

• For a p-subgroup Q of G and z a p-element of N_{G}(Q), (Q, p_{C_{G}(Q)}) ≤ (⟨z, Q⟩, p_{C_{G}(⟨z,Q⟩)}). Then by induction,

• For p-subgroups Q ≤ Q′ of G (Q, p_{C_{G}(Q)}) ≤ (Q′, p_{C_{G}(Q′)}).

The theorem follows from the uniqueness of inclusion of Brauer pairs and the fact that the principal block of kH for any finite group H is invariant under any automorphism of H. □

5.2. Relative projectivity and representation type.

The next result gives various characterisations of defect groups, each of which has been used as a definition of defect groups in the literature. We only give versions over k, but there are analogous characterisations over O. Note that if b is a central idempotent of kG, then kGb is invariant under left and right multiplication by G and hence kGb is a k[G × G]-module via (x, y).a = xay^{−1} for x, y ∈ G and a ∈ kGb = bkGb. For any H ≤ G, let ΔH = {(x, x) : x ∈ G} ≤ G × G be the diagonally embedded copy of H in G × G.

Proposition 5.10. Let b be a block of kG and let P be a p-subgroup of G. The following are equivalent.

(a) P is a defect group of b.

(b) br_{P} (b) ≠ 0 and P is maximal amongst p-subgroups of G with this property.

(c) b ∈ Tr_{P}^{G}((kG)^{P}) and P is minimal amongst subgroups with this property.

(d) The k[G × G]-module kGb is relatively ΔP -projective and ΔP is minimal amongst subgroups of G × G with this property.

(e) Any kGb-module is relatively P -projective and P is minimal amongst subgroups of G with this property.

Proof. The equivalence of (a), (b) and (c) is immediate from Theorem 2.20 (recall the identification of the Br^{kG}P with br_{P} as given in Proposition 3.12). The equivalence of (c) and (d) is given in [Be1, Proposition 6.1.2], and the equivalence of (d) and (e) is in [Be1, Proposition 6.3.3]. The subtle part of the equivalence of (d) and (e) is to deduce, from (d), the existence of a kGb-module which is not relatively Q-projective for any proper subgroup Q of G. □

Combining the equivalence of (a) and (e) above for P = {1} with the structure theory of finite dimensional algebras gives the following.

Theorem 5.11. Let b be a block of kG and let P be a defect group of kGb. If P = 1, kGb is isomorphic to a full matrix algebra over k. If P is non-trivial, then J(kGb) ≠ 0.

By the equivalence of (a) and (e) in Proposition 5.10 we see that the defect groups of a block b of kG measure how far away kGb-modules are from being projective, and this in turn is a measure of the complexity of the module category of the block algebra. The representation type of a finite dimensional algebra is another measure of the complexity of the module category of the algebra. For the definition of representation type, we refer the reader to [Be1, Section 4.4]. Here we just note that there are three possibilities for representation type: finite, tame, and wild. Roughly speaking, algebras of wild type have the most complex module categories, while algebras of finite type have the least complicated structure.

Theorem 5.12. ([Bre], [BonD], [H]) Let b be a block of kG and let P be a defect group of kGb. The block algebra kGb and the group algebra kP have the same representation type. Further,

(a) kP is of finite representation type if P is cyclic.

(b) kP is of tame representation type if p = 2 and P is the Klein 4-group, or a generalized quaternion, dihedral or semi-dihedral group.

(c) In all other cases kP is of wild representation type.

Blocks of finite representation type, i.e., blocks with cyclic defect groups, are well understood by work of many authors, notably Brauer, Thompson, Dade, Janusz, Green and Linckelmann. Blocks of tame representation type are also well understood by work of Erdmann. By contrast blocks of wild representation type are little understood. The differences in the state of knowledge for the various types will become apparent in the discussion regarding the status of various block theoretic conjectures. Many of the standard reference books on representation theory contain an exposition of the theory of cyclic blocks; Erdmann’s book [Er] gives an account of tame blocks.

5.3. Finiteness conjectures.

Definition 5.13. Let P be a finite group. A P -block is a block b of kH, for H some finite group, such that the defect groups of kHb are isomorphic to P.

One of the themes of block theory is this : Given a finite p-group P, to what extent are the representation theoretic invariants of P -blocks bounded by a function of |P |?

Theorem 5.14. (Brauer) Let b be a block of kG and let P be a defect group of kGb. The group R_{k}(G, b)/P r_{k}(G, b) has exponent |P |.

Proof. We sketch the idea behind the proof given in [NT, Chapter 3, Theorem 6.35]. Let |P | = p^{d}. One proves the following block-wise version of Theorem 4.22 (b) :

• With the notation of Theorem 4.22 (b), for any Φ ∈ R_{k}(G, b), p^{d}Φ^{∗} ∈ Pr_{O}(G, b).

The result follows from this and the fact that the decomposition map induces an isomorphism between Pr_{O}(G, b) and Pr_{k}(G, b) (Theorem 4.15 (a)). □

Theorem 5.15. (Brauer-Feit) Let b be a block of kG and let P be a defect group of kGb. Then,

|Irr_{K}(G, b)| ≤ |P |^{2}.

Proof. We sketch the idea behind the proof given in [NT, Chapter 3, Theorem 6.39]. The starting point is the result highlighted in the proof of the previous theorem. So, in particular, p^{d}(Res_{G_{p′}}^{G} χ)^{∗} ∈ R_{K}(G, b) for all χ ∈ Irr_{K}(G, b). Then one proves:

• For any χ ∈ Irr_{k}(G, b), writing

p^{d}(Res_{G_{p′}}^{G} χ)^{∗} = ∑_{τ∈Irr_{K}(G,b)} n_{τ} τ,

we have

∑_{τ∈Irr_{K}(G,b)} n_{τ}^{2} ≤ p^{2d}.

This is an easy consequence of the orthogonality relations for characters.

• There exists a χ ∈ Irr_{k}(G, b) such that writing p^{d}(Res_{G_{p′}}^{G} χ)^{∗} as above, we have that n_{τ} ≠ 0 for all τ ∈ Irr_{K}(G, b).

Since each n_{τ} above is an integer, the proof follows. □

Actually, the proof above yields the better upper bound of \frac{1}{4}|P |^{2} + 1. This is still quadractic in |P |. The following conjecture, due to Brauer, predicts that |P | itself is an upper bound for |Irr_{K}(G, b)|.

Conjecture 5.16 (Brauer’s k(b) conjecture). Let b be a block of kG and let P be a defect group of kGb. Then,

|Irr_{K}(G, b)| ≤ |P |.

By Theorem 4.21, |Irr_{k}(G, b)| ≤ |Irr_{K}(G, b)|, so it follows from Theorem 5.15 that the Cartan matrix of a P -block has size at most |P |^{2}. By Theorem 5.14, the elementary divisors of the Cartan matrix of a P -block are of the form p^{m} ≤ |P |. Thus there are only finitely many possibilities for the size and the elementary divisors of the Cartan matrix of a P -block. The following conjecture says that the same is true of the entries of the Cartan matrix.

Conjecture 5.17 (Weak Donovan Conjecture). [Al2] Let P be a finite p-group. There are only finitely many possibilities for Cartan numbers of P -blocks.

For a commutative ring R with identity, and A an R-algebra, denote by mod(A), the R-linear category of finitely generated A-modules. Two Ralgebras A and B are said to be Morita equivalent if mod(A) and mod(B) are equivalent as R-linear categories. The above conjecture is implied by the following conjecture, also due to Donovan. For the statement, we identify a block with the corresponding block algebra.

Conjecture 5.18 (Strong Donovan conjecture). [Al2] Let P be a finite p-group. Up to Morita equivalence, there are only finitely many P -blocks.

Conjecture 5.18 is know to be true if P is cyclic by results of Brauer, Dade, Green and Janusz. Erdmann settled the case P ≅ C_{2} × C_{2} and P ≅ Q_{8}. For the other tame cases, Erdmann’s work nearly settles the conjecture, but there are some outstanding rationality questions; Conjecture 5.17 is known to hold for all tame blocks. By Puig’s nilpotent block theorem (see Theorem 5.40), Conjecture 5.18 also holds for those p-groups P with the property that the only saturated fusion system on P is the system F_{P} (P). Both conjectures are open for all other P.

The isomorphism class of the center Z(A) of a finite dimensional kalgebra is an invariant of the Morita equivalence class of A. In this context, we have the following result.

Theorem 5.19. Let P be a finite p-group. Up to isomorphism, there are only finitely many commutative k-algebras that occur as centers of P -blocks.

Proof. Let b be a block of kG with defect group P. Since dim_{k}(Z(kGb)) = dim_{K}(Z(KGb)) = |Irr_{K}(G, b)|, it follows from Theorem 5.15, that the kdimension of Z(kGb) is bounded by a: = |P |^{2}. By a result of Cliff, Plesken and Weiss [CPW], Z(kGb) has an 𝔽_{p}-form, that is Z(kGb) has a k-basis such that the multiplicative constants of Z(kGb) with respect to this basis are all in 𝔽_{p}. Thus there are at most p^{a^{3}}-possibilities for the isomorphism type of Z(kGb). □

5.4. Source algebras and Puig’s conjecture.

The concept of source algebras is due to Puig, who also proved their main properties. For a detailed account see [Th, Section 38].

Definition 5.20. Let b a block of kG and let P be a defect group of kGb.

• A source idempotent of kGb (or of b)with respect to P is a primitive idempotent i of (kG)^{P} such that i ≤ b and br_{P} (i) ≠ 0.

• A source algebra of kGb (or of b) is an interior P -algebra (ikGi, ι) where i is a source idempotent of kGb and ι is defined by ι(x) = ixi for x ∈ P.

By Lemma 1.19 (a), it is clear that source algebras exist. They are unique up to G-conjugation, as we now explain. Let P be a defect group of b and let i be a source idempotent of b with respect to P. Let e(i) be the unique block of kC_{G}(P) such that i is associated to e(i) (see Lemma 2.8) . For any u ∈ (A^{P})^{∗}, ^{u}e is a source idempotent of kGb and e(^{u}i) = e(i).

Proposition 5.21. With the notation above, the mapping (P, i) → (P, e(i)), induces a bijection between pairs (P, γ) such that P is a defect group of b and γ is an (A^{P})^{∗}-conjugacy class of source idempotents of b with respect to P and the set of maximal (kG, b, G)-Brauer pairs.

Consequently, if ikGi and jkGj are source algebras of b, where i and j are source idempotents with respect to defect groups P and Q respectively, then there exists g ∈ G and an isomorphism of k-algebras f : jkGj → ikGi such that P = ^{g}Q and f(jxj) = i^{g}xi for all x ∈ Q.

Proof. See [Th, Proposition 18.3]. □

Thus, for a given defect group P, a source algebra of b is unique up to twisting the interior P -algebra structure by an outer automorphism of P.

Proposition 5.22. Let b be a block of kG and let i be a source idempotent of b with respect to some defect group of kGb. The algebras ikGi and kGb are Morita equivalent.

Proof. See [Th, Proposition 38.2]. □

The k-dimension of a source algebra ikGi of kGb is usually much smaller than that of kGb, so the above result reduces the study of the representation theory of kGb to a smaller algebra. A source algebra of kGb is not in any sense ”the smallest” amongst subalgebras of kGb which are Morita equivalent to kGb. For instance, if I is a primitive decomposition of a source idempotent i in ikGi, and J ⊆ I is such that J contains at least one kG-conjugate of each element of I, then setting j = J^{+}, the algebra jkGj is also Morita equivalent to kGb. The advantage of source algebras is that they contain information about blocks which is finer than that captured by Morita equivalence. For instance, the next result shows that the fusion system of kGb can be read off from a source algebra of kGb.

