FUSION SYSTEMS AND LOCALITIES

Andrew Chermak

Kansas State University

August 2011 (revised January 2013)

Abstract. We introduce objective partial groups, of which the linking systems and p-local finite groups of Broto, Levi, and Oliver, the transporter systems of Oliver and Ventura, and the F-localities of Puig are examples, as are groups in the ordinary sense. As an application we show that if F is a saturated fusion system over a finite p-group then there exists a centric linking system L having F as its fusion system, and that L is unique up to isomorphism. The proof relies on the classification of the finite simple groups in an indirect and - for that reason - perhaps ultimately removable way.

Introduction

Let S be a finite p-group, p a prime. A fusion system on S is a category whose objects are the subgroups of S, and whose morphisms are injective group homomorphisms, among which are all of those which are induced by conjugation by elements of S. A fusion system F on S is saturated if it satisfies some further conditions, such as would be found to hold if S were a Sylow subgroup of a finite group G and if the morphisms in F were the homomorphisms between subgroups of S induced by conjugation within G.

Saturated fusion systems were introduced by Lluis Puig (as “Frobenius categories”) in notes which, although widely influential, remained unpublished for some years. Puig’s formalism provided a setting for the Brauer Theory of blocks of characters of finite groups, in which no ambient finite group need be assumed. Somewhat later, David Benson [Be] suggested the possibility of associating a “classifying space” to each Frobenius category. The notion of such a classifying space was then formulated in a rigorous way by Carles Broto, Ran Levi, and Bob Oliver in [BLO], thereby providing a generalized setting for the homotopy theory of p-completed classifying spaces of finite groups. Here also, as in Puig’s setup, no ambient finite group is required. Instead, what is required is a “linking system” (or “p-local finite group”) attached to a given saturated fusion system, and

Typeset by AMS-TEX

1991 Mathematics Subject Classification. AAA.

Key words and phrases. Fusion systems, linking systems, finite groups.

which has a richer and, in many respects, a more “group-like” structure than the fusion system alone.

More recently, linking systems and their homotopy-theoretic correlatives have been further generalized by Bob Oliver and Joana Ventura [OV] to “transporter systems”. More recently still, the notions of linking system and transporter system have been treated by Puig in his book [P2], where they are called “F-localities” - but where the homotopical context is absent (as will also be the case in the present work).

This paper is intended, in part, as a step toward providing a setting for the methods of the so-called “p-local analysis” from finite group theory, in which no ambient finite group is required. The formalism developed here turns out to be equivalent in a technical sense to that of [BLO] and [OV], but it is pitched in a completely different language - one which involves nothing of categories and functors - and it has a more recognizably finite group-like flavor. Partly for this reason, and partly because the “p-local” in “plocal finite groups” already has a meaning for finite group theorists, we have chosen to adopt Puig’s terminology; and so this paper involves the study of what we call localities. We retain the terminology from [BLO] for the special sort of locality known as a linking system. The aim is to establish some basic structural properties of localities in general, and to prove the following result.

Main Theorem. Let F be a saturated fusion system on the finite p-group S, p a prime. Then there exists a centric linking system L such that F is the fusion system generated by the conjugation maps in L between subgroups of S. Further, L is uniquely determined by F, up to an an isomorphism which restricts to the identity map on S.

We remark that if L is a centric linking system on S, then it is straightforward to show that the fusion system F_{S}(L) generated by the conjugation maps in L between subgroups of S is saturated (see 2.17(a), below). Thus, the effect of the Main Theorem is that there is a one-to-one correspondence, up to a rigid notion of isomorphism, between saturated fusion systems and centric linking systems.

In this introduction we shall outline our proof of the Main Theorem, and point out the indirect way in which it relies on the classification of the finite simple groups (hereinafter referred to as the CFSG).

A group may be regarded as a set G together with an “inversion map” and a multivariable “product” Π : W → G, where W = W(G) is the free monoid on G. The usual definition of a group is easily formulated in terms of Π instead of the binary multiplication. To obtain the notion of “partial group”, one drops the requirement that Π be defined on all words in W, and one places certain conditions on the subset D of W on which Π is defined, while retaining the essential properties that one expects from a product.

Once the definition is written down in 2.1, partial analogs of basic group-theoretic notions immediately suggest themselves, including the notions of homomorphism and subgroup. A partial subgroup of a partial group may in fact be a group. Moreover, it may be the case for a given partial group M that there is a collection Δ of subgroups which determines the domain D of the product Π. Namely, it may happen that a word w = (f_{1}, ⋯, f_{n}) is in the domain D if and only if there exists a sequence (X_{0}, ⋯, X_{n}) of “objects” (i.e. members of Δ) such that X_{i−1} is conjugated by f_{i} to X_{i} for all i, 1 ≤ i ≤ n. If such is the case, and if moreover, any subgroup of an object containing a conjugate of an object is again an object, then the pair (M, Δ) is an objective partial group.

Our interest is in objective partial groups L = (M, Δ) such that the set Δ of objects has a unique maximal member S with respect to inclusion, and where S is a finite p-group which is maximal (though not necessarily uniquely so) in the poset of all p-subgroups of M. When these conditions are met, and M is finite, then the triple L = (M, Δ, S) is a locality. A locality L is a Δ-linking system if, for any object P ∈ Δ, the centralizer subgroup C_{L}(P) is just the center Z(P) of P. If, moreover, Δ is the set of all subgroups P of S such that C_{S}(Q) = Z(Q) for every L-conjugate Q of P with Q ≤ S, then L is a centric linking system.

To any locality L = (M, Δ, S) there is associated a fusion system F := F_{S}(L) on S, whose morphisms are those maps ϕ from one subgroup of S into another, such that ϕ is a composition of restrictions of L-conjugation maps between objects. We find that for any locality L, the pair (L, F_{S}(L)) is essentially the same thing as a “transporter system” in the sense of Oliver and Ventura [OV], and we show that all transporter systems arise from localities in this way. The proof is given in an Appendix, so as not to interrupt the flow of the development. The Appendix includes also a proof that the Main Theorem implies the corresponding result for “centric linking systems” taken in the sense of [BLO].

Section 1 introduces saturated fusion systems (and the notion of “fully normalized” subgroup) in an unconventional way, in analogy to the way in which one defines a scheme as a gluing-together of affine schemes. In this analogy, the “affine” things are the fusion systems F_{R}(H) of finite groups H at a Sylow p-subgroup R, where H has the property that C_{H}(O_{p}(H)) ≤ O_{p}(H). A fusion system F on a finite p-group S is saturated provided that F is locally affine, F is “generated” by its affine subsystems, and every F-centric subgroup of S has a fully normalized F-conjugate. Another way to say this is that our definition of saturation is based on the notion, due to Aschbacher [As], of a “model” of a “constrained” fusion system. In any case, our formulation is equivalent to those found in [BLO] and elsewhere. Readers who are already familiar with fusion systems and with models will find little new after 1.4. Indeed, the purpose of section 1 is primarily to fix terminology and notation - and to state a result (Proposition 1.10) announced by Aschbacher and proved by Oliver - which lies at the foundation of this work.

The definitions pertaining to partial groups, objective partial groups, and localities are introduced, and a few of their elementary consequences are derived, in section 2. Among these consequences is the basic one (Proposition 2.10) that for any locality L = (M, Δ, S), and any f ∈ M, the set S_{f} of elements x ∈ S such that the product x^{f} := f^{−1}xf is defined and is an element of S, is in fact an object, and hence a subgroup of S. As a corollary, we obtain the result (Proposition 2.21) that every subgroup of M is contained in the normalizer of an object, and that all p-subgroups of M are conjugate to subgroups of S.

Section 3 introduces homomorphisms of partial groups and of partial normal subgroups. A few very basic consequences of the definitions are derived, but it isn’t until the focus is restricted to localities, in section 4, that these concepts begin to bear fruit. We make no attempt in this paper to formulate a general notion of homomorphism of localities beyond an obvious notion of isomorphism.

Section 4 provides some basic computational tools for working with a locality L = (L, Δ, S) and a partial normal subgroup N ⊴ L. The “Frattini lemma” (4.6) says that every element f ∈ L can be written as a product xh (or hy := hx^{h}) with x, y ∈ N and with h ∈ N_{L}(S ∩ N). The “splitting lemma” (4.10) shows that it is possible to choose the pairs (x, h) so that (S_{f})^{x} ≤ S. The section ends with a result (4.11) on extending an automorphisms of a sub-locality of a finite group to an automorphism of the group itself.

It is in section 5 that the proof of the Main Theorem begins to take shape. The main results (Theorems 5.14 and 5.15) yield a concrete procedure for constructing a locality L^{+} from a locality L having a smaller set of objects. Thus: suppose that one is given a locality L with the set Δ of objects, and maximal object S; and suppose that one is given also a fusion system F on S such that L is “F-natural”, in the sense that for any object P, the set of L-conjugation maps from P into S is equal to the set of F-homomorphisms of P into S. Now suppose further that one is given a subgroup T of S, such that T is not in Δ, but with the property that every pair of distinct F-conjugates of T in S generates a member of Δ. One may assume (upon replacing T by a suitable L-conjugate) that T is fully normalized in F, in the sense of 1.2. There are then two questions to consider. First: under what conditions is it possible to regard L as the “restriction” to Δ of an F-natural locality L^{+} whose set Δ^{+} of objects is the union of Δ with the set of overgroups in S of F-conjugates of T ? Second: under what conditions are two such “extensions” of L to Δ^{+} “rigidly isomorphic” (i.e. isomorphic via an isomorphism which restricts to the identity map on S)? Theorems 5.14 and 5.15 provide a complete answer to these questions; and in doing so they provide a blueprint for the proof of the Main Theorem.

In brief, Theorem 5.14 says that there exists an F-natural locality L^{+} extending L in the prescribed manner, provided that there exists

(1) a finite group M containing N_{S}(T) as a Sylow p-subgroup, and with fusion system F_{N_{S}(T)}(M) equal to N_{F}(T), and

(2) a rigid isomorphism λ from the normalizer locality N_{L}(T) to a locality L_{Δ_{T}} (M)

contained in M, where Δ_{T} is the set of objects Q ∈ Δ such that T ≤ Q ≤ N_{S}(T), and L_{Δ_{T}} (M) is the locality obtained by restricting the group M (itself viewed as a locality) to Δ_{T}. Further, Theorem 5.15 says that if λ and λ′ are two isomorphisms as in (2), then the resulting localities L^{+}(λ) and L^{+}(λ′) are rigidly isomorphic if and only if the composition λ^{−1} followed by λ′ extends to an automorphism of M.

Theorem 5.17 establishes that every locality L = (M, Δ, S) can be constructed in the above way, by an iterative procedure. For example, one may begin with the group N_{L}(S), regarded as the restriction of L to a locality with a single object. One then proceeds (via the “+-operation” outlined above) to construct the restriction of L to larger and larger sets of objects, until the set Δ has been exhausted. At that point L itself will have been recovered as a “filtration” of its restrictions to an increasing sequence of subsets of Δ.

Section 6 provides a proof of the Main Theorem modulo a technical condition on localities in finite groups which is proved in section 7 as Proposition 7.1. In somewhat more detail: the proof of the Main Theorem depends on being able to produce an iterative procedure, of the kind described in the preceding paragraph, by which to create a linking system rather than to recover one, starting only with a saturated fusion system F on S and with the set Δ = F^{c} of F-centric subgroups of S. The procedure begins with the linking system L_{0} of N_{F}(R) for some suitably chosen R ∈ Δ; and where the existence and uniqueness of L_{0} is given by a result (see 1.10 below), obtained independently by Bob Oliver and Lluis Puig, which lies at the foundation of this paper. The difficulty, in going from one step to the next via the +-construction, lies in showing that what has already been constructed (and constructed uniquely, up to rigid isomorphism) yields an essentially unique rigid isomorphism λ at the local level required for the next step. This requires finding a good way to descend, step by step, through Δ - and this is what is achieved in section 6. The argument focuses on properties of one version of the Thompson J-subgroup J(R) of a finite p-group R, and on properties of finite groups G such that R is a Sylow p-subgroup of G, F^{∗}(G) = O_{p}(G), and J(R) is not a normal subgroup of G. Thus, section 6 provides a method of “descent”, while Proposition 7.1 enables the argument in section 6 and completes the proof of the Main Theorem. By ordering things in this way, all of the non-elementary finite group theory involved in the proof of the Main Theorem is pushed to the very end.

Proposition 7.1 concerns so-called F F -pairs (G, V), where G is a finite group such that O_{p}(G) = 1, and where V is a faithful G-module over the field of p elements, having the following property:

(*) There exists a non-identity abelian p-subgroup A of G (called a “best offender”

in G on V) such that |A||C_{V} (A)| ≥ |B||C_{V} (B)| for every subgroup B of A, and where G is generated by the set of such best offenders. The classification of such pairs (G, V) has been carried out piecemeal, over a period of many years, by many authors. It has only very recently been given a complete treatment (including the determination of the best offenders in the case where V is irreducible and G is almost simple) by Meierfrankenfeld and Stellmacher [MS2], as part of the project initiated by Meierfrankenfeld to provide an alternative approach to the classification of finite simple groups of local characteristic p. Parts of the classification of F F -pairs (for example the decomposition into “J-components”) are elementary, but as things stand at this date, the determination of the possible J-components themselves relies on the CFSG. Though we have attempted to organize the arguments on the basis of general principles where possible (see for example 7.7), any proof based on the CFSG, by its very nature, is opportunistic to some degree, and not entirely principled.

We should alert those readers who are familiar with arguments involving the Thompson J-subgroup J(P) of a finite p-group P, that in this paper J(P) is not defined in the way that has gained currency over the course of the decades since Thompson first introduced his version of J(P). That is, we define J(P) to be the subgroup of P that is generated by the abelian subgroups of P of maximal order (as in [Th]), rather than the elementary abelian subgroups of maximal order. This is the definition which is needed here, for reasons that will become clear from the arguments in sections 6 and 7.

Remark. Our tendency is toward right-hand notation for mappings, in any discussion which may involve composition of mappings. In particular, if C is a category, and X, Y, Z are objects of C, then composition defines a mapping

Mor_{C}(X, Y) × Mor_{C}(Y, Z) → Mor_{C}(X, Z).

Consistent with this policy, conjugation within any group G is taken in the right-handed sense, so that x^{g} = g^{−1}xg for any x, g ∈ G.

Acknowledgements: First, to my friends at Christian-Albrechts Universität zu Kiel, I wish to extend my thanks for their kind hospitality during my visits, over the course of many years, and for their patience in the face of lectures in which some tentative efforts were made to frame fusion systems and linking systems in a group-theoretic way. Special thanks are due to Bob Oliver for his hospitality in the fall of 2007, at Paris XIII, where the ideas that led to this paper were first conceived, and for his guidance past some fundamental misconceptions. My most heartfelt thanks go to Bernd Stellmacher, not least for his insightful reading of portions of earlier versions of this paper. His comments and suggested revisions, often given in great detail, have led to the simplification and clarification of many arguments, and to corrections of errors too embarassing to mention. In particular, the definition of “partial group” owes a great deal to his intervention, as does much of section 4. The remaining errors and infelicities are all my own.

Section 1: Fusion systems, saturation, and models

This section is, in part, a review of the basic notions pertaining to fusion systems and saturation; but the definitions of “fully normalized subgroup” and of saturation that turn out to be most convenient for the task at hand are not the standard ones. Still, the ideas are due to Puig [P1], while the terminology that we employ is that of [BLO], which has gained broad currency.

Let p be a prime, G a finite group, and S a Sylow p-subgroup of G. For subgroups P and Q of S, set

N_{G}(P, Q) = {g ∈ G | P^{g} ≤ Q}.

Here P^{g} is the set of elements x^{g} := g^{−1}xg, for x ∈ P. Set

Hom_{G}(P, Q) = {c_{g} : P → Q | g ∈ N_{G}(P, Q)},

where c_{g} : P → Q is the conjugation map x ↦ x^{g} induced by g. The fusion system F_{S}(G) induced on S by G is the category whose objects are the subgroups of S, and where the set of morphisms P → Q is Hom_{G}(P, Q). More generally:

Definition 1.1. Let S be a finite p-group. A fusion system on S is a category F, whose objects are the subgroups of S, and whose morphisms satisfy the following two conditions.

(a) Hom_{S}(P, Q) ⊆ Hom_{F}(P, Q) for all subgroups P and Q of S.

(b) Every F-homomorphism can be factored in F as an F-isomorphism followed by an inclusion map, and every F-isomorphism is an isomorphism of groups.

Example. For any finite p-group S there is the total fusion system \overline{F}(S), characterized by

Hom_{\overline{F}(S)}(P, Q) = Inj(P, Q),

where Inj(P, Q) is the set of all injective group homomorphisms P → Q.

Let F be a fusion system on S and let P ≤ S be a subgroup of S. A subgroup Q ≤ S is an F-conjugate of P if Q = P ϕ for some F-isomorphism ϕ.

Definition 1.2. Let F be a fusion system on S. A subgroup P of S is fully normalized in F provided that, for each F-conjugate Q of P, there exists an F-homomorphism ψ : N_{S}(Q) → N_{S}(P) such that Qψ = P.

Example. If F = F_{S}(G), G a finite group, and S ∈ Syl_{p}(G), then every subgroup of S has a fully normalized F-conjugate, by Sylow’s Theorem.

Definition 1.3. Let S be a finite p-group and let F be a subset of Hom(\overline{F}(S)) (i.e. a subset of the set of morphisms of the total fusion system on S) such that F contains Hom(F_{S}(S)). The fusion system on S generated by F is the category whose objects are the subgroups of S, and whose morphisms are the homomorphisms ϕ : P → Q such that ϕ is a composition of restrictions of members of F.

We note that it is immediate from definition 1.1 that the “fusion system generated by F” is in fact a fusion system on S.

Example. Let F be a fusion system on S and let T ≤ S be a subgroup of S, with T fully normalized in F. Define N_{F}(T) to be the fusion system on N_{S}(T) generated by the set of all F-homomorphisms ϕ : P → N_{S}(T) such that T ⊴ P and such that T ϕ = T.

A collection Δ of subgroups of S is closed under F-conjugation (or, is F-invariant) if P ϕ ∈ Δ whenever P ∈ Δ and ϕ ∈ Hom_{F}(P, S). We say that Δ is overgroup closed if Q ∈ Δ whenever Q is a subgroup of S which contains a member of Δ.

Example. For any fusion system F on S, let F^{c} be the largest F-invariant collection Δ of subgroups P of S such that C_{S}(P) ≤ P for all P ∈ Δ. Then S ∈ F^{c}, and F^{c} is overgroup closed in S. The members of F^{c} are the F-centric subgroups of S.

Definition 1.4. Let F be a fusion system on S and let Δ be a non-empty collection of subgroups of S, such that Δ is both overgroup closed and closed under F-conjugation. Then F is Δ-saturated if the following two conditions hold.

(A) Every member of Δ has a fully normalized F-conjugate.

(B) For each P ∈ Δ ∩ F^{c} such that P is fully normalized in F, there exists a finite

group M such that N_{S}(P) ∈ Syl_{p}(M), and with N_{F}(P) = F_{N_{S}(P)}(M). If F is F^{c}-saturated, and F is generated by the union of its subsystems N_{F}(P) as P ranges over the fully normalized members of F^{c}, then F is saturated.

Remark 1.5. (a) The above definition of saturation is equivalent to the (by now) standard one given in [BLO], and hence also to the various equivalent formulations found in [5A1] and [Sta]. Actually, in view of the Main Theorem, one may be satisfied to know that a fusion system satisfying the standard definition of saturation satisfies the conditions of 1.4. That the standard definition implies 1.4(A) is an easy exercise, while (B) follows from [2.4 and 2.5 in As]. The reverse implication (that 1.4 really does define saturation in the standard sense) is given by [Theorem A in 5A1].

(b) For any finite group G with Sylow p-subgroup S, the fusion system F_{S}(G) is Δ-saturated, for any non-empty, overgroup closed, F_{S}(G)-invariant collection Δ of subgroups of S.

Definition 1.6. Let F be a fusion system on S, and let T ≤ S be a subgroup of S. Then T is normal in F if F = N_{F}(T). The (unique) largest subgroup of S which is normal in F is denoted O_{p}(F). More generally, T is strongly closed in F if P ϕ ≤ T whenever P ≤ T and ϕ ∈ Hom_{F}(P, S). More generally still, T is weakly closed in F if T ϕ = T for all ϕ ∈ Hom_{F}(T, S).

Lemma 1.7. Let F be a saturated fusion system on S, let P ≤ S be a subgroup of S such that P is fully normalized in F, and let U be a subgroup of P such that N_{S}(P) ≤ N_{S}(U). Then there exists ϕ ∈ Hom_{F}(P, S) such that both P ϕ and Uϕ are fully normalized in F.

Proof. By 1.4(A) there exists ϕ ∈ Hom_{F}(N_{S}(U), S) such that V := Uϕ is fully normalized in F. Set Q = P ϕ. As N_{S}(P) ≤ N_{S}(U), and P is fully normalized, ϕ restricts to an isomorphism N_{S}(P) → N_{S}(Q). Now let ψ ∈ Hom_{F}(Q, S) and set R = Qψ. Then R is an F-conjugate of P, and so there exists η ∈ Hom_{F}(N_{S}(R), N_{S}(P)) with Rη = P. Composing η with ϕ yields an F-homomorphism N_{S}(R) → N_{S}(Q), so Q is fully normalized in F. □

Definition 1.8. Let F be a saturated fusion system over S. Then F is constrained if O_{p}(F) is F-centric.

The following terminology is taken from [As].

Definition 1.9. Let F be a constrained fusion system over S, and let M be a finite group. Then M is a model for F provided that:

(1) S is a Sylow p-subgroup of M, (2) F = F_{S}(M), and

(3) C_{M}(O_{p}(M)) ≤ O_{p}(M).

Notice that if M is a model for F then O_{p}(M) = O_{p}(F).

The definition of model in [As] (or, equivalently, of “localizer” in [P2]) is somewhat more flexible than the one we have given here; but 1.9 will suffice for our purposes. The following quoted result may be interpreted as saying that the Main Theorem holds in the case that F is constrained. This special result lies at the foundation of our proof of the Main Theorem.

Proposition 1.10. Let F be a constrained fusion system over the finite p-group S. Then the following hold.

(a) There exists a model M for F.

(b) Let M_{1} and M_{2} be models for F. Then there exists an isomorphism β : M_{1} → M_{2} such that β restricts to the identity map on S. Moreover, if β′ is any other such isomorphism, then the automorphism β^{−1} ∘ β′ of M_{2} is an inner automorphism c_{z}, given by conjugation by an element z ∈ Z(S). In particular:

(c) If M is a model for F then {c_{z} | z ∈ Z(S)} is the set of automorphisms of M which restrict to the identity map on S.

Proof. Point (a), and the uniqueness of M up to isomorphism, appear as proposition 4.3 in [5A1]. A different treatment, along with the “strong uniqueness” of M in point (b), is due to Puig [P2, Theorem 18.6]. There is also a subsequent (and independent) proof by Bob Oliver - including the important point (b) [Theorem 5.10 in section III of AKO]. □

Lemma 1.11. Let M be a model of the saturated, constrained fusion system F over S, and let E be a saturated fusion system on S such that the set Hom(E) of E-homomorphisms is contained in Hom(F). Then M contains a unique model H for E.

Proof. Set T = O_{p}(M), and let H be the set of all g ∈ M such the conjugation automorphism c_{g} of T is in E. The set of all such c_{g} with g ∈ H is equal to Aut_{E}(T), so H is a subgroup of M. Moreover, S ≤ H as F_{S}(S) ⊆ E.

Let E′ be the fusion system F_{S}(H). Then

Λ := Aut_{E′}(T) = Aut_{H}(T) = Aut_{E}(T).

Fix λ ∈ Λ, let h ∈ H with c_{h} = λ, and let P_{λ} be the largest subgroup P of S such that Aut_{P} (T)^{λ} ≤ Aut_{S}(T). Set P = P_{λ} and let Q be the pre-image in S of Aut_{P} (T)^{λ}. As conjugation by h induces an automorphism of Aut_{H}(T), the natural isomorphism of Aut_{H}(T) → H/Z(T) yields P^{h} = Q. That is, α extends to an E′-isomorphism ϕ : P → Q. Since E is constrained, also E has a model, and so α extends also to an E-isomorphism ψ : P → Q. Then ϕ ∘ ψ^{−1} is an F-automorphism which restricts to the identity on T, and so ψ = ϕ ∘ c_{z} for some z ∈ Z(T). Since F_{S}(S) ⊆ E ∩ E′, we conclude that Iso(E) = Iso(E′). Then Hom(E) = Hom(E′) by 1.1(b). Thus, E = E′, and H is a model for E.

Now suppose that there is another subgroup K of M which is a model for E. Let c : M → Aut(T) be the map which sends g ∈ M to the automorphism c_{g} of T. Then Ker(c) = Z(T) ≤ H ∩ K, and Kc = Aut_{K}(T) = Aut_{H}(T) = Hc, so K = H. □

By a group of Lie type in characteristic p we mean a finite group O^{p′}(C\overline{K}(σ)), where \overline{K} is a semisimple algebraic group over the algebraic closure \overline{𝔽}_{p} of the field of p elements, and where σ is a Steinberg endomorphism of \overline{K}. The following well-known result will play an important role in section 7.

Lemma 1.12. Let G be a group of Lie type in characteristic p, let S ∈ Syl_{p}(G) be a Sylow p-subgroup of G, and let X be a parabolic subgroup of G containing S. Then O_{p}(X) is weakly closed in F_{S}(G).

Proof. Let Φ be the root system (or twisted root system) associated with G, and let Φ^{+} be the set of positive roots, taken so that S is generated by the set of root subgroups U_{α} for α ∈ Φ^{+}. Set Q = O_{p}(X), set B = N_{G}(S), and let H be a complement to S in B. For any subset Δ of Φ^{+} let Δ′ be the set of roots −α such that α ∈ Φ^{+} and α ∉ Δ. Standard results concerning the structure of parabolic subgroups (see [Theorem 2.6.5 in GLS]) yield the existence of a subset Δ := Δ(X) of Φ^{+}, such that

(*) Q = ⟨U_{δ} | δ ∈ Δ⟩, O^{p′}(X) = ⟨U_{γ} | γ ∈ Φ^{+} ∪ Δ′⟩, and X = O^{p′}(X)H.

Let g ∈ N_{G}(Q, S). By Alperin’s fusion theorem there is a sequence (R_{1}, ⋯, R_{n}) of subgroups of S, and elements g_{i} ∈ N_{G}(R_{i}), such that g_{i} ∈ N_{G}(R_{i}), Q ≤ R_{1}, Q^{g_{1}···g_{i}} ≤ R_{i} for all i, and such that g = hg_{1} ⋯ g_{n} for some h ∈ C_{G}(Q). Moreover, the groups R_{i} may be chosen so that R_{i} = O_{p}(N_{G}(R_{i})) and N_{S}(R_{i}) ∈ Syl_{p}(N_{G}(R_{i})), and then a theorem of Borel and Tits [Theorem 3.1.3 in GLS] yields the result that each N_{G}(R_{i}) is a parabolic subgroup of G over S. Thus, in order to prove that Q′ = Q, and hence that Q is weakly closed in F_{S}(G), it suffices to consider the case where g = g_{1} ∈ Y for some parabolic subgroup Y = N_{G}(R) of G over S, with Q ≤ R = O_{p}(Y).

Set Γ = Δ(N_{G}(R)). Then Δ ⊆ Γ and Γ′ ⊆ Δ′. Applying (*) to both N_{G}(R) and X, we obtain N_{G}(R) ≤ X. Thus g ∈ N_{G}(Q), as required. □

Section 2: Partial groups, objective partial groups, and localities

For any set X we write W(X) for the free monoid on X. Thus, an element of W(X) is a finite sequence of (or word in) the elements of X, and the multiplication in W(X) consists of concatenation of sequences (denoted u∘v). The use of the same symbol “∘” for concatenation of sequences and for composition of functions should cause no confusion.