If b is a block of kG and i is a source idempotent of b with respect to P, then since xi = ix for all x ∈ P, ikGi is invariant under left and right multilplication by elements of P. Thus, ikGi is a k[P × P]-module via (x, y).a = xay^{−1} for x, y ∈ P and a ∈ ikGi. This k[P × P]-module structure determines the fusion system of b. We make this more precise. For a finite group H, and an automorphism φ: H → H, we denote by _{φ}kH the k[H × H]-module defined by _{φ}kH = kH as k-space and where the k[H × H]-module structure is given by (g, h).m = φ(g)mh for g, h ∈ H and m ∈ kH.

Proposition 5.23. Let b be a block of kG, let P be a defect group of kGb and let i be a source idempotent of b with respect to P. Let e_{P} be the block of kC_{G}(P) such that i is associated to e_{P} and let F = F_{(P,e_{P})}(kG, b, G). Then F is generated by the set of inclusions between subgroups of P and those automorphisms φ of subgroups Q of P such that _{φ}kQ is isomorphic to a direct summand of ikGi as k[Q × Q]-module.

Proof. This is essentially proved in [P2]. In the the formulation above, the result appears in [Li5, Corollary 5.3]. □

Source idempotents and source algebras are also defined over O. Let b̂ be the block of OG lifting b and let P be a defect group of kGb. A source idempotent of OGb̂ (or of b̂) with respect to a defect group P of b is a primitive idempotent i of (OG)^{P} such that i ≤ b̂ and br_{P} (i + 𝔭) ≠ 0 and the corresponding source algebra is iOGb̂i considered as interior P -algebra via the map x → ixi, for x ∈ P. A primitive idempotent i of (OG)^{P} is a source idempotent if and only if ī: = i + 𝔭 is a source idempotent of kGb and in that case īkGī = iOGi/𝔭iOGi . Conversely, by idempotent lifting, for any source idempotent i of kGb, there exists a source idemotent î of OGb with î + 𝔭 = i. From the point of view of representation theory it is better to study source algebras over O since this allows us to keep track of the ordinary character theory of blocks as well. On the other hand, the structure of a source algebra over O is determined largely by the structure over k.

Proposition 5.24. Let b be a block of kG and b̂ the lift of b in OG. Let P be a defect group of kGb, let i be a source idempotent of OGb and let ī = i+𝔭. Then, iOGi has an O-basis stable under left and right multiplication by P and īkGī = iOGi/𝔭iOGi. Further, if B is a interior P -algebra over O having an O-basis which is stable under left and right multiplication by P and such that īkGī is isomorphic to B/𝔭B as interior P -algebras, then B is isomorphic to iOGi as interior P -algebras.

Proof. See [Th, Proposition 38.8]. □

Definition 5.25. Let G and H be finite groups, b a block of kG and c a block of kH. We say that b and c aresource algebra equivalent if b and c have a common defect group P and there are source idempotents i of b and j of c with respect to P such that there is an isomorphism of interior P -algebras algebras ikGi ≅ jkHj.

From Propositions 5.22 and 5.23 it is immediate that two source algebra equivalent blocks have equivalent module categories and isomorphic fusion systems. We describe one well-known situation of a source algebra equivalence between blocks of different finite groups.

Proposition 5.26. Let H be a subgroup of G. Suppose that c is a central idempotent of kH such that Tr_{H}^{G}(c) is a central idempotent of kG and c^{g}c = 0 for all g ∈ G − H. Set d = Tr_{H}^{G}(c). Then the map e → Tr_{H}^{G}(e) is a bijection between the set of blocks e of kH such that e ≤ c and the set of blocks f of kG such that f ≤ d. Further, if e is a block of kH such that e ≤ c, then the block algebras kHe and kGTr_{H}^{G}(e) are source algebra equivalent.

Proof. See [Th, Lemma 16.1, Proposition 16.6, Proposition 16.9] □

Conjecture 5.27 (Puig’s conjecture). Let P be a finite p-group. Up to source algebra equivalence, there are only finitely many P -blocks.

The above conjecture implies the Donovan conjectures. The conjecture is known to be true when P is cyclic ([Li1]). Recently, using the classification of finite simple groups it has been settled for the case P ≅ C_{2} × C_{2} ([CrEKL]). It is open for all other p-groups.

5.5. Külshammer-Puig classes.

Let n be a natural number and A = Mat_{n}(k). We recall the following standard facts about A.

• (Noether-Skolem theorem) Aut(A) = Inn(A).

• Z(A) consists of scalar matrices. Hence, under the natural identification of the group of invertible scalar matrices with the group of non-zero elements k^{∗} of k, we have a group isomorphism Inn(A) ≅ A^{∗}/k^{∗}.

Now suppose N is a finite group and that A = Mat_{n}(k) is an N-algebra. Suppose further that C is a normal subgroup of N and that the restriction of the N-algebra structure of A to C is interior, that is suppose that there is a group homomorphism ι: C → A^{∗} such that for all x ∈ C, the action of x on A is given by conjugation by ι(x). We use this data to describe a certain distinguished element α_{A,N,C} of H^{2}(N/C, k^{×}). For each y ∈ N, the map a → ^{y}a for a ∈ A defines a k-algebra automorphism of A and by the Noether-Skolem theorem every algebra automorphism of A is an inner automorphism. Define a function s: N → A^{∗} as follows. Choose a set of coset representatives, say I of C in N containing 1. For each x ∈ N, set s(x) = ι(x) and for each z ∈ I − N choose s(z) ∈ A^{∗} such that ya = s(y)as(y)^{−1} for all a ∈ A. Then, for y ∈ N − C, set s(y) to be equal to ι(x)s(z), where y = xz, x ∈ C, z ∈ I.

For all y ∈ N, ^{y}a = s(y)as(y)^{−1} for all a ∈ A. So, for any y, z ∈ N, and any a ∈ A,

s(yz)as(yz)^{−1} = ^{yz}a = s(y)(^{z}a)s(y)^{−1} = s(y)s(z)as(z)^{−1}s(y)^{−1}.

Hence, s(z)^{−1}s(y)^{−1}s(yz) ∈ Z(A) ∩ A^{∗}. Since Z(A) consists of scalar matrices, it follows that s(yz) = α(y, z)s(y)s(z) for some element α(y, z) ∈ k^{∗}. It is a straightforward check that the map α: N × N → k^{∗} sending an element (y, z) to α(y, z) is a 2-cocycle. Further, for any y, z ∈ N and any x, x′ ∈ C, α(yx, zx′) = α(y, z). Hence α induces a 2-cocycle, say ᾱ of N/C. If α, α′ are 2-cocycles of N resulting from different choices of s above, then the corresponding 2-cocycles ᾱ and ᾱ′ of N/C are cohomologous. Thus, we can make the following definition.

Definition 5.28. With the notation above, α_{A,N,C} is the class in H^{2}(N/C, k^{∗}) of the 2-cocycle N/C × N/C → k^{∗} which sends an element (yC, zC) of N/C × N/C to the element α(y, z) of k^{∗}.

In the special case that A = k, the N-action on A is trivial and the interior C-algebra structure on A is defined by a homomorphism ι: C → k^{∗}.

Proposition 5.29. Suppose that A = k, and let N, C and ι be as above. Denote also by ι the induced map from C/ ker(ι) → k^{∗} and let ι_{∗}: H^{2}(N/C, C/ ker(ι)) → H^{2}(N/C, k^{∗}) denote the map induced by ι. Suppose that ker(ι) is normal in N and that C/ ker(ι) is central in N/ ker(ι). Then, α_{A,N,C} = ι_{∗}(β) where β ∈ H^{2}(N/C, C/ ker(ι)) corresponds to the central extension

1 → C/ ker(ι) → N/ ker(ι) → N/C → 1.

Proof. In the recipe given for constructing α_{A,N,C} chooose s(z) = 1_{A} for every element of the chosen system of coset representatives of C in N. □

Before proceeding we record the following fact.

Proposition 5.30. Let Q be a central p-subgroup of a finite group H. Then the canonical k-algebra surjection

kH → k(H/Q)

induces a bijection between blocks of kH and blocks of kH/Q. If R is a defect group of a block of kH, then Q ≤ R and R/Q is a defect group of the corresponding block of kH/Q.

Proof. This is in [NT, Chapter 5, Theorem 8.11]. □

Let b be a block of kG, (P, e_{P}) a maximal (kG, b, G)-Brauer pair and F = F_{(P,e_{P})}(kG, b, G). Suppose that Q is an F-centric subgroup of P. Let \overline{C_{G}(Q)} = C_{G}(Q)/Z(Q) and let \overline{e_{Q}} be the image of e_{Q} in k\overline{C_{G}(Q)} under the canonical surjection of kC_{G}(Q) → k\overline{C_{G}(Q)}. By Theorem 3.20, Z(Q) is a defect group of the block e_{Q}. Since Z(Q) is central in C_{G}(Q), by Proposition 5.30, \overline{e_{Q}} is a block of k\overline{C_{G}(Q)} with trivial defect groups. In particular, k\overline{C_{G}(Q)}\overline{e_{Q}} is a full matrix algebra over k. Set A = k\overline{C_{G}(Q)}\overline{e_{Q}}. Regard \overline{C_{G}(Q)} as a subgroup of N_{G}(Q, e_{Q})/Q through the natural isomorphism \overline{C_{G}(Q)} ≅ QC_{G}(Q)/Q. Regard A as an N_{G}(Q, e_{Q})/Q algebra through the conjugation action of N_{G}(Q, e_{Q}) on kC_{G}(Q) and regard A as interior \overline{C_{G}(Q)}-algebra through the map ι:\overline{C_{G}(Q)} → A^{∗} which sends an element x of \overline{C_{G}(Q)} to x\overline{e_{Q}}. Let α_{Q} be the element α_{A,N,C,ι} of Definition 5.28 with N = N_{G}(Q, e_{Q})/Q and C = \overline{C_{G}(Q)}. We regard α_{Q} as an element of H^{2}(Out_{F}(Q), k^{∗}) through the identification Out_{F}(Q) = N_{G}(Q, e_{Q})/QC_{G}(Q) = N/C. We will denote also by α_{Q} the element of H^{2}(Aut_{F}(Q), k^{∗}) obtained from α_{Q} through the canonical homomorphism Aut_{F}(Q) → Out_{F}(Q).

Definition 5.31. Let b be a block of kG, (P, e_{P}) a maximal (kG, b, G)- Brauer pair, and F = F_{(P,e_{P})}(kG, b, G). Let Q be an F-centric subgroup of P. With the notation above, the Külshammer-Puig class at (Q, e_{Q}) is the element α_{Q} of H^{2}(Out_{F}(Q), k^{∗}).

We describe Külshammer-Puig classes in some cases.

Proposition 5.32. With the notation of Definition 5.31, if b is the principal block of kG, then α_{Q} is trivial for all Q ∈ F^{c}.

Proof. By Brauer’s Third Main Theorem (see Theorem 5.9) P is a Sylow p-subgroup of G, F = F_{P} (G) and e_{Q} is the principal block of kC_{G}(Q) for all Q ≤ P. Thus \overline{e}_{Q} is the principal block of k\overline{C_{G}(Q)} for all Q ≤ P.

Suppose that Q is F-centric. Since F = F_{P} (G), Z(Q) is a Sylow psubgroup of C_{G}(Q), C_{G}(Q) = Z(Q)×O_{p′}(C_{G}(Q)) and \overline{C_{G}(Q)} ≃ O_{p′}(C_{G}(Q)). Since k is of characteristic p, by the Fourier inversion formula, the principal block idempotent, \overline{e_{Q}} of k\overline{C_{G}(Q)} is given by the formula

ē_{Q} = \frac{1}{|k\overline{C_{G}(Q)}|} ∑_{x∈k\overline{C_{G}(Q)}} x.

Hence, x\overline{e}_{Q} = \overline{e}_{Q} for all x ∈ \overline{C_{G}(Q)}. It follows that k\overline{C_{G}(Q)}\overline{e}_{Q} is a onedimensional algebra, and ι: \overline{C_{G}(Q)} → k^{∗} is the trivial map. So α_{Q} is trivial by Proposition 5.29. □

Certain defect groups also support only trivial Külshammer-Puig classes.

Proposition 5.33. With the notation of Definition 5.31, suppose that either P is cyclic or p = 2 and P is a Klein 4-group, or a dihedral, semidihedral or quaternion group. Then α_{Q} is trivial for all Q ∈ F^{c}.

Proof. Let Q be a subgroup of P. We will show that H^{2}(Out_{F} (Q), k^{×}) = 0 for any subgroup Q of P. Since H^{2}(H, k^{×}) = 0 if H is cyclic or if H is a p-group, we may assume that Out_{F}(Q) is neither cyclic nor a p-group and hence that Aut(Q) is neither cyclic nor a p-group. Since the automorphism group of a cyclic group of odd prime power is cyclic, we may assume that p = 2 and that Aut(Q) is not a 2-group. By Example I.3.8, Q is isomorphic to one of Q_{8} or C_{2}^{2}. Now Aut(C_{2}^{2}) is cyclic, so we may assume that Q ≅ Q_{8}. In this case, Out_{F}(Q) is isomorphic to one of C_{3}, C_{2} or S_{3} and H^{2}(C_{3}, k^{∗}) = H^{2}(C_{2}, k^{∗}) = H^{2}(S_{3}, k^{∗}) = 0. □

In contrast to the above two results, we describe one source for nontrivial Külshammer-Puig classes. Recall the definition of constrained fusion systems from before Theorem I.4.9.

Definition 5.34. For a saturated fusion system F on a finite p-group and a block b of kG, we say that b is an F-block if F ≅ F_{(P,e_{P})}(kG, b, G) for some maximal (kG, b, G)-Brauer pair (P, e_{P}).

Proposition 5.35. Let F be a constrained fusion system on a finite pgroup P and let Q ⊴ F be an F-centric subgroup of P. Then for any element α ∈ H^{2}(Out_{F}(Q), k^{∗}), there exists a finite group G and an Fblock b of kG such that the corresponding Külshammer-Puig cocycle at Q is α.

Proof. Let L be a model for F (see Theorem I.4.9) and identify L/Q with Out_{F}(Q). Let C be the cyclic subgroup of order |L|_{p′} of k^{∗} and let ι: C → k^{∗} be the inclusion of C in k^{∗}. The group C exists since k is algebraically closed. Since k is of characteristic p, the induced map ι_{∗}: H^{2}(L/Q, C) → H^{2}(L/Q, k^{∗}) is surjective. Let β ∈ H^{2}(L/Q, C) be such that α = ι_{∗}(β). Denote also by β the element of H^{2}(L, C) obtained from β through the canonical homomorphism L → L/Q and let

1 → C → G → L → 1

and

1 → C → G/Q → L/Q → 1

be the corresponding central extensions. For each p-subgroup R of G, identify R with the Sylow p-subgroup of the full inverse image of R in G. So P is a Sylow p-subgroup of G and F_{P} (G) = F_{P} (L) ≅ F.

Set

b = \frac{1}{|C|} ∑_{z∈C} ι(z)^{−1}z.

So by the Fourier inversion formula, b is the block of kC corresponding to the linear representation ι of C. We claim that b is a block of kG. Indeed, since C is central in G, b is a central idempotent of kG. On the other hand, since Q is a normal p-subgroup of G, by Lemma 3.17 any central idempotent of kG is an element of kC_{G}(Q) and by Proposition 4.26, any central idempotent of kG is a k-linear combination of p′-elements of G. Thus any central idempotent of kG is a k-linear combination of p′-elements of C_{G}(Q). Since L is strongly p-constrained, C_{G}(Q) = C × Z(Q). Thus any central idempotent of kG is in kC. Since by definition, b is primitive in Z(kC), it follows that b is primitive in Z(kG), proving the claim.

Let R be a p-subgroup of G. Since C ≤ Z(G) ≤ C_{G}(R), and b ∈ kC, we have br_{R}(b) = b and (R, b) is a (kG, b, G)-Brauer-pair. In particular, (P, b) is a maximal (kG, b, G)-Brauer pair, (R, b) ≤ (P, b) for any subgroup R of G and N_{G}(R, b) = N_{G}(R) for any R ≤ P. In particular, F_{(P,b)}(kG, b, G) = F, that is b is an F-block.

Now we show that the Külshammer-Puig class α_{Q} equals α. Note that N_{G}(Q, b) = G and C_{G}(Q) = C × Z(Q). Now

k\overline{C_{G}(Q)}\overline{b} = k[C × Z(Q)/Z(Q)]\overline{b} ≅ kCb ≅ k.

Identify \overline{C_{G}(Q)} with C through the canonical isomorphism \overline{C_{G}(Q)} ≅ C. Then the interior \overline{C_{G}(Q)}-algebra structure of k\overline{C_{G}(Q)}\overline{b} corresponds to ι:C → k^{∗}, ι is faithful and C is central in G/Q. Thus Proposition 5.29 applies. By definition the central extension

1 → C → G/Q → L/Q

corresponds to the element β ∈ H^{2}(C, k^{∗}). Thus, by Proposition 5.29, α_{Q} = ι_{∗}(β). This proves the result as by definition ι_{∗}(β) = α. □

Definition 5.36. Let R be a a commutative ring with 1 and let α ∈ H^{2}(G, R^{∗}). Let α̃: G×G → R^{∗} be a 2-cocycle in the class of α. The twisted group algebra of G with respect to α is the R-algebra R_{α}G which is equal to RG as R-module and with multiplication ∗ satisfying g ∗g′ = α̃(g, g′)gg′, for g, g′ ∈ G.

It is easy to check that up to isomorphism, the R-algebra R_{α}G is independent of the choice of 2-cocycle α̃. Also, if α is the trivial class then R_{α}G is isomorphic to the usual group algebra RG.

A twisted group algebra of G over k corresponds to a factor of the untwisted group algebra of a central extension of G by a p′-group:

Proposition 5.37. Let α ∈ H^{2}(G, k^{∗}). Then there exists a central extension of finite groups

1 → Z → H → G → 1,

with Z cyclic of order prime to p such that k_{α}G is isomorphic to kHd for some central idempotent d of kH.

Proof. This is very similar to the first part of the proof of Proposition 5.35. Let Z be a cyclic group of order |G|_{p′} and let ι: Z → k^{∗} be an injective group homomorphism. The induced map ι_{∗}: H^{2}(G, Z) → H^{2}(G, k^{∗}) is surjective. Choose β ∈ H^{2}(G, Z) with ι_{∗}(β) = α and let

1 → Z → G → H → 1

be the central extension corresponding to β. Set

d = \frac{1}{|Z|} ∑_{z∈Z} ι(z)^{−1}z

Then d is a block of kZ and hence a central idempotent of kH. Choose a section s: G → H and let φ: k_{α}G → kHd be the k-linear map which sends g to s(g)d for each g ∈ G. Then, φ is an isomorphism of k-algebras. □

5.6. Nilpotent blocks and extensions.

Let G be a finite group, b a block of kG and let (P, e_{P}) be a maximal (kG, b, G)-Brauer pair. Set F: = F_{(P,e_{P})}(G, b) and let b̂ denote the block of of OG lifting b, that is b is the image of b̂ under the surjection OG → kG.

Recall Frobenius’s theorem: For a finite group G and S a Sylow psubgroup of G, F_{S}(G) = F_{S}(S) if and only if G has a normal p-complement, that is if and only if G = O_{p′}(G) ⋊ S.

Definition 5.38. The block b is nilpotent if F = F_{P} (P).

The above definition, due to Broué and Puig [BP1] is a natural generalization of Frobenius’s condition as by Brauer’s Third Main Theorem, if b is the principal block of kG, then b is nilpotent if and only if F_{S}(G) = F_{S}(S) for any Sylow p-subgroup S of G. It was shown in [BP1] that the ordinary character theory of a nilpotent block is analogous to that of a p-nilpotent group. In [P3], Puig gave a structure theorem for nilpotent blocks, which we state below. For a detailed exposition, we refer the reader to [Th, Chapter 7]. Recall that if V is a kP -module then End_{k}(V) is naturally an interior P -algebra (see Example 1.12). We will refer to this interior P -algebra structure below. Recall also that any interior P -algebra inherits a natural P -algebra structure. Further, note that if (A, ι) and (B, κ) are interior P -algebras over k, then A ⊗_{k} B is naturally an interior P -algebra via the map which sends an element x of P to the element ι(x) ⊗ κ(x) of A ⊗_{k} B.

Definition 5.39. A Dade P -algebra is an interior P -algebra S such that

• As k-algebra S ≅ Mat_{n}(k); and

• S is a p-permutation P -algebra with Br_{P}^{S} (S^{P}) ≠ 0.

A primitive Dade P -algebra is a Dade P -algebra S such that 1_{S} is a primitive idempotent of S^{P}.

Theorem 5.40. Suppose that b is a nilpotent block of kG, and let i be a source idempotent of b with respect to P. There exists a primtive Dade P -algebra S such that

ikGi ≅ S ⊗_{k} kP

as interior P -algebras.

In particular, the k-algebra kGb is Morita equivalent to kP and up to isomorphism, kGb has a unique simple module.

Proof. This is the main result of [P3]. In fact, [P3] gives a structure theorem over O. The lifting of the above result from k to O is a deep result, even given the rigidity of source algebras over O described in Proposition 5.24. The main difficulty is in proving that the kP -module V above has a lift to an O-free OP -module. The paper [KulOW] gives a simplification of Puig’s proof over k. □

In [KulP] Külshammer and Puig proved an extension theorem for nilpotent blocks. We describe a special case of their result which gives the structure of the normaliser block algebras of centric Brauer correspondents. Suppose that Q ∈ F^{c} is fully F-normalized. By Proposition 3.17, e_{Q} is a block of kN_{G}(Q, e_{Q}), and by Theorem 3.19, (N_{P} (Q), e_{N_{P} (Q)}) is a maximal (kN_{G}(Q, e_{Q}), e_{Q}, N_{G}(Q, e_{Q}))-Brauer pair, and the corresponding fusion system is N_{F}(Q). Since Q is F-centric, QC_{F}(Q) = F_{Q}(Q) and hence N_{F}(Q) is constrained. Consequently, N_{F}(Q) has a model (see Theorem I.4.9). In [KulP], Külshammer and Puig described the source algebra of kN_{G}(Q, e_{Q})e_{Q} in terms of a model for N_{F}(Q) and the Kulshammer-Puig cohomology class α_{Q} (the existence and uniqueness of the model for N_{F}(Q) in this context is also a part of their proof).

Theorem 5.41. With the notation above, let Q ∈ F^{c} be fully F-normalized and let α_{Q} be as in Definition 5.31. There exists a strongly p-constrained group L_{Q}, unique up to isomorphism, with N_{P} (Q) as Sylow p-subgroup, Q = O_{p}(L_{Q}), F_{N_{P} (Q)}(L_{Q}) ≅ N_{F}(Q) and a primitive Dade P -algebra S, such that denoting also by α_{Q} the element of H^{2}(L_{Q}, k^{∗}) obtained by restriction through the surjection of L_{Q} onto L_{Q}/Q ≅ Out_{F}(Q), there is an isomorphism of interior P -algebras

ikN_{G}(Q, e_{Q})i ≅ S ⊗_{k} k_{α_{Q}}L_{Q},

where i is a source idempotent of kN_{G}(Q, e_{Q})e_{Q}. In particular, the kalgebra kN_{G}(Q, e_{Q})e_{Q}) is Morita equivalent to k_{α_{Q}}L_{Q})

Proof. See [KulP, Theorem 1.12, 1.20.3]. □

Theorems 5.40 and 5.41 show that the module categories of certain blocks are determined to a large extent by the corresponding fusion systems. Another example of such control is the following consequence of Glauberman’s Z^{∗}-theorem (see I.1):

Theorem 5.42. Suppose that p = 2, that b is the principal block of kG and that Z(F) contains a subgroup Q of order 2. Then kGb and kC_{G}(Q)e_{Q} are isomorphic k-algebras.