The length ℓ(w) of the word w = (x_{1}, ⋯, x_{n}) is n. The “empty word” is the word (∅) of length 0. We shall make no careful distinction between the set X and the set of words of length 1. That is to say, we regard X as a subset of W(X) via the identification x ↦ (x).

Definition 2.1. Let M be a non-empty set, and let W = W(M) be the free monoid on M. Let D be a subset of W such that

(1) M ⊆ D, and

u ∘ v ∈ D ⟹ u, v ∈ D.

(Notice that (1) implies that also the empty word is in D.) A mapping Π : D → M is a product if:

(2) Π restricts to the identity map on M, and

(3) u ∘ v ∘ w ∈ D ⟹ u ∘ (Π(v)) ∘ w ∈ D, and Π(u ∘ v ∘ w) = Π(u ∘ (Π(v)) ∘ w).

An inversion on M consists of an involutory bijection f ↦ f^{−1} on M, together with the mapping u ↦ u^{−1} on W given by

(f_{1}, ⋯, f_{n}) ↦ (f_{n}^{−1}, ⋯ f_{1}^{−1}).

A partial group consists of a product Π : D → M, together with an inversion (−)^{−1} on M, such that:

(4) u ∈ D ⟹ u^{−1} ∘ u ∈ D and Π(u^{−1} ∘ u) = 1,

where 1 denotes the image of the empty word under Π.

We list some elementary consequences of the definition, as follows.

Lemma 2.2. Let M (with D, Π, and inversion) be a partial group.

(a) Π is D-multiplicative. That is, if u ∘ v is in D then the word (Π(u), Π(v)) of length 2 is in D, and

Π(u ∘ v) = Π(u)Π(v),

where Π(u)Π(v) is an abbreviation for Π((Π(u), Π(v)).

(b) Π is D-associative. That is:

u ∘ v ∘ w ∈ D ⟹ Π(u ∘ v)Π(w) = Π(u)Π(v ∘ w).

(c) If u ∘ v ∈ D then u ∘ (1) ∘ v ∈ D and Π(u ∘ (1) ∘ v) = Π(u ∘ v).

(d) If u ∘ v ∈ D then both u^{−1} ∘ u ∘ v and u ∘ v ∘ v^{−1} are in D, Π(u^{−1} ∘ u ∘ v) = Π(v), and Π(u ∘ v ∘ v^{−1}) = Π(u).

(e) The cancellation rule: If u ∘ v, u ∘ w ∈ D, and Π(u ∘ v) = Π(u ∘ w), then Π(v) = Π(w) (and similarly for right cancellation).

(f) If u ∈ D then u^{−1} ∈ D, and Π(u^{−1}) = Π(u)^{−1}. In particular, 1^{−1} = 1.

(g) The uncancellation rule: Let u, v, w ∈ W, and suppose that both u ∘ v and u ∘ w are in D and that Π(v) = Π(w). Then Π(u ∘ v) = Π(u ∘ w). (Similarly for right uncancellation.)

Proof. Let u ∘ v ∈ D. Then 2.1(3) applies to (∅) ∘ u ∘ v and yields the result that (Π(u)) ∘ v ∈ D with Π(u ∘ v) = Π((Π(u)) ∘ v). Now apply 2.1(3) to (Π(u)) ∘ v ∘ (∅), to obtain (a).

Let u ∘ v ∘ w ∈ D. Then u ∘ v and w are in D by 2.1(1), and D-multiplicativity yields Π(u ∘ v ∘ w) = Π(u ∘ v)Π(w). Similarly, Π(u ∘ v ∘ w) = Π(u)Π(v ∘ w), and (b) holds.

Notice that point (c) is immediate from 2.1(3).

Assume u ∘ v ∈ D. Then v^{−1} ∘ u^{−1} ∘ u ∘ v ∈ D by 2.1(4), and then also u^{−1} ∘ u ∘ v ∈ D. Multiplicativity then yields

Π(u^{−1} ∘ u ∘ v) = Π(u^{−1} ∘ u)Π(v) = 1Π(v) = Π(∅)Π(v) = Π(∅ ∘ v) = Π(v).

As (w^{−1})^{−1} = w for any w ∈ W, one obtains w ∘ w^{−1} ∈ D for any w ∈ D, and Π(w ∘ w^{−1}) = 1. From this one easily completes the proof of (d).

Now let u ∘ v and u ∘ w be in D, with Π(u ∘ v) = Π(u ∘ w). Then (d) (together with multiplicativity and associativity, which will not be explicitly mentioned hereafter) yield

Π(v) = Π(u^{−1} ∘ u ∘ v) = Π(u^{−1})Π(u)Π(v) = Π(u^{−1})Π(u)Π(w) = Π(u^{−1} ∘ u ∘ w) = Π(w),

and (e) holds.

Let u ∈ D. Then u ∘ u^{−1} ∈ D, and then Π(u)Π(u^{−1}) = 1. But also (Π(u), Π(u)^{−1}) ∈ D, and Π(u)Π(u)^{−1} = 1. Now (f) follows by cancellation.

Let u, v, w be as in (g). Then u^{−1} ∘ u ∘ v and u^{−1} ∘ u ∘ w are in D by (d). By two applications of (d), Π(u^{−1} ∘u∘v) = Π(v) = Π(w) = Π(u^{−1} ∘u∘w), so Π(u∘v) = Π(u∘w) by (e). That is, Π(u)Π(v) = Π(u)Π(w), and (g) holds. □

Lemma 2.3. Let M be a partial group, and write xy for Π(x, y) when (x, y) ∈ D.

(a) For each x ∈ M, both (x, 1) and (1, x) are in D, and 1x = x1.

(b) For each x ∈ M, both (x^{−1}, x) and (x, x^{−1}) are in D, and x^{−1}x = 1 = xx^{−1}.

(c) If W(M) = D then M is a group via the binary operation (x, y) ↦ xy.

Proof. As x = ∅ ∘ x = x ∘ ∅, as Π(x) = x by 2.1(2), and since Π(∅) = 1, point (a) follows from 2.2(a). Point (b) is immediate from 2.1(4). Thus, 1 is an identity element for M by (a), and x^{−1} is an inverse for x by (b). Finally, if M × M × M ⊆ D then the operation (x, y) ↦ xy is associative by 2.2(b). In particular, (c) holds. □

2.4 Examples.

1. The first example is the basic one, in which M is a group G, 1 is the identity element of G, g^{−1} is the inverse of g in G, D = W(G), and Π is the (multi-variable) product in G. Let “·” be the binary operation given by restricting Π to M × M. Then (M, ·) is a group by 2.3(c), and visibly that group is equal to G. Conversely, if (M, D, Π) is a partial group in which D = W then (M, ·) is a group, again by 2.3(c).

2. Let G be a group and let Δ be a collection of subgroups of G. For X ∈ Δ and g ∈ G write X^{g} for the subgroup g^{−1}Xg ≤ G. One then obtains a partial group M = M(G, Δ), for which D is the set of all words w = (g_{1}, ⋯, g_{n}) ∈ W(G) such that there exists X ∈ Δ with X^{g_{1}···g_{i}} ∈ Δ for all i (1 ≤ i ≤ n). Take Π to be the restriction to D of the multivariable product in G, inversion as the restriction to M of inversion in G, and 1 as the identity element of G. Notice that if there exists X ∈ Δ with X^{g} ∈ Δ for all g ∈ G, then all products are defined, and one then recovers G as a bona fide group.

3. Here is a special case of example (2). Let G be the group O_{4}^{+}(2) (or equivalently, the wreath product S_{3} ≀ C_{2}). Thus, G is a group of order 72, with a normal elementary abelian subgroup A of order 9, and with a dihedral Sylow 2-subgroup S acting faithfully on A. Let Δ be the set of subgroups of S of order 2. Then, as a set, the partial group M (defined as in example 2) is equal to G, since every element of G fuses some involution of S into S. But D(M) is a proper subset of W(M), so M is not a group.

It is often convenient to eliminate the symbol “Π” and to speak of “the product f_{1} ⋯ f_{n}”. More generally, if {X_{i}}_{1≤i≤n} is a collection of subsets of M then the “product set X_{1} ⋯ X_{n}” is by definition the image under Π of the set of words (f_{1}, ⋯, f_{n}) ∈ D such that f_{i} ∈ X_{i} for all i. If X_{i} = {f_{i}} is a singleton then we may write f_{i} in place of X_{i} in such a product. Thus, for example, the product Xfg stands for the set of all Π(x, f, g) with (x, f, g) ∈ D, and with x ∈ X.

A word of urgent warning: in writing products in the above way one may be led, mistakenly, into imagining that “associativity” holds in a stronger sense than that which is given by 2.2(b). For example, one should not suppose, if (f, g, h) ∈ W, and both (f, g) and (fg, h) are in D, that (f, g, h) is in D. That is, it may be that “the product fgh” is undefined, even though the product (fg)h is defined. Of course, one is tempted to simply extend the domain D to include such triples (f, g, h), and to “define” the product fgh to be (fg)h. The trouble is that it may also be the case that gh and f(gh) are defined (via D), but that (fg)h ≠ f(gh).

Let M be a partial group and let H be a non-empty subset of M. Then H is a partial subgroup of M if H is closed under inversion (f ∈ H implies f^{−1} ∈ H) and with respect to products. The latter condition means that Π(w) ∈ H whenever w ∈ W(H) ∩ D. If in fact W(H) is contained in D, then H is a subgroup of M (i.e. a partial subgroup which is a group) by 2.3.

For M a partial group and f ∈ M, write D(f) for the set of all x ∈ M such that the product f^{−1}xf is defined. There is then a mapping

c_{f} : D(f) → M

given by x ↦ f^{−1}xf (and called conjugation by f). Since our preference is for “righthand” notation, we write

x ↦ (x)c_{f} or x ↦ x^{f}

for conjugation by f.

At this early point, and in the context of arbitrary partial groups, one can say very little about the maps c_{f}. The cancellation rule 2.2(e) implies that each c_{f} is injective, but beyond that, the following lemma may be the best that can be obtained.

Lemma 2.5. Let M be a partial group and let f ∈ M. Then the following hold. (a) 1 ∈ D(f) and 1^{f} = 1.

(b) D(f) is closed under inversion, and (x^{−1})^{f} = (x^{f})^{−1} for all x ∈ D(f).

(c) c_{f} is a bijection D(f) → D(f^{−1}), and c_{f^{−1}} = (c_{f})^{−1}.

(d) M = D(1), and x^{1} = x for each x ∈ M.

Proof. By 2.1(4), f ∘ ∅ ∘ f^{−1} = f ∘ f^{−1} ∈ D, so 1 ∈ D(f) and then 1^{f} = 1 by 2.3(a). Thus (a) holds. Now let x ∈ D(f) and set w = (f^{−1}, x, f). Then w ∈ D, and w^{−1} = (f^{−1}, x^{−1}, f) by definition in 2.1. Then 2.1(4) yields w^{−1} ∘ w ∈ D, and so w^{−1} ∈ D by 2.1(1). This shows that D(f) is closed under inversion. Also, 2.1(4) yields 1 = Π(w^{−1} ∘ w) = (x^{−1})^{f}x^{f}, and then (x^{−1})^{f} = (x^{f})^{−1} by 2.2(f). This completes the proof of (b).

As w ∈ D, 2.2(d) implies that f ∘ w and then f ∘ w ∘ f^{−1} are in D. Now 2.1(3) and two applications of 2.2(d) yield

fx^{f}f^{−1} = Π(f, f^{−1}, x, f, f^{−1}) = Π((f, f^{−1}, x) ∘ f ∘ f^{−1}) = Π(f, f^{−1}, x) = x.

Thus x^{f} ∈ D(f^{−1}) with (x^{f})^{f^{−1}} = x, and thus (c) holds.

Finally, 1 = 1^{−1} by 2.2(f), and ∅ ∘ x ∘ ∅ = x ∈ D for any x ∈ M, proving (d). □

If X is a subgroup of M with X ⊆ D(f), write X^{f} for {x^{f} | x ∈ X}. Example 2.4(3), with X a fours group contained in S, and with f a suitable element of order 3, shows that X^{f} need not be a group with respect to the product Π.

For subgroups X and Y of M, set

N_{M}(X, Y) = {f ∈ M | X ⊆ D(f) and X^{f} ≤ Y },

and set

N_{M}(X) = {f ∈ M | X ⊆ D(f) and X^{f} = X}.

In practice, all of the objective partial groups that will be encountered in this paper will have the property that their objects are finite, so we will always have N_{M}(X, X) = N_{M}(X) by 2.5(c). Write C_{M}(X) for the set of all f ∈ N_{M}(X) such that x^{f} = x for all x ∈ X.

Henceforth, if X is a subgroup of a partial group M, any statement involving the expression “X^{f}” should be understood as being based on the tacit hypothesis that X ⊆ D(f).

Example 2.4(2) may be formalized and generalized as follows.

Definition 2.6. Let M be a partial group and let Δ be a collection of subgroups of M. Let D_{Δ} be the set of all w = (f_{1}, ⋯, f_{n}) ∈ W(M) such that:

(*) there exists (X_{0}, ⋯, X_{n}) ∈ W(Δ) with (X_{i−1})^{f_{i}} = X_{i} for all i (1 ≤ i ≤ n). Then (M, Δ) is an objective partial group (in which Δ is the set of objects), if the following two conditions hold. (O1) D = D_{Δ}. (O2) Whenever X, Z ∈ Δ, Y ≤ Z is a subgroup of Z, and f ∈ M with X^{f} ≤ Y, then

Y ∈ Δ. In particular, X^{f} ∈ Δ.

We say that a word w = (f_{1}, ⋯, f_{n}) is in D via (X_{0}, ⋯, X_{n}) if the condition (*) in 2.6 applies specifically to w and (X_{0}, ⋯, X_{n}). We may also say, more simply, that w is in D via X_{0}, since the sequence (X_{0}, ⋯, X_{n}) is determined by w and X_{0}.

Remark. Notice that in the preceding definition, one needs to already have D in order to know what D_{Δ} is, since D_{Δ} is defined in terms of conjugation in the partial group defined by D. In practice, when one tries to construct an objective partial group, it’s often easy to decide on a suitable D which yields the partial group that one wants, and which has the property that D ⊆ D_{Δ}. But it can then be very difficult to establish the reverse inclusion D ⊇ D_{Δ}. In fact, much of this paper is built around three such exercises: one of them in the Appendix (in order to establish that Oliver-Ventura “transporter systems” give rise to localities), and one each in sections 4 and 5.

Remark. Condition (O2) in 2.6 has been stated in the form appropriate for this paper, where objects will always be finite p-groups, from 2.9 on. A more general formulation would be: (O2)′ Whenever X, Z ∈ Δ, Y ≤ Z is a subgroup of Z, and f ∈ M with X^{f} ⊆ Y, then

N_{Y} (X^{f}) ∈ Δ.

Lemma 2.7. Let (M, Δ) be an objective partial group.

(a) N_{M}(X) is a subgroup of M for each X ∈ Δ.

(b) Let f ∈ M and let X ∈ Δ with X^{f} ∈ Δ. Then N_{M}(X) ⊆ D(f), and

c_{f} : N_{M}(X) → N_{M}(X^{f})

is an isomorphism of groups.

(c) Let w = (f_{1}, ⋯, f_{n}) ∈ D via (X_{0}, ⋯, X_{n}). Then

c_{f_{1}} ∘ ⋯ ∘ c_{f_{n}} = c_{Π(w)}

as maps from X_{0} to X_{n}.

Proof. We first prove (c). Thus, let w and (X_{0}, ⋯, X_{n}) be as in (c). For any x ∈ X_{0} set u_{x} = w^{−1} ∘ (x) ∘ w. Then u_{x} ∈ D_{Δ} via X_{n}. Setting f = Π(w), and recalling that Π(w^{−1}) = f^{−1} (by 2.2(f)), we get

Π(f^{−1}, x, f) = Π(u_{x}) = ((⋯ (x)^{f_{1}}) ⋯)^{f_{n}},

by repeated application of 2.2(a). This yields (c).

Let X ∈ Δ and set L = N_{M}(X). Then L is non-empty since 1 ∈ L by 2.5(b). Further, L is closed with respect to inversion by 2.5. For any w ∈ W(L), the condition (O1) in 2.6 implies that w ∈ D via X, and then Π(w) ∈ L by (c). Now (a) follows from 2.3(c).

Let f ∈ M with X ⊆ D(f) and with X^{f} ∈ Δ. Let x, y ∈ L and set u = (f^{−1}, x, f, f^{−1}, y, f). Then u ∈ D_{Δ} via X^{f}. Thus u ∈ D by (O1), and then 2.2(a) yields Π(u) = x^{f}y^{f}. We note also that 2.1(3) and 2.3(b) yield

Π(x, f, f^{−1}, y) = Π(x ∘ 1 ∘ y)

and so Π(x, f, f^{−1}, y) = xy by 2.3(b). Then

Π(u) = Π(f^{−1} ∘ (x, f, f^{−1}, y) ∘ f) = (xy)^{f}

by 2.1(3), and thus c_{f} : L → L^{f} is a homomorphism of groups. Then c_{f} an isomorphism by 2.5(c), proving (b). □

Remark 2.8. We mention two structures associated with a given objective partial group (M, Δ).

1. There is a category C = Cat(M, Δ) whose set of objects is Δ, whose morphisms are triples (f, X, Y) with X, Y ∈ Δ and with f ∈ N_{M}(X, Y), and where composition of morphisms is given by the product in M:

(f, X, Y) ∘ (g, Y, Z) = (fg, X, Z),

(in right-hand notation). The morphisms (1, X, Y) with X ≤ Y are called inclusion morphisms. Notice that every morphism in C can be factored in a unique way as an isomorphism followed by an inclusion morphism.

2. There is a category F = F(M, Δ), to be called the fusion system of (M, Δ), and defined as follows. First, the objects of F are the groups U such that U ≤ X for some X ∈ Δ. Then, the morphisms in F from U to V are taken to be the group homomorphisms ϕ : U → V such that ϕ can be factored as a composition of restrictions of conjugation homomorphisms c_{f} : X → Y between objects.

3. There is another category F^{∗} = F^{∗}(M, Δ), in which Ob(F^{∗}) = Ob(F(M, Δ)), but where Hom_{F^{∗}}(U, V) is the set (containing Hom_{F}(U, V)) of all homomorphisms ϕ : U → V such that ϕ is a composition of restrictions of conjugation homomorphisms c_{f} : X → Y between subgroups X and Y of M. Here X and Y are not assumed to be objects. It appears to be a highly non-trivial question, as to whether the fusion systems F and F^{∗} are necessarily equal - even in the case of localities or linking systems (defined below).

Here is the main definition.

Definition 2.9. Let p be a prime, let L be a partial group, and let S be a finite psubgroup of L. Then (L, S) is a locality if L is finite, and provided that there exists a set Δ of subgroups of S, such that S ∈ Δ, and such that the following two conditions hold.

(L1) (L, Δ) is objective.

(L2) S is maximal in the poset (ordered by inclusion) of finite p-subgroups of L.

We say also that L is a locality on S via Δ.

There are a number of special sorts of localities that deserve special names. In order to assign names to them in a way that is consistent with established usage, we define F := F_{S}(L) to be the fusion system on S whose homomorphisms are the compositions of restrictions of conjugation maps in L from one object to another. That is, F is the fusion system F(L, Δ) defined in 2.8(2).

A locality L is a Δ-linking system if C_{L}(P) ≤ P for each P ∈ Δ. If moreover Δ is the set of all F-centric subgroups of S then L is a centric linking system.

Example/Lemma 2.9.1. Let M be a finite group, let S be a Sylow p-subgroup of M, set F = F_{S}(M), and let Γ be a non-empty F-invariant collection of subgroups of S, such that Γ is overgroup closed in S. Define L to be the set of all g ∈ M such that S ∩ S^{g} ∈ Γ, and set D = D_{Γ} (as defined in 2.6). Then L is a partial group via the restriction of the multivariable product in M to D. Moreover, (L, S) is a locality via Γ; to be denoted L_{Γ}(M).

Proof. If g ∈ L then (S ∩ S^{g^{−1}})^{g} = S ∩ S^{g} ∈ Γ, and then (S ∩ S^{g^{−1}}) ∈ Γ since Γ is F-invariant. Thus L ⊆ D, and L is contained in the partial group M = M(M, Γ) given by example 2.4(2). In that example, M is the set of all g ∈ M such that there exists P ∈ Γ with P^{g} ∈ Γ. Such an element g has the property that S ∩ S^{g} ∈ Γ since Γ is overgroup closed, and so L = M. Example 2.4(2) now shows that L is a partial group with respect to the multivariable product and the inversion in G. The condition (O1) for objectivity is given by the definition of D, while (O2) is immediate from the assumption that Γ is overgroup closed and F-invariant. Thus, (L, Γ) is objective. All members of Γ are subgroups of S, and S is maximal in the poset of p-subgroups of G, so (L, S) is a locality via Γ. □

For any locality (L, S), let Ω(L, S) be the set of all collections Δ of subgroups of S, such that S ∈ Δ and such that (L, Δ) is objective. We say that (L, S) is complete if it satisfies the following condition.

(*) For each Δ ∈ Ω(L, S), and each f ∈ L, the set S_{f} = {s ∈ S | s^{f} ∈ S} is a member of Δ. In particular, S_{f} is a subgroup of S.

Proposition 2.10. Every locality is complete.

Proof. Let L be a locality on S, let Δ ∈ Ω(L, S), and let f ∈ L. The word (f) of length 1 is in D := D(L), so there exists P ∈ Δ with Q := P^{f} ∈ Δ. Let a ∈ S_{f}, and set b = a^{f}. Then {a, a^{−1}, f} ⊆ N_{L}(P, S), while b ∈ N_{L}(Q, S). Thus (a^{−1}, f, b) ∈ D via P^{a}. Then also (f, b) ∈ D, while (a, f) ∈ D via P^{a^{−1}}. From f^{−1}af = b we get af = fb by 2.2(e), and hence

a^{−1}fb = a^{−1}(fb) = a^{−1}(af) = f,

by D-associativity. Since a^{−1}fb conjugates P^{a} into S, we conclude that:

(1) P^{a} ≤ S_{f} for all a ∈ S_{f}, and for all P ∈ Δ for which P^{f} ≤ S.

In order to show that S_{f} is a subgroup of S it suffices to show that xy ∈ S_{f} for all x, y ∈ S_{f}, since by 2.5(b) S_{f} is closed under inversion. From (1), both P^{x} and (P^{x})^{y} are subgroups of S, and hence in Δ by (O2). Further, (1) yields P^{xf} and (P^{xy})^{f} in Δ. Thus

w := (f^{−1}, x, f, f^{−1}, y, f) ∈ D via (P^{f}, P, P^{x}, P^{xf}, P^{x}, P^{xy}, (P^{xy})^{f}).

Then (f^{−1}xf)(f^{−1}yf) = f^{−1}(xy)f, and since x^{f}y^{f} ∈ S we get (xy)^{f} ∈ S. That is, xy ∈ S_{f}, and S_{f} is a subgroup of S. As S_{f} contains a member of Δ, (O2) then yields S_{f} ∈ Δ. □

Corollary 2.11. Let L be a locality on S. There is then a unique smallest collection Γ of subgroups of S such that (L, Γ) is objective.

Proof. Set Ω = Ω(L, S) and set Γ = ⋂ Ω. That is, Γ is the set of all P such that P ∈ Δ for all Δ ∈ Ω. Let w = (f_{1}, ⋯, f_{n}) ∈ D. Then for each Δ ∈ Ω there exists P_{Δ} ∈ Δ such that w ∈ D via Δ, by (O1). Set Q_{0} = ⟨P_{Δ} | Δ ∈ Ω⟩. Then 2.10 shows that Q_{0} ≤ S_{f_{1}} and that there is a well-defined sequence (Q_{0}, ⋯, Q_{n}) of subgroups of S such that Q_{i} = (Q_{i−1})^{f_{i}} for all i with 1 ≤ i ≤ n. Each Q_{i} is in Γ by (O2), so (L, Γ) satisfies (O1). Now let X, Y ∈ Γ and let f ∈ L with X^{f} ≤ Y. As Y ∈ Δ for all Δ ∈ Ω, (O2) implies that the same holds for X^{f}, and so X^{f} ∈ Γ. That is, (L, Γ) satisfies (O2), and (L, Γ) is objective. □

Lemma 2.12. Let L be a locality on S. Then there is a unique largest set Γ of subgroups of S such that (L, Γ) is objective.

Proof. Let Δ_{1}, Δ_{2} ∈ Ω := Ω(L, S) and set Δ = Δ_{1} ∪ Δ_{2}. It will suffice to show that Δ ∈ Ω.

Let (P_{0}, ⋯, P_{n}) ∈ W(Δ), and let (f_{1}, ⋯, f_{n}) ∈ W(L) such that P_{k−1}^{f_{k}} = P_{k} for all k from 1 to n. Since all objects are subgroups of S, and S ∈ Δ_{i} for all i, the condition (O2) on (L, Δ_{i}) implies that if P_{0} ∈ Δ_{i} then also each P_{k} is in Δ_{i}. Thus,

D_{Δ} ⊆ D_{Δ_{1}} ∪ D_{Δ_{2}} = D(L) ⊆ D_{Δ},

and so D_{Δ} = D(L). That is, (L, Δ) satisfies the condition (O1).

It remains to show that (L, Δ) satisfies (O2). So, let X, Y ∈ Δ and let f ∈ L with X^{f} ≤ Y. Then X^{f} ≤ S ∈ Δ_{1} ∩ Δ_{2}. If X ∈ Δ_{i} then (O2) applied to (L, Δ_{i}) yields X^{f} ∈ Δ_{i}, so X^{f} ∈ Δ and the proof is complete. □

For any word w in W(L), L a locality on S, we have also the notion of S_{w}, treated in the following lemma.

Lemma 2.13. Let L be a locality on S, set D = D(L), and let Δ ∈ Ω(L, S). Let w = (f_{1}, ⋯, f_{n}) ∈ W(L), and define S_{w} to be the set of all elements s_{0} ∈ S such that there is a sequence (s_{0}, s_{1}, ⋯, s_{n}) of elements of S given by (s_{i−1})^{f_{i}} = s_{i} (1 ≤ i ≤ n). Then the following hold.

(a) S_{w} is a subgroup of S, and S_{w} ∈ Δ if and only if w ∈ D.

(b) Let w, w′ ∈ D with Π(w) = Π(w′), and with S_{w} = S_{w′}. Let u, v ∈ W. Then

u ∘ w ∘ v ∈ D ⟺ u ∘ w′ ∘ v ∈ D.

Proof. (a): Let x_{0}, y_{0} ∈ S_{w}, and define x_{i} recursively by x_{i} = (x_{i−1})^{f_{i}} (1 ≤ i ≤ n). Similarly define y_{i}. Then x_{i−1}y_{i−1} ∈ S_{f_{i}} by 2.10, and (x_{i−1}y_{i−1})^{f_{i}} = x_{i}y_{i} by 2.7(b). Thus S_{w} is closed under multiplication. Since 2.5(b) shows that S_{w} is closed under inversion, and since 1 ∈ S_{w}, S_{w} is then a subgroup of S. If S_{w} ∈ Δ then w ∈ D by (O1). Conversely, if w ∈ D via some P ∈ Δ P ≤ S_{w} and (O2) yields S_{w} ∈ Δ.

(b): Set a = u∘w∘v and b = u∘w′ ∘v, and assume that a ∈ D. Then (S_{a})^{Π(u)} ≤ S_{w∘v}, and

S_{w∘v} = {s ∈ S_{w} | s^{Π(w)} ∈ S_{v}} = {s ∈ S_{w′} | s^{Π(w′)} ∈ S_{v}} = S_{w′∘v}.

Thus (S_{a})^{Π(u)} ≤ S_{w′∘v} and b ∈ D via S_{a}. □

For any locality (L, S), we write F_{S}(L) for the fusion system on S generated by the conjugation maps in L between objects. Notice that F_{S}(L) does not depend on the choice Δ of the set of objects, since 2.10 shows that S_{f} is independent of Δ for f ∈ L.