Proof. By Brauer’s Third Main Theorem, e_{Q} is the principal block of kC_{G}(Q), P is a Sylow p-subgroup of G and F = F_{P} (G). So, by Glauberman’s Z^{∗}-theorem, G = C_{G}(Q)O_{p′}(G). The isomorphism of block algebras follows by combining Glauberman’s theorem with the following fact about principal blocks (see [NT, Chapter 5, Theorem 8.1]):

• If b is the principal block of kG, then xa = a for all x ∈ O_{p′}(G) and all a ∈ kGb.

□

5.7. Counting Conjectures.

Let G be a finite group, b a block of kG and let (P, e_{P}) be a maximal (kG, b, G)-Brauer pair. Set F: = F_{(P,e_{P})}(G, b) and let b̂ denote the block of OG lifting b. For a finite dimensional algebra A over k, let l(A) denote the number of isomorphism classes of simple A-modules. So, l(kGb) = |Irr_{k}(G, b)|. Let z(A) denote the number of isomorphism classes of A-modules which are both simple and projective. In this subsection, we consider the question: to what extent does F determine |Irr_{k}(G, b)| and |Irr_{K}(G, b)|? For instance, if F = F_{P} (P), then by Theorem 5.40, |Irr_{k}(G, b)| = 1. If kP has tame representation type, then also the number of isomorphism classes of simple kGb-modules is completely determined by F:

• If P is cyclic, a Klein 4-group or a quaternion group of order 8, then |Irr_{k}(G, b)| = |Out_{F}(P)|.

• If P is a dihedral, semidihedral or quaternion group of order 2^{n} ≥ 16 as in Examples I.2.7 and I.3.8, then

- |Irr_{k}(G, b)| = 1 if F = F_{00},

- |Irr_{k}(G, b)| = 2 if F = F_{10} or F = F_{01}; and

- |Irr_{k}(G, b)| = 3 if F = F_{11}.

The above results are due to Brauer ([Br1]) and Dade ([Da1]) when P is cyclic, to Brauer ([Br2], [Br3] ) when P is a Klein 4 group or dihedral and to Olsson ([Ols]) when P is semidihedral or quaternion.

Example 5.43. Let p = 3 and let P be an elementary abelian group of order 9 with generators x, y. Let E = ⟨σ, τ:σ^{4} = τ^{4} = 1, στσ^{−1} = τ^{−1}⟩ be a quaternion group of order 8. The group E acts on P via

σx = x^{2} ^{τ} x = x, ^{σ}y = y ^{τ} y = y^{2}.

Set

G: = P ⋊ E, b_{0} = \frac{1}{2}(1 + σ^{2}), b_{1} = \frac{1}{2}(1 − σ^{2}).

By the Fourier inversion formula, b_{0} is the principal block of k⟨σ^{2}⟩ and b_{1} is the block of k⟨σ^{2}⟩ containing the non-trivial one dimensional k⟨σ^{2}⟩- module. Now, ⟨σ^{2}⟩ = C_{G}(P) and is a central subgroup of G; thus by Lemma 3.17, b_{0} and b_{1} are blocks of kG.

Since br_{P} (b_{0}) = b_{0}, br_{P} (b_{1}) = b_{1} and b_{0} and b_{1} are also blocks of kC_{G}(P), it follows that (P, b_{0}) is a maximal b_{0}-Brauer pair, (P, b_{1}) is a maximal b_{1}-Brauer pair. So, N_{G}(P, b_{0}) = N_{G}(P, b_{1}) = N_{G}(P). Since P is abelian it follows that F_{(P,b_{0})}(G, b_{0}) = F_{(P,b_{1})}(G, b_{1}) = F_{P} (G).

We count the number of simple modules in b_{0} and b_{1}. For this, note that as b_{0} + b_{1} = 1_{kG}, b_{0} and b_{1} are the only blocks of kG and hence any simple kGb-module belongs to exactly one of b_{0} or b_{1}. Also, note that if N is a normal subgroup of a finite group H, then a k[H/N] -module V is a kH-module through restriction along the natural map H → H/N and V is simple as k[H/N]-module if and only if V is simple as kH-module.

Further, a kH-module W is the inflation of a k[H/N]-module as above if and only if xw = w for all x ∈ N and all w ∈ W.

Since P is normal in G, by the first part of the proof of Lemma 3.16, P acts trivially on any simple kG-module. It follows that simple kG-modules are just inflations of simple k[G/P]-modules. Now G/P is isomorphic to E, and |E| is relatively prime to 3 = char(k). Thus, by Theorem 4.5, |Irr_{k}(G)| = |Irr_{k}(G/P)| equals the k-dimension of E, which in turn equals the number of conjugacy classes of E, that is |Irr_{k}(G/P)| = 5. Now, if V is a simple kE -module, it is easy to see that V regarded as kG-module belongs to the block b_{0} if and only if σ^{2} acts trivially on V, that is if and only if V is the inflation of a simple kE/⟨σ^{2}⟩-module. Since E/⟨σ^{2}⟩ = C_{2}^{2} and Irr_{k}(C_{2}^{2}) = 4, it follows that |Irr_{k}(G, b_{0})| = 4 and |Irr_{k}(G, b_{1})| = 1.

The above example shows that |Irr_{k}(G, b)| is not in general completely determined by F. However, in [Al3], Alperin gave a conjectural formula for |Irr_{k}(G, b)| which implies that |Irr_{k}(G, b)| is described completely by F and the Külshammer-Puig classes associated to F-centric-radical subgroups of P.

For a p-subgroup Q of G, and an element a ∈ kN_{G}(Q), denote by ā the image of an element a of kN_{G}(Q) under the canonical surjection

kN_{G}(Q) → kN_{G}(Q)/Q.

Recall that either br_{Q}(b) = 0 or br_{Q}(b) is a central idempotent of N_{G}(Q). Consequently, either \overline{br_{Q}(b)} = 0 or \overline{br_{Q}(b)} is a central idempotent of kN_{G}(Q)/Q.

Definition 5.44. A weight of kGb is a pair of the form (Q, w), where Q is a p-subgroup of G, and w is a block of kN_{G}(Q)/Q such that w has trivial defect group and such that w\overline{Br_{Q}(b)} = w.

Note that G acts on the set of kGb-weights by conjugation.

Conjecture 5.45. (Alperin’s Weight Conjecture) [Al3] The number of isomorphism classes of simple kGb-modules equals the number of G-orbits of kGb-weights.

The number of G-orbits of kGb-weights can be expressed in terms of the fusion system F and the Külshammer-Puig classes. Recall from Definition I.3.1 that F^{cr} denotes the full subcategory of F whose objects are the F-centric-radical subgroups of P.

Proposition 5.46. For Q ∈ F^{cr}, let α_{Q} denote the element of H^{2}(Out_{F}(Q), k^{∗}) as in Definition 5.31 and let F/F^{cr} denote a set of representatives of the F- isomorphism classes of objects of F^{cr}.

The number of G-orbits of kGb-weights is equal to

∑_{Q∈F^{cr}/F} z(k_{α_{Q}}Out_{F}(Q)).

Thus, conjecture 5.45 is equivalent to the statement

|Irr_{k}(G, b)| = ∑_{Q∈F^{cr}/F} z(k_{α_{Q}}Out_{F}(Q)).

Proof. See [K1, Proposition 5.4]. □

If F = F_{P} (P), then P is the only centric-radical subgroup of P, and Out_{F}(P) = 1. Hence in this case, by Proposition 5.46, Conjecture 5.45 is equivalent to the equality |Irr_{k}(G, b)| = 1, which is known to hold by Theorem 5.40.

By Proposition 5.33, if P is cyclic or p = 2 and P is a Klein 4-group, or a dihedral, semi-dihedral or quaternion group, then α_{Q} is trivial for all Q in F^{c}. Then, using Proposition 5.46, it is an easy exercise to check that the results of Brauer, Dade and Olsson described at the beginning of this subsection are consistent with Alperin’s weight conjecture. Hence, the conjecture is true if kP is of finite or tame representation type. It has also been proved in the case that P is an abelian 2-group of rank 2 ([PUs],[Sa]). Using the classification of finite simple groups, Conjecture 5.45 has been proved when P is an elementary abelian 2-group of order 8 [KKoL].

We consider the statement of Conjecture 5.45 in some special cases. First suppose that b is the principal block of kG. By Proposition 5.32, α_{Q} is trivial for all Q ∈ F^{c}, hence by Proposition 5.46, Alperin’s conjecture in this case is the equality

|Irr_{k}(G, b)| = ∑_{Q∈F^{cr}/F} z(kOut_{F}(Q)).

In particular, two principal blocks with isomorphic fusion systems have conjecturally the same number of simple modules.

Now suppose P is abelian. Then P is the only F-centric subgroup of P and the sum in the right hand side of the equality in Proposition 5.46 reduces to a single term. By Proposition 5.37, the twisted group algebra k_{α_{P}} Out_{F} (P) is isomorphic to kHd, where H is a central extension of Out_{F}(P) by a cyclic p′-group, and d is a central idempotent of kH. Since Out_{F}(P) is a p′-group, H is a p′-group, and it follows from Theorem 4.5, that every simple k_{α_{P}} Out_{F} (P)-module is projective. Thus, if P is abelian, Conjecture 5.45 is the equality

|Irr_{k}(G, b)| = l(k_{α_{P}} Out_{F}(P)).

Let L_{P} be a model for N_{F}(P) and denote by α_{P} the element of H^{2}(L_{P} , k^{∗}) obtained from the restriction of α_{P} through the canonical surjection of L_{P} onto Out_{F}(P). Since L_{P} is an extension of Out_{F}(P) by a p-group, using Proposition 5.37, and the fact that a normal p-subgroup of a finite group H acts trivially on each simple kH-module (see the first part of the proof of Lemma 3.16), it follows that l(k_{α_{P}} L_{P}) = l(k_{α}Out_{F}(P)). Hence, if P is abelian, then Conjecture 5.45 is the equality

|Irr_{k}(G, b)| = l(k_{α_{P}} L_{P}).

The number of isomorphism classes of simple modules of a finite dimensional k-algebra is an invariant of the bounded derived category D^{b}(mod(A)) of the module category of A (see [KoZ] for an account of derived categories in the context of group representation theory). In the case that P is abelian, Conjecture 5.45 is therefore a consequence of the following.

Conjecture 5.47 (Broué’s Abelian Defect Group Conjecture). [B2] Suppose that P is abelian. Let α_{P} ∈ H^{2}(Out_{F}(P), k^{∗}) be as in Definition 5.31 and let L_{P} be a model for N_{F}(P). Denote also by α_{P} the element of H^{2}(L_{P} , k^{∗}) obtained by restriction of the surjection L_{P} onto L_{P} /P ≃ Out_{F}(P). Then, there is an equivalence of triangulated categories D^{b}(mod(kGb)) ≅ D^{b}(mod(k_{α_{P}} L_{P})).

By Theorem 5.41, the block algebra kN_{G}(P, e_{P})e_{P} is Morita equivalent to k_{α_{P}} L_{P}. On the other hand, by Proposition 5.26 and by Brauer’s First Main Theorem it follows that kN_{G}(P, e_{P})e_{P} is source algebra equivalent to kN_{G}(P)br_{P} (b). Since source alegbra equivalence implies Morita equivalence and Morita equivalence implies derived equivalence, the above conjecture has the following reformulation, which is the version in which it was originally stated.

Conjecture 5.48. Suppose that P is abelian. Then, there is an equivalence of triangulated categories D^{b}(mod(kGb)) ≅ D^{b}(mod(kN_{G}(P)br_{P} (b))).

Conjecture 5.47 is known to be true when P is cyclic ([Ri1, Theorem 4.2]) or P ≅ C_{2} × C_{2}([Ri2, Section 3]).

In [KR], Knörr and Robinson showed that Conjecture 5.45 can be reformulated in terms of alternating sums indexed by certain chains of psubgroups of G. Using that Irr_{K}(G, b) − Irr_{k}(G, b) is locally determined (see Theorem 5.7), they also gave a version of the conjecture which involves ordinary irreducible characters. Various refinements of the weight conjecture have appeared since (see for instance [Da2], [Da3], [Da4] , [Ro1],

[Ro2] [Bo], [Li4], [Li2], [Un]), [Wb2]). We describe one such version, the ordinary weight conjecture, due to Robinson.

It is a standard fact of ordinary character theory that if V is a simple KG-module, then dim_{K}(V) is a divisor of |G| (see [NT, Chapter 3, Theorem 2.4]); the p-defect of V is defined to be the non-negative integer d such that p^{d} is the highest power of p dividing \frac{|G|}{dim_{K}(V)}. For a non-negative integer d, we denote by Irr_{K}^{d}(G) the subset of Irr_{K}(G) consisting of the images [V] in R_{K}(G) of those simple KG-modules V whose p-defect equals d and we denote by Irr_{K}^{d}(G, b) the set Irr_{K}^{d}(G) ∩ Irr_{K}(G, b).

Let Q ≤ P be F-centric and F-radical. Let N_{Q} denote the set of strictly increasing chains

σ: 1 < X_{1} < ⋯ < X_{l}

of p-subgroups of Out_{F}(Q) such that for each i, 1 ≤ i ≤ l, X_{i} is normal in X_{l}. For each σ as above, let |σ| = l + 1 (so if σ is the chain consisting solely of the trivial subgroup, then |σ| = 1).

Clearly, Out_{F}(Q) acts on N_{Q} by conjugation. For σ ∈ N_{Q}, denote by I(σ) the stabilizer in Out_{F}(Q) of σ. There is a natural action of Aut(Q) on Irr_{K}(Q), where for φ ∈ Aut(Q) and a simple KQ-module V, ^{φ}V is the kQ-module which equals V as K-vector space and where x.v: = φ(x)v for x ∈ Q, and v ∈ V. If φ ∈ Inn(Q), then for any simple KQ-module V, φV ≅ V. Further, dim_{K}(^{φ}V) = dim_{K}(V) for any simple KQ-module V and any φ ∈ Aut(Q). Thus the action of Aut(Q) on Irr_{K}(Q) induces an action of Out_{F}(Q) on Irr_{K}(Q) and on Irr_{K}^{d}(Q) for any non-negative integer d. For σ ∈ N_{Q}, and μ ∈ Irr_{K}(Q), let I(σ, μ) denote the intersection of I(σ) with the stabilizer in Out_{F}(Q) of μ; denote by α_{Q} also the restriction of the Külshammer-Puig class α_{Q} to I(σ, μ) (see definition 5.31) and define the integer ω(Q, σ, μ) by

ω(Q, σ, μ): = z(k_{α_{Q}}I(σ, μ)).

For a non-negative integer d, set

w(Q, d): = ∑_{σ∈N_{Q}/Out_{F} (Q)}(−1)^{|σ|+1} ∑_{μ∈Irr_{K}^{d}(Q)/I(σ)} ω(Q, σ, μ).

Conjecture 5.49 (Ordinary Weight Conjecture). [Ro2] Let d be a nonnegative integer. Then,

|Irr_{K}^{d}(G, b)| = ∑_{Q∈F^{cr}/F} w(Q, d).

Summing over all non-negative integers in the above gives a formula for the number, |Irr_{K}(G, b)|, of simple KGb-modules.

The Ordinary Weight Conjecture is known to hold for all blocks of tame representation type (see Example after [Ro2]), for all P which admit only F_{P} (P) as saturated fusion system and for P = C_{2} × C_{2} × C_{2}.

We state two other famous counting conjectures-these are related to the conjectures stated above and have interesting refinements (see [IsNa], [Tu]). We say that an element χ ∈ Irr_{K}(G, b) is a height zero character if the p-defect of χ is maximal amongst all elements of Irr_{K}(G, b).

Conjecture 5.50 (Brauer’s height zero conjecture). All elements of Irr_{K}(G, b) are of height zero if and only if P is abelian.

Conjecture 5.51 (Alperin-McKay Conjecture). The number of height zero elements in |Irr_{K}(G, b)| and in |Irr_{K}(N_{G}(P), br_{P} (b))| is the same.

Remark. The conjectures of this section as well as the finiteness conjectures of the previous sections have been established for only a few isomorphism types of defect groups or fusion systems. However, work on four families of finite groups: p-solvable groups, Weyl groups and their covers, finite groups of Lie type, and sporadic simple groups has yielded a huge amount of evidence for these conjectures. Understanding blocks in the different families requires the use of very different and often deep techniques and we cannot give an exposition here which would do the justice to the state of the art. There has also been a lot of progress, especially recently, on reducing the counting conjectures to statements about quasi-simple groups. Again it is not possible to give a proper account of these results here.

6. Block fusion systems and normal subgroups.

Let N be a normal subgroup of G and consider kN as a G-algebra via the conjugation action of G. As noted in Section 3.4, the G-action on kN induces an action of G on the set of blocks of kN. For a block c of kN, we let I(c) be the stabilizer in G of c. By Proposition 3.10, if c is a block of kN, then Tr_{I(c)}^{G}(c) is an idempotent contained in (kN)^{G} and the set {Tr_{I(c)}^{G}(c)} as c runs over a set of representatives of the G-orbits of the blocks of kN is the primitive idempotent decomposition of 1 in (kN)^{G}. Since (kN)^{G} = kN ∩ Z(kG) ⊆ Z(kG), the set {Tr_{I(c)}^{G}(c)} is an idempotent decomposition of 1 in Z(kG). By the uniqueness of block decompositions, it follows that for each block b of kG, there exists a unique G-orbit, say X, of blocks of kN such that bTr_{I(c)}^{G}(c) ≠ 0 for some (any) c ∈ X and for this c, bTr_{I(c)}^{G}(c) = b = Tr_{I(c)}^{G}(c)b.

Definition 6.1. Let b be a block of kG and let c be a block of kN. We say b covers c if bTr_{I(c)}^{G}(c) ≠ 0.

It is immediate from the definition that for any block b of kG, the set of blocks of kN covered by b is a G-orbit of blocks of kN. The following is an easy characterization of covering blocks.

Proposition 6.2. Let b be a block of kG and let c be a block of kN. Then b covers c if and only if bc ≠ 0.

Proof. Let X be the G-orbit of c. Then, Tr_{I(c)}^{G}(c) = ∑_{d∈X} d and the elements of X are pairwise orthogonal idempotents of kN and hence of kG. Since b is a central idempotent of kG, it follows that bTr_{I(c)}^{G}(c) = 0 if and only if bd = 0 for all d ∈ X. Let d ∈ X. By definition, d = ^{x}c for some x ∈ G. Since b is central in kG, we have

bd = b^{x}c = ^{x}(bc).

Thus, bc = 0 if and only if bd = 0 for all d ∈ X. The result follows. □

If b is a block of kG and V is a kG-module lying in the block b, then the restriction of V to kN is a direct sum of kN-modules, W_{d}, where d runs over the set of blocks of kN covered by b. The relationship between the representation theory of a block and of the blocks it covers, often referred to as Clifford theory, is an important part of block theory. So, it makes sense to investigate what happens to fusion systems on passage to covered blocks. The following proposition shows that for this (as for Clifford theory), one may always reduce to the case where there is a unique covered block.

Proposition 6.3. Let N be a normal subgroup of G, and c a block of kN. Set I = I(c), the stabilizer of c in G. The map e → Tr_{H}^{G}(e) is a bijection between the set of blocks of kI covering c and the set of blocks of kG covering c. Further, if e is a block of kI covering c, then the block algebras kIe and kGTr_{H}^{G}(e) are source algebra equivalent. In particular, if b is a block of kG covering c, then the fusion system of kGb is isomorphic to the fusion system of some block of kI.

Proof. By definition of I(c), c ∈ (kN)^{I} ⊆ Z(kI). Thus, c is a central idempotent of kI. If g ∈ G − I, then again by definition of I, ^{x}c is a block of kN different from c, hence c^{g}c = 0. Also note that since N is normal in I, the condition e ≤ c for a block of e of kI is equivalent to the condition that e covers c, and similarly for blocks of kG. The first part of the statement is now immediate from Proposition 5.26. The second part follows from the fact that source algebra equivalent blocks have isomorphic fusion systems (see Proposition 5.23). □

Now let b be a block of kG covering c and assume that I(c) = G, so that bc = b and c ∈ (kN)^{G}. Then, by Proposition 3.14, (kN, c, G), (kN, c, N) and (kG, b, G) are triples of saturated type and thus by Theorem 3.2, the corresponding fusion systems are saturated, the latter two triples yielding the fusion systems of kNc and kGb respectively. Proposition 3.9 describes the relationship between the fusion systems associated to the triples (kN, c, G) and (kN, c, N). From the point of view of block theory however, one wants to understand the relationship between the fusion systems associated to the triples (kN, c, G) and (kG, b, G).

Theorem 6.4. Let N be a normal subgroup of G and c a block of kN which is G-stable. Let b be a block of kG covering c and let (P, e_{P}) be a maximal (kG, b, G)-Brauer pair.

(a) There exists a maximal (kN, c, G)-Brauer pair (S, e′_{S}) such that P ≤ S

and

F_{(P,e_{P})}(kG, b, G) ⊆ F_{(S,e′_{S})}(kN, c, G).

(b) For each Q ≤ S, let (Q, e′_{Q}) be the unique (kN, c, G)-Brauer pair contained in (S, e′_{S}). Then, P ∩ N = S ∩ N and (S ∩ N, e′_{S∩N}) is a maximal (kN, c, N)-Brauer pair. Further, F_{(S∩N,e′_{S∩N})}(kN, c, N) is a normal subsystem of F_{(S,e′_{S})}(kN, c, G).

Proof. Part (a) is proved in [KS, Theorem 3.5]. Part (b) is just Proposition 3.9, applied with A = kG, S replacing P and c replacing b. A version of Part (b) with “weakly normal” replacing normal is proved in [KS, Theorem 3.5]. □

The above theorem gives an indirect relationship between a fusion system of a block and a covered block through a third saturated fusion system. By analogy with finite group fusion systems, one might expect a more direct relationship, namely that a fusion system of a covered block is a normal subsystem of the fusion system of the covering block. The next example shows that such a straightforward relationship does not hold and it can even be the case that a fusion system of the covering block is a subsystem of the covered block.

Example 6.5. The details of the arguments for this example are similar to those used in Example 5.43 and have been ommitted. Let p = 3 and let P be the cyclic group of order 3 on generator r. Let T = ⟨x, y⟩ be a quaternion group of order 8. Let T act on P via

xr = r^{2} ^{y}r = r.

Let G = P ⋊ T and let N be the normal subgroup P ⟨x⟩ of G. Consider

b = \frac{1}{2}(1 − x^{2}) = \frac{1}{2}(1 − y^{2}).