Definition 2.14. Let L = (L, Δ, S) be a locality and let P ∈ Δ be an object. Then P is centric in L if C_{L}(P)/Z(P) is a p′-group; radical in L if P = O_{p}(N_{L}(P)); and essential in L provided that

(i) P is centric in L,

(ii) N_{S}(P) ∈ Syl_{p}(N_{L}(P)), and

(iii) N_{L}(P)/P has a strongly p-embedded subgroup. Notice that condition (iii) implies that P is radical in L.

Definition 2.15. Let L = (L, Δ, S) be a locality, let Δ^{e} be the set of objects Q ∈ Δ such that Q is essential in L, and set A = A(L) = Δ^{e} ∪ {S}. Let f ∈ L. Then f is A-decomposable if there exists w = (g_{1}, ⋯ g_{n}) ∈ D(L) such that the following hold.

(i) S_{f} ≤ S_{w}, and f = Π(w).

(ii) For all i: S_{g_{i}} is in A, and either g_{i} ∈ O^{p′}(N_{L}(S_{g_{i}})) or S_{g_{i}} = S.

The Alperin-Goldschmidt fusion theorem [Gd] implies that in a locality L = L_{Γ}(M) of a finite group G, an element f ∈ L is A-decomposable provided that C_{G}(S_{f}) ≤ S_{f}. In particular, each f ∈ L is A-decomposable if C_{M}(O_{p}(M)) ≤ O_{p}(M). The following result provides a generalization to linking systems. Recall the definition of Δ-linking system preceding 2.9.1.

Proposition 2.16. Let L = (L, Δ, S) be a Δ-linking system and define A(L) as above. Let f ∈ L. Then f is A(L)-decomposable.

Proof. Among all f for which the lemma fails to hold, choose f with P := S_{f} as large as possible. Then P ≠ S. Set P′ = P^{f} and set A = A(L).

Let Q be a fully normalized L-conjugate of P (and hence also of P′), and let g, h ∈ L with Q = P^{g} = (P′)^{h}. Thus, N_{S}(Q) ∈ Syl_{p}(N_{L}(Q)). By 2.7(b) and Sylow’s Theorem, g and h may be chosen so that N_{S}(Q) contains both N_{S}(P)^{g} and N_{S}(P′)^{h}. The maximality of P then implies that g and h are A-decomposable. Then g^{−1} and h^{−1} are A-decomposable via the inverses of words which yield A-composability for g and h.

Set f′ = g^{−1}fh, M = N_{L}(Q), and R = N_{S}(Q). Then f′ ∈ M, and u := (g, f′, h^{−1}) ∈ D via P, and Π(u) = f. If f′ is A-decomposable then so is f, and thus we may assume that f = f′ and P = Q. That is, we are reduced to establishing the proposition for the finite group M rather than the locality L. If P ∈ A then f is A-decomposable by definition, so we may assume otherwise. Applying the Alperin-Goldschmidt theorem to M, c_{f} ∈ Aut(P) is a composition c_{f} = c_{g_{1}} ∘ ⋯ ∘ c_{g_{n}} with g_{i} ∈ N_{M}(E_{i}) for some E_{i} ∈ A(M). As P ∉ A, |P | < |E_{i}| for all i. The maximality of P then implies that each g_{i} is A-decomposable, and hence also g := g_{1} ⋯ g_{n} is A-decomposable. Finally, z := fg^{−1} ∈ C_{M}(P) = Z(P), so f = zg is A-decomposable. □

Proposition 2.17. Let (L, S) be a locality via Δ, and let F := F_{S}(L) be the associated fusion system on S. Then the following hold.

(a) F is Δ-saturated, and if F^{c} ⊆ Δ then F is saturated.

(b) For each P ∈ Δ, the map N_{L}(P) → Aut_{F}(P) given by f ↦ c_{f} is a surjective homomorphism with kernel C_{L}(P).

(c) If P ∈ Δ and P is fully normalized in F then N_{S}(P) is a Sylow p-subgroup of N_{L}(P).

(d) If P ∈ Δ and ϕ ∈ Hom_{F}(P, S) then ϕ = c_{f} for some f ∈ N_{L}(P, S).

Proof. We first show:

(1) For each P ∈ Δ there exists f ∈ N_{L}(P, S) such that N_{S}(P^{f}) is a Sylow psubgroup of N_{L}(P^{f}).

By (L2), (1) holds for P = S (and for f = 1). Among all P ∈ Δ for which (1) fails, choose P so that |S : P | is as small as possible. We are free to replace P with any L-conjugate of P in S, so we may assume that |N_{S}(P)| is maximal among all such conjugates. Set R = N_{S}(P), and let R^{∗} be a Sylow p-subgroup of N_{L}(P) containing R. Then R is a proper subgroup of R^{∗}, and hence also a proper subgroup of N_{R^{∗}}(R). By the minimality of |S : P |, there exists an L-conjugate Q := R^{f} of R such that N_{S}(Q) is a Sylow psubgroup of N_{L}(Q). Without loss, we may replace f with fg for any g ∈ N_{L}(Q) since any such product fg is defined via (R, Q, Q). By Sylow’s Theorem, we may therefore assume that N_{R^{∗}}(R)^{f} ≤ N_{S}(Q). But N_{R^{∗}}(R)^{f} normalizes P^{f}, and we thereby contradict the maximality of |N_{S}(P)|. Thus, (1) is proved.

Next, let P ∈ Δ and let ϕ ∈ Hom_{F}(P, S). By definition, ϕ is a composite ϕ = ϕ_{1} ∘ ⋯ ∘ ϕ_{n}, where ϕ_{i} is given by conjugation by an element h_{i} of L, and where

(2)  P (ϕ_{1} ∘ ⋯ ∘ ϕ_{i}) ≤ S

for all i with 1 ≤ i ≤ n. Then the word w = (h_{1}, ⋯, h_{n}) is in D via P, and 2.7(c) yields P ϕ = P^{h} where h = Π(w). Thus (d) holds, and one observes that point (b) follows immediately from (d).

We may now complete the proof of (a) and (c). Namely, let P ∈ Δ and let Q = P^{f} be an L-conjugate of P, as in (1), so that N_{S}(Q) ∈ Syl_{p}(N_{L}(Q). As N_{S}(P)^{f} ≤ N_{L}(Q), there then exists g ∈ N_{L}(Q) such that (N_{S}(P)^{f})^{g} ≤ N_{S}(Q). As c_{f} ∘ c_{g} ∈ F, we conclude that Q is fully normalized in F, in the sense of definition 1.2. Thus, F satisfies the condition 1.4(A) for Δ-saturation. On the other hand, suppose that P itself is fully normalized in F. Then, by (2), there exists h ∈ L such that N_{S}(Q)^{h} = N_{S}(P) and with Q^{h} = P. This shows that N_{S}(P) ∈ Syl_{p}(N_{L}(P)) (and thus (c) holds).

Set M = N_{L}(P). By definition, each ϕ in N_{F}(P) extends to an F-homomorphism which maps P to P. Then (d) implies that ϕ = c_{f} for some f ∈ M. Thus F_{N_{S}(P)}(M) = N_{F}(P), so that F satisfies the condition 1.4(B) for Δ-saturation. This completes the proof that F is Δ-saturated.

Suppose that L is a centric linking system. Then Δ is the set of F-centric subgroups of S, by definition. By 2.16, F is generated by the fusion systems N_{F}(P) for P ∈ Δ in this case, so by definition 1.4, F is saturated. This completes the proof of (a), and of the lemma. □

Recall the notion of normalizer from 2.7.

Lemma 2.18. Let (L, S) be a locality via the set Δ of objects, let T be a subgroup of S, and set Δ_{T} = {N_{P} (T) | T ≤ P ∈ Δ}.

(a) N_{L}(T) is a partial subgroup of L.

(b) If Δ_{T} ⊆ Δ, then (N_{L}(T), Δ_{T}) is an objective partial group.

(c) If Δ_{T} ⊆ Δ, and |N_{S}(T)| ≥ |N_{S}(U)| for every L-conjugate U of T in S, then (N_{L}(T), N_{S}(T)) is a locality via Δ_{T}.

Proof. Let w = (f_{1}, ⋯, f_{n}) ∈ W(N_{L}(T)), and suppose that w ∈ D := D(L) via a sequence (P_{0}, ⋯, P_{n}) of objects. Then ⟨P_{i−1}, T ⟩ ≤ S_{f_{i}} for all i, by completeness, and then

⟨P_{i−1}, T ⟩^{f_{i}} = ⟨P_{i}, T ⟩.

Thus, T ≤ S_{w}, and we may assume for the sake of simplicity that T ≤ P_{i} for all i. Set f = Π(w). Then 2.7(c) yields T^{f} = T, and so N_{L}(T) is closed under products. One observes that if f ∈ N_{L}(T) and x ∈ T, with (f^{−1}, x, f) ∈ D via P ∈ Δ, then (f, x^{−1}, f^{−1}) ∈ D via P^{x^{f}}. Since an analogous statement holds when x is replaced by x^{−1}, it follows that N_{L}(T) is closed under inversion, and so (a) is proved.

For the remainder of the proof, we may assume that Δ_{T} ⊆ Δ. Set

D_{T} = D_{Δ} ∩ W(N_{L}(T))

(where D_{Δ} is defined in 2.6). With w and (P_{0}, ⋯, P_{n}) as in the proof of (a), we may then replace P_{i} with N_{P_{i}}(T), and this shows that D_{T} is contained in the subset D_{Δ_{T}} of W(N_{L}(T)). The reverse inclusion is obvious, so (N_{L}(T), N_{S}(T)) satisfies the condition (O1) for objectivity. Any overgroup in N_{S}(T) of an element of Δ_{T} is again in Δ_{T}, so the condition (O2) is satisfied, and (N_{L}(T), Δ_{T}) is an objective partial group. Thus, (b) holds.

Now assume further that T has been chosen so that |N_{S}(T)| ≥ |N_{S}(U)| for each L-conjugate U of T in S. In order to show that (N_{L}(T), N_{S}(T)) is a locality via Δ_{T}, it suffices to show that N_{S}(T) is maximal in the poset of p-subgroups of N_{L}(T). Set R = N_{S}(T), let R_{1} be a p-subgroup of N_{L}(T) containing R, and set R_{2} = N_{R_{1}}(R). As R ∈ Δ, there exists f ∈ L such that Q := R^{f} is fully normalized in F_{S}(L), by 2.17(a). Then N_{S}(Q) is a Sylow p-subgroup of N_{L}(Q), and so there exists g ∈ N_{L}(Q) such that (R_{2})^{fg} ≤ N_{S}(Q). But (R_{2})^{fg} ≤ N_{S}(T^{fg}), and the maximality condition on R then yields R = R_{2} and R = R_{1}. This completes the proof of (c). □

Definition 2.19. Let (L, Δ, S) be a locality, and let Γ ⊆ Δ be a non-empty subset such that Γ is both overgroup-closed in S and F_{S}(L)-invariant. Set D = D(L), set

D |_{Γ}:= {w ∈ D | S_{w} ∈ Γ},

and let L |_{Γ} be the set of words of length 1 in D |_{Γ}, regarded as a subset of L. The restriction of L to Γ consists of L |_{Γ} together with the restriction to D |_{Γ} of the product in L, and the restriction to L |_{Γ} of the inversion in L.

Lemma 2.20. Let (L, Δ, S) be a locality, and let Γ be a non-empty subset of Δ, such that Γ is both overgroup-closed in S and F_{S}(L) − invariant.

(a) D |_{Γ} is the set D_{Γ} of 2.6, and (L |_{Γ}, Γ, S) is a locality.

(b) If L is a group M, then L |_{Γ} is the locality L_{Γ}(M) given by 2.9.1.

Proof. Set M = L |_{Γ}. For any w ∈ W, the condition that S_{w} be in Γ is the defining condition for D |_{Γ}, and in view of 2.13(a) it is also the defining condition for D_{Γ}. These subsets of W are therefore identical, and (M, Γ) satisfies the condition (O1) for objectivity. Condition (O2) is given by the assumption that Γ is closed in F_{S}(L), so (M, Γ) is objective. All members of Γ are subgroups of S, and S is maximal in the poset of p-subgroups of M since the corresponding statement holds in L. As L is finite, so is M, so M is a locality, and (a) holds.

Suppose that L is in fact a group M, and set K = L_{Γ}(M). By definition, an element g of M is in K if and only if S ∩ S^{g} ∈ Γ. The latter condition means that S_{g} = S ∩ S^{g^{−1}}, so g ∈ K if and only if S_{g} ∈ Γ. Similarly, w ∈ D(K) if and only if S_{w} ∈ Γ. This shows that D(K) = D_{Γ}, and then (b) follows from (a). □

We shall refer to the locality (L |_{Γ}, Γ, S) as the restriction of L to Γ.

The following proposition gives two applications of completeness to localities.

Proposition 2.21. Let L be a locality on S and let Δ ∈ Ω(L, S). Then the following hold.

(a) Every subgroup of L is a Δ-local subgroup. That is: for any subgroup H of L, there exists U ∈ Δ such that H ≤ N_{L}(U).

(b) Every p-subgroup of L is conjugate to a subgroup of S.

Proof. (a) Let w = (h_{1}, ⋯, h_{n}) ∈ W(H) be chosen so that the sequence (g_{1}, ⋯, g_{n}), in which g_{i} = h_{1} ⋯ h_{i}, includes all of the elements of H. As H is a subgroup of L we have W(H) ⊆ D (all products in H are defined), and so w ∈ D. Thus, there exists P ∈ Δ such that P^{g_{i}} ∈ Δ for all i. Set U = ⟨P^{g_{i}} | 1 ≤ i ≤ n⟩; a subgroup of S. As H = {g_{i}}_{1≤i≤n}, U = ⟨P^{H}⟩, and so H ≤ N_{L}(U). Here U ∈ Δ as Δ is overgroup closed in S.

(b) Let Q be a p-subgroup of L. Then Q is finite, as L is. By (a) there exists U ∈ Δ with Q ≤ N_{L}(U). By 2.17(a) there is an L-conjugate V = U^{f} of U such that N_{S}(V) is a Sylow p-subgroup of N_{L}(V). By Sylow’s Theorem, there then exists g ∈ N_{L}(V) such that Q^{fg} ≤ N_{S}(V). □

Section 3: Homomorphisms and partial normal subgroups

We introduce homomorphisms of partial groups and their kernels, and we tentatively define homomorphisms of objective partial groups. The “correct definition” of a homomorphism of localities, and of linking systems, is elusive, and we shall make no attempt here to formulate such a definition, other than to introduce an obvious notion of isomorphism, and a perhaps less obvious notion of projection.

Whenever M and M′ are partial groups, we write W for W(M) and W′ for W(M′). Similarly for D and D′, for Π and Π′, and for 1 and 1′. We shall make no such careful distinction regarding the inversion maps for M and M′.

Definition 3.1. Let M and M′ be partial groups, let β : M → M′ be a mapping, and let β^{∗} : W → W′ be the induced mapping. Then β is a homomorphism (of partial groups) if: (H1) Dβ^{∗} ⊆ D′, and (H2) (Π(w))β = Π′(wβ^{∗}) for all w ∈ D. The kernel of β is the set Ker(β) of all g ∈ M such that gβ = 1′. We say that β is an isomorphism if there exists a homomorphism β′ : M′ → M such that β ∘ β′ and β′ ∘ β are identity mappings.

Lemma 3.2. Let β : M → M′ be a homomorphism of partial groups. Then 1β = 1′, and (f^{−1})β = (fβ)^{−1} for all f ∈ M.

Proof. Since 11 = 1, (H1) and (H2) yield 1β = (11)β = (1β)(1β), and then 1β = 1′ by left or right cancellation. Since (f, f^{−1}) ∈ D for any f ∈ M, by 2.3(b), (H1) yields (fβ, (f^{−1})β) ∈ D′, and then 1β = (ff^{−1})β = (fβ)((f^{−1})β) by (H2). As 1β = 1′ = (fβ)(fβ)^{−1}, left cancellation yields (f^{−1})β = (fβ)^{−1}. □

Lemma 3.3. Let β : M → M′ be a homomorphism of partial groups, and set N = Ker(β). Then N is a partial subgroup of M, and f^{−1}N f ⊆ N for all f ∈ M. That is, g^{f} ∈ N whenever g ∈ N ∩ D(f).

Proof. By 3.2 N is closed under inversion. If w is in W(N) ∩ D then the map β^{∗} : W → W′ induced by β sends w to a word of the form (1′, ⋯, 1′). Then Π′(wβ^{∗}) = 1′, and thus Π(w) ∈ N. This shows that N is a partial subgroup of M. Now let f ∈ M and let g ∈ N ∩ D(f). Then

(f^{−1}, g, f)β^{∗} = ((fβ)^{−1}, 1′, fβ) (by 3.2),

so that

(g^{f})β = Π′((f^{−1}, g, f)β^{∗}) = Π′(fβ)^{−1}, 1′, fβ) = 1′.

□

Definition 3.4. Let M be a partial group and let N be a partial subgroup of M. Then N is a partial normal subgroup of M if f^{−1}N f ⊆ N for all f ∈ M. (That is, x^{f} ∈ N whenever x ∈ N ∩ D(f).)

We may write N ⊴ M to indicate that N is a partial normal subgroup of M.

Definition 3.5. Let L = (L, Δ, S) and L′ = (L′, Δ, S) be localities having the same set of objects. An isomorphism β : L → L′ of partial groups is rigid (over S) if β restricts to the identity map S → S.

Lemma 3.6. Let (L, Δ, S) and (L′, Δ, S) be localities having the same set Δ of objects, and let β : L → L′ be a surjective homomorphism of partial groups. Suppose:

(1) S_{f} = S_{fβ} for all f ∈ L, and

(2) Ker(β) = 1. Then β is an isomorphism.

Proof. Let h ∈ L′ and let f, g ∈ L with fβ = gβ = h. Then S_{f} = S_{g} by (1), so (f^{−1}, g) ∈ D via (S_{f})^{f}, and (f^{−1}g)β = 1. Thus f = g by (2), and β is a bijection.

Let w′ = (h_{1}, ⋯, h_{n}) ∈ D(L′), set g_{i} = h_{i}β^{−1}, and set w = (g_{1}, ⋯, g_{n}). Then w ∈ D via S_{w′} by (1), and Π(w)β = Π′(w′) as β is a homomorphism. Thus Π′(w′)β^{−1} = Π(w′(β^{−1})^{∗})), and β^{−1} is a homomorphism. □

Lemma 3.7. Let L = (L, Δ, S) be a locality and let N be a partial normal subgroup of L. Set Γ = {P ∩ N | P ∈ Δ} and suppose that Γ ⊆ Δ. Then (N , Γ, S ∩ N) is a locality.

Proof. Let w = (f_{1}, ⋯, f_{n}) ∈ D via P ∈ Δ. Then also w ∈ D via Q := P ∩ N, and hence (N , Γ) is objective. Each member of Γ is a subgroup of T := S ∩ N, and N is finite, so it only remains to show that T is maximal in the poset of p-subgroups of N, in order to conclude that (N , Γ, T) is a locality.

Let R be a p-subgroup of N containing T. As T ≤ S_{x} for each x ∈ S, it follows from the definition 3.4 of partial normal subgroup that T ⊴ S. As T ∈ Γ, S is then a Sylow p-subgroup of the group N_{L}(T), and hence N_{R}(T)^{g} ≤ S for some g ∈ N_{L}(T). The definition of partial normal subgroup then yields N_{R}(T)^{g} ≤ N, so N_{R}(T)^{g} = T and N_{R}(T) = T. Thus R = T, as required. □

Lemma 3.8. Let (L, Δ, S) be a Δ-linking system, and let β be a rigid automorphism of L. Let (K, Γ, R) be a locality such that K is a partial subgroup of L, and such that Γ ⊆ Δ. Then β restricts to a rigid automorphism of (K, Γ, R).

Proof. Let f ∈ K and set P = R_{f}. That is, P is the largest subgroup of R which is conjugated by f into R (obtained by applying 2.10 to the locality K). Then P ∈ Γ by 2.10, and hence P ∈ Δ. As L is a Δ-linking system, C_{K}(P) = C_{L}(P) = Z(P). Now 2.16 implies that f is A-decomposable, where A is the union of {R} with the set of K-essential objects in Γ. Thus f = Π(w) where w = (g_{1}, ⋯, g_{n}) ∈ D(K), each g_{i} normalizes some Q_{i} ∈ A, Q_{i} = R_{g_{i}}, and P ≤ S_{w}.

Since β is rigid, each Q_{i} is β-invariant, and β then restricts to an automorphism γ_{i} on each N_{L}(Q_{i}) by 2.7(b). But also β centralizes Q_{i}, and C_{L}(Q_{i}) = Z(Q_{i}). Taking commutators in the subgroup Aut_{L}(Q_{i})⟨β⟩ of Aut(Q_{i}), we then obtain

[N_{L}(Q_{i}), β] ≤ Z(Q_{i}) ≤ N_{K}(Q_{i}).

We have thus shown that each of the groups N_{K}(Q_{i}) is β-invariant. Then

fβ = (Π(w))β = Π(wβ^{∗}) = Π(g_{1}β, ⋯, g_{n}β) ∈ K,

and so K is β-invariant. The same holds for β^{−1}, so β is restricts to an automorphism β_{K} of K. As R ≤ S, β_{K} is rigid. □

Lemma 3.9. Let M be a finite group, and let K ≤ M be a subgroup. Let S be a Sylow p-subgroup of M, set F = F_{S}(M), and let Γ be a non-empty F-invariant set of subgroups of S, such that Γ is overgroup closed in S. Let L := L_{Γ}(M) be the locality given by 2.9.1, and set K = K ∩ L. Then K is a partial subgroup of L, and is a partial normal subgroup if K ⊴ M.

Proof. One observes first of all that K is closed under the inversion in M, which is the inversion in L. Let w = (x_{1}, ⋯, x_{n}) ∈ D(L) ∩ W(K). Then Π(w) ∈ L ∩ K, and so K is a partial subgroup of L.

Now assume that K ⊴ M, let f ∈ L and let x ∈ K ∩ D(f). Then x^{f} ∈ L and x^{f} ∈ K, so x^{f} ∈ K. Thus K is a partial normal subgroup of M. □

Recall from 2.15 the notion of L-essential subgroup.

Lemma 3.10. Let (L, Δ, S) be a Δ-linking system and let β be an endomorphism of the partial group L such that β restricts to the identity automorphism on O^{p′}(N_{L}(R)) for each L-essential subgroup R ≤ S, and restricts to the identity automorphism also on N_{L}(S). Then β is the identity automorphism of L.

Proof. Let f ∈ L and set Q = S_{f}. Let A be the union of {S} with the set of all Lessential subgroups of S. Then f is A-decomposable by 2.16. In particular, f = Π(w) for some w = (g_{1}, ⋯, g_{n}) ∈ D having the property that g_{i} ∈ N_{L}(R_{i}) for some R_{i} ∈ A. It is then immediate from definition 3.1 and from the hypothesis concerning β, that fβ = f. □

Section 4: The Frattini Lemma and the Splitting Lemma

This section develops two of the main computational tools that will enable the later arguments. We obtain an analog of the Frattini Lemma in 4.6, which shows if N is a partial normal subgroup of a locality L, then each element of L may be written as a product of an element f ∈ N and an element g ∈ N_{L}(T), where T = S ∩ N. The “splitting lemma” (4.10) refines the choice of g. We end with an important application (4.11) which provides a criterion for extending an automorphism of a linking system in a finite group to an automorphism of the group itself.

The notation S_{f} and S_{w}, defined in 2.10 and 2.13, will be employed without further comment.

The following hypothesis (and notation) will be assumed throughout this section.

Hypothesis 4.1. There is given a locality L = (L, Δ, S) and a partial normal subgroup N of L. Set T = S ∩ N.

Lemma 4.2. The following hold.

(a) T is strongly closed in F_{S}(L), and T is maximal in the poset of all p-subgroups of N.

(b) If P ∈ Δ and x ∈ N with P ≤ S_{x}, then P T = P^{x}T.

(c) If T = 1, then N_{N} (P, S) = C_{N} (P) for all P ∈ Δ.

Proof. Let x ∈ T and let ϕ ∈ F := F_{S}(L) such that x lies in the domain of ϕ. As ϕ is a composition of restrictions of conjugation maps between objects, it suffices, in proving (a), to consider only the case where xϕ = x^{f} for some f ∈ L; and in that case we have xϕ ∈ N. Thus xϕ ∈ S ∩ N = T, and so T is strongly closed in F. Now let R be a p-subgroup of N containing T. By 2.21(b) we may choose R with R ≤ S, and then R = T. Thus (a) holds.

Next, let g ∈ P ∈ Δ and let x ∈ N_{N} (P, S). Then the word w = (x^{−1}, g, x, g^{−1}) is in D via P^{x}, and then Π(w) = x^{−1}x^{g^{−1}} = g^{x}g^{−1}. Thus Π(w) ∈ N ∩ S = T, and g^{x} ∈ gT. In particular, this proves (c), and it shows that P^{x} ≤ P T. Upon replacing (P, x) with (P^{x}, x^{−1}), the same argument shows that P ≤ P^{x}T, and this yields (b). □

Definition 4.3. Let L ∘ Δ be the set of all pairs (f, P) ∈ L × Δ such that P ≤ S_{f}. Define a relation ↑ on L ∘ Δ by (f, P) ↑ (g, Q) if there exist elements x ∈ N_{N} (P, Q) and y ∈ N_{N} (P^{f}, Q^{g}) such that xg = fy.

This relation may be indicated by means of a commutative diagram:

(*)  \begin{matrix}Q & ⟶^{g} & Q^{g} \\ ^{x}↑ & & ↑^{y} \\ P & ⟶^{f} & P^{f} \end{matrix}

of conjugation maps, labeled by the conjugating elements, and in which the horizontal arrows are isomorphisms and the vertical arrows are injective homomorphisms. The relation (f, P) ↑ (g, Q) may also be expressed by:

w := (x, g, y^{−1}, f^{−1}) ∈ D via P, and Π(w) = 1.

It is easy to see that ↑ is a reflexive and transitive relation on L∘Δ. We say that (f, P) is maximal in L ∘ Δ if (f, P) ↑ (g, Q) implies that |P | = |Q|. As S is finite there exist maximal elements in L ∘ Δ. Since (f, P) ↑ (f, S_{f}) for (f, P) ∈ L ∘ Δ, we have P = S_{f} for every maximal element (f, P). For this reason, we introduce the following terminology.

Definition 4.4. Let f ∈ L. Then f is ↑-maximal in L if (f, S_{f}) is maximal in L ∘ Δ.

The first main result of this section is as follows.

Proposition 4.5. Let f ∈ L and suppose that f is ↑-maximal. Then T ≤ S_{f}.

The proof requires two preliminary lemmas.

Lemma 4.5.1. Let (g, Q), (h, R) ∈ L ∘ Δ with (g, Q) ↑ (h, R), and suppose that T ≤ R. Then there exists a unique y ∈ N with g = yh. Moreover:

(a) y ∈ N_{N} (Q, R), and Q ≤ S_{(y,h)}.

(b) If N_{T} (Q^{g}) ∈ Syl_{p}(N_{N} (Q^{g})), then N_{T} (Q^{y}) ∈ Syl_{p}(N_{N} (Q^{y})).

Proof. By the definition of ↑, there exist elements u ∈ N_{N} (Q, R) and v ∈ N_{N} (Q^{g}, R^{h}) such that (u, h, v^{−1}, g^{−1}) ∈ D via Q, and such that Π(w) = 1.

R ⟶^{h} R^{h}
u↑  ↑^{v}
Q ⟶_{g} Q^{g}

In particular, uh = gv. Since T ≤ R, points (a) and (b) of 4.2 yield

T = T^{h}, Q^{u}T = QT ≤ R, and Q^{g}T = Q^{gv}T ≤ R^{h}.

Then

w := (u, h, v^{−1}, h^{−1}) ∈ D via (Q, Q^{u}, Q^{uh}, Q^{uhv^{−1}} = Q^{g}, Q^{gh^{−1}}).

Set y = Π(w). Then y = u(v^{−1})^{h^{−1}} ∈ N_{N} (Q, R). Since (u, h, v^{−1}, h^{−1}, h) and (g, v, v^{−1}) are in D (as L is a partial group), we get yh = uhv^{−1} = g. This yields (a), and the uniqueness of y is given by right cancellation.