Then b is a block of kG as well as of kN. Now C_{G}(P) = P ×⟨y⟩ Thus, br_{P} (b) is a sum of two blocks, \frac{1}{4}(1+iy−y^{2}−iy^{3}) and \frac{1}{4}(1−iy−y^{2}+iy^{3}) of kC_{G}(P), where i is a primitive fourth root of unity. Set e_{P}: = \frac{1}{4}(1 + iy − y^{2} − iy^{3}). Then (P, e_{P}) is a maximal (kG, b, G)-Brauer pair

On the other hand, C_{N}(P) = P × ⟨x^{2}⟩ whence br_{P} (b) is a block of kC_{N}(P) and (P, br_{P} (b)) is a maximal (kN, b, N)-Brauer pair. Let F = F_{(P,e_{P})}(kG, b, G) and F′ = F_{(P,br_{P} (b))}(kG, b, G). We have Aut_{F}(P) = 1 and Aut_{F′}(P) is cyclic of order 2.

If N is a normal p′-subgroup of G, and S is a Sylow p-subgroup of G, then S is a Sylow p-subgroup of G/N under the natural identification of an element x of S with its coset xN in G/N and F_{S}(G) = F_{S}(G/N). In block theory, the analogue of a normal p′-subgroup is a covered block having trivial defect group. The equality of fusion systems in the group theoretic situation has an analogue in the block theory setting. Note that if N is a normal subgroup of G and c is a block of kN which is G-stable and such that kNc has trivial defect groups, then kNc is a matrix algebra, which is both an interior kN-algebra and a kG-algebra.

Theorem 6.6. Let N be a normal subgroup of G and c a G-stable block of kN. Suppose that {1} is a defect group of c. Let α = α_{kNc,N,G} be as in Definition 5.28 and let s: G → (kNc)^{∗} induce α as in Definition 5.28.

(a) Let

φ: kNc ⊗_{k} k_{α}G/N → kGc

be the k-linear function satisfying x⊗gN → xs(g)^{−1}g for x ∈ kNc and g ∈ G. Then φ is a k-algebra isomorphism, with φ^{−1}(gc) = s(g) ⊗ gN for g ∈ G.

(b) Let b be a block of G covering c and let (P, e_{P}) be a maximal (kG, b, G)- Brauer pair. Then N ∩ P = 1 and there exists a a central extension

1 → Z → H → G/N → 1,

where Z is a cyclic p′-group and a block d of kH such that identifying P with the Sylow p-subgroup of the inverse image in H of P N/N, there is a maximal (kH, d, H)-Brauer pair of the form (P, f_{P}) with F_{(P,e_{P})}(kG, b, G) = F_{(P,f_{P})}(kH, d, H).

Proof. The first assertion is a straight forward verification. The second is deduced from the first through Proposition 5.37 and relies on some properties of endo-permutation modules. The result is implicit in the [P3]; the details may be found in [Ke1, Section 3]. □

By Proposition I.6.2, if F is a simple saturated fusion system and F = F_{P} (G), for a finite group G and a Sylow p-subgroup P of G, then F = F_{S}(H) for some simple group H. In other words, any realizable simple saturated fusion system (see remarks after Theorem I.2.3) is realizable by a finite simple group, i.e., is the fusion system of a finite simple group.

Definition 6.7. A saturated fusion system F over a p-group P is blockrealizable if there exists a finite group G and a block b of kG such that there is a maximal (kG, b, G)-Brauer pair of the form (P, e_{P}) with F ≅ F_{(P,e_{P})}(kG, b, G); otherwise, F is said to be block-exotic.

It is not known whether any block-realizable simple saturated fusion system is realizable by a block of a finite simple or quasi-simple group. Recall that a finite group G is quasi-simple if G/Z(G) is simple and G = [G, G]. Combining Proposition 6.3, Theorem 6.4 and Theorem 6.6 yields the following partial result.

Theorem 6.8. Let F be a simple saturated fusion system on a finite pgroup P. Suppose further that P contains no proper, non-trivial strongly F-closed subgroup and that there does not exist a saturated fusion system G on P containing F as a proper subsystem. If F is block-realizable, then there exists a quasi-simple group G with Z(G) a p′-group and a block b of kG having a maximal (kG, b, G)-Brauer pair of the form (P, e_{P}) such that F ≅ F_{(P,e_{P})}(kG, b, G).

Proof. This is a special case of [KS, Theorem 4.2]. □

By Brauer’s Third Main Theorem, a realizable saturated fusion system is block realizable. The converse is open. The above theorem (or some variant of it) reduces the question of the block-exoticity of certain saturated fusion systems to the question of whether these occur as fusion systems of blocks of finite quasi-simple groups. Even though the fusion systems of blocks of quasi-simple groups are not completely understood, it is possible to show that some of the known exotic saturated fusion systems are block-exotic.

Theorem 6.9. Let F be the exotic saturated fusion system F_{Sol}(q) at p = 2 (see Theorem III.6.7) or one of the three Ruiz-Viruel fusion systems at p = 7 (see Section III.6.2). Then F is block-exotic.

Proof. The case F = F_{Sol}(3) is proved in [Ke1], the case F_{Sol}(q) for general q is in [Cr2] and the case that F is a Ruiz-Viruel system is in [KS]. To check that these systems do not occur as block fusion systems in quasi-simple groups, one relies heavily on the modular versions of Deligne-Lusztig theory of representations of finite groups of Lie type developed by many authors, notably Bonnafe, Broué, Cabanes, Dignes, Enguehard, Fong, Geck, Hiss, Malle, Michel, Rouquier and Srinivasa. □

7. Open Problems

The block theory conjectures mentioned in the previous sections are the big open problems in modular representation theory. We list additionally a few questions related to block fusion systems.

1. Are all exotic fusion systems also block-exotic? At the moment, other than in the situation of principal blocks, there seems no way to use directly the information that a given saturated fusion system is a block fusion system in order to conclude that it must be the fusion system of a finite group. However, Theorem 6.9 seems to suggest that it may be possible to use the classification of finite simple groups to answer the above question, at least partially. The next two questions in our list form part of this approach.

2. Improve the reduction result of Theorem 6.8. Is any blockrealizable simple saturated fusion system realizable by a block of a quasisimple group?

3. Describe the fusion systems of blocks quasi-simple groups. In particular, does there exist a block of a quasi-simple group whose fusion system is exotic?

4. Does there exist a solution to the gluing problem for blocks? Recall from Definition 5.31 that if F is the saturated fusion system of a block b of kG, then to each F-centric subgroup Q of P is associated an element α_{Q} of H^{2}(Aut_{F}(Q), k^{∗}). The gluing problem for blocks, first stated in [Li2, Conjecture 4.2]) is:

Does there exist a second cohomology class α ∈ H^{2}(F^{c}, k^{∗}) whose restriction to Aut_{F}(Q) equals α_{Q} for each centric subgroup Q of P, and if so is there a canonical choice for α? For instance if b is the principal block of kG, then by Proposition 5.32, α may be taken to be the zero class. As another example, if F is constrained with Q = O_{p}(F), then restriction induces an isomorphism H^{2}(F^{c}, k^{∗}) ≅ H^{2}(Aut_{F}(Q), k^{∗}), and hence there is a unique choice for α (see [Li6, Proposition 6.4 and 6.5]). In [Pa1] S.Park has shown that α is not unique in general. If the gluing problem has a solution, then one obtains interesting structural reformulations of Alperin’s weight conjecture ([Li2, 4.5, 4.7], [Li4, Theorem 4.3]) involving centric linking systems.

5. Find interesting classes of p-permutation G-algebras to which Theorem 3.2 applies. Block algebras are a particular class of p-permutation algebras, and it is possible that there are other interesting classes which might provide a source of saturated fusion systems. In [KKuM], saturated triples arising from p-permutation modules are considered. Finding other classes of triples of saturated type might be interesting for exoticity questions. For instance the following question is open: Given a saturated fusion system on a finite p-group P, does there exist a finite group G, a p-permutation algebra A, and a primitive idempotent b of A^{G} such that

304 MICHAEL ASCHBACHER, RADHA KESSAR, AND BOB OLIVER

(A, b, G) is a saturated triple and such that the corresponding fusion system is isomorphic to F?

Appendix A. Background facts about groups

We collect here, for reference, some of the standard results in group theory which have been used throughout the book. We begin with some of the standard elementary properties of p-groups and Sylow subgroups.

Lemma A.1. For any pair of p-groups P ≤ Q, if N_{Q}(P) = P, then Q = P.

Proof. This is a general property of nilpotent groups (finite or not). If G is nilpotent and H < G, then by definition, there is K > H such that [G, K] ≤ H. Hence H < K ≤ N_{G}(H). That all p-groups are nilpotent is shown, e.g., in [A4, 9.8] or [G1, Theorem 2.3.3]. □

Lemma A.2. Fix a p-group P, and an automorphism α ∈ Aut(P) of order prime to p. Assume 1 = P_{0} ⊴ P_{1} ⊴ ⋯ ⊴ P_{m} = P is a sequence of subgroups all normal in P, such that for each 1 ≤ i ≤ m, α|_{P_{i}} ≡ Id_{P_{i}} (mod P_{i−1}). Then α = Id_{P}.

Proof. See, for example, [G1, Theorem 5.3.2]. It suffices by induction to prove this when m = 2, and when the order of α is a prime q ≠ p. In this case, for each g ∈ P, α acts on the coset gP_{1} with fixed subset of order ≡ |gP_{1}| (mod q). Since |gP_{1}| = |P_{1}| is a power of p, this shows that α fixes at least one element in gP_{1}. Thus α is the identity on P_{1} and on at least one element in each coset of P_{1}, and so α = Id_{P}. □

Lemma A.3. Assume that H ⊴ G are finite groups and S ≤ G is a psubgroup. Then S ∈ Syl_{p}(G) if and only if S ∩ H ∈ Syl_{p}(H) and SH/H ∈ Syl_{p}(G/H).

Proof. Since |S| = |S ∩ H|·|SH/H|, [G:S] is prime to p if and only if [H:S∩H] and [G/H:SH/H] are both prime to p. □

Proposition A.4 (Frattini argument). Fix a group G and a normal subgroup H ⊴ G.

(a) If H is finite, then for any T ∈ Syl_{p}(H), G = H·N_{G}(T).

(b) More generally, if G acts on a set X in such a way that H acts transitively on X, then for any x ∈ X with stabilizer subgroup Stab_{G}(x), G = H·Stab_{G}(x).

Proof. Fix g ∈ G. In the situation of (b), there is h ∈ H such that gx = hx (since H acts transitively), and so g = h(h^{−1}g) where h^{−1}g ∈ Stab_{G}(x). Point (a) now follows by setting X = Syl_{p}(H). □

We next look at minimal normal subgroups.

Lemma A.5. If M is a minimal nontrivial normal subgroup of G, then M is a product of simple groups which are pairwise isomorphic to each other.

Proof. This is shown in [A4, 8.2–8.3], but we also sketch the proof here. Let 1 ≠ L ⊴ M be a minimal normal subgroup, and let L = L_{1}, …, L_{k} be the distinct subgroups of M which are G-conjugate to L. For i ≠ j, L_{i} ∩L_{j} is also normal in M, and hence [L_{i}, L_{j}] ≤ L_{i} ∩ L_{j} = 1 by the minimality of L. Also, ⟨L_{1}, …, L_{k}⟩ ≤ M is normal in G, and hence equal to M since M is a minimal normal subgroup of G. In particular, any subgroup normal in L is also normal in M (since the other L_{i} commute with L), and so the L_{i} are simple by the minimality of L again.