Suppose now that N_{T} (Q^{g}) ∈ Syl_{p}(N_{N} (Q^{g})). As N_{T} (Q^{y})^{h} = N_{T} (Q^{g}), it follows from 2.7(b) that N_{T} (Q^{y}) ∈ Syl_{p}(N_{N} (Q^{y})). □

Lemma 4.5.2. Suppose that f is ↑-maximal, and let y ∈ N_{N} (S_{f}, S). Then |T ∩ S_{f}| = |T ∩ (S_{f})^{y}|, and (f, S_{f}) ↑ (y^{−1}f, (S_{f})^{y}). In particular, y^{−1}f is ↑-maximal.

Proof. Set P = S_{f}. Then P^{y}T = P T, by 4.2(b). Then

|P^{y} : P^{y} ∩ T | = |P^{y}T : T | = |P T : T | = |P : P ∩ T |,

and so |T ∩ P | = |T ∩ P^{y}|. The following diagram

P^{y} ⟶^{y^{−1}f} P^{f}
y↑  ↑^{1}
P ⟶_{f} P^{f}

shows that (f, P) ↑ (y^{−1}f, P^{y}). □

Proof of Proposition 4.5. Let f be ↑-maximal. Set P = S_{f} and Q = P^{f}, and suppose first that N_{T} (P) ∈ Syl_{p}(N_{N} (P)). Then N_{T} (P)^{f} ∈ Syl_{p}(N_{N} (Q)), by 2.7(b), and there exists x ∈ N_{N} (Q) such that N_{T} (Q) ≤ (N_{T} (P)^{f})^{x}. Here (f, x) ∈ D via P, so (N_{T} (P)^{f})^{x} = N_{T} (P)^{fx}, and then (f, P) ↑ (fx, N_{T} (P)P). As f is ↑-maximal, we conclude that N_{T} (P) ≤ P, and hence T ≤ P. Thus T ≤ S_{f} if N_{T} (P) ∈ Syl_{p}(N_{N} (P)). Assuming that f provides a counterexample to 4.5, we conclude:

(1) N_{T} (P) ∉ Syl_{p}(N_{N} (P)).

Among all counterexamples to 4.5, choose f so that first |P ∩ T | and then |P | are as large as possible. Choose g ∈ N_{L}(Q, S) so that Q^{g} is fully normalized in F_{S}(L), and set h = fg and R = P^{h}. As R = Q^{g} is fully normalized we have N_{S}(R) ∈ Syl_{p}(N_{L}(R)), and then N_{T} (R) ∈ Syl_{p}(N_{N} (R)). Let (h^{−1}, R) ↑ (h′, S_{h′}), where h′ is ↑-maximal, and set R′ = S_{h′} and P′ = (R′)^{h′}. Thus, there exist y, z ∈ N such that yh′ = h^{−1}z, R^{y} ≤ R′, and P^{z} ≤ P′, as indicated in the following diagram.

R′ ⟶^{h′} P′
y↑  ↑^{z}
R ⟶_{h^{−1}} P

Then (T ∩ R)^{y} ≤ T ∩ R′.

Suppose that T ≰ R′. The conditions on the choice of f then yield (T ∩ R)^{y} = T ∩ R′ and R^{y} = R′. Since N_{T} (R) ∈ Syl_{p}(N_{N} (R)), we get N_{T} (R)^{y} ∈ Syl_{p}(N_{N} (R′)), and so there exists x ∈ N_{N} (R′) such that (N_{T} (R)^{y})^{x} = N_{T} (R′). Replacing y and h′ with yx and x^{−1}h′, we then obtain N_{T} (R)^{y} = N_{T} (R′). But then T ≤ R′ by (1), in any case, and then also T ≤ P′.

Evidently (h, P) ↑ ((h′)^{−1}, P′), so by 4.5.1 there exists ỹ ∈ N_{N} (P, S) such that h = ỹ(h′)^{−1}, P ≤ S_{(ỹ,(h′)^{−1})}, and N_{T} (P ỹ) ∈ Syl_{p}(N_{N} (P ỹ)). Then 4.5.2 applies to (f, P) and ỹ, and yields the result that y^{−1}f is ↑-maximal and S_{y^{−1}f} = P ỹ. Then (1) implies that T ≤ P ỹ, and then also T ≤ P. □

Recall from 2.18(c) that the partial group N_{L}(T) is a locality via the set Δ_{T} of objects Q ∈ Δ with Q ≤ T.

Corollary 4.6 (Frattini Lemma). Let L = (L, Δ, S) be a locality, let N be a partial normal subgroup of L, and set T = S ∩ N. Then L = N N_{L}(T) as a product of partial subgroups of L.

Proof. Let f ∈ L, set P = S_{f}, and choose (g, Q) ∈ L ∘ Δ so that (f, P) ↑ (g, Q) and so that g is ↑-maximal. By transitivity of ↑, we may take Q = S_{g}. Then T ≤ Q by 4.5, and then by 4.5.1 there exists y ∈ N_{N} (P, Q) with f = yg. Here g ∈ N_{L}(T) by 4.2(a). □

The next few results concern the decomposition of the partial group L into “cosets” of N.

Lemma 4.7. Let (L, Δ, S) be a locality, let N ⊴ L, and set T = S ∩ L. Let f ∈ N_{L}(T) and let x, y ∈ N with (x, f) and (f, y) ∈ D. Then the following hold.

(a) (f, f^{−1}, x, f) ∈ D, xf = fx^{f}, and S_{(x,f)} = S_{(f,x^{f})} = S_{x} ∩ S_{f}.

(b) (f, y, f^{−1}, f) ∈ D, fy = y^{f^{−1}}y, and S_{(f,y)} = S_{(y^{f^{−1}} ,y)} = S_{y^{f^{−1}}} ∩ S_{f}.

Proof. For point (a): Set Q = S_{(x,f)} and note that T ≤ S_{f} by hypothesis. We have Q^{x}T = QT by 2.5(b), so Q ≤ S_{f}. Thus Q ≤ P := S_{x} ∩ S_{f}. But also P^{x}T = P X, so P = Q. Moreover, we now have (f, f^{−1}, x, f) ∈ D via Q, and then Π(f, f^{−1}, x, f) = xf = fx^{f}. Thus, (a) holds.

For point (b): Set R = S_{(f,y)}. Then R^{fy}T = R^{f}T ≤ S_{f^{−1}}, so (f, y, f^{−1}, f) ∈ D via R, and fy = y^{f^{−1}}f. The remainder of (b) now follows as an application of (a) to (y^{f^{−1}}, f). □

We have the following immediate corollary.

Corollary 4.8. Let f ∈ N_{L}(T). Then the following hold.

(a) N f = fN.

(b) If x, y ∈ N, with (x, f), (f, y) ∈ D, and with xf = fy, then S_{(x,f)} = S_{(f,y)}.

□

Recall that if {X_{i}}_{i=1}^{n} is a collection of subsets of a partial group M, and {f_{i}}_{i=1}^{n} is a set of elements of M, then the product X_{1}f_{1} ⋯ X_{n}f_{n} is defined to be the set of all products Π(x_{1}, f_{1}, ⋯, x_{n}, f_{n}) with x_{i} ∈ X_{i} and with (x_{1}, f_{1}, ⋯, x_{n}, f_{n}) ∈ D.

Corollary 4.9. Let w = (f_{1}, ⋯, f_{n}) ∈ D, and assume that each f_{i} is ↑-maximal. Then

(*)  N f_{1} ⋯ N f_{n} = N Π(w).

Explicitly, if

u := (x_{1}, f_{1}, ⋯, x_{n}, f_{n}) ∈ D with x_{i} ∈ N,

then

(**)  Π(u) = Π(v), where v = (y_{1}, ⋯, y_{n}, f_{1}, ⋯, f_{n}) ∈ D,

and where y_{1} = x_{1}, and y_{i} = (x_{i})^{(f_{1}···f_{i−1})^{−1}} ∈ N for all i with 1 < i ≤ n.

Proof. Set A = N f_{1} ⋯ N f_{n} and B = N Π(w). Then A ⊇ B since 1 ∈ N. Now let x_{i} ∈ N with u := (x_{1}, f_{1}, ⋯, x_{n}, f_{n}) ∈ D. By 4.8(b) there exists y := (x_{n})^{f_{n}−1} ∈ N, with f_{n−1}x_{n} = yf_{n−1}. Then also S_{(f_{n−1},x_{n})} = S_{(y,f_{n−1})} by 4.8(b). Then 4.7(b) yields:

v := (x_{1}, f_{1}, ⋯, x_{n−1}, y, f_{n−1}, f_{n}) ∈ D and Π(u) = Π(v).

Iteration of this process yields (**), and hence A ⊆ B. □

The following result concerns a key relationship between N and the set of ↑-maximal elements of L.

Lemma 4.10 (Splitting Lemma). Let f ∈ N_{L}(T) be ↑-maximal and let x ∈ N with (x, f) ∈ D. Then S_{xf} = S_{(x,f)}. Similarly, if y ∈ N with (f, y) ∈ D then S_{fy} = S_{(f,y)}.

Proof. If g = xf with x ∈ N then 4.7(a) yields g = fy where y = x^{f}. On the other hand, if we begin with y ∈ N and g = fy then 4.7(b) yields g = xf where x = y^{f^{−1}}. Further, 4.7 yields S_{(x,f)} = S_{(f,y)}, so it will suffice to show that S_{xf} = S_{(x,f)} in the case that x ∈ N and (x, f) ∈ D.

Set Q = S_{(x,f)} and set y = x^{f}. Then Q ≤ S_{g} as g = xf. Thus, it suffices to show that S_{g} ≤ Q. Among all counterexamples (x, f) (with S_{g} ≰ Q), let (x, f) be chosen so that |Q| is as large as possible. Set S_{0} = N_{S_{f}} (Q), S_{1} = N_{S_{g}}(Q), and P = ⟨S_{0}, S_{1}⟩. Since Q ≤ S_{g} and Q ≠ S_{g}, we get S_{1} ≰ Q.

Set R = S_{0} ∩ S_{1}. Then R^{f} ≤ N_{S}(Q^{f}) and R^{g} ≤ N_{S}(Q^{g}). Here Q ≤ S_{(f,y)} as (Q^{f})^{y} = Q^{g}, so also (R^{f})^{y} = R^{g}, and R ≤ S_{(f,y)}. Then R ≤ Q, and thus R = Q. Since S_{1} ≰ Q, we obtain:

(1) S_{1} ≰ S_{f}.

Case 1 The case x ∈ N_{N} (T).

Here T ≤ Q as f ∈ N_{L}(T), and then x ∈ N_{N} (Q) by 4.2(b). Then Q^{g} = Q^{xf} = Q^{f}. Since x^{−1}g = Π(x^{−1}, x, f) = f, we get c_{f} = c_{x}^{−1} ∘ c_{g} as an isomorphism N_{L}(Q) → N_{L}(Q^{f}), and hence:

S_{1}^{f} = (S_{1}^{x^{−1}})^{g} ≤ (S_{1}N_{N} (Q))^{g} ≤ N_{S}(Q^{g})N_{N} (Q^{g}) = N_{S}(Q^{f})N_{N} (Q^{f}).

This yields

P^{f} ≤ N_{S}(Q^{f})N_{N} (Q^{f}).

As T ≤ Q we have T ≤ Q^{f}, and hence N_{N} (Q^{f})/T is a p′-group. Then N_{S}(Q^{f}) ∈ Syl_{p}(N_{S}(Q^{f})N_{N} (Q^{f}), and there exists z ∈ N_{N} (Q^{f}) such that P^{fz} ≤ N_{S}(Q^{f}). Thus P ≤ S_{fz}. Here z ∈ N_{N} (T), so S_{0}^{fz} = S_{0}^{f} by 4.2(b). Thus:

(2) S_{0} ≤ S_{(f,z)}.

By 4.8 there exists z′ ∈ N_{N} (T) such that

(*)  fz = z′f and S_{(f,z)} = S_{(z′,f)}.

If S_{(z′,f)} = S_{z′f} then P ≤ S_{(z′,f)} = S_{(f,z)} ≤ S_{f}, and we contradict (1). Thus S_{(z′,f)} ≠ S_{z′f}, and (z′, f) is a counterexample to the lemma. The maximality of Q then yields Q = S_{(z′,f)}, and then S_{0} ≤ Q by (2). Then Q = S_{f} and S_{f} ≤ S_{fz}. As f is ↑-maximal we conclude that S_{f} = S_{fz}. As S_{1} ≤ P ≤ S_{fz} we again contradict (1). Case 2 The general case.

Let h be ↑-maximal with (g, S_{g}) ↑ (h, S_{h}). Then g = rh for some r ∈ N, and S_{g} ≤ S_{(r,h)}, by 4.5.1(a). Then Q ≤ S_{(r,h)}, and we obtain (f^{−1}, x^{−1}, r, h) ∈ D via Q^{g}. But now Π(f^{−1}, x^{−1}, r, h) = Π(g^{−1}, g) = 1, so

f = x^{−1}rh and h = r^{−1}xf.

Since f, h ∈ N_{L}(T), it follows that x^{−1}r and r^{−1}x are in N_{N} (T). Now Case 1, as applied to (r^{−1}x, f), yields S_{h} ≤ S_{f}. Since S_{1} ≤ S_{g} ≤ S_{h}, we again contradict (1). □

Lemma 4.11. Let M be a finite group, let S be a Sylow p-subgroup of M, and let K be a normal subgroup of M. Set F = F_{S}(M) and let Γ be a non-empty, overgroup closed, F-invariant collection of subgroups of S. Let L := L_{Γ}(M) be the locality given by 2.9.1, and let β be a rigid automorphism of L. Assume that the following three conditions hold.

(1) Q ∩ K ∈ Γ for all Q ∈ Γ.

(2) C_{M}(O_{p}(M)) ≤ O_{p}(M) ≤ K.

(3) Γ is a set of F-centric subgroups of S. Set Φ = {Q ∩ K | Q ∈ Γ}, and set K = L_{Φ}(K). Then:

(a) β restricts to a rigid automorphism κ of K, and

(b) β extends to an automorphism of M if and only if κ extends to an automorphism of K.

Proof. Set Y = O_{p}(M) and let P ∈ Γ. Since C_{M}(Y) ≤ Y by (2), the Thompson A × B Lemma [5.3.4 in Gor] implies that C_{M}(P) is a p-group. Then C_{M}(P) = C_{L}(P) = Z(P), as P is F-centric by (3). Point (a) then follows from 3.8. Further, 2.16 yields:

(4) Every f ∈ L is a product Π(f_{1}, ⋯, f_{n}), where f_{i} is in a normalizer N_{L}(R_{i}) for some R_{i} ∈ Γ.

Let (M, β) be a counterexample to (b) with |M| as small as possible. Let K_{0} be the subgroup of K generated by the subset K of K. Let g ∈ K ∩ L. Then S_{g} ∈ Γ and S_{g} ∩ K ∈ Φ, so g ∈ K. Thus K ∩ L ⊆ K. The reverse inclusion is given by the definition of K, so K = K ∩ L. Then K is a partial normal subgroup of L by 3.10. Set T = S ∩ K and observe that for any h ∈ N_{M}(T) we have (h^{−1}, g, h) ∈ D via (S_{g} ∩ T)^{h}, and hence g^{h} ∈ K. Thus, K is N_{M}(T)-invariant, so also K_{0} is N_{M}(T)-invariant.

Set M_{0} = N_{M}(T)K_{0} and set L_{0} = L_{Γ}(M_{0}). We next show:

(5) N_{M}(P) ≤ M_{0} for all P ∈ Γ.

Among all P for which (5) fails to hold, choose P so that |P | is as large as possible. Suppose that P is not fully normalized in F, and let P′ be a fully normalized F-conjugate of P. Then Alperin’s theorem yields a sequence w = (g_{1}, ⋯, g_{n}) of elements of M and a sequence (R_{1}, ⋯, R_{n}) of fully normalized F-centric subgroups of S, such that P_{0} := P ≤ R_{1}, P_{i} := P^{g_{1}···g_{i}} ≤ R_{i} for all i, and P′ = P^{Π(w)}. One may assume that n is minimal for these conditions, and hence P_{i} ≠ R_{i} for any i. The maximality of |P | in the choice of P then yields N_{M}(R_{i}) ≤ M_{0} for all i, and hence Π(w) ∈ M_{0}. Without loss, then, we may assume that P = P′.

With P fully normalized in F we obtain N_{T} (P) ∈ Syl_{p}(N_{K}(P)). As N_{K}(P) ⊴ N_{M}(P) the Frattini Lemma yields

(*)  N_{M}(P) = N_{K}(P)(N_{M}(N_{T} (P)) ∩ N_{M}(P)).

If T ≤ P then (*) yields N_{M}(P) ≤ K_{0}N_{M}(T) = M_{0}, contrary to the choice of P. Thus T ≰ P, and hence N_{T} (P) ≰ P. Set Q = N_{T} (P)P. Then N_{M}(Q) ≤ M_{0} by the maximality of |P |, and then (*) again implies that N_{M}(P) ≤ M_{0}. This completes the proof of (5). Now (4) yields L ⊆ M_{0}. Thus:

(6) L_{0} = L.

Suppose next that K_{0} is a proper subgroup of K. Then K ∩ M_{0} = N_{K}(T)K_{0} = K_{0}, and so M_{0} is a proper subgroup of M. Since L_{0} = L, the minimality of |M| then yields an extension of β to an automorphism γ of M_{0}. The condition (2), together with 1.10(c) then implies that γ = c_{z} is conjugation by some z ∈ Z(S). Since c_{z} is also an automorphism of M, we have an extension of β to an automorphism of M in this case, so we conclude that K_{0} = K.

Let h, \overline{h} ∈ K, let x, \overline{x} ∈ N_{M}(T), and suppose that (h, x) and (\overline{h}, \overline{x}) are in D(L)

with hx = \overline{h}\overline{x}. Set P = S_{h} ∩ K and \overline{P} = S\overline{h} ∩ K, and set Q = P^{h} and \overline{Q} = \overline{P} \overline{h}. Then (h, x, \overline{x}^{−1}) ∈ D via P, and Π(h, x, \overline{x}^{−1}) = \overline{h}. It follows that P = \overline{P} and that (h^{−1}, \overline{h}) ∈ D via Q. Then

(**)  Π(h^{−1}β, \overline{h}β) = (h^{−1}\overline{h})β = (x\overline{x}^{−1})κ.

By hypothesis, there exists an extension η of κ to an automorphism of K. It follows from (**) that there is a well-defined mapping γ : M → M given by γ : hx ↦ (hη)(xβ) for h ∈ K and x ∈ N_{M}(T).

In order to show that γ is a homomorphism, it suffices to show that (hη)^{xβ} = (h^{x})η for all h ∈ K and x ∈ N_{M}(T). As K_{0} = K we may write h as a product Π_{K}(h_{1}, ⋯, h_{n}) with h_{i} ∈ K. Then

(h^{x})η = (h_{1}^{x} ⋯ h_{n}^{x})η = (h_{1}^{x})η ⋯ (h_{n}^{x})η
= (h_{1}η)^{xβ} ⋯ (h_{n}η)^{xβ} = (h_{1}η ⋯ h_{n}η)^{xβ} = (hη)^{xβ},

as required.

We check that Ker(γ) = 1. Namely, if (hη)(xβ) = 1 with h and x as above, then x ∈ N_{K}(T) and xβ = xη, and then hx = 1 as η is injective. Thus γ is injective, and is therefore an automorphism of M. □

Section 5: Filtrations

Recall that for any partial group M and subgroups X and Y of M, N_{M}(X, Y) is the set of all f ∈ M such that X ⊆ D(f) and X^{f} ⊆ Y. Write Hom_{M}(X, Y) for the set of all conjugation maps c_{f} : X → Y with f ∈ N_{M}(X, Y).

Definition 5.1. Let S be a finite p-group, let F be a fusion system on S, and let Δ be a non-empty, F-invariant collection of subgroups of S, closed with respect to overgroups in S. Let L be a partial group such that Δ is a set of subgroups of L, and such that D(L) = D_{Δ} in the sense of (O1) in 2.6. Then L is F-natural if Hom_{L}(P, Q) = Hom_{F}(P, Q) for all P, Q ∈ Δ.

Definition 5.2. Let M be a finite group, let S be a Sylow p-subgroup of M, and set F = F_{S}(M). Let Γ be an F-invariant, overgroup-closed collection of subgroups of S. Let L = L_{Γ}(M) be the locality given by 2.9.1. (Equivalently, by 2.20(b), L is the locality obtained by restricting the group M, itself viewed as a locality on the set of all subgroups of S, to the set Γ). Let Aut_{0}(L) be the set of rigid automorphisms of L, and let γ, γ′ ∈ Aut_{0}(L). Then γ and γ′ are M-equivalent if γ^{−1} ∘ γ′ extends to an automorphism of the group M.

Notice that M-equivalence is in fact an equivalence relation on Aut_{0}(L).

Hypothesis 5.3. Assume given:

(1) a fusion system F on the finite p-group S,

(2) an F-natural locality (L, Δ, S),

(3) a subgroup T of S, fully normalized in F, and having the property that ⟨U, V ⟩ ∈ Δ for every pair of distinct F-conjugates U, V of T,

(4) a finite group M such that T ⊴ M, N_{S}(T) ∈ Syl_{p}(M), and N_{F}(T) = F_{N_{S}(T)}(M); and

(5) a rigid isomorphism λ : N_{L}(T) → L_{Δ_{T}} (M), where Δ_{T} is the set of all P ∈ Δ such that T ⊴ P (and where L_{Δ_{T}} (M) is the locality given by 2.9.1).

Hypothesis 5.3 will be assumed throughout the remainder of this section. The symbols W, D, and Π will always refer to L, while Π_{M} denotes the multivariable product in the group M.

Lemma 5.4. Let U be an F-conjugate of T. Then the following hold.

(a) N_{P} (U) ∈ Δ for every object P ∈ Δ such that U ≤ P.

(b) There exists x ∈ L such that T^{x} = U, and such that N_{S_{x}}(T)^{x} = N_{S}(U).

Proof. (a) Let P ∈ Δ with U ≤ P. If U ⊴ P then there is nothing to prove, while if U is not normal in P then N_{P} (U) contains an L-conjugate of U^{g} ≠ U of U, where g ∈ N_{P} (N_{P} (U)). As Hom_{S}(U, S) ⊆ Hom_{F}(U, S), U^{g} is an F-conjugate of T, and then 5.3(3) yields ⟨U, U^{g}⟩ ∈ Δ. As Δ is overgroup closed in S, we obtain (a).

As T is fully normalized in F, there exists ψ ∈ Hom_{F}(N_{S}(U), S) such that Uψ = T. Here N_{S}(U) ∈ Δ by (a). As L is F-natural, ψ is given by conjugation by an element x′ ∈ L. Setting x = (x′)^{−1}, (b) follows. □

Let Θ be the set of all triples

θ = (x^{−1}, g, y) ∈ L × M × L

which satisfy the following set (1) of conditions.

(1) T ≤ S_{x} ∩ S_{y}, N_{S_{x}}(T)^{x} = N_{S}(T^{x}), and N_{S_{y}}(T)^{y} = N_{S}(T^{y}).

Define a relation ∼_{0} on Θ as follows.

(2) (x^{−1}, g, y) ∼_{0} (\overline{x}^{−1}, \overline{g}, \overline{y}) if

(i) T^{x} = T \overline{x}, T^{y} = T \overline{y}, and

(ii) (\overline{x}x^{−1})λ · g = \overline{g} · (\overline{y}y^{−1})λ (as elements of M).

Notice that (2)(ii) makes sense. Namely, taking U := T^{x} = T \overline{x} (by (i)), we get (\overline{x}, x^{−1}) ∈ D via (N_{S\overline{x}}(T), N_{S}(U), N_{S_{x}}(T)) by (1) and 5.4(a), and hence \overline{x}x^{−1} ∈ N_{L}(T). Similarly, \overline{y}y^{−1} ∈ N_{L}(T).

One may depict the relation ∼_{0} by means of a diagram, as follows:

U ⟶^{x^{−1}} T ⟶^{g} T ⟶^{y} V
∥ ^{(\overline{x}x^{−1})λ}↑  ↑^{(\overline{y}y^{−1})λ} ∥
U ⟶_{\overline{x}^{−1}} T ⟶_{\overline{g}} T ⟶_{\overline{y}} V

where V = T^{y} = T \overline{y}. As L is F-natural, the conjugation maps c_{x^{−1}} : S_{x^{−1}} → S and c_{y^{−1}} : S_{y^{−1}} → S are in F, and thus U and V are F-conjugates of T.

Lemma 5.5. ∼_{0} is an equivalence relation on Θ.

Proof. It is evident that ∼_{0} is reflexive and symmetric. Let θ_{i} = (x_{i}^{−1}, g_{i}, y_{i}) ∈ Θ (1 ≤ i ≤ 3) with θ_{1} ∼_{0} θ_{2} ∼_{0} θ_{3}. Then T^{x_{1}} = T^{x_{3}} and T^{y_{1}} = T^{y_{3}}. Notice that

(x_{3}, x_{2}^{−1}, x_{2}, x_{1}^{−1}) ∈ D via N_{S}(U)^{x_{3}^{−1}} and,
(y_{3}, y_{2}^{−1}, y_{2}, y_{1}^{−1}) ∈ D via N_{S}(V)^{y_{3}^{−1}}.

Then

(x_{3}x_{1}^{−1})λ · g_{1} = (x_{3}x_{2}^{−1})λ · (x_{2}x_{1}^{−1})λ · g_{1}
= (x_{3}x_{2}^{−1})λ · g_{2} · (y_{2}y_{1}^{−1})λ
= g_{3} · (y_{3}y_{2}^{−1})λ · (y_{2}y_{1}^{−1})λ
= g_{3} · (y_{3}y_{1}^{−1})λ,

which completes the proof. □

Define a relation ⊢ from L to Θ, by taking f ⊢ (x^{−1}, g, y) if g ∈ Im(λ), (x^{−1}, gλ^{−1}, y) ∈ D, and f = Π(x^{−1}, gλ^{−1}, y). Let ∼_{1} be the symmetrization of ⊢, and let ≈ be the weakest equivalence relation on L ∪ Θ containing the union of ∼_{0} and ∼_{1}. The ≈-class C of an element θ = (x^{−1}, g, y) of Θ may be denoted [x^{−1}, g, y].

The following lemma is immediate from the definition of ∼_{0}.

Lemma 5.6. Let Σ be a ∼_{0}-equivalence class in Θ, let θ = (x^{−1}, g, y) ∈ Σ, and set U = T^{x} and V = T^{y}. Then the pair (U, V) depends only on Σ, and not on the choice of representative θ. □

Lemma 5.7. Let f ∈ L, and suppose that S_{f} contains an F-conjugate U of T. Set V = U^{f}, and let Ξ = Ξ(f, U, V) be the set of all θ ∈ Θ such that U = T^{x}, V = T^{y}, and f ∼_{1} θ. Then the following hold.

(a) Ξ is a ∼_{0}-class of Θ.

(b) If \overline{f} ∈ L, and \overline{f} ∼_{1} θ for some θ ∈ Ξ, then f = \overline{f}.

Proof. As T is fully normalized in F, and since L is F-natural, there exist elements x, y ∈ L such that both N_{S}(U)^{x^{−1}} and N_{S}(V)^{y^{−1}} are contained in N_{S}(T), and such that T^{x} = U and T^{y} = V. Then (x, f, y^{−1}) ∈ D via N_{S_{f}} (U)^{x^{−1}}, and the product h = xfy^{−1} is an element of N_{L}(T). Set g = hλ. Then (x^{−1}, g, y) ∈ Γ.

Set E = N_{S_{f}} (U), and set A = E^{x^{−1}}, B = A^{g}, and F = E^{f}. Then B = F^{y^{−1}}, and each of E, A, B, F is in Δ by 5.4(a). Let Σ be the ∼_{0}-class containing θ, and let \overline{θ} = (\overline{x}^{−1}, \overline{g}, \overline{y}) ∈ Δ. Then U = T \overline{x} and V = T \overline{Y}, by 5.6. By the definition of Θ, we have E ≤ S\overline{x}^{−1} and the group \overline{A} := E^{\overline{x}^{−1}} is contained in N_{S}(T). Similarly, F ≤ S\overline{y}^{−1} and \overline{B} := F^{\overline{y}^{−1}} ≤ N_{S}(T). These facts, together with the rigidity of λ, result in a sequence of conjugation maps between objects in Δ, in which the conjugating elements are as indicated in the following diagram.

(*)  E ⟶^{x^{−1}} A ⟶^{x\overline{x}^{−1}} \overline{A} ⟶^{(\overline{x}x^{−1})λ} A ⟶^{g} B ⟶^{(y\overline{y}^{−1})λ} \overline{B} ⟶^{\overline{y}y^{−1}} B ⟶^{y} F.

As θ ∼_{0} \overline{θ}, we have

(\overline{x}x^{−1})λ · g · (y\overline{y}^{−1})λ = \overline{g}

35 and thus (*) yields

E ⟶^{\overline{x}^{−1}} \overline{A} ⟶^{\overline{g}} \overline{B} ⟶^{\overline{y}} F.

As \overline{A} and \overline{B} are in Δ, it follows that \overline{g} = \overline{h}λ for some \overline{h} ∈ N_{L}(T). But also,

f = Π(x^{−1}, h, y) = Π(x^{−1}, x, \overline{x}^{−1}, \overline{x}x^{−1}, h, y\overline{y}^{−1}, \overline{y}, y^{−1}, y)
= Π(\overline{x}^{−1}, \overline{x}x^{−1}, h, y\overline{y}^{−1}, \overline{y})
= Π(\overline{x}^{−1}, ((\overline{x}x^{−1})λ · g · (y\overline{y}^{−1})λ)λ^{−1}, \overline{y})
= Π(\overline{x}^{−1}, (\overline{g})λ^{−1}, \overline{y}) = Π(\overline{x}^{−1}, \overline{h}, \overline{y}).

Thus f ∼_{1} \overline{θ}, and (a) holds. If \overline{f} ∈ L with \overline{f} ∼_{1} θ, then f = Π(x^{−1}, h, y) = \overline{f}, and we have (b). □

Let L^{+} be the set (L ∪ Θ)/ ≈ of equivalence classes, and let L_{0}^{+} be the set of all C ∈ L^{+} such that C ∩ Θ ≠ ∅.

Lemma 5.8. Let C ∈ L^{+}.

(a) If C ∩ L = ∅ then C is a ∼_{0}-class in Θ.

(b) If C ∩ Θ = ∅ then C = {f} for some f ∈ L.

Proof. Immediate from the definition of ∼_{0} and ∼_{1}. □

Lemma 5.9.

(a) Let f, g ∈ L. Then f ≈ g if and only if f = g.

(b) If θ ∈ Θ and f ∈ L with f ≈ θ, then f ∼_{1} θ.

Proof. (a) As f ≈ g there is a sequence

(f = f_{0}, θ_{1}, \overline{θ}_{1}, f_{1}, ⋯, f_{n−1}, θ_{n}, \overline{θ}_{n}, f_{n} = g)

such that θ_{i} and \overline{θ}_{i} are in Θ, f_{i} is in L, and with f_{i−1} ∼_{1} θ_{i} ∼_{0} \overline{θ}_{i} ∼_{1} f_{i}. Then f_{i−1} = f_{i} by 5.7(b), and so f = g.

(b) Let θ ∈ Θ and f ∈ L with f ≈ θ. Point (a) together with 5.7(a) and 5.6, then yields a sequence

f ∼_{1} θ_{1} ∼_{1} f ∼_{1} ⋯ ∼_{1} f ∼_{1} θ_{n} = θ,

and this proves (b). □

Define D_{0}^{+} to be the set of words w = (C_{1}, ⋯, C_{n}) ∈ W(L_{0}^{+}) such that, for some choice of representatives (x_{i}^{−1}, g_{i}, y_{i}) of the classes C_{i}, the products y_{i}x_{i+1}^{−1} are defined in L and lie in N_{L}(T) (1 ≤ i < n). For such a word w, and such a choice of representatives, set

w_{0} = (g_{1}, (y_{1}x_{2}^{−1})λ, g_{2}, ⋯, (y_{n−1}x_{n}^{−1})λ, g_{n}).

We now wish to define a mapping Π_{0}^{+} : D_{0}^{+} → L_{0}^{+} by taking

(*)  Π_{0}^{+}(w) = [x_{1}^{−1}, Π_{M}(w_{0}), y_{n}].

Of course, we will define Π_{0}^{+}(∅) to be [1, 1_{M}, 1].

Lemma 5.10. There is a well-defined mapping Π_{0}^{+} : D_{0}^{+} → L_{0}^{+}, given by (∗).

Proof. By induction on word-length, we need only show that Π_{0}^{+} is well-defined on words w = (C_{1}, C_{2}) ∈ D_{0}^{+} of length 2. Let D(w) be the set of all pairs (θ_{1}, θ_{2}) ∈ C_{1} × C_{2}, where θ_{i} = (x_{i}^{−1}, g_{i}, y_{i}), and such that (y_{1}, x_{2}^{−1}) ∈ D and T^{y_{1}} = T^{x_{2}}. That is, D(w) is the set of all (θ_{1}, θ_{2}) for which it is possible to form a “product” as in (∗). The problem is to show that [x_{1}^{−1}, g_{1} · (y_{1}x_{2}^{−1})λ · g_{2}, y_{2}] is independent of the choice of representatives θ_{i} ∈ C_{i}.

Fix (θ_{1}, θ_{2}) ∈ D(w) and set U_{0} = T^{x_{1}}, U_{1} = T^{y_{1}} = T^{x_{2}}, and U_{2} = T^{y_{2}}. Suppose first that C_{i} ∩ L ≠ ∅ for both i = 1 and 2, and that (f_{1}, f_{2}) ∈ D, where f_{i} is the unique element (see 5.8(b)) of C_{i} ∩ L. Set v = (f_{1}, f_{2}), set E_{1} = N_{(S_{v})^{f_{1}}} (U_{1}), and set E_{0} = (E_{1})^{f_{1}^{−1}} and E_{2} = (E_{1})^{f_{2}}. Then v ∈ D via E_{0}. Using the definition of Θ, one observes that A_{1} := (E_{0})^{x_{1}^{−1}} ≤ N_{S}(T), and that B_{1} := (A_{1})^{g_{1}} = (E_{1})^{y_{1}^{−1}}. Similarly, one has A_{2} := (E_{1})^{x_{2}^{−1}} ≤ N_{S}(T), and B_{2} := (A_{1})^{g_{2}} = (E_{2})^{y_{2}^{−1}}. Thus, the groups A_{i}, and B_{i} are in Δ, and g_{1} · (y_{1}x_{2}^{−1})λ · g_{2} ∈ Im(λ). Setting h_{i} = g_{i}λ^{−1}, one then has (h_{1}, y_{1}x_{2}^{−1}, h_{2}) ∈ D via A_{1}, and

(x_{1}^{−1}, g_{1} · (y_{1}x_{2}^{−1})λ · g_{2}, y_{2}) ∼_{1} f_{1}f_{2}.

The result is independent of the choice of (θ_{1}, θ_{2}) ∈ D(w), so the lemma holds in this case. Thus, we may assume that no such pair (f_{1}, f_{2}) exists.

We now aim to show that (U_{0}, U_{1}, U_{2}) is independent of the choice of (θ_{1}, θ_{2}) ∈ D(w). This is given by 5.6 and 5.8(a) if C_{i} ∩ L = ∅ for both i = 1 and 2. Suppose next that C_{1} ∩ L = ∅ ≠ C_{2} ∩ L. Here C_{1} uniquely determines (U_{0}, U_{1}), and then U_{1} = T^{x_{2}} since y_{1}x_{2}^{−1} ∈ N_{L}(T). Setting U_{2} = (U_{1})^{f_{2}}, for f_{2} ∈ C_{2} ∩ L, it follows from 5.9(a) that, again, (U_{0}, U_{1}, U_{2}) depends only on (C_{1}, C_{2}) and not on the choice of representatives. The next case, where C_{1} ∩ L ≠ ∅ = C_{2} ∩ L, evidently yields the same result. By assumption, we have v = (f_{1}, f_{2}) ∉ D, and so S_{v} ∉ Δ. There is then a unique L-conjugate U_{0} of T with U_{0} ≤ S_{v}. Set U_{1} = (U_{0})^{f_{1}}, and U_{2} = (U_{1})^{f_{2}}. Then each U_{i} is an L-conjugate of T; and (U_{0}, U_{1}, U_{2}) is uniquely determined by (C_{1}, C_{2}), as desired.

Let (\overline{θ}_{1}, \overline{θ}_{2}) ∈ D(w), with \overline{θ}_{i} = (\overline{x}_{i}^{−1}, \overline{g}_{i}, \overline{y}_{i}). The result of the preceding paragraph, taken with 5.7(a) and 5.8(a) then yields θ_{i} ∼_{0} \overline{θ}_{i}. The definition of ∼_{0} then yields a commutative diagram as follows.

U_{0} ⟶^{x_{1}^{−1}} T ⟶^{g_{1}} T ⟶^{y_{1}} U_{1} ⟶^{x_{2}^{−1}} T ⟶^{g_{2}} T ⟶^{y_{2}} U_{2}
∥  ^{r_{1}}↓  ↓^{h_{1}}  ∥  ^{r_{2}}↓  ↓^{h_{2}}  ∥
U_{0} ⟶_{\overline{x}_{1}^{−1}} T ⟶_{\overline{g}_{1}} T ⟶_{\overline{y}_{1}} U_{1} ⟶_{\overline{x}_{2}^{−1}} T ⟶_{\overline{g}_{2}} T ⟶_{\overline{y}_{2}} U_{2}

(Here r_{i} = (x_{i}\overline{x}_{i}^{−1})λ and h_{i} = (y_{i}\overline{y}_{i}^{−1})λ.) The “middle” portion of this diagram leads at once to a commutative diagram in M, as follows.

T ⟶^{g_{1}} T ⟶^{(y_{1}x_{2}^{−1})λ} T ⟶^{g_{2}} T
r_{1}↓  ^{h_{1}}↓  ↓^{g_{2}}  ↓^{h_{2}}
T ⟶_{\overline{g}_{1}} T ⟶_{(\overline{y}_{1}\overline{x}_{2}^{−1})λ} T ⟶_{\overline{g}_{2}} T

The result is a diagram

U_{0} ⟶^{x_{1}^{−1}} T ⟶^{Π_{M} (w_{0})} T ⟶^{y_{2}} U_{2}
∥  ^{r_{1}}↓  ↓^{h_{2}}  ∥
U_{0} ⟶_{\overline{x}_{1}^{−1}} T ⟶_{Π_{M} (\overline{w}_{0})} T ⟶_{\overline{y}_{2}} U_{2}

which establishes that Π_{0}^{+} is well defined. □

Let u = (f_{1}, ⋯, f_{n}) ∈ W and let v = (C_{1}, ⋯, C_{n}) ∈ W(L_{0}^{+}). We shall write u ≈ v to indicate that f_{i} ∈ C_{i} for all i. Write D_{0}^{+} ∩ D for the set of all v ∈ D_{0}^{+} such that there exists u ∈ D with u ≈ v.

Lemma 5.11. Π_{0}^{+} and Π agree on D_{0}^{+} ∩ D.

Proof. Let w = (C_{1}, ⋯, C_{n}) ≈ (f_{1}, ⋯, f_{n}) be in D_{0}^{+} ∩ D, and let θ_{i} = (x_{i}^{−1}, g_{i}, y_{i}) ∈ C_{i} be chosen so that f_{i} ∼_{1} θ_{i}. Let (U_{0}, ⋯, U_{n}) be the sequence of L-conjugates of T given by T^{x_{i}} = U_{i−1} and T^{y_{i}} = U_{i}. As C_{i} ≈ f_{i} we have g_{i} ∈ Im(λ), and (x_{i}^{−1}, (g_{i})λ^{−1}, y_{i}) is in D via a subgroup of N_{S_{g_{i}}} (U_{i−1}). This shows that when w is viewed as an element of D, one has U_{0} ≤ S_{w}. Setting P_{0} = N_{S_{w}}(U_{0}) we get P_{0} ∈ Δ by 5.4(a), and w ∈ D via P_{0}.

Set h_{i} = (g_{i})λ^{−1}, and set

v = (x_{1}^{−1}, h_{1}, y_{1}, ⋯, x_{n}^{−1}, h_{n}, y_{n}) and v_{0} = (h_{1}, y_{1}x_{2}^{−1}, ⋯, y_{n−1}x_{n}^{−1}, h_{n}).

Set P_{i} = P_{0}^{f_{1}···f_{i}}. Then 5.4(a) implies that v ∈ D via P_{0}, since each P_{i−1} is a member of Δ contained in S_{f_{i}} ∩N_{S}(T). Then also v_{0} ∈ D via (P_{0})^{x_{1}^{−1}}, and since also v_{0} ∈ W(N_{L}(T)), the isomorphism λ : N_{L}(T) → L_{Δ_{T}} (M) sends Π(v_{0}) to Π_{M}(w_{0}), where

w_{0} = (g_{1}, λ(y_{1}x_{2}^{−1}), ⋯, λ(y_{n−1}x_{n}^{−1}), g_{n}).

We now obtain

Π_{0}^{+}(C_{1}, ⋯, C_{n}) = [x_{1}^{−1}, Π_{M}(w_{0}), y_{n}] ≈ Π(f_{1}, ⋯, f_{n})

since (x_{1}^{−1}, Π(v_{0}), y_{n}) ∈ D (via P_{0}). This yields the lemma. □

By 5.8 and 5.9(a) we may identify L_{0}^{+} and L with their images in L^{+} via ≈. Set L_{0} = L_{0}^{+} ∩ L and set L_{1} = L − L_{0}. Thus, L^{+} is the disjoint union of L_{0}^{+} and L_{1}. Set D^{+} = D_{0}^{+} ∪ D. By 5.11 there is a “product”

Π^{+} = Π_{0}^{+} ∪ Π : D^{+} → L^{+}

whose restriction to D_{0}^{+} is Π_{0}^{+}, and whose restriction to D is Π. Set 1^{+} = [1, 1_{M}, 1] (or equivalently, via ≈, 1^{+} = 1). We now define an “inversion map” on L_{0}^{+} by [x^{−1}, f, y] ↦ [y^{−1}, f^{−1}, x]. (That this is indeed a well-defined involutory bijection of L_{0}^{+} will be shown in the proof of lemma 5.12, immediately below.) We extend the inversion on L_{0}^{+} to all of L^{+} in the obvious way, by forming the union with the inversion on L.

Lemma 5.12. L^{+}, with the above product, identity element, and inversion, is a partial group.

Proof. That D^{+} contains L^{+} as words of length 1 is immediate from the definition, as is the fact that u ∘ v ∈ D^{+} implies u, v ∈ D^{+}. Thus 2.1(1) holds for L^{+}. That Π^{+} restricts to the identity map on L^{+}, and that Π^{+} is multiplicative is immediate from the definition of Π^{+}, and so points (2) and (3) of 2.1 hold for L^{+}. It remains to check that inversion is well defined and that L^{+} satisfies 2.1(4).

Let θ = (x^{−1}, g, y) ∈ Θ and set θ^{−1} = (y^{−1}, g^{−1}, x). The conditions on x (immediately following 5.4) which define θ as being in Θ are that T ≤ S_{x} and that T^{x} ≤ S and N_{S_{x}}(T)^{x} = N_{S}(T^{x}); and these are the same conditions on x that are required in order that θ^{−1} be in Θ. The analogous set of conditions applies to y, by symmetry, so we obtain θ^{−1} ∈ Θ. Now let \overline{θ} = (\overline{x}^{−1}, \overline{f}, \overline{y}) ∈ Θ, with θ ∼_{0} \overline{θ}. Thus T^{x} = T \overline{x}, T^{y} = T \overline{y}, and

(*)  (\overline{x}x^{−1})λ · g = \overline{g} · (\overline{y}y^{−1})λ.

The condition (*) concerns multiplication in the group M, where inversion and straightforward manipulation yields

(\overline{y}y^{−1})λ · g^{−1} = \overline{g}^{−1} · (\overline{x}x^{−1})λ.

This shows that θ^{−1} ∼_{0} \overline{θ}^{−1}. Now suppose that f ∈ L and that θ ∼_{1} f. This means that g ∈ Im(λ), (x^{−1}, gλ^{−1}, y) ∈ D, and f = Π(x^{−1}, gλ^{−1}, y). Then f^{−1} = Π(y^{−1}, g^{−1}λ^{−1}, x) by 2.2(f), and so θ^{−1} ∼_{1} f^{−1}. This completes the verification that there is a well-defined mapping [x^{−1}, g, y] ↦ [y^{−1}, g^{−1}, x] on ≈-classes, and which agrees with the inversion map from L on L_{0}^{+} ∩ L. Evidently, this inversion map on L^{+} is then an involutory bijection. One readily verifies that u^{−1} ∘u ∈ D^{+} if u ∈ D^{+}, and that then Π^{+}(u^{−1} ∘u) = 1^{+}. Thus 2.1(4) holds for L^{+}, and L^{+} is a partial group. □

Let Δ^{+} be the union of Δ with the set of subgroups P of S such that P contains an F-conjugate of T. We now have a candidate, in the partial group L^{+}, for a locality whose set of objects is Δ^{+}. In order to establish the conditions (O1) and (O2) for objectivity in 2.5, it must be shown that if A is an object in Δ^{+}, and C ∈ L^{+} with A^{C} ≤ S, then either C ∈ L or else C is of the form [x^{−1}, g, y] with T^{x} = A and T^{y} = A^{C}. This is not immediate, since the statement “A^{C} ≤ S” merely says that for each a ∈ A, we have (C^{−1}, a, C) ∈ D^{+} and a^{C} ∈ S. The following lemma addresses this issue.

Lemma 5.13. Let θ = (x^{−1}, g, y) ∈ Θ, let C be the ≈-class of θ, and let A be an F-conjugate of T such that A^{C} ≤ S.

(a) If θ ≈ f ∈ L, then A ≤ S_{f}.

(b) If C ∩ L = ∅, then A = T^{x} and A^{C} = T^{y}.

Proof. (a): Let a ∈ A and set b = a^{C}. Set P = S_{f} and Q = P^{f}. Then (a^{−1}, f, b) ∈ D via (P^{a}, P, Q, Q^{b}), since a and b are elements of S, and similarly (a, f) and (f, b) are in D.

Set w = (C^{−1}, a, C). As w ∈ D^{+} we have also (a, C) ∈ D^{+} and the axioms 2.1 for a partial group yield

Π^{+}(a, C) = Π^{+}((C) ∘ w) = Π^{+}(C, Π^{+}(w)) = Π^{+}(C, b).

(In particular, (C, b) ∈ D^{+}.) Since Π^{+}(C^{−1}, a, C) = b = Π(f^{−1}, f, b) = Π^{+}(C^{−1}, C, b) by 5.11, left cancellation and 5.11 yield Π(a, f) = Π(f, b). Then

f = Π(a^{−1}, a, f) = Π(a^{−1}, f, b),

by 2.2(e). Since conjugation by a^{−1}fb carries P^{a} to Q^{b}, we conclude that P^{a} ≤ S_{f}. That is, P^{a} = P, and then (f^{−1}, a, f) ∈ D (via Q). Apply 5.11 to get a^{f} = b in the partial group L. Thus a ∈ S_{f}, and (a) holds.

(b): Here C is a ∼_{0}-class by 5.8(a). Set U = T^{x} and V = T^{y}, and recall that U and V depend only on C, by 5.6. Again let a ∈ A and set b = a^{C}. As (C^{−1}, a, C) ∈ D^{+}, it follows that (C^{−1}, a, C) ∈ D^{+} via (V, U, U, V), and thus a ∈ N_{S}(U). Similarly, b ∈ N_{S}(V). Then, by (1) in the definition of Θ, both a^{x^{−1}} and b^{y^{−1}} are in N_{S}(T).

Set w = (a^{−1}x^{−1}, 1, xa^{2}) ∈ W(L), and observe that w ∈ D via N_{S}(U), and that also w ∈ D^{+} via (U, T, T, U). In particular, we get a ∼_{1} D := [a^{−1}x^{−1}, 1_{M}, xa^{2}]. The representatives (y^{−1}, g^{−1}, x) of C^{−1}, (a^{−1}x^{−1}, 1_{M}, xa^{2}) of D, and (x^{−1}, g, y) of C have the property that Π^{+}(C^{−1}, D, C) may be given in terms of these representatives. That is, we have

Π^{+}(C^{−1}, a, C) = Π^{+}(C^{−1}, D, C) = [y^{−1}, g^{−1}(xax^{−1})g, y)] = [y^{−1}, (a^{x^{−1}})^{g}, y],

since the products x(a^{−1}x^{−1}) and (xa^{2})x^{−1} are defined in L and lie in N_{L}(T). As a^{C} = b, we then have

b ≈ (y^{−1}, (a^{x^{−1}})^{g}, y).

The reader may verify that y ∼_{1} (y, 1_{M}, 1) and that

v := ((1, 1_{M}, y), (y^{−1}, (a^{x^{−1}})^{g}, y), (y^{−1}, 1_{M}, 1)) ∈ D^{+},

and further that

Π^{+}(v) = [1, (a^{x^{−1}})^{g}, 1].

Then, appealing to 5.9(b), we get

(1, b^{y^{−1}}, 1) ∼_{1} b^{y^{−1}} ∼_{1} (1, (a^{x^{−1}})^{g}, 1).

Now (1, b^{y^{−1}}, 1) ∼_{0} (1, (a^{x^{−1}})^{g}, 1) by 5.7(a). A glance at the diagram following the definition of ∼_{0} will now convince the reader that b^{y^{−1}} = (a^{x^{−1}})^{g}. Thus (A^{x^{−1}})^{g} = B^{y^{−1}}, and we obtain a sequence of conjugation maps (between subgroups of S) as follows.

UA ⟶^{x^{−1}} T A^{x^{−1}} ⟶^{g} T B^{y^{−1}} ⟶^{y} V B.

Since C ∉ L, it follows that UA ∉ Δ, whence A = U and B = V. This completes the proof of (b). □

The following two theorems, along with proposition 1.10 above, form the foundation for our proof of the Main Theorem.

Theorem 5.14. Assume hypothesis 5.3 and let Δ^{+} be the union of Δ with the set of subgroups P of S such that P contains an F-conjugate of T. Then (L^{+}, Δ^{+}, S) is an F-natural locality, and moreover:

(a) The isomorphism λ : N_{L}(T) → L_{Δ_{T}} (M) extends in a unique way to an isomorphism λ^{+} : N_{L^{+}}(T) → M of groups, such that

(*)  [x^{−1}, g, y]λ^{+} = Π_{M}(x^{−1}λ, g, yλ)

for any x, y ∈ N_{L}(T).

(b) If L is a Δ-linking system and C_{M}(T) ≤ T, then L^{+} is a Δ^{+}-linking system.

Proof. Let w = (C_{1}, ⋯, C_{n}) ∈ W(L^{+}) and (U_{0}, ⋯, U_{n}) ∈ W(Δ^{+}), with U_{i−1}^{C_{i}} = U_{i} for all i, (1 ≤ i ≤ n). Suppose first that U_{0} (and hence each U_{i}) is an L-conjugate of T. Then 5.13 implies that w ∈ D_{0}^{+}. On the other hand, if U_{0} ∈ Δ then each U_{i} is in Δ, and w ∈ D. Thus, either way, we get w ∈ D^{+}, so that (L^{+}, Δ^{+}) satisfies the condition (O1) in the definition 2.6 of objectivity.

We next check that L^{+} is F-natural. For P and Q in Δ we have Hom_{L^{+}}(P, Q) = Hom_{L}(P, Q) by construction, so Hom_{L^{+}}(P, Q) = Hom_{F}(P, Q) in this case. Now let U and V be L-conjugates of T, and let C ∈ N_{L^{+}}(U, V). If C ≈ f for some f ∈ L then U^{f} = V; and since c_{f} : S_{f} → S is an F-homomorphism we conclude that c_{C} : U → V is an F-homomorphism. On the other hand, suppose that C ∈ L^{+} −L, and let (x^{−1}, g, y) ∈ C. Then 5.13 yields T^{x} = U and T^{y} = V. Here F_{N_{S}(T)}(M) = N_{F}(T) by hypothesis, so each of c_{x}, c_{gλ^{−1}}, and c_{y} is an F-homomorphism, and then so is c_{C}. Thus Hom_{L^{+}}(U, V) ⊆ F, and L^{+} is F-natural. Then also L^{+} satisfies the condition (O2) for objectivity, and so (L^{+}, Δ^{+}) is an objective partial group.

By definition, S is the unique maximal member of Δ^{+}. Since N_{L^{+}}(S) = N_{L}(S) it follows that also S is maximal in the poset of finite p-subgroups of L^{+}, and so L^{+} satisfies the conditions (L1) and (L2) in definition 2.9. As L and M are finite, so is the set Θ of triples (x^{−1}, g, y), and thus L^{+} is a locality.

Set H = N_{L^{+}}(T) and K = N_{L}(T). Let C = [x^{−1}, g, y] ∈ H. Then x, y ∈ K, and C = [1, (x^{−1}λ)g(yλ), 1]. Because of this, it is now readily verified that there is a well-defined mapping λ^{+} : H → M given by C ↦ (x^{−1}λ)g(yλ), that λ^{+} coincides with λ on K, and that λ^{+} is a homomorphism of groups. Suppose that C ∈ Ker(λ^{+}). Then (x^{−1}λ)g(yλ) = 1_{M}, so g = (xy^{−1})λ ∈ Im(λ), and then (x^{−1}, g^{−1}, y) ∼_{1} 1. Thus Ker(λ^{+}) = 1, and λ^{+} is then an isomorphism since M is finite. The uniqueness of λ^{+} is immediate from the condition (*), so (a) holds.

Suppose next that L is a Δ-linking system and that C_{M}(T) ≤ T. Let U ∈ Δ^{+} with U ∉ Δ. There is then a unique F-conjugate U_{0} of T contained in U, and hence N_{L^{+}}(U) ≤ N_{L^{+}}(U_{0}). As L^{+} is F-natural, there exists C ∈ L^{+} such that (U_{0})^{C} = T. Conjugation by C induces an isomorphism of N_{L^{+}}(U_{0}) with N_{L^{+}}(T), and hence with M. Since C_{M}(X) ≤ X for any p-subgroup X of M containing T, it follows that C_{L^{+}}(U) ≤ U. By construction, N_{L^{+}}(P) = N_{L}(P) for P ∈ Δ, so we conclude that L^{+} is a Δ^{+}-linking system. Thus (b) holds. □

We write also L^{+}(λ) for the locality constructed by 5.14, in order to emphasize its dependence on the isomorphism λ : N_{L}(T) → L_{Δ_{T}} (M). In the same vein, we may write L_{0}^{+}(λ) for the partial subgroup L_{0}^{+} of L^{+}.

Theorem 5.15. Assume Hypothesis 5.3, and let Δ^{+} be the union of Δ with the set of all subgroups of S which contain an F-conjugate of T.

(a) Let L^{∗} be a locality via the set Δ^{+} of objects, let L′ be the restriction of L^{∗} to Δ, let β : L → L′ be a rigid isomorphism, and let β_{T} : N_{L}(T) → N_{L′}(T) be the isomorphism induced by restriction of β. Assume that there is given an isomorphism μ : M → N_{L^{∗}}(T) of groups, such that μ restricts to the identity map on N_{S}(T). Let μ_{0} be the restriction of μ to L_{Δ_{T}} (M), and set λ = β_{T} ∘ μ_{0}^{−1}. Then there exists a unique isomorphism β^{+} : L^{+}(λ) → L^{∗} such that β^{+} restricts to β on L and to λ^{+} ∘ μ on N_{L^{+}(λ)}(T).

(b) Let (L, Δ, S) and (L′, Δ, S) be localities having the same set Δ of objects, let β : L → L′ be a rigid isomorphism, and let β_{T} : N_{L}(T) → N_{L′}(T) be the rigid isomorphism given by restriction of β. Further, let λ : N_{L}(T) → L_{Δ_{T}} (M) and λ′ : N_{L′}(T) → L_{Δ_{T}} (M) be rigid isomorphisms, and let μ_{0} be the automorphism λ^{−1} ∘ β_{T} ∘ λ′ of L_{Δ_{T}} (M).