If |L| = p for some prime p, then M is an elementary abelian p-group. If not, then the L_{i} are nonabelian and commute pairwise. Hence the product of the inclusions is a surjective homomorphism L_{1}×⋯×L_{k} ⟶ M, whose kernel intersects trivially with each factor. We leave it as an exercise to check that this kernel must be trivial, and hence that M = L_{1}×⋯×L_{k}. □

We next look at strongly embedded subgroups of a finite group.

Definition A.6. Fix a finite group G and a prime p. A proper subgroup H of G is strongly p-embedded if p||H|, and for each x ∈ G∖H, H ∩^{x}H has order prime to p.

The properties of strongly embedded subgroups which were used to prove Theorem I.3.5 (the version of Alperin’s fusion theorem stated here) are listed in the following proposition.

Proposition A.7. Fix a finite group G. For each S ∈ Syl_{p}(G), set

H_{S} = ⟨x ∈ G |^{x}S ∩ S ≠ 1⟩ .

Then the following hold.

(a) Each strongly p-embedded subgroup H < G contains some S ∈ Syl_{p}(G).

(b) For each S ∈ Syl_{p}(G), either H_{S} = G and G contains no strongly p-embedded subgroups; or else H_{S} < G, H_{S} is strongly p-embedded, and each strongly p-embedded subgroup H < G which contains S also contains H_{S}.

(c) If G contains a strongly p-embedded subgroup, then O_{p}(G) = 1.

Proof. (a) Assume H < G is strongly p-embedded. Since p||H| by assumption, there is g ∈ H of order p. Choose any S ∈ Syl_{p}(G) such that g ∈ S. For x ∈ Z(S) of order p, g ∈ H ∩^{x}H implies x ∈ H. Hence for all y ∈ S, x ∈ H ∩^{y}H implies y ∈ H. Thus S ≤ H.

(b) Assume S ≤ H < G, where S ∈ Syl_{p}(G) and H is strongly pembedded. For each x ∈ G such that S ∩^{x}S ≠ 1, p||H ∩^{x}H|, and hence x ∈ H. Thus H_{S} ≤ H, and H_{S} < G. Clearly, if H_{S} < G, then H_{T} < G for each T ∈ Syl_{p}(G). In other words, by (a), if H_{S} = G for some S ∈ Syl_{p}(G), then G contains no strongly p-embedded subgroup.

To prove (b), it remains to show that H_{S} is strongly p-embedded if it is a proper subgroup of G. So assume this holds, and fix x ∈ G such that p||H_{S} ∩^{x}H_{S}|. Choose g ∈ H_{S} ∩^{x}H_{S} of order p. Since g,^{x^{−1}}g ∈ H_{S} and S ∈ Syl_{p}(H_{S}), there are elements y, z ∈ H_{S} such that ^{y}g ∈ S and zx^{−1}g ∈ S. Then ^{y}g ∈ S ∩ ^{yxz^{−1}}S, so yxz^{−1} ∈ H_{S}, and x ∈ H_{S} since y, z ∈ H_{S} by assumption. Thus H_{S} is strongly p-embedded.

(c) Set Q = O_{p}(G). Assume H < G is strongly p-embedded, and fix S ∈ Syl_{p}(G) such that S ≤ H. Thus Q ≤ S and Q ⊴ G. Fix x ∈ G∖H. Then H ∩^{x}H contains Q and has order prime to p, so Q = 1. □

Theorem A.8 (Burnside’s Fusion Theorem). Let G be a finite group, p a prime, and assume a Sylow p-subgroup S of G is abelian. Then N_{G}(S) controls fusion in S.

Proof. This is a result of Burnside, appearing in [Bu]. Or see for example [A4, 37.6]. Here, “control of fusion” is interpreted in the strong sense of paragraph one of the introduction; equivalently, F_{S}(G) = F_{S}(N_{G}(S)) by Exercise II.2.2. □

Definition A.9. Fix a group G.

• For any x, y ∈ G, the commutator of x and y is the element

[x, y] = x^{−1}y^{−1}xy ∈ G

(cf. Section 8 in [A4]).

• For subgroups X and Y of G, the commutator of X and Y is the sub-

group

[X, Y] = ⟨[x, y] | x ∈ X, y ∈ Y ⟩ ≤ G .

• The commutator subgroup of G is the subgroup [G, G]. This subgroup is also called the derived subgroup of G.

• The derived series of G is defined recursively by G^{(0)} = G, and for n > 0, G^{(n)} = [G^{(n−1)}, G^{(n−1)}].

• G is solvable if and only if G^{(n)} = 1 for some positive integer n (cf. [A4, 9.1]).

• G is perfect if G = [G, G].

Observe that [X, Y] = 1 if and only if each element of X commutes with each element of Y.

Observe that the Puig series of a saturated fusion system, defined in Definition II.12.2, is analogous to the derived series of a finite group, so that Puig’s definition of a “solvable fusion system” (which we call Puig solvable) is quite natural.

Later we will need:

Lemma A.10 (3-Subgroups Lemma). Let A, B, C be subgroups of a group G, and set [A, B, C] = [[A, B], C]. If two of the subgroups [A, B, C], [B, C, A], and [C, A, B] are trivial, then so is the third.

Proof. This follows from the relation

[[a, b^{−1}], c]^{b}·[[b, c^{−1}], a]^{c}·[[c, a^{−1}], b]^{a} = 1

for any triple of elements a, b, c in a group G. See [A4, 8.7] for more details. □

Definition A.11. Let G be a finite group.

• A subgroup H of G is subnormal in G if there exists a series

H = H_{0} ⊴ H_{1} ⊴ ⋯ ⊴ H_{n} = G

(cf. [A4, Section 7]). Thus subnormality is the transitive extension of the normality relation.

• G is quasisimple if G is perfect (cf. Definition A.9) and G/Z(G) is simple.

• The components of G are its nontrivial subnormal quasisimple subgroups. Also, E(G) is the subgroup of G generated by the components of G (cf. [A4, Section 31]).

• The Fitting subgroup of G, denoted F (G), is the largest nilpotent normal subgroup of G.

• F^{∗}(G) = E(G)F (G) is the generalized Fitting subgroup of G.

Recall (cf. [A4, 9.11]) that a finite group is nilpotent if and only if it is the direct product of its Sylow subgroups. Hence F (G) is the product of the groups O_{p}(G), taken over all primes p||G|.

Lemma A.12. If L is a component of G and H is a subnormal subgroup of G, then either [L, H] = 1, or L is a component of H.

Proof. Let G be a minimal counterexample. Thus L ≰ H and [L, H] ≠ 1. Let X = ⟨L^{G}⟩ and Y = ⟨H^{G}⟩ be the normal closures of L and H, respectively, in G. If G = L, then since L ≠ H and each proper normal subgroup of L is contained in Z(L), H ≤ Z(L), contradicting the assumption [L, H] ≠ 1. Thus L < G, and so X < G. Also, H ≠ G since L ≰ H, so Y < G.

Set K = X ∩ Y. If L ≤ K, then L, H ≤ Y < G, contrary to the minimality of G. Thus L is a component of X < G, K ⊴ X, and L ≰ K, which by the minimality of G implies [L, K] = 1. Now, [L, Y] ≤ [X, Y] ≤ K, so [L, Y, L] = [Y, L, L] ≤ [K, L] = 1. Then by the three subgroups lemma (Lemma A.10), [L, L, Y] = 1, and so [L, Y] = 1 as L is perfect. Thus [L, H] ≤ [L, Y] = 1, contradicting the assumption G is a counterexample.

□

Theorem A.13. Let G be a finite group. Then the following hold.

(a) Distinct components of G commute elementwise. Thus E(G) is the central product of the components of G, and these are permuted by G via conjugation.

(b) [F (G), E(G)] = 1.

(c) C_{G}(F^{∗}(G)) = Z(F^{∗}(G)).

Proof. This is shown, for example, in 31.5, 31.12, and 31.13 in [A4], but we will reproduce the proof here.

As F (G) is solvable, it contains no component of G, so point (b) follows from Lemma A.12. Similarly if L and K are components of G then by Lemma A.12, either [K, L] = 1, or K ≤ L ≤ K. This proves point (a).

To prove (c), let H = C_{G}(F^{∗}(G)). Then H ∩ F^{∗}(G) = Z(F^{∗}(G)) ≤ Z(H) is abelian. If L is a component of H, then L is also a component of G (since H ⊴ G), so L ≤ H ∩ F^{∗}(G), which is impossible since L cannot be both abelian and quasisimple. Thus F^{∗}(H) = F (H) is nilpotent and normal in G, so F^{∗}(H) ≤ F (G), and hence F^{∗}(H) = Z(H) ≤ F (G) (since F (H) ≥ Z(H) for any H). If H is abelian, then H = C_{G}(F^{∗}(G)) ≤ F (G), and we are done.

So assume H is nonabelian, and choose a minimal (nontrivial) normal subgroup M/Z(H) of H/Z(H). Then by Lemma A.5, M/Z(H) = S_{1} × ⋯ × S_{n} is the direct product of pairwise isomorphic simple groups. If S_{1} is of prime order, then [M, M] ≤ Z(H), so M is nilpotent and hence M ≤ F^{∗}(H) = Z(H), a contradiction. Thus S_{1} is nonabelian. Let X be the preimage of S_{1} in M, and set Y = [X, X]. Then Y is perfect with Y/Z(Y) ≅ S_{1}, so Y is a component of G, Y ≤ H ∩ F^{∗}(G) ≤ Z(H), which is a contradiction. □

Note in particular the following consequence of Theorem A.13. We say that a subgroup H ≤ G is centric in G if C_{G}(H) ≤ H.

Corollary A.14. For any finite group G, F (G) is centric in G if and only if F (G) = F^{∗}(G) (equivalently, E(G) = 1). For any prime p||G|, O_{p}(G) is centric in G if and only if O_{p}(G) = F (G) = F^{∗}(G).

Definition A.15. Let S be a p-group for some prime p.

• The Thompson subgroup of S is the subgroup generated by the elementary abelian subgroups (ie. abelian subgroups of exponent p) of S of maximal order (cf. [A4, Section 32]). We write J(S) for the Thompson subgroup of S.

• The Baumann subgroup of S is C_{S}(Ω_{1}(Z(J(S)))). We denote the Baumann subgroup of S by Baum(S).

Recall that for any p-group X, Ω_{1}(X) = ⟨x ∈ X | x^{p} = 1⟩ (cf. [A4, Section 1]).

For more information about the Baumann subgroup, see [ASm, Section B.2].