(i) There exists an isomorphism β^{+} : L^{+}(λ) → (L′)^{+}(λ′) extending β if and only if μ_{0} extends to an automorphism μ of M.

(ii) Let μ be an extension of μ_{0} to M. Then there is a unique isomorphism β^{+} : L^{+}(λ) → (L′)^{+}(λ′) having the property that β^{+} restricts to β on L and to λ^{+} ∘ μ ∘ ((λ′)^{+})^{−1} on N_{L^{+}}(T). Moreover, β^{+} is then given explicitly on L_{0}^{+}(λ) by

[x^{−1}, g, y] ↦ [x^{−1}β, gμ, yβ]

for (x^{−1}, g, y) ∈ Θ.

(c) Suppose that there exists a rigid isomorphism N_{L}(T) → L_{Δ_{T}} (M), and that all rigid automorphisms of L_{Δ_{T}} (M) are M-equivalent (as defined in 5.2). Then, up to rigid isomorphism, there exists a unique locality (L^{∗}, Δ^{+}, S) whose restriction to Δ is L, and having the property that N_{L^{∗}}(T) is equal to M.

We now prove the points (b), (a), and (c) of 5.15, in that order.

Proof of 5.15(b). Set N = N_{L}(T) and N′ = N_{L′}(T). Let Θ′ be the subset of L′ ×M ×L′ defined by the conditions immediately following 5.4, but now with respect to λ′. In order to avoid confusion, we shall distinguish the relations ∼_{0}, ∼_{1} and ≈ in the two cases, by the following sort of notational device. Thus, for example, if ϕ = (x^{−1}, g, y) and \overline{ϕ} = (\overline{x}^{−1}, \overline{g}, \overline{y}) are in Θ′, then we shall write, for example,

ϕ ∼_{0} \overline{ϕ} (rel λ′)

to indicate that the products \overline{x}x^{−1} and \overline{y}y^{−1} exist, and are elements of N′, and satisfy the condition

(\overline{x}x^{−1})λ′ · g = \overline{g} · (\overline{y}y^{−1})λ′.

The expressions “f ∼_{1} θ (rel λ′)”, and “f ≈ θ (rel λ′)” should be understood similarly. The relations ∼_{0}, ∼_{1}, and ≈ on Θ will be provided with a corresponding “rel λ” in order to lend emphasis to this distinction.

Set μ_{0} = λ^{−1} ∘ β_{T} ∘ λ′ and assume that μ_{0} extends to an automorphism μ of M. Let α be the mapping on Θ given by

α : (x^{−1}, g, y) ↦ ((xβ)^{−1}, gμ, yβ).

Since β is rigid, T β = T, and from this it is easily verified that Im(α) ⊆ Θ′. There is an obvious inverse to α, so in fact α is a bijection Θ → Θ′.

We claim that α sends ∼_{0}-classes (rel λ) to ∼_{0}-classes (rel λ′). Namely, let θ and \overline{θ} be elements of Θ, written in the usual way, and assume that θ ∼_{0} \overline{θ} (rel λ). Setting a = \overline{x}x^{−1} and b = \overline{y}y^{−1}, we then have (aλ) · g = \overline{g} · (bλ) in M. Applying μ, we obtain (aβ)λ′ · gμ = \overline{g}μ · (bβ)λ′, and thus ((xβ)^{−1}, gμ, yβ) ∼_{0} ((\overline{x}β)^{−1}, \overline{g}μ, \overline{y}β) (rel λ′). This proves the claim.

Now let f ∈ L, and suppose that f ∼_{1} θ (rel λ). That is, suppose that there exists h ∈ N with g = hλ, that (x^{−1}, h, y) ∈ D, and that f = Π(x^{−1}, h, y). Then ((xβ)^{−1}, hβ, yβ) ∈ D′ and fβ = Π′((xβ)^{−1}, hβ, yβ). Since (hβ)λ′ = gμ, we conclude that α sends ∼_{1}-classes (rel λ) into ∼_{1}-classes (rel λ′). By 5.8 and 5.9, any ≈-class C relative to λ is either a ∼_{0}-class or is of the form {f}∪X where X is the set of all θ ∈ Θ such that f ∼_{1} θ. Since the same is true for ≈-classes relative to λ′, we conclude that α respects the ≈-relations relative to λ and λ′.

Set L^{+} = L^{+}(λ) and (L′)^{+} = (L′)^{+}(λ′), and similarly define L_{0}^{+} and (L′_{0})^{+}. Thus α induces a mapping

(*)  γ : L_{0}^{+} → (L′_{0})^{+}, ([x^{−1}, g, y] ↦ [(xβ)^{−1}, gμ, yβ]),

and fγ = fβ for all f ∈ L such that f ∼_{1} θ for some θ ∈ Θ. Since L^{+} is the union of L with L_{0}^{+}, α extends to a mapping

γ : L^{+} → (L′)^{+},

which restricts to the identity map on L. Further, since all of the above arguments can be carried out with α^{−1} in place of α, γ is a bijection.

We next show that γ is a homomorphism of partial groups. Set D(λ) = D((L′)^{+}(λ)), and similarly define D(λ′). Let Π^{+} and (Π′)^{+} be the corresponding products. Let w = (C_{1}, ⋯, C_{n}) ∈ D(λ) and set Q = S_{w}. If Q contains no L-conjugate of T then C_{i} ∈ L for all i, wγ^{∗} = wβ^{∗}, and hence (Π′)^{+}(wγ^{∗}) = (Π^{+}(w))γ. On the other hand, suppose that Q contains an L-conjugate of T. Then C_{i} = [x_{i}^{−1}, g_{i}, y_{i}] for some (x_{i}^{−1}, g_{i}, y_{i}) ∈ Θ, and Π_{λ′}^{+}(wγ^{∗}) = [(x_{1}β)^{−1}, Π_{M}(u_{0}), y_{n}β], where

u_{0} = (g_{1}μ, (y_{1}x_{2}^{−1})βλ′, ⋯, (y_{n−1}x_{n}^{−1})βλ′, g_{n}μ).

One observes that u_{0} = w_{0}μ^{∗}, where

w_{0} = (g_{1}, (y_{1}x_{2}^{−1})λ, ⋯, (y_{n−1}x_{n}^{−1})λ, g_{n}),

and hence (Π′)^{+}(wγ^{∗}) = (Π^{+}(w))γ, and γ is a homomorphism. Since the roles of (L, λ) and (L′, λ′) can be reversed γ is then an isomorphism, and γ is rigid since β is rigid.

Set N^{+} = N_{L^{+}}(T) and (N′)^{+} = N_{(L′)^{+}(λ′)}(T), and suppose that there is given an isomorphism σ : L^{+}(λ) → (L′)^{+}(λ′) such that σ restricts to β on L. Let σ_{T} : N^{+} → (N′)^{+} be the isomorphism induced by σ. Then μ_{0} extends to the automorphism ν = (λ^{+})^{−1}∘σ_{T} ∘(λ′)^{+} of M, and this completes the proof of (bi). In order to obtain (bii), one need only observe that the formula (*) defines the unique mapping L^{+}(λ)_{0} → L^{+}(λ′)_{0} which, in union with β, defines a homomorphism β^{+} : L^{+} → (Λ′)^{+} which restricts to β on L and to λ^{+} ∘ μ ∘ ((λ′)^{+})^{−1} on N^{+}. □

Proof of 5.15(a). Let λ′ : N_{L′}(T) → L_{Δ_{T}} (M) be the restriction of μ^{−1} to N_{L′}(T). To simplify the notation, we shall write L(λ) for L^{+}(λ), and L′(λ′) for (L′)^{+}(λ′). Set L_{T} = N_{L}(T) and L(λ)_{T} = N_{L(λ)}(T), and similarly define L′_{T} and L′(λ′)_{T}. Also, set L_{T}^{∗} = N_{L^{∗}}(T), and let μ_{0} : L_{Δ_{T}} (M) → L′_{T} be the restriction of μ.

Suppose first that 5.15(a) holds in the case where L = L′ and where β is the identity map on L. We shall show that 5.15(a) then holds in generality. Thus, taking (L′, λ′) in the role of (L, λ), the assumed special case of 5.15(a) yields an isomorphism ϕ : L′(λ′) → L^{∗} such that ϕ restricts to the identity on L′ and to (λ′)^{+} ∘ μ on L′(λ)_{T}.

Since λ = β_{T} ∘ μ_{0}^{−1} by hypothesis, we have

λ^{−1} ∘ β_{T} ∘ λ′ = μ_{0} ∘ β_{T}^{−1} ∘ β_{T} ∘ μ_{0}^{−1} = id_{L_{Δ_{T}}} (M).

Thus λ^{−1} ∘ β_{T} ∘ λ′ extends to the identity automorphism of M. By 5.15(b) (proved above) there then exists an isomorphism γ : L(λ) → L′(λ′) whose restriction to L is β, and whose restriction to L(λ)_{T} → L′(λ′) is λ^{+} ∘ ((λ′)^{+})^{−1}. Now set β^{+} = γ ∘ ϕ. Then β^{+} restricts to β : L → L′, and on L(λ)_{T} to

(λ^{+} ∘ ((λ′)^{+})^{−1}) ∘ ((λ′)^{+} ∘ μ) = λ^{+} ∘ μ.

Thus, β^{+} fulfills the requirements of the statement of 5.15(a). The uniqueness of β^{+} subject to the given conditions follows in the usual way (for example, as in the proof of uniqueness in 5.15(b)), and is omitted.

We assume for the remainder of the proof that L = L′ and that β is the identity automorphism of L. Then also β_{T} is the identity automorphism of L_{T}, and λ^{−1} = μ_{0}.

Let θ = (x^{−1}, g, y) ∈ Θ. Then (x^{−1}, gμ, y) ∈ D^{∗} via the object T^{x}, and one checks that the product Π^{∗}(x^{−1}, gμ, y) in L^{∗} remains unchanged when θ is replaced by any \overline{θ} such that θ ∼_{0} \overline{θ}. In some detail: let \overline{θ} = (\overline{x}^{−1}, \overline{g}, \overline{y}) with θ ∼_{0} \overline{θ}. Then T^{x} = T \overline{x} by 5.6. Since μ_{0} = λ^{−1} we get

Π^{∗}(x^{−1}, gμ, y) = Π^{∗}(\overline{x}^{−1}, \overline{x}x^{−1}, gμ, y\overline{y}^{−1}, \overline{y}) = Π^{∗}(\overline{x}^{−1}, \overline{g}, \overline{y}).

Suppose next that f ∈ L with f ∼_{1} θ. Then g = hλ for some h ∈ N_{L}(T), (x^{−1}, h, y) ∈ D, and f is equal to the product x^{−1}hy in L. This yields

Π^{∗}(x^{−1}, gμ, y) = Π^{∗}(x^{−1}, h, y) = f,

since β is the identity automorphism of L. Now 5.9(b) implies that Π^{∗}(x^{−1}, gμ, y) depends only on the ≈-class of θ, and we have a well-defined mapping γ_{0} : L_{0}^{+} → L^{∗} given by

γ_{0} : [x^{−1}, g, y] → Π^{∗}(x^{−1}, gμ, y)

and which restricts to the identity map on L_{0}^{+} ∩ L. One also observes that

[x^{−1}, g, y]λ^{+} = Π_{M}(x^{−1}λ, g, yλ),

for [x^{−1}, g, y] ∈ N_{L^{+}}(T).

We now define the mapping γ : L^{+} → L^{∗} to be the union of the identity map on L with γ_{0}. Notice that if C = [x^{−1}, g, y] ∈ N_{L^{+}}(T), then

Cλ^{+} = (x^{−1}λ · g · yλ) and (x^{−1}λ · g · yλ)μ = Π^{∗}(x^{−1}, gμ, y).

Thus the restriction of γ to N_{L^{+}}(T) is λ^{+} ∘ μ.

We next show:

(1) Let C ∈ L^{+}, set f^{∗} = Cγ, and let a ∈ S. Then

a^{C} ∈ S ⟺ (a^{f^{∗}} ∈ S and a^{C} = a^{f^{∗}}).

The proof is as follows. First, let C = [x^{−1}, g, y] ∈ L_{0}^{+} − L, and let (U, V) be the pair of F-conjugates of T, uniquely determined by C, such that U = T^{x} and V = T^{y}. Then U^{x^{−1}} = T = T^{g} = V^{y^{−1}}. For any a ∈ U we then get

a^{C} = ((a^{x^{−1}})^{g})^{y}

(*) = (((a^{x^{−1}})^{g})μ)^{y} (as μ is rigid)

= ((a^{x^{−1}})^{gμ})^{y} (again as μ is rigid)
= a^{Cγ}

as Cγ = Π^{∗}(x^{−1}, gμ, y).

More generally, we find that (*) holds for C ∈ L_{0}^{+} and any a ∈ S_{C}. Namely, if C ∩L = {f} and a ∈ S_{f}, then a = aγ and f = fγ, and hence a^{C} = a^{f} = (a^{f})γ = a^{fγ} = a^{Cγ}.

One observes that (1) and (*) may be read “in reverse”. Namely, if f^{∗} ∈ L^{∗}, and f^{∗} = Cγ with C ∈ L_{0}^{+} − L, then a^{f^{∗}} = a^{C} for any a ∈ S such that a^{f^{∗}} ∈ S. Finally, in the case that C ∈ L^{+} with C ∩ L ≠ ∅, there is really nothing to show since S ≤ L. Thus (1) holds.

We may now show that γ is a homomorphism of partial groups. Namely, let γ^{∗} be the map W(L^{+}) → W(L^{∗}) induced by γ. Let w ∈ D^{+}, set X = S_{w}, and suppose first that X ∈ Δ. Then w ∈ D(L) via X, and w = wγ^{∗}. On the other hand, suppose that X ∉ Δ. Then w = (C_{1}, ⋯, C_{n}) where C_{i} ∈ L_{0}^{+}, and there is a uniquely determined sequence (U_{0}, ⋯, U_{n}) of F-conjugates of T such that U_{i−1}^{C_{i}} = U_{i} for all i from 1 to n. Set f_{i}^{∗} = C_{i}γ and set w^{∗} = (f_{1}^{∗}, ⋯, f_{n}^{∗}). Then (1) yields U_{i−1}^{f_{i}^{∗}} = U_{i} for all i, and thus w^{∗} ∈ D^{∗}. The verification that

Π^{∗}(w^{∗}) = (Π^{+}(w))γ,

and thus that γ is a homomorphism, is now a formality. We treat the case n = 2 in detail; and for this it suffices to consider the case where w = (C_{1}, C_{2}) ∈ D_{0}^{+}, since the restriction of γ to L is the identity map. Let θ_{i} = (x_{i}^{−1}, g_{i}, y_{i}) ∈ C_{i}, with (y_{1}, x_{2}^{−1}) ∈ D and with y_{1}x_{2}^{−1} ∈ N_{L}(T). Then Π^{+}(w) = [x_{1}^{−1}, g_{1} · (y_{1}x_{2}^{−1})λ · g_{2}, y_{2}], and

Π^{+}(w)γ = Π^{∗}(x_{1}^{−1}, (g_{1} · (y_{1}x_{2}^{−1})λ · g_{2})μ, y_{2})
= Π^{∗}(x_{1}^{−1}, g_{1}μ · (y_{1}x_{2}^{−1})λμ · g_{2}μ, y_{2})
= Π^{∗}(x_{1}^{−1}, g_{1}μ · y_{1}x_{2}^{−1} · g_{2}μ, y_{2})
= Π^{∗}(x_{1}^{−1}, g_{1}μ, y_{1}, x_{2}^{−1}, g_{2}μ, y_{2})

where the last line is obtained by observing that (1) implies that

(x_{1}^{−1}, g_{1}μ, y_{1}, x_{2}^{−1}, g_{2}μ, y_{2}) ∈ D^{∗}.

Now

Π^{∗}(x_{1}^{−1}, g_{1}μ, y_{1}, x_{2}^{−1}, g_{2}μ, y_{2}) = Π^{∗}(x_{1}^{−1}(g_{1}μ)y_{1}, x_{2}^{−1}(g_{2}μ)y_{2}) = Π^{∗}(w^{∗}).

The case w = (C_{1}, ⋯, C_{n}) with n > 2 differs in no essential way from the case n = 2, so the above argument establishes that γ is a homomorphism.

The next step will be to show:

(2) For each P, Q ∈ Δ^{+}, the mapping γ_{P,Q} : N_{L^{+}}(P, Q) → N_{L^{∗}}(P, Q) is surjective. Of course, we may assume that N_{L^{∗}}(P, Q) is non-empty. If P ∈ Δ then Q ∈ Δ and γ_{P,Q} = β_{P,Q} is bijective. So assume that P ∉ Δ. Then P contains a unique L-conjugate U of T, and hence U ⊴ P. Let f^{∗} ∈ N_{L^{∗}}(P, Q) and set V = U^{f^{∗}}. By 5.4(b) there exist elements x, y ∈ L with T^{x} = U and T^{y} = V, and such that N_{S_{x}}(T)^{x} = N_{S}(U) and N_{S_{y}}(T)^{y} = N_{S}(V). Then (x, f^{∗}, y^{−1}) ∈ D^{∗} via T, and we set h^{∗} = Π^{∗}(x, f^{∗}, y^{−1}). Then (x^{−1}, x, f^{∗}, y^{−1}, y) ∈ D^{∗} via U, so f^{∗} = Π^{∗}(x^{−1}, h^{∗}, y) by 2.1(3). Moreover, we have h^{∗} ∈ N_{L^{∗}}(T). Now [x^{−1}, h^{∗}μ^{−1}, y] is mapped to f^{∗} by γ, and so (2) holds.

Next, we show:

(3) Ker(γ) = 1. As γ |_{L}= id, the set of non-identity elements of Ker(γ) is contained in L_{0}^{+}. Let C = [x^{−1}, g, y] ∈ Ker(γ), and let U, V be the conjugates of T associated with C by 5.6. Then 1^{∗} = Cγ = Π^{∗}(x^{−1}, gμ, y), and (1) implies that U = V = T. Then C = [1, x^{−1}λ · g · yλ, 1] ∈ N_{L^{+}}(T). Since μ is injective, we conclude that C = 1^{+}, and thus (3) holds.

Since S_{h} ∈ Δ for all h ∈ L′ it is immediate from (2) that γ is surjective, and from (1) that S_{f} = S_{fγ} for all f ∈ L. Then (3) and 3.6 imply that γ is an isomorphism, and hence a rigid isomorphism by construction. Thus, it remains only to establish the uniqueness of γ, subject to the conditions that γ |_{L}= id_{L} and that γ |_{N_{L}(T)}= λ^{+} ∘ μ. Let γ′ : L^{+} → L^{∗} be another such isomorphism. Then, for any C = [x^{−1}, g, y] ∈ L_{0}^{+}, we get

Cλ′ = ([x^{−1}, 1_{M}, 1][1, g, 1][1, 1_{M}, y])λ′ = Π^{∗}(x^{−1}, gμ, y) = Cλ,

and this completes the proof. □

Proof of 5.15(c). Assuming that there exists a rigid isomorphism N_{L}(T) → L_{Δ_{T}} (M), 5.14 yields the existence of a locality (L^{∗}, Δ^{+}, S) with the required properties, and 5.5(b) implies that L^{∗} is rigidly isomorphic to some L^{+}(λ). Assuming further that all rigid automorphisms of L_{Δ_{T}} (M) are M-equivalent, 5.15(b) yields the uniqueness of L^{∗} up to rigid isomorphism. That is, 5.15(c) holds, and the proof is complete. □

This completes the proof of theorem 5.15.

It will be convenient, for the applications in the next two sections, to state a corollary concerning a special case of 5.15.

Corollary 5.16. Assume hypothesis 5.3, and let (L^{∗}, Δ^{+}, S) be a locality such that the restriction of L^{∗} to Δ is equal to L, and with N_{L^{∗}}(T) = M. Let β be a rigid automorphism of L and let λ := β_{T} be the automorphism of N_{L}(T) given by restricting β.

(a) There exists a unique rigid isomorphism α : L^{+}(λ) → L^{∗} such that β is the restriction of α to L, and such that λ^{+} is the restriction of α to N_{L^{+}(λ)}(T).

(b) β extends to an automorphism of L^{∗} if and only if λ extends to an automorphism of M.

Proof. The locality L^{∗} is rigidly isomorphic to a locality of the form L^{+}(λ), by 5.15(a) (and with μ the identity automorphism of M). Then also 5.15(a) yields an isomorphism α : L^{+}(λ) → L^{∗} with the properties required in point (a). Point (b) is then given by 5.15(b).

Definition 5.17. Let (L, Δ, S) be a locality with fusion system F = F_{S}(L). Let (Δ_{i})_{i=0}^{N} be a sequence of subsets of Δ, and let (R_{i})_{i=0}^{N} be a sequence of subgroups of S, such that each R_{i} is fully normalized in F. Then (Δ_{i}, R_{i})_{i=0}^{N} is an F-filtration of Δ if the following conditions hold.

(1) R_{0} is weakly closed in F, and Δ_{0} is the set of overgroups of R_{0} in S.

(2) For i > 0, Δ_{i} is the union of Δ_{i−1} with the set R_{i} of subgroups of S which contain an F-conjugate of R_{i}.

(3) For any U, V ∈ R_{i}, ⟨U, V ⟩ ∈ R_{i−1} if and only if U ≠ V.

(4) Δ = Δ_{N}.

Lemma 5.18. Let (L, Δ, S) be a locality, let F = F_{S}(L) be its fusion system, and let R ∈ Δ be weakly closed in F. Then there exists an F-filtration F = (Δ_{i}, R_{i})_{i=0}^{N} for L, such that R_{0} = R, and such that, for all i > 0, R_{i} is of maximal order among subgroups of S in Δ_{i} − Δ_{i−1}.

Proof. Take R_{0} = R and let Δ_{0} be the set of all overgroups of R in S. Then Δ_{0} is F-invariant as R is weakly closed in F. Now let m be an index with 1 ≤ m ≤ N. Now suppose that Δ_{m−1} has been given, and that Δ_{m−1} is F-invariant and overgroup closed in S, and that Δ_{m−1} ≠ Δ. Choose Q ∈ Δ − Δ_{m−1} so that |Q| is as large as possible, and let R_{m} be a fully normalized F-conjugate of Q. Define Δ_{m} to be the union of Δ_{m−1} with the set of subgroups of S which contain an F-conjugate of R_{m}. Then Δ_{m} is Finvariant and overgroup closed, so the process may be iterated until arriving at an index N with Δ_{N} = Δ. The points (1), (2), and (4) of definition 5.17 are given at once by this construction, while point (3) is immediate from the maximality in the choice of R_{m}. □

Section 6: The reduction to F F -pairs

Our aim in this section is to establish the Main Theorem, modulo a result (Proposition 6.10) on localities in finite groups. In order to state that result (to be proved in the following section), we begin by reviewing some notions from finite group theory.

For any finite p-group P, the set of elements z ∈ Z(P) such that z^{p} = 1 is a characteristic subgroup of P, often denoted Ω_{1}(Z(P)), but which we shall write as Z_{P}. A p-group V is elementary abelian if V = Z_{V}. Equivalently, V is elementary abelian if V is the underlying group of a finite-dimensional vector space Ṽ over the field 𝔽_{p} of p elements.

Let A be a group, and let D be a group on which A acts (from the right). Then [D, A] is by definition the subgroup of D generated by the set of commutators [g, a] = g^{−1}a^{−1}ga (g ∈ D, a ∈ A). If [D, A] ≤ C_{D}(A) one says that A acts quadratically on D, and one also expresses this condition by writing [D, A, A] = 1.

We begin with two elementary (and well-known) results.

Lemma 6.1. Let D be an abelian p-group admitting action (from the right) by a group X, and set V = Z_{D}. Let A be the set of elements a ∈ X of order p such that [D, a, a] = 1, and suppose that G = ⟨A⟩. Then [D, G] ≤ V.

Proof. Let g ∈ D, let a ∈ A, and set h = [g, a]. Then h ∈ C_{D}(a), and so h = h^{a^{k}} for all integers k. Thus:

h = g^{−1}g^{a} = (g^{a})^{−1}g^{a^{2}} = ⋯ = (g^{a^{p−1}})^{−1}g^{a^{p}}.

One then observes that h^{p} = g^{−1}g^{a^{p}}, and then h^{p} = 1 since a^{p} = 1.

Since D is abelian, we have [g, a]^{−1} = [g^{−1}, a], and [g_{1}g_{2}, a] = [g_{1}, a][g_{2}, a] for all g_{1}, g_{2} ∈ D. Thus

V ≥ {[g, a] | g ∈ D, a ∈ A} = [D, G].

□

Lemma 6.2. Let X be a finite group, let S be a Sylow p-subgroup of X, and let D be an abelian p-group. Assume that there is given an action X → Aut(D) of X on D. Set V = Z_{D}, and set

W = [V, O^{p}(X)]C_{V} (X)/C_{V} (X).

Then the following hold.

(a) C_{D}(S) ≤ [D, O^{p}(X)]C_{D}(X).

(b) Suppose that X = KS, where K ⊴ X is generated by elements that act quadratically on D. Then C_{D}(S) = C_{V} (S)C_{D}(X).

(c) Suppose that [C_{W} (S), X] = 1. Then [C_{D}(S), X] = 1.

Proof. As X = O^{p}(X)S there exists a right transversal {x_{1}, ⋯, x_{r}} for S in X such that each x_{i} is in O^{p}(X). Thus Ω = {Sx_{1}, ⋯, Sx_{r}} is the set of right cosets of S in X, and each x ∈ X defines a permutation of Ω, by right multiplication. That is, (Sx_{i})x = Sx_{j} for some j. Let g ∈ C_{D}(S), and set

h = g^{x_{1}} ⋯ g^{x_{r}}.

Then h^{x} = h for all x ∈ X, while also

h = g^{r}[g, x_{1}] ⋯ [g, x_{r}] = g^{r}d,

where d ∈ [D, O^{p}(X)]. As (p, r) = 1 and D is a p-group, we get h^{n} = gd^{n} for some n, and thus g = d^{−n}h^{n} ∈ [D, O^{p}(X)]C_{D}(X). That is, point (a) holds.

We continue the preceding setup in order to prove:

(*) If [C_{V} (S), X] = 1 then [C_{D}(S), X] = 1. Suppose by way of contradiction that [C_{V} (S), X] = 1 but that [C_{D}(S), X] ≠ 1. The element g in the proof of (a) may then be chosen so that g ∉ C_{D}(X) and with g^{p} ∈ C_{D}(X). Then g^{p} ≠ 1. As (g^{x})^{p} = (g^{x})^{p} = g^{p} for all x ∈ X, we get h^{p} = g^{pr} (where h is defined as in the proof of (a)), and then h^{p} ≠ 1 as r is relatively prime to p. Set D_{0} = ⟨d^{p} | d ∈ D⟩. We may assume that D = ⟨g^{X}⟩, so D_{0} = ⟨g^{p}⟩ = ⟨h^{p}⟩ is cyclic. Then D = V ⟨h⟩, and C_{D}(S) = C_{V} (S)⟨h⟩ = C_{D}(X). This contradiction completes the proof of (*).

Now suppose that [C_{W} (S), X] = 1. Applying (a) with V in the role of D, we obtain C_{V} (S) ≤ [V, X]C_{V} (X). Then [C_{V} (S), X] ≤ C_{V} (X) by the definition of W, and so [C_{V} (S), O^{p}(X)] = 1. As X = O^{p}(X)S, (c) is proved.

Finally, assume the hypothesis of (b). Then O^{p}(X) ≤ K, and [D, O^{p}(X)] ≤ V by 6.1. Then (a) yields C_{D}(S) ≤ C_{D}(X)V, so

C_{D}(S) = C_{D}(S) ∩ C_{D}(X)V = C_{D}(X)(C_{D}(S) ∩ V) = C_{D}(X)C_{V} (S),

and (b) holds. □

Definition 6.3. Let M be a finite group, let S a Sylow p-subgroup of M, and set Y = O_{p}(M). Then M is p-reduced, and (M, S, Y) is a reduced setup, if

C_{M}(Y) ≤ Y, C_{S}(Z(Y)) = Y and O_{p}(M/C_{M}(Z(Y))) = 1.

Lemma 6.4. Let (M, S, Y) be a reduced setup, and set D = Z(Y), V = Z_{Y}, and G = M/C_{M}(Z(Y)). Let A be an abelian p-subgroup of G. Then V = Z_{D}, and the following hold.

(a) C_{M}(D) = C_{M}(V).

(b) If A acts quadratically on V then A is elementary abelian. In particular, if A acts quadratically on D, then A is elementary abelian.

Proof. Evidently V = Z_{D}, and C_{M}(D) ≤ C_{M}(V) ⊴ M. But also O^{p}(C_{M}(V)) ≤ C_{M}(D), by [Theorem 5.3.10 in Gor]. Thus, the image of C_{M}(V) in M/C_{M}(D) is a normal psubgroup of M/C_{M}(V), and then since M is p-reduced we obtain point (a). Point (b) is given by [9.1.1(c) in KS]. □

The following result shows how to isolate a reduced setup from any finite group M such that C_{M}(O_{p}(M)) ≤ O_{p}(M).

Lemma 6.5. Let M be a finite group with C_{M}(O_{p}(M)) ≤ O_{p}(M), and let S be a Sylow p-subgroup of M. Then there exists a unique largest (with respect to inclusion) subgroup D of Z(O_{p}(M)) such that Z(S) ≤ D ⊴ M and such that O_{p}(M/C_{M}(D)) = 1. Moreover, the following hold for Y := C_{S}(D), H := N_{M}(Y), and F := F_{S}(M).

(a) (H, S, Y) is a reduced setup.

(b) Y is strongly closed in F.

Proof. We aim first of all to define subgroups Y_{k} and D_{k} of S, for all k ≥ 0, with the following properties:

(1) Y_{k} is strongly closed in F, and C_{M}(Y_{k}) ≤ Y_{k}.

(2) D_{k} = Z(Y_{k}) ⊴ M. The conditions (1) and (2) are satisfied by Y_{0} := O_{p}(M) and D_{0} = Z(Y_{0}). We shall define Y_{k} and D_{k} for k ≥ 1, in the following recursive way. Take Y_{k} to be the pre-image in S of O_{p}(M/C_{M}(D_{k−1})) and take D_{k} = Z(Y_{k}). We now check that the conditions (1) and (2) hold for Y_{k} and D_{k} under the asssumption that they hold for Y_{k−1} and D_{k−1}.

As D_{k−1} ⊴ M, also C_{M}(D_{k−1}) ⊴ M, and then C_{M}(D_{k−1})P ⊴ M where P is defined to be the pre-image in S of O_{p}(M/C_{M}(D_{k−1})). As P is a Sylow p-subgroup of the normal subgroup C_{M}(D_{k−1})P of M, P is strongly closed in F. Again as P is Sylow in C_{M}(D_{k−1})P, we have C_{S}(D_{k−1}) ≤ P, and so Y_{k−1} ≤ P. As C_{M}(Y_{k−1}) ≤ Y_{k−1} we obtain C_{M}(P) ≤ P, and Z(P) ≤ Z(Y_{k−1}) = D_{k−1}. Here M = C_{M}(D_{k−1})N_{M}(P) by the Frattini Lemma, so Z(P) ⊴ M. Thus (1) and (2) hold where Y_{k} = P and D_{k} = Z(P).

As M is finite there exists n minimal subject to D_{n} = D_{n+1}. Set D = D_{n} and Y = Y_{n+1}. Then C_{S}(D) ≤ Y and D = Z(Y), so Y = C_{S}(D) is a Sylow p-subgroup of C_{M}(D). The Frattini Lemma yields M = C_{M}(D)H where H = N_{M}(Y). Also Y C_{M}(D)/C_{M}(D) = O_{p}(M/C_{M}(D)), so as Y ≤ C_{M}(D) we have O_{p}(M/C_{M}(D)) = 1. Since M = C_{M}(D)H we get M/C_{M}(D) ≅ H/C_{H}(D), so also O_{p}(H/C_{H}(D)) = 1. As O_{p}(H)C_{H}(D)/C_{H}(D) ≤ O_{p}(H/C_{H}(D)) = 1 it follows that O_{p}(H) ≤ C_{S}(D) = Y, so O_{p}(H) = Y. Since C_{H}(Y) ≤ Y by (1), (H, S, Y) is then a reduced setup.

We now establish the uniqueness and maximality of D. Thus, let U be a subgroup of Z(Y_{0}) such that Z(S) ≤ U ⊴ M and such that O_{p}(M/C_{M}(U)) = 1. It will suffice to show that U ≤ D. Assuming otherwise, we have [U, Y] ≠ 1. Since O_{p}(M/C_{M}(U)) = 1 we have U ≤ Z(O_{p}(M)). That is, U ≤ D_{0}, and so there is a largest index n such that U ≤ D_{n}. Then C_{M}(U)Y_{n+1}/C_{M}(U) is a non-identity normal p-subgroup of M/C_{M}(U), contrary to O_{p}(M/C_{M}(U)) = 1. We conclude that U ≤ D, as required. □

In what follows, the group H = N_{M}(Y) in the preceding lemma will be called the reduced core of M with respect to S.

We next review the definition of a certain characteristic subgroup of an arbitrary finite p-group S, and of some terms related to it. All of the themes, and much of the terminology and notation that will be introduced here, have their origin in early work of John Thompson, and they have been of fundamental importance to the p-local viewpoint in finite group theory ever since.

Let d(S) be the maximum, taken over all abelian subgroups A of S, of the numbers |A|. As in [Th], we take A(S) to be the set of all abelian subgroups A of S such |A| = d(S), and we set

(*)  J(S) := ⟨A(S)⟩.

Notice that J(S) is the unique subgroup of S which is isomorphic to J(S). Because of this, the operator ”J” has the following inheritance property: If R is a subgroup of S and J(S) ≤ R, then J(S) = J(R). In particular, J(S) is weakly closed in any fusion system on S. One observes that J(S) has the further property that it is centric in any fusion system on S.

Remark. The tendency in the last 30 years or more, has been to define A(S) to be the set of elementary abelian subgroups of S of maximal order, and then to define J(S) by the formula (*). But the formulation that we have chosen is the one which is needed for the task at hand.

Let G be a finite group, let D be a finite abelian p-group, and suppose that there is given a faithful group action (from the right, as always) of G on D. An abelian psubgroup A of G is an offender (on D, in G) if |A||C_{D}(A)| ≥ |D|. An offender A is non-trivial if A ≠ 1, and A is a best offender if |A||C_{D}(A)| ≥ |B||C_{D}(B)| for every subgroup B of A. A quadratic offender is an offender A such that [D, A, A] = 1. Write A_{D}(G) for the set of best offenders in G on D, and set J_{D}(G) = ⟨A_{D}(G)⟩.

We shall most often be interested in the situation where D is a normal abelian psubgroup of a finite group M, and where G = M/C_{M}(D). In this case, we say that (G, D) is an F F -pair if O_{p}(G) = 1 and J_{D}(G) = G. The structure of F F -pairs in the case that D is elementary abelian is analyzed in [MS2], using the CFSG (the classification of the finite simple groups), and the preceding terminology is adapted from [MS2].

Lemma 6.6. Let D be a finite abelian p-group and let G be a finite group acting faithfully on D.

(a) Every non-trivial offender in G (on D) contains a non-trivial best offender.

(b) Every non-trivial best offender in G contains a non-trivial quadratic best offender.

(c) If A is a best offender in G, and U is an A-invariant subgroup of D, then A/C_{A}(U) is a best offender in N_{G}(U)/C_{G}(U) on U.

Proof. Point (a) is a triviality: If A ≠ 1 is an offender on D, one has only to choose a subgroup B ≠ 1 of A so as to maximize |B||C_{D}(B)|, in order to obtain a best offender on D.

Point (b) is essentially given by the Timmesfeld replacement theorem [Ti]; but here it must be noted that in the hypothesis of Timmesfeld’s theorem the groups D and A are assumed to be elementary abelian. A one-page proof of this result (with the extra, but in fact extraneous, hypothesis concerning D and A) is given as [Theorem 2 in Ch]. At no point in the proof is the extra hypothesis used.

The argument for (c) is again a case of asking the reader to check a very short proof (about half a page) in which there is an extraneous hypothesis as mentioned in the proof of (b). In this case, the relevant result is [Lemma 2.5(c) in MS1]. □

Lemma 6.7. Let M be a finite group, let S ∈ Syl_{p}(M), let D be a normal abelian psubgroup of M, and set G = M/C_{M}(D). Let A ∈ A(S), and set Y = C_{S}(D). Then the following hold.

(a) The image of A in G is a best offender on D.

(b) J(S) ≰ Y ⟺ J(S) ≠ J(Y) ⟹ J_{D}(G) ≠ 1.

Proof. We provide the standard argument for the convenience of the reader. Let \overline{A} be the image of A in G, and suppose that \overline{A} is not a best offender on D. Thus, there exists a subgroup \overline{B} ≤ \overline{A} such that

(1) |\overline{B}||C_{D}(\overline{B})| > |\overline{A}||C_{D}(\overline{A})|.

Let B be the preimage of \overline{B} in A, and set B^{∗} = C_{D}(B)B. Then:

(2) C_{B}(D) = C_{A}(D).

As A ∈ A(S) we have C_{D}(A) = D ∩ A, and then also D ∩ A = C_{D}(B) ∩ A. Thus:

(3) C_{D}(B) ∩ A = D ∩ A = C_{D}(A).

We now obtain:

|B^{∗}| = |C_{D}(B)||B|/|C_{D}(B) ∩ B|
= |C_{D}(\overline{B})||\overline{B}||C_{B}(D)|/|C_{D}(B) ∩ B|
≥ |C_{D}(\overline{B})||\overline{B}||C_{B}(D)|/|C_{D}(B) ∩ A|
> |C_{D}(\overline{A})||\overline{A}||C_{A}(D)|/|C_{D}(B) ∩ A| by (1) and (2)
= |C_{D}(\overline{A})||\overline{A}||C_{A}(D)|/|C_{D}(A)| by (3)
= |C_{D}(A)||A|/|C_{D}(A)| = |A|.

This contradicts the maximality of |A| among the abelian subgroups of S, and completes the proof of (a). Point (b) follows from (a) and from the inheritance property of the J-operator, mentioned above. □

This completes the background material for this section. The next result lists some criteria for extending rigid automorphisms of localities within finite groups to automorphisms of the groups themselves.

Lemma 6.8. Let M be a finite group, let S ∈ Syl_{p}(M), set X = O_{p}(M), and assume that C_{M}(X) ≤ X. Set F = F_{S}(M), and let Γ be a non-empty, F-invariant, overgroupclosed set of subgroups of S, such that X ≤ Q for all Q ∈ Γ. Set L = L_{Γ}(M).

(a) Suppose that M = C_{M}(Z(X))S. Then L is the unique F-natural Γ-linking system, up to a unique rigid isomorphism. In particular, the identity automorphism is the unique rigid automorphism of L.

(b) Let H be the reduced core of M with respect to S, set Y = O_{p}(H), and let Γ_{Y} be the set of all Q ∈ Γ such that Y ≤ Q. Let γ be a rigid automorphism of L, let γ_{H} be the restriction of γ to L_{Γ_{Y}} (H) (see 3.9), and suppose that γ_{H} extends to an automorphism of H. Then γ extends to an automorphism of M.

(c) Set D = Z(X) and assume that C_{M}(D)/X is a p′-group. Assume also that there exists Q ∈ Γ such that M = C_{M}(D)N_{M}(Q), and an automorphism γ of L such that γ restricts to the identity automorphism of N_{M}(Q). Then γ is the identity automorphism of L.

Proof. (a) Let M be a minimal counterexample, and choose an F-filtration (Γ_{k}, T_{k})_{k=0}^{N} for Γ, with Γ_{0} = {S} (see 5.17). Let L_{k} be the restriction of L to Γ_{k}. Then L_{0} = N_{M}(S). As Z(S) ≤ Z(X), we have Z(S) ≤ Z(M), and then 1.10(b) shows that for any F-natural Δ_{0}-linking system K_{0}, there is a unique rigid isomorphism K_{0} → L_{0}.

Let n be the largest index such that L_{n} is unique up to a unique rigid isomorphism, in the preceding sense. Then, without loss of generality we may assume that n = N − 1. Let K be an F-natural Γ-linking system, and let K_{n} be the restriction of K to Γ_{n}. There is then no loss of generality in taking K_{n} = L_{n}. Set T = T_{N}, and set K_{T} = N_{K}(T) and L_{T} = N_{L}(T). Observe that

O^{p}(N_{M}(T)) ≤ O^{p}(M) ≤ C_{M}(Z(X)) ≤ C_{M}(Z(T)).

Thus, the hypothesis of 6.8(a) holds with N_{M}(T) in place of M, and with Γ_{T} := {Q ∈ Γ | T ⊴ Q} in place of Γ. As M is a minimal counterexample, we conclude that the identity automorphism of L_{T} is the unique rigid automorphism of L_{T}. Now 5.15(b) yields a rigid isomorphism γ : K → L. Since γ restricts to the identity automorphism on L_{n} and on N_{M}(T) so γ is uniquely determined, by 5.16(a).

(b) Set E = Z(Y) and note that M = C_{M}(E)H by 6.5. Let β_{H} be an extension of γ_{H} to an automorphism of H. Then β_{H} = c_{z} for some z ∈ Z(S), by 1.10(c). Now let β be the automorphism c_{z} of M. In order to complete the proof of (b) it suffices to show that β restricts to γ on L. Both c_{z} and γ restrict to the identity map on C_{M}(E)S, by (a). Set K = C_{L}(D). Then K is a partial normal subgroup of L by 3.9, and then the Frattini Lemma 4.6 for localities yields the result that each f ∈ L is a product f = gh, with g ∈ K and h ∈ H. Then

fγ = (gh)γ = (gγ)(hγ) = g(hβ) = gh^{z} = (gh)^{z} = f^{z} = fβ,

as required.

(c) Let A_{0} be the set of L-essential subgroups of S (as defined in 2.14), and set A = A_{0} ∪ {S}. By hypothesis, each member of Γ contains X, so C_{M}(P) ≤ P for all P ∈ Γ. By 2.16, each f ∈ L is A-decomposable, so in order to prove (c) it suffices to show that γ restricts to the identity map on O^{p′}(N_{M}(P)) for each P ∈ A_{0}, and to the identity map on N_{M}(S).

Fix P ∈ A, and set N = O^{p′}(N_{M}(P)) (or N = N_{M}(S) if P = S). Suppose first that Q ≤ P and let g ∈ N. By hypothesis, g = g_{1}g_{2} (where the product is taken in M), and where g_{1} ∈ N_{M}(Q) and g_{2} ∈ C_{M}(D). Then Q^{g_{2}} = Q^{g} ≤ P (and thus (g_{1}, g_{2}) ∈ D(L) via P). As C_{M}(D)/X is a p′-group, and X ≤ Q, it follows that Q^{g_{2}} = Q, and thus N ≤ N_{M}(Q). As γ restricts to the identity map on N_{M}(Q), there is no more to prove in this case. In particular, we have shown that γ restricts to the identity map on N_{M}(S).

We are now reduced to the case where P ∈ A_{0}, and where Q ≰ P. Then N_{Q}(P) ≰ P. On the other hand, N_{Q}(P)C_{M}(D) ⊴ C_{M}(D)N, and thus N_{Q}(P)C_{N}(D)P/C_{N}(D) is a non-identity normal p-subgroup of N/C_{N}(D), properly containing C_{N}(D)P/C_{N}(D). Since N/P has a strongly p-embedded subgroup, it follows that N/C_{N}(D) is a p-group. Thus N = C_{N}(D)N_{S}(P), and point (a) implies that γ restricts to the identity map on N. □

Lemma 6.9. Let F be a constrained fusion system on S, let M be a model for F, let H be the reduced core of M with respect to S, and set Y = O_{p}(H). Suppose that there is given an F-natural Γ-linking system (L, Γ, S) such that Y ∈ Γ, and such that O_{p}(M) ≤ P for all P ∈ Γ. Suppose also that there is given an isomorphism β : N_{L}(Y) → H of groups, such that β restricts to the identity map on S. Then β extends to an isomorphism L → L_{Γ}(M).

Proof. As Y is weakly closed in F, by 6.5(b) 5.18 implies that there is an F-filtration F = (Δ_{i}, T_{i})_{i=0}^{N} of Γ, in which Δ_{0} is the set of overgroups of Y in S. For each k, 1 ≤ k ≤ N, let L_{k} be the restriction of L to Γ_{k}, and set M_{k} = L_{Γ_{k}}(M). As Y ∈ Γ, β is an isomorphism L_{0} → M_{0} = H. Let n be the largest index such that β extends to an isomorphism β_{n} : L_{n} → M_{n}. Thus n ≥ 0, and we may assume that n < N as otherwise there is nothing to prove. There is then no loss of generality in assuming further that n = N − 1.

Set M = L_{Γ}(M). Then C_{M}(O_{p}(M)) ≤ O_{p}(M) as M is a model for the constrained fusion system F = F_{S}(M). Since each P ∈ Γ contains O_{p}(M), by assumption, M is then a Γ-linking system. Set T = T_{N}, K = N_{M}(T), Σ = {P ∈ Γ_{n} | T ⊴ P } and K = L_{Σ}(K). As T ∈ Γ, both N_{L}(T) and K are models for F_{N_{S}(T)}(K), so 1.10(b) yields an isomorphism γ : N_{L}(T) → K which restricts to the identity map on N_{S}(T). We may therefore apply 5.15(a) with K in the role of M, and find that there are rigid isomorphisms λ and λ′ from N_{L_{n}}(T) to K such that L and M are rigidly isomorphic to L^{+}(λ) and L^{+}(λ′), respectively. Moreover, λ is given explicitly as β_{T} ∘ γ^{−1} where β_{T} is the restriction of β_{n} to N_{L_{n}}(T), while λ′ is the composition β_{T} ∘ ι where ι is the identity map on K.

Let α be the rigid automorphism λ^{−1} ∘ λ′ of K. By 5.15(b) it suffices to show that α extends to an automorphism of K in order to conclude that β extends to an isomorphism L → M.

Since T ∉ Γ_{0} we have Y ≰ T, so T < N_{Y} (T)T, and then N_{Y} (T)T ∈ Γ_{n} by the construction of F. Thus α restricts to an automorphism α_{0} of N_{K}(N_{Y} (T)T) which centralizes N_{S}(T), and then 1.10(c) yields α_{0} = c_{z} for some z ∈ Z(N_{S}(T)). We now observe that N_{Y} (T) = N_{S}(T) ∩ C_{S}(D) is a Sylow subgroup of C_{K}(D), and hence

K = C_{K}(D)N_{K}(N_{Y} (T)) = C_{K}(D)N_{K}(N_{Y} (T)T)

by the Frattini Lemma. By 3.9, C_{K}(D) is a partial normal subgroup of K, and evidently N_{S}(T) ∩ C_{K}(D) = N_{Y} (T). The Frattini Lemma for localities (4.6) then yields

K = C_{K}(D)N_{K}(N_{Y} (T)) = C_{K}(D)N_{K}(N_{Y} (T)T),

since N_{K}(N_{Y} (T)) = N_{K}(N_{Y} (T)T), and since N_{Y} (T)T ∈ Σ.

Let C be the locality L_{Σ}(C_{K}(D)N_{S}(Y)). Then the identity automorphism of C is the unique rigid automorphism of C by 6.8(a). In particular, the restriction α_{1} of α to C is the identity automorphism, and so c_{z} induces α_{1} on C. By 4.6 each f ∈ K is a product gh taken in K, where g ∈ C_{K}(D) and h ∈ N_{K}(N_{Y} (T)). Since Y ≤ T we have z ∈ C_{M}(Y) = D, and so

fα = (gh)α = (gα_{1})(hα_{0}) = gh^{z} = (gh)^{z} = f^{z},

and thus α extends to the automorphism c_{z} of K. As remarked earlier, this suffices to complete the proof. □

Let (M, S, Y) be a reduced setup, and set D = Z(Y) and V = Z_{Y}. Recall from 6.4(a) that C_{M}(V) = C_{M}(D). Set G = M/C_{M}(V), and recall that, for any subgroup K of G, J_{D}(K) is defined to be the subgroup of K generated by the best offenders in K on D, and similarly for J_{V} (K). For the remainder of this paper, whenever such a setup is given, and whenever H is a subgroup of M, we write J(H, D) for the preimage in H of J_{D}(H/C_{H}(D)). We define J(H, V) analogously, relative to J_{V} (H/C_{H}(V)).

The proof of the following proposition will be postponed to the next (concluding) section.

Proposition 6.10. Let (M, S, Y) be a reduced setup, and set D = Z(Y). Let Γ be the set of all overgroups Q of Y in S such that J(Q, D) ≠ Y, and assume that S ∈ Γ. Set L = L_{Γ}(M), and let γ be a rigid automorphism of L. Then γ extends to an automorphism of M.

Proposition 6.11. Let M be a finite group, and assume that 6.10 holds for all reduced setups (M′, S′, Y′) with |M′| < |M|. Let S be a Sylow p-subgroup of M, and let X be a normal p-subgroup of M with C_{M}(X) ≤ X. Set Y = O_{p}(M), set F = F_{S}(M), D = Z(Y), and let Γ be an F-invariant, overgroup-closed collection of overgroups of X in S such that

Q ∈ Γ ⟹ J(Q, D) ∈ Γ.

Assume that J(S, D) ∈ Γ. Then every rigid automorphism of L_{Γ}(M) extends to an automorphism of M.

Among all pairs (M, Γ) for which 6.11 fails, choose one so that first |M| is as small as possible, and then so that |X| is as large as possible. Set L = L_{Γ}(M), and fix a rigid automorphism γ of L such that γ has no extension to an automorphism of M. Set V = Z_{D}.

6.11.1. X = Y, and Y ∉ Γ.

Proof. If Y ∈ Γ then L_{Γ}(M) = M, and the conclusion of 6.11 holds trivially. Thus Y ∉ Γ. Now suppose that X is a proper subgroup of Y, and let Γ_{Y} be the set of all Q ∈ Γ such that Y ≤ Q. The maximality of |X| in the choice of (M, Γ) then implies that every rigid automorphism of L_{Γ_{Y}} (M) extends to an automorphism of M. Let γ_{Y} be the restriction of γ to L_{Γ_{Y}} (M) and let β be an extension of γ_{Y} to an automorphism of M.

Let Q ∈ Γ. Then QY ∈ Γ_{Y} and N_{M}(Q) ≤ N_{M}(QY), so γ and β agree on N_{M}(Q) for all Q ∈ Γ. By 3.8, β restricts to an automorphism β_{0} of L_{Γ}(M), and now 3.10 shows that β_{0} ∘ γ is the identity automorphism of L_{Γ}(M). Thus, β is an extension of γ to an automorphism of M. □

6.11.2. (M, S, Y) is a reduced setup. In particular, C_{M}(D) = C_{M}(V).

Proof. Suppose false, let H be the reduced core of M with respect to S, set R = O_{p}(H), and set L_{H} = L_{Γ}(H). If R ∈ Γ then L_{H} = N_{M}(R) = H, and then γ extends to an automorphism of M by 6.8(b). We conclude that R ∉ Γ, and hence no member of Γ is contained in R.

Let Γ_{R} be the set of all products RP with P ∈ Γ. Thus, Γ_{R} also has the usual meaning. Namely, Γ_{R} is the set of members Q of Γ such that R ⊴ Q. We note that since R = O_{p}(H) and H is reduced, we have R = C_{S}(Z(R)). For Q ∈ Γ_{R} write J(Q, Z(R)) for the preimage in Q of J_{Z(R)}(Q/R). Since Y ≤ R, and since J(Q, D) ∈ Γ for all Q ∈ Γ by hypothesis, 6.6(c) implies that J(Q, Z(R)) ∈ Γ_{R} for each Q ∈ Γ_{R}. Since R ∉ Γ_{R}, by the preceding paragraph, the hypothesis of 6.11 is fulfilled with (H, Γ_{R}) in place of (M, Γ). Here H is a proper subgroup of M as (M, S, Y) is not a reduced setup, so we conclude that the restriction γ_{H} of γ to L_{Γ_{R}}(H) extends to an automorphism of H. We appeal again to 6.8(b) and conclude that γ extends to an automorphism of M, contrary to our choice of (M, γ). Thus, (M, S, Y) is reduced, and then C_{M}(D) = C_{M}(V) by 6.4(a). □

6.11.3. M = J(M, D).

Proof. Set K = J(M, D). Thus, K is the preimage in M of the subgroup of M/C_{M}(D) generated by best offenders on D, so K ⊴ M. Set S_{0} = S ∩ K and let Φ be the set of all Q ∈ Γ with Q ≤ S_{0}. Then Φ is F-invariant. Since Q ∈ Γ implies J(Q, D) ∈ Γ, by the hypothesis in 6.11, we get Q ∩ S_{0} ∈ Φ for all Q ∈ Γ. In particular, S_{0} ∈ Φ since (by hypothesis) S ∈ Γ. Then Φ is a non-empty, overgroup-closed collection of subgroups of S_{0}.

Set K = L_{Φ}(K). Then β restricts to a rigid automorphism κ of K, by 3.8. Assume now that K ≠ M, and set X_{0} = X ∩ K. The hypothesis of 6.11 is satisfied with (K, S_{0}, X_{0}, F_{S_{0}}(K), Φ) in place of (M, S, X, F, Γ), and hence, by the minimality of |M| as a counterexample to 6.11, κ extends to an automorphism λ of K. But then γ extends to an automorphism of M by 4.15, and contrary to the choice of (M, Γ). Thus M = K. □

Let Φ now denote the set of all subgroups P of S such that Y ≤ P and such that J(P, D) ≠ Y. If Φ = Γ then 6.11.1 through 6.11.3 yield the hypothesis of 6.10, and then 6.10 yields an extension of γ to an automorphism of M. Thus Γ is a proper subset of Φ. Choose T ∈ Φ − Γ so that:

(1) T is fully normalized in F,

(2) |J(T, D)| is as large as possible subject to (1), and

(3) |T | is as small as possible subject to (2).

6.11.4. Y is a proper subgroup of T, and N_{M}(T) is a proper subgroup of M.

Proof. By assumption, Y ≤ T, and since Φ ≠ Γ we know that Y ≠ T. Then T ≰ Y, and so T is not normal in M. □

6.11.5. The following hold.

(a) T = J(T, D).

(b) If P and P′ are distinct F-conjugates of T, then ⟨P, P′⟩ ∈ Γ.

Proof. Since T ∈ Φ we have J(T, D) ≠ Y, and then J(T, D) ∈ Φ by definition. Since J(T, D) has a fully normalized F-conjugate, point (a) then follows from the minimality condition (3) in the choice of T. Then also P = J(P, D) for any F-conjugate P of T. For F-conjugates P, P′ of T, the definition of the “J(−, D)”-operator then yields ⟨P, P′⟩ = J(⟨P, P′⟩, D). For P ≠ P′ we get |J(⟨P, P′⟩, D)| > |J(T, D)|, and then (b) follows from the condition (2) in the choice of T. □

Set M_{T} = N_{M}(T), set R = N_{S}(T), and let Γ^{+} be the union of Γ with the set of subgroups of S which contain an F-conjugate of T. Let Γ_{T} be the set of all Q ∈ Γ with T ≤ Q ≤ R, and set L_{T} = N_{L}(T). Thus:

L_{T} = N_{L}(T) = L_{Γ_{T}} (M_{T}).

6.11.6. Let γ_{T} be the restriction of γ to L_{T}. Then γ_{T} does not extend to an automorphism of M_{T}.

Proof. Suppose γ_{T} extends to an automorphism of M_{T}. Then 5.16(b), with M_{T} in the role of M and with γ in the role of β, yields an extension of γ to an automorphism of M. Thus, as (M, γ) is a counterexample to 6.11, no such extension of γ_{T} exists. □

Let H be the reduced core of M_{T} with respect to R, set X = O_{p}(H), and set U = Z(X). Thus H = N_{M_{T}} (X), and the Frattini Lemma yields

(*)  M_{T} = C_{M_{T}} (U)H.