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FUSION SYSTEMS IN ALGEBRA AND TOPOLOGY 319

List of notation

c_{g}, ^{g}x, x^{g}, ^{g}X, X^{g}, 4 ⟨X⟩, 4 φα, β^{φ}, 4 H ≤ G, H < G, H ⊴ G, 4 Aut_{G}(H), 4 Inn(H), Out(H), 4 Syl_{p}(G), 4 O_{π}(G), O^{π}(G), O(G), 4 Ω_{1}(P), 4 Hom_{G}(H, K), 10 F_{S}(G), 10 Hom_{F}(P, Q), Aut_{F}(P), 12 P^{F}, 12 N_{φ}, N_{φ}^{F}, 12 x^{F}, 13 Out_{F}(P), Out_{S}(P), 16 F^{c}, F^{cr}, 16 F|_{H}, 20 O_{p}(F), 23 Z(F), 23 G(F), 26 N_{S}^{K}(Q), 26 N_{F}^{K}(Q), N_{F}(Q), C_{F}(Q), 28 O^{p}(F), 38 O^{p′}(F), 41 F/E, 55 F^{f}, 57 C_{F}(P), 57 N_{F}(P), 57 F^{frc}, 59 G(F), 59

F/T, 61
Θ_{F,T}, 64
F_{T}^{f}, 65
F_{T}^{fc}, F_{T}^{c}, 70
𝔈(A), 71
E ∧ D, 73
ED, 73
C_{F}(E), 73
RE, 74
E(F), 74
F^{∗}(F), 74
O^{p′}(F), 75
[T^{F}], 77
𝔖𝔲𝔟(S), 78
CF (F), 82
X_{p}^{∧}, 108
π_{1}(X, x_{0}), π_{1}(X), 112
X_{p}^{∧}, 118
Out(X), 124
B(G), E(G), 129
BG, EG, 130
T_{S}(G), 138
L_{S}^{c}(G), 138
F_{S}^{c}(G), T_{S}^{c}(G), 138
F_{S,f}(X), 140
L_{S,f}(X), 141
L_{S}^{q}(G), 143
T_{H}(S), 145
F^{q}, 148
Aut(S, F), Out(S, F), 157
Out_{typ}(L), 157

320 MICHAEL ASCHBACHER, RADHA KESSAR, AND BOB OLIVER

Aut_{typ}^{I}(L), 158 L/L_{0}, 160 O(G), O_{S}(G), 173 O(F), O_{S}(F^{c}), 173 C-mod, 175 C^{n}(C; F), C^{n}(C; F), 176 Z^{n}(C; F), B^{n}(C; F), 180 𝔛(S), 193 O_{p}(G), 194 Λ^{∗}(Γ; M), 194 S_{p}(G), 197 𝔯𝔢𝔡(F), 215 Orb_{X}(B), 227 B^{X}, 227 C_{X}(a), 227 C^{+}, 227 f ≤ e, 229 A^{H}, 232 Res_{L}^{H}, 232 A_{L}^{H}, 232 c_{g,H}, 232 A_{H}^{L}, 233 A/A_{<H}^{H}, 233 Br_{H}^{A} , 233 g(Br_{H}^{A}(a)), 233 gA(H), 233 Br_{P,Q}, 239 b_{P,Q}^{A}, 240 (P, e), 240 (Q, f) ≤ (P, e), 241 (Q, f) ⊴ (P, e), 245 x(P, e), 245 Hom_{X}((Q, f), (R, u)), 248 F_{(P,e_{P})}(A, b, G), 248 Mor_{F_{(P,e_{P} )}(A,b,G)}(Q, R), 248 Hom_{(A,G,e_{P})}(Q, R), 248 C_{G}(Q, e), N_{G}(Q, e), 249 br_{P}, 258 K, O, k, 𝔭, 266 R_{F} (G), Irr_{F} (G), IPr_{F} (G), P r_{F} (G), 268 IPr_{O}(G), P r_{O}(G), 268 [V], 268 c_{Φ,φ}, 268 d_{χ,φ}, 269 R_{K}(G, b), Irr_{K}(G, b), 270 P r_{O}(G, b), IPr_{O}(G, b), 270 R_{k}(G, b), Irr_{k}(G, b), P r_{k}(G, b), IPr_{k}(G, b), 270 χ_{M}, 271 P/F, 278 mod(A), 282 φkH, 284 α_{A,N,C}, 286 H^{2}(N/C, k^{∗}), 286 α_{Q}, 287 H^{2}(Out_{F}(Q), k^{∗}), H^{2}(Aut_{F}(Q), k^{∗}), 287 H^{2}(G, R^{∗}), 290 l(A), 292 z(A), 293 Irr_{K}^{d}(G), Irr_{K}^{d}(G, b), 297 N_{Q}, 297 I(σ), 297 I(σ, μ), 297 ω(Q, σ, μ), 297 w(Q, d), 297 I(c), 298

E(G), F (G), F^{∗}(G), 308

J(S), 310

Baum(S), 310

Index

3-subgroups lemma, 308

admissible set, 77

height of an element, 77 Alperin’s fusion theorem, 10

for fusion systems, 18, 19 Alperin’s weight conjecture, 294, 295 axioms

for a linking system

axioms (A1), (A2), (B), (C), 145 for normality

extension condition, 31 Frattini condition, 31 invariance condition, 31

for saturation extension axiom, 14 Sylow axiom, 14

Baumann pair, 100 Bender group, 83 block, 230

covering another block, 298 decomposition of, 230 lift of, 270 nilpotent, 290 principal, 278

Bousfield-Kan p-completion, 118 Brauer

(A, b, G)-Brauer pair, 246 (A, G)-Brauer pair, 240 characterization of characters, 266 correspondent, 276 first main theorem, 275 homomorphism, 233, 238, 258 inclusion of Brauer pairs, 241 pair, 240 second main theorem, 276 third main theorem, 278

Broué’s Abelian Defect Group

Conjecture, 296 Burnside’s fusion theorem, 11, 307

Campbell pair, 100, 101 Cartan map, 268 Cartan matrix, 268 Cartan number, 268 category/categories

bar resolution of, 176 equivalence of, 128 geometric realization of, 127 nerve of, 127 central idempotent, 230 character(s), 271

Brauer, 272 irreducible Brauer, 272 modular irreducible, 271 ordinary irreducible, 271 height zero, 298

characteristic biset, 47–50, 168, 170 class function, 271 classifying space of a group, 108 CW complex, 114

Dade P -algebra, 291

primitive, 291 decomposition

map, 269 matrx, 269 number, 269

defect group, 260 Donovan conjecture

strong, 282 weak, 282

F-block, 288 F-conjugate elements, 13 F-isomorphic elements, 278 finite group

centric linking category of, 138 component of, 308 fusion category of, 10, 121 p-constrained, 26 quasisimple, 308 transporter category of, 138

focal subgroup theorem, 11, 38 Fourier inversion formula, 273 Frobenius normal p-complement

theorem, 11 Frobenius Relations, 233 fusion preserving isomorphism, 121 fusion subsystem, 31

centralizer, 73 condition (N1), 67 F-Frattini, 66 F-invariant, 31, 65 generalized Fitting subsystem, 74 normal, 31, 67 of index prime to p, 38–43, 165 of p-power index, 38–39, 165 weakly normal, 31, 160

fusion system(s), 12 of b, 260 of kGb, 260 automorphism of, 157 Baumann characteristic two, 94 Baumann component type, 94 center of, 23 central product, 73 centralizer, 57 centric subgroup, 57 characteristic p-type, 92 component, 74 component type, 92 composition factor, 82 composition series, 78 constrained, 26, 58, 59, 180

model for, 26 even characteristic, 94 even component type, 94 exotic, 13, 214 F-characteristic subgroup, 100 F-invariant map, 71

constricted, 71 factor system, 61 fundamental group and covering

spaces of, 165 generated by certain morphisms or

subsystems, 18 H-generated, 20 H-saturated, 20 indecomposable, 217 Jordan-Hölder Theorem, 81 L-balance, 74 local CK-system, 98 local subsystem, 57 local theory, 54 model, 59 morphism of, 57

surjective, 57 N-system, 104 normal map, 71 normal subgroup, 58 normalizer, 57 normalizer fusion subsystem, 28 product of, 34 Puig solvable, 85 pushing up, 99 quasisimple, 74 realizable, 13 reduced, 215 S_{3}-free, 103 saturated, 12 simple, 31, 74 solvable, 85

tame, 216

G-algebra, 231 interior, 231

generalised decomposition number, 277 Glauberman’s Z^{∗}-theorem, 10, 23 Glauberman-Niles pair, 100 Goldschmidt group, 83 Green’s indecomposability theorem,

277 Grothendieck group, 268 group

p-solvable, 298 B-Conjecture, 97 characteristic p-type, 92 classifying space of, 108 component type, 92 cover of, 298 discrete p-toral, 172 finite group of Lie type, 298 N-group, 104 p-Bender, 88 p-Goldschmidt, 87 p-solvable, 85 section, 103 S_{n}-free, 103 sporadic, 298 standard component, 96 Unbalanced Group Conjecture, 97 Weyl, 298 group of Lie type

fundamental subgroup, 95

Higman’s criterion, 264 homotopic maps, 111 homotopy colimit, 202 homotopy decompositions, 202–209,

212–213 centralizer decomposition, 213 normalizer decomposition, 213 subgroup decomposition, 205

homotopy equivalence, 111 homotopy functor, 207 rigidification of, 207 homotopy type, 111 hyperfocal subgroup theorem, 11, 38

idempotent(s), 228 (primitive)decomposition of, 228 lifting, 228 associated to, 240 containment of, 229 orthogonal, 228

primitive, 228

Jacobson radical, 228

k(b) conjecture, 282 K-group, 98 Külshammer-Puig class, 287, 297

linking subsystem

normal, 160 linking system(s)

automorphism group of, 158, 168,

189 centric, 145 existence and uniqueness of, 187,

207 extension of, 162 geometric realization of

cohomology of, 168 covering space of, 163, 165 fundamental group of, 164

isomorphism of, 151 of group, 138 quasicentric, 150

linking systems, 145 local ring, 229

Mackey Formula, 232 Martino-Priddy conjecture, 108, 123,

144, 192 Maschke’s theorem, 265 minimal parabolic, 100 model theorem for constrained fusion

systems, 26, 184 module

indecomposable, 227 projective, 227 relatively H-projective, 264 simple, 227

Morita equivalent, 282

nilpotent element, 228 ideal, 228

Noether-Skolem theorem, 285

obstruction to pushing up, 98 orbit category

of a fusion system, 173 of a group, 173

Ordinary Weight Conjecture, 297

P -block, 281 p-defect, 297 p-local compact group, 172 p-local finite group(s), 147

isomorphism of, 168 morphism of, 221 wreath product of, 222

p-modular system, 266 p-permutation algebra, 237 P -stable, 237 principal block, 275, 278

qrc-Lemma, 102 quaternion fusion packet, 95

Lie packet, 95

relative trace, 232, 238 representation type

finite, 280 tame, 280 wild, 280

Rosenberg’s Lemma, 229

saturated fusion system block-exotic, 301 block-realizable, 301

saturated triple, 250 simplicial category, 125 simplicial set, 125

geometric realization, 125 homology of, 126 product, 127 simplicial map, 125

source algebra, 283, 284 source algebra equivalent blocks, 285 source idempotent, 283, 284 space

classifying space, 115 contractible, 111 covering space of, 112, 133 CW complex, 114 fundamental group of, 112, 131, 132 fusion category of, 140, 170 homology of, 115, 118 linking category of, 141, 170 p-complete, 119 p-completion of, 108, 118 p-good, 119 path in, 111 simply connected, 112 singular simplicial set of, 127

spectrum, 170

classifying spectrum of a fusion

system, 170 splitting field, 264 Stellmacher qrc-Lemma, 102 subgroup(s)

Baumann, 310 central, 22 commutator, 307 F-centric, 16 F-conjugate, 12 F-essential, 16 F-quasicentric, 148 F-radical, 16 Fitting, 308 focal, 37 fully automized, 12 fully centralized, 13 fully normalized, 13 generalized Fitting, 308 hyperfocal, 37 K-normalizer, 27 normal, 22 O_{p}(F), 23 p-centric, 138, 146 p-quasicentric, 143, 150 receptive, 12 strongly closed, 22, 56 strongly embedded, 306 subnormal, 308 Thompson, 193, 310 tightly embedded, 96 weakly closed, 22 Z(F), 23

Sullivan conjecture, 108, 123 supranormal series, 78

pivot in, 78

TI-subgroup, 83 transporter systems, 147 twisted group algebra, 290

weight of a block, 294

Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA

E-mail address: asch@caltech.edu

Institute of Mathematics, Fraser Noble Building, University of Aberdeen, Aberdeen, AB243UE, UK.

E-mail address: kessar@maths.abdn.ac.uk

LAGA, Institut Galilée, Av. J-B Clément, 93430 Villetaneuse, France

E-mail address: bobol@math.univ-paris13.fr