Set H = N_{L_{T}} (X). Then C_{L_{T}} (U) is a partial normal subgroup of L_{T} by 3.9, and then 4.6 yields

(**)  L_{T} = C_{L_{T}} (U)H.

6.11.7. Let β be the restriction of γ_{T} to H. Then β does not extend to an automorphism of H. In particular, X ∉ Γ.

Proof. If β extends to an automorphism of H then γ_{T} extends to an automorphism of M_{T}, by 6.8(b), and contrary to 6.11.6. Thus no such extension of β exists. If X ∈ Γ then H = N_{H}(X) = N_{H}(X) = H, so we conclude that X ∉ Γ. □

proof of 6.11.. Since H is a proper subgroup of M by 6.11.4, the minimality of M in the choice of a counterexample to 6.11 implies that the conclusion of 6.11 holds with with (H, Γ_{X}) in place of (M, Γ). This contradicts 6.11.7, and so the proof is complete. □

Lemma 6.12. Let M, S, Y = X, D, F, and Γ be as in 6.11, and assume that 6.10 holds for all reduced setups (M′, S′, Y′) with |M′| < |M|. Then every F-natural Γ-linking system is rigidly isomorphic to L_{Γ}(M).

Proof. Set L = L_{Γ}(M) and let L′ be any other F-natural Γ-linking system. Set T_{0} = J(S, D) and set L_{0} = N_{L}(T_{0}) and L′_{0} = N_{L′}(T_{0}). Then L_{0} and L′_{0} are isomorphic groups, via an isomorphism which restricts to the identity map on S, by 1.10(b).

Let Γ_{0} be the set of overgroups of T_{0} in S. Among all groups T_{1} ∈ Γ − Γ_{0} such that T_{1} is fully normalized in F, choose T_{1} so as first to maximize |J(T_{1}, D)| and then so as to minimize |T_{1}|. Then T_{1} = J(T_{1}, D) and, as in the proof of 6.11.5(b), we find that any two distinct F-conjugates P and P′ of T_{1} generate a member of Γ_{0}. Let Γ_{1} be the union of Γ_{0} with the set of subgroups X of S such that X contains an F-conjugate of T_{1}, and then iterate this procedure, so as to obtain an F-filtration F = (Γ_{i}, T_{i})_{i=0}^{N} of Γ. Let L_{i} and L′_{i} be the restrictions of L and L′ to Γ_{i}, and let n be an index such that there exists a rigid isomorphism L_{n} → L′_{n}. Then n < N, or else there is nothing to prove. Set T = T_{n+1}, and set Δ = Γ_{n}.

Set K = L_{n} and K_{T} = N_{K}(T), and similarly define K′ and K′_{T}. Then K_{T} = L_{Δ_{T}} (N_{M}(T)), and L_{n+1} is rigidly isomorphic to K^{+}(ι), where ι is the identity automorphism of K_{T}, by 5.15(a). But also, L′_{n+1} is rigidly isomorphic to (K′)^{+}(λ) for some rigid isomorphism λ : K_{T} → L_{Δ_{T}} (N_{M}(T)), again by 5.15(a). Now, by 5.15(b)(i), it suffices to show that all rigid automorphisms of L_{Δ_{T}} (N_{M}(T)) extend to automorphisms of N_{M}(T), in order to complete the proof.

Set M_{T} = N_{M}(T), Y_{T} = O_{p}(M_{T}), D_{T} = Z(Y_{T}), and set R = N_{S}(T). Then R ∈ Δ by 5.4(a). Then J(R, D) ≠ T, by construction of the filtration F. Then, again by construction of F, we get J(R, D) ∈ Δ. If J(R, D) ≤ Y_{T} then Y_{T} ∈ Δ and L_{Δ_{T}} (M_{T}) = M_{T}. Since there is nothing to prove in that case, we may assume that J(R, D) ≰ Y_{T}. Then also J(R, D_{T}) ≰ Y_{T}, by 6.6(c). In fact, the preceding argument shows that J(Q, D_{T}) ≰ Y_{T} for any Q ∈ Δ. We may then apply 6.11, with (M_{T} , R, Δ) in place of (M, S, Γ), in order to conclude that all rigid automorphisms of L_{Δ_{T}} (M_{T}) extend to automorphisms of M_{T}, and to thereby complete the proof. □

Recall from 2.9 that a locality (L, Δ, S) is a centric linking system if it is a Δ-linking system, where Δ is the set of all F_{S}(L)-centric subgroups of P. Recall also from 2.17(a) that if L is a centric linking system then F_{S}(L) is saturated.

On the other hand, let F be a given saturated fusion system on S. By an F-centric linking system we mean a centric linking system (L, Δ, S) such that F = F_{S}(L). Thus, Δ is the set of F-centric subgroups of S if L is an F-centric linking system.

Assume now that the Main Theorem is false. We express this as a hypothesis, as follows.

Hypothesis 6.13. Proposition 6.10 holds, and F is a saturated fusion system on the p-group S such that one of the following holds.

(i) There exists no F-centric linking system on S.

(ii) There exist F-centric linking systems on S which are not rigidly isomorphic.

Set X_{0} = J(S), and define Δ_{0} to be the set of all overgroups of X_{0} in S. Then Δ_{0} is closed under F-conjugation since J(S) is weakly closed in F, and since J(S) = J(Q) for all Q ∈ Δ_{0}. As remarked earlier, it is immediate from the definition of J(S) that J(S) ∈ F^{c}, and hence Δ_{0} ⊆ F^{c}.

Now suppose that Δ_{0} ≠ F^{c}, and let X = X_{1} be the set of all X ∈ F^{c} − Δ_{0} such that X is fully normalized in F. Among all X ∈ X_{1}, choose X so that:

(1) d(X) is as large as possible,

(2) |J(X)| is as large as possible (subject to (1)),

(3) J(X) ∈ F^{c}, if possible, (subject to (1) and (2)), and (4) Subject to the conditions (1) through (3), |X| is as small as possible if J(X) ∈ F^{c},

and otherwise |X| is as large as possible. Set X_{1} = X, and define Δ_{1} to be the union of Δ_{0} with the set of subgroups of S which contain an F-conjugate of X_{1}. If Δ_{1} ≠ F^{c} we then repeat the above procedure, taking X_{2} to be the set of all X ∈ F^{c} − Δ_{1} such that X is fully normalized in F, and choosing X_{2} ∈ X_{2} according to the rules (1) through (4). By iteration, we arrive at a sequence of pairs

F = (Δ_{i}, X_{i})_{i=0}^{N}

where Δ_{N} = F^{c}. Recall now the notion of F-filtration from 5.17.

Lemma 6.14. F is an F-filtration of F^{c}. Moreover, each X_{i} may be chosen so that J(X_{i}) is fully normalized in F.

Proof. As X_{0} is weakly closed in F we have 5.17(1), while points (2) and (4) of 5.17 (with F^{c} in the role of Δ) hold by construction. Assuming now that F is not an F-filtration of F^{c}, we conclude that 5.17(3) fails to hold. There is then a smallest index n such that there exist F-conjugates P and P′ of X := X_{n} such that P ≠ P′ and such that ⟨P, P′⟩ ∉ Δ_{n−1}.

Set Q = ⟨P, P′⟩. Then d(Q) = d(X), and |J(Q)| = |J(X)|, and thus J(P) = J(Q) = J(P′). If J(P) ∈ F^{c} then X = J(X) by the minimality condition in (4). But in that case we obtain also P = J(P) and P′ = J(P′), and so P = P′, contrary to hypothesis. Thus J(X) ∉ F^{c}. But then P = Q = P′ by the maximality condition in (4), and again contrary to hypothesis. Thus, F is an F-filtration of F^{c}. The second part of the lemma follows from 1.7. □

Lemma 6.15. Let n be an index with 1 ≤ n ≤ N. Suppose that J(X_{n}) is F-centric, and let Q ∈ Δ_{n−1} with X_{n} ≤ Q. Then J(Q) ∈ Δ_{n−1}.

Proof. Suppose false, and let n be the smallest index for which the lemma fails. Set X = X_{n} and set Δ = Δ_{n−1}. As J(X) is F-centric, by assumption, condition (4) in the choice of X implies that X = J(X). If d(Q) > d(X), or if d(Q) = d(X) but J(Q) > J(X), then J(Q) ∈ Δ by the maximality conditions (1) and (2) in the choice of X. So, we conclude that J(Q) = J(X), and then J(Q) = X.

As Q ∈ Δ, there exists an index m with 0 ≤ m < n such that Q contains an Fconjugate U of X_{m}. Then the construction of F yields d(U) ≥ d(X). But d(U) ≤ d(Q) = d(X), so in fact d(U) = d(X). Similarly, we obtain |J(U)| = |J(X)|, and hence J(U) ≅ J(X). Here J(U) is F-centric by condition (3) in the construction of Δ_{m}, so U = J(U) = J(X) = X. This is contrary to m < n, and completes the proof. □

By 1.10(a) there exists a model M_{0} for the fusion system N_{F}(X_{0}), and M_{0} may then be viewed as an F-natural Δ_{0}-linking system. Any two such linking systems are rigidly isomorphic by 1.10(b), so there is a largest index n such that there exists an F-natural Δ_{n} linking system, and such that all such linking systems are rigidly isomorphic. By 6.13, we have n < N. Set Δ = Δ_{n}, and let L be the unique (up to rigid isomorphism) F-natural Δ-linking system.

Set X = X_{n+1}, set R = N_{S}(X), let M_{X} be a model for N_{F}(X), and let H be the reduced core of M_{X} with respect to R. Set Y = O_{p}(H), and set D = Z(Y). In view of 6.14 we may assume that J(X) is fully normalized in F.

Lemma 6.16. Suppose that Y ∉ Δ, and let Δ_{X} be the set of all P ∈ Δ with X ⊴ P. Then the following hold.

(a) Δ_{X} is the set of all subgroups Q of R, properly containing X, and such that J(Q) ≠ X.

(b) X = J(X) = J(Y).

(c) R = N_{S}(Y) ∈ Δ.

(d) Y is fully normalized in F, and H is a model for N_{F}(Y).

Proof. Let Q be a subgroup of R containing X, and suppose that Q ∉ Δ. The condition (1) in the choice of X then yields d(X) = d(Q), and thus A(X) ⊆ A(Q) and J(X) ≤ J(Q). Now 6.15 will complete the proof of (a), once it is shown that J(X) is F-centric.

Set B = C_{S}(J(X)). Then B is X-invariant, and

N_{B}(X) = C_{R}(J(X)) ≤ C_{R}(Z(X)) ≤ C_{R}(D) = Y,

and thus N_{B}(X) = C_{Y} (J(X)). But J(X) = J(Y) as Y ∉ Δ, and so

N_{B}(X) = C_{Y} (J(Y)) ≤ J(Y) ≤ X.

Thus B ≤ X, and since J(X) is fully normalized in F we conclude that J(X) is Fcentric. Then X = J(X) by condition (4) in the choice of X. This completes the proof of (a) and of (b).

Suppose that R ∉ Δ. Then J(R) = J(X) = X, by (a) and (b). Then

N_{S}(R) ≤ N_{S}(J(R)) = N_{S}(X) = R,

and hence R = S. Then X = J(S), and Y ∈ Δ_{0}, contrary to Y ∉ Δ. Thus, R ∈ Δ. We have Y ⊴ R by 6.5. Since N_{S}(Y) ≤ N_{S}(J(Y)) and J(Y) = X, we conclude that R = N_{S}(Y), completing the proof of (c).

Let ϕ ∈ Hom_{F}(R, S) be chosen so that Y′ := Y ϕ is fully normalized in F, and set X′ = Xϕ. Then N_{S}(Y′) ≤ N_{S}(X′) by (b), and since X is fully normalized it follows that |N_{S}(Y′)| ≤ |R|. Thus N_{S}(Y′) = Rϕ, and so Y is fully normalized in F. Point (d) then follows from 1.11. □

Lemma 6.17. Y ∈ Δ.

Proof. Suppose Y ∉ Δ. We check that the hypothesis of 6.11 (and hence also of 6.12) holds, with the role of (M, S, Y, F_{S}(M), Γ) being taken by (M_{X}, R, X, N_{F}(X), Δ_{X}). First, C_{M_{X}} (X) ≤ X as M_{X} is a model of the constrained fusion system N_{F}(X). Next, by 6.16(a), Q ∈ Δ_{X} implies J(Q) ≰ X, while 6.16(b) implies that X = J(X) (and hence that J(X) is F-centric). Then 6.15 yields J(Q) ∈ Δ_{X}, and so J(Q, Z(X)) ∈ Δ_{X} by 6.7(b). In particular, J(R, Z(X)) ∈ Δ_{X}, and so the claim has been verified. Since we are assuming 6.10, we are free to apply 6.12 to the setup with M_{X}; and so L_{Δ_{X}} (M_{X}) is the unique F_{Δ_{X}} (M_{X})-natural linking system, up to rigid isomorphism. Then all rigid isomorphisms N_{L}(X) → L_{Δ_{X}} (M_{X}) are M_{X}-equivalent, by 5.15(b). Now 5.15(c) applies with M_{X} in the role of M, and we conclude that there exists an F-natural Δ_{n+1}-linking system, and that any two such are rigidly isomorphic. This contradicts the maximality of n. □

Lemma 6.18. Y ∉ Δ.

Proof. Suppose Y ∈ Δ. Then Y ∈ Δ_{X} as X ≤ Y. Set H = L_{Δ_{X}} (H). Then H = N_{H}(Y) = H, and thus every rigid automorphism of H is in fact an automorphism of H. Then 6.8(b) applies, with M_{X} in the role of M, and so every rigid automorphism of L_{D_{X}} (M_{X}) extends to an automorphism of M_{X}. That is, all rigid automorphisms of L_{Δ_{X}} (M_{X}) are M_{X}-equivalent. As in the proof of 6.18, we conclude via 5.15(b) and 5.15(c) that there exists an F-natural Δ_{n+1}-linking system, that any two such are rigidly isomorphic, and thereby contradict the maximality of n. □

With 6.17 and 6.18 we now have a contradiction to Hypothesis 6.13. This contradiction provides a proof of the Main Theorem modulo Proposition 6.10. Thus, in order to complete the proof of the Main Theorem it remains to prove 6.10.

Section 7: The Main Theorem

Our aim in this section is to give a proof of Proposition 6.10, using the CFSG (the Classification of the Finite Simple Groups). As was pointed out at the end of the preceding section, this will complete the proof of the Main Theorem.

We continue that terminology and notation relating to F F -pairs. In particular, it is important to recall that our definition of J(P), for P a p-group, is given in terms of abelian (and not elementary abelian) subgroups of P of maximal order.

For ease of reference, we re-state 6.10, as follows.

Proposition 7.1. Let (M, S, Y) be a reduced setup, and set D = Z(Y). Let Γ be the set of all overgroups Q of Y in S such that J(Q, D) ≠ Y, and assume that S ∈ Γ. Set L = L_{Γ}(M), and let γ be a rigid automorphism of L. Then γ extends to an automorphism of M.

Among all (M, γ) satisfying the hypothesis of 7.1, and such that γ does not extend to an automorphism of M, fix (M, γ) so that |M| is as small as possible. We note that proposition 6.11 may then be applied to groups of order less than |M|.

Set F = F_{S}(L), and recall the definition of F-essential subgroup of S, from 2.14.

Lemma 7.2. Let A be the union of {S} with the set of F_{S}(L)-essential subgroups of S, and set M_{0} = ⟨{N_{M}(Q) | Q ∈ A⟩. Then M_{0} = M. In particular, there exists no proper subgroup of M which contains the partial subgroup L of M.

Proof. Suppose that M_{0} is a proper subgroup of M. We first show:

(1) L ⊆ M_{0}.

Indeed, in order to establish (1) it suffices, by 2.16, to show that C_{M}(P) ≤ P for all P ∈ Γ. But C_{M}(P) ≤ C_{M}(Y) = D = Z(Y) as (M, S, Y) is a reduced setup. Thus (1) holds.

Since L = L_{Γ}(M) we have also L = L_{Γ}(M_{0}). We now claim that the hypothesis of 6.11 is fulfilled with (M_{0}, Y, Γ) in the role of (M, X, Δ). Set Y_{0} = O_{p}(M_{0}), D_{0} = Z(Y_{0}), and D = Z(Y). Then Y ≤ Y_{0} and so D_{0} ≤ D. We must show that J(Q, D_{0}) ∈ Γ for each Q ∈ Γ. Set M̃ = M/C_{M}(D). As Q ∈ Γ, J(Q, D) ≠ Y, so there exists A with Y ≤ A ≤ Q such that the image Ã of A in M̃ is a best offender on D. By 6.6(c), Aut_{A}(D_{0}) is a best offender on D_{0}, so J(Q, D)C_{Q}(D_{0}) ≤ J(Q, D_{0}). Thus J(Q, D_{0}) ∈ Γ, as required.

Now 6.11 yields an extension of γ to an automorphism β of M_{0}. Then β = c_{z} for some z ∈ Z(S) by 1.10(c), and c_{z} is then also an extension of γ to an automorphism of M. □

In what follows, set V = Z_{Y}. That is, V is the subgroup (to be regarded as a vector space over the field 𝔽_{p} of p elements) of D consisting of those elements x ∈ D such that x^{p} = 1. Set G = M/C_{M}(D), and recall from 6.4(a) that also G = M/C_{M}(V).

Lemma 7.3. G is generated by quadratic best offenders on D, and any quadratic best offender on D is also a quadratic best offender on V.

Proof. Let G_{0} be the subgroup of G generated by the set of all subgroups A of G such that A is a quadratic best offender on D. Let M_{0} be the preimage of G_{0} in M, and set S_{0} = S ∩ M_{0}. Then Y ≤ O_{p}(M_{0}). The reverse inclusion holds since M_{0} ⊴ M, so Y = O_{p}(M_{0}).

Let Q ∈ Γ, and set P = Q ∩ S_{0}. Here Y = C_{S}(D) ≤ S_{0}, and the image Q̃ of Q in G contains a best offender on D, so by 6.6(b) there is a non-trivial quadratic best offender B̃ ≤ Q̃. The preimage B of B̃ in S is contained in P, by the definition of M_{0}, so P̃ ≠ 1. Now let Φ be the set of all subgroups Q ∩ S_{0} of S_{0} with Q ∈ Γ. Thus:

(*) Φ ⊆ Γ.

Let K be the partial normal subgroup M_{0} ∩ L of L, as given by 3.9. Then 3.10 gives K the structure of a locality K = L_{Φ}(M_{0}). Further, the hypothesis of 6.12 holds with (M_{0}, S_{0}, Γ_{0}) in the role of (M, S, Γ). By 4.15(a), γ restricts to an automorphism γ_{0} of K, and if M_{0} ≠ M then γ_{0} extends to an automorphism β_{0} of M_{0} by 6.12. Then γ extends to an automorphism of M by 4.15(b), contrary to hypothesis, and completing the proof. □

Any group that acts faithfully and quadratically on an elementary abelian p-group is itself elementary abelian, by [9.1.1(c) in KS]. For that reason the preceding lemma effects the transition from “abelian offenders on abelian p-groups” to “elementary abelian offenders on vector spaces over 𝔽_{p}” that is needed in order to apply the results of Meierfrankenfeld and Stellmacher [Theorems 1 and 2 in MS2] on F F -pairs.

The set of all non-identity subgroups X of G such that X = [X, G] is a poset, with respect to inclusion, consisting of normal subgroups of G. Define X to be the set of minimal elements of this poset. The elements of X are the J-components of G. The product of the set X of J-components is then a normal subgroup of G, henceforth to be denoted by G_{0}. Set V = Z_{Y} and set

W = [V, G_{0}]C_{V} (G_{0})/C_{V} (G_{0}).

For any X ∈ X set W_{X} = [W, X] and V_{X} = [V, X].

It will turn out that for each X ∈ X, either X is quasisimple, or p ∈ {2, 3} and X is isomorphic to the commutator subgroup of SL_{2}(p) (a group of order 3 if p = 2, and a quaternion group of order 8 if p = 3). It will be convenient to set up some further notation, in order to accomodate such solvable J-components. Thus, let X_{sol} be the set of all subgroups X of G such that X is a direct factor of G, X ≅ SL_{2}(p) (with p = 2 or 3), [X, X] ∈ X, and |V_{X}| = p^{2}. Let X^{∗} be the union of X_{sol} with the set of non-solvable J-components of G. The elements of X^{∗} will be referred to as the J^{∗}-components of G. Set G_{0}^{∗} = ⟨X^{∗}⟩.

Theorem 7.4.1. The following hold.

(a) Each J^{∗}-component of G is normal in G.

(b) G_{0}^{∗} is the direct product of the J-components of G.

(c) Let A ≤ G be a best offender on V. Then A is a best offender on every A-invariant subspace of V and on every A-invariant subspace of W.

(d) If A ≤ G is a best offender on V, and X ∈ X, then C_{A}(X) = C_{A}(V_{X}) = C_{A}(W_{X}), and either [X, A] = 1 or [X, A] = X.

(e) W is the direct sum of its subspaces W_{X} for X ∈ X^{∗}, and [V, X_{1}, X_{2}] = 0 whenever X_{1} and X_{2} are distinct members of X^{∗}.

(f) G = G_{0}R where R is the image of S in G.

Proof. See [Theorem 1 in MS2]. We remark that points (a) through (e) are “elementary” in that they are proved without appealing to the CFSG. □

In 7.4.2 and 7.4.3 we sum up the remaining parts of the General F F -module Theorem in the form that will be needed here, and eliminate some special cases. As in [MS2] we write U^{(r)} for the direct sum of r copies of a module U.

Theorem 7.4.2. Suppose that G has only one J^{∗}-component. Then one of the following holds (where q is a power of p).

(1) G is a linear group SL_{n}(q), and W is a direct sum U^{(r)} ⊕ (U^{∗})^{(s)}, where U is a natural module for G and U^{∗} is the dual of U. Moreover, if both r and s are non-zero then n ≥ 4.

(2) G is a classical group (unitary, symplectic, or orthogonal) in characteristic p, and

W is a direct sum U^{(r)} of natural modules for G. Specifically:

(i) G ≅ Sp_{2n}(q) (n ≥ 2), (ii) G ≅ SU_{n}(q) (n ≥ 4), (iii) p is odd, G ≅ Ω_{2n+1}(q) (n ≥ 2), (iv) G ≅ Ω_{2n}^{ϵ}(q) (n ≥ 3), or

(v) p = 2, G ≅ O_{2n}^{ϵ}(q) (n ≥ 3).

(3) p = 2, G is a symmetric group of degree n (n ≥ 5), and W is a natural module for G.

Proof. Let R be the image of S in G. In the list of groups and their modules given by [MS2], all but the three types listed above are eliminated by 7.2. In detail: by Theorem 2 in [MS2] G_{0} is either a group of Lie type in characteristic p or an alternating group. If G_{0} is of Lie type in characteristic p then, in the cases other than the above three, [MS2] states that either R contains a unique quadratic best offender A or that G ≅ Spin_{7}(q) and that W is a spin module of order q^{8}. In the case of a unique quadratic offender A, it follows from 6.6(b) that the image in G of the normalizer in M of any object in Γ is contained in N_{G}(A). Since every element of L is a product of elements of normalizers of objects, by 2.16, it follows that L is contained in the proper subgroup C_{M}(D)N_{M}(B), where B is the preimage of A in S; and we thereby obtain a contradiction to 7.2. In the case where G ≅ Spin_{7}(q), it is pointed out in [Theorem 2 in MS2] that every quadratic best offender A has the same commutator space on W. Then L is contained in the M-stabilizer of that subspace, and so in either case we contradict 7.2.

Finally, if G_{0} = G is an alternating group, then [MS2] (see 7.4.3 immediately following) says that R contains a unique quadratic best offender, leading again to a contradiction to 7.2. □

Proposition 7.4.3. Suppose that G is a symmetric group Sym(n) (n ≥ 5), and that W is a natural module for G. Let R be the image of S in G. Then every offender in G on W is a best offender, and one of the following holds.

(1) n is odd, and each offender A is generated by transpositions.

(2) n is even, and for any quadratic offender A ≤ R, there exists a set {t_{1}, ⋯, t_{k}} of pairwise commuting transpositions in R such that one of the following holds.

(a) A = ⟨t_{1}, ⋯, t_{k}⟩.

(b) n = 2k and A = ⟨t_{1}t_{2}, …, t_{l−1}t_{l}⟩×⟨t_{l+1}, ⋯, t_{k}⟩, for some l with 1 < l ≤ k.

(c) n = 2k and A = ⟨t_{1}t_{2}, s_{1}s_{2}⟩ × ⟨t_{3}, ⋯, t_{k}⟩, where s_{1} and s_{2} are commuting transpositions distinct from t_{1} and t_{2}, and where Supp(s_{1}s_{2}) = Supp(t_{1}t_{2}).

(d) n = 8 = |A| = |W/C_{W} (A)|, and A acts regularly on the standard G-set. Moreover, if A is a quadratic offender and |A| > |W/C_{W} (A)|, then n is even, and A is generated by the set of all transpositions in R.

Proof. See [Theorem 2 in MS2]. □

Lemma 7.5. G has a unique J^{∗}-component.

Proof. Suppose false, let \overline{K} := \overline{K}_{1} be a J^{∗}-component of G, and let \overline{K}_{2} be the product of the J^{∗}-components other than \overline{K}. Let K_{i} be the preimage of \overline{K}_{i} in M. Then K_{i} is S-invariant by 7.4.1(a). Set W_{i} = C_{W} (K_{i}). Then W_{i} is S-invariant, as is C_{M}(W_{i}). Set M_{i} = C_{M}(W_{i})S. Then M_{i} = K_{i}S by 7.4.1(e).

Set L_{i} = L_{Γ}(M_{i}), and set L = L_{Γ}(M). Further, define C_{i} to be the set of all g ∈ L such that [W_{i}, g] = 1. Then C_{i} is a partial subgroup of L, and in fact a partial normal subgroup since each W_{i} is M-invariant. Since Γ is S-invariant, an element g of M is in L_{1} if and only if g = hs for some h ∈ C_{M}(W_{2}) such that S_{g} ∈ Γ, and thus L_{i} = C_{i}S. Also, for any h ∈ C_{i} we have hγ ∈ C_{i}, since γ centralizes V by rigidity. Thus L_{i} is a γ-invariant locality contained in L. Let γ_{i} be the restriction of γ to L_{i}.

We have M_{i} = K_{i}S by 7.4.1(e), so M_{i} is a proper subgroup of M. We may then apply 6.11 with (M_{i}, Y) in place of (M, X), and thereby conclude that γ_{i} extends to an automorphism β_{i} of M_{i}. Then β_{i} centralizes S, and since C_{M}(Y) ≤ Y it follows from 1.10(c) that β_{i} is conjugation by z_{i} for some z_{i} ∈ Z(S).

Set D_{i} = [D, K_{i}]. Then Z(S) ≤ C_{D}(K_{i})D_{i} by 6.2(a), and we may therefore take z_{i} ∈ D_{i}. We now claim that z_{1} centralizes K_{2} (and by symmetry of argument, that z_{2} centralizes K_{1}). To prove the claim, we recall that [W, K_{1}] centralizes K_{2}. Observe also that D_{1} is K_{2}-invariant, since K_{1} ⊴ M. Set V_{1} = V ∩ D_{1} and set U_{1} = [V_{1}, K_{1}]/C_{[V_{1},K_{1}]}(K_{1}). Then U_{1} is M-isomorphic to W_{1}, and so [U_{1}, K_{2}] = 1. We now apply 6.2(c) with D_{1}, V_{1}, and K_{2} in place of D, V, and X, and conclude that [C_{D_{1}}(S), K_{2}] = 1. Thus, [z_{1}, K_{2}] = 1 as claimed, and similarly [z_{2}, K_{1}] = 1.

Since M = M_{1}M_{2} = K_{1}K_{2}S it now follows that g extends to the automorphism c_{z_{1}z_{2}} of M. This contradicts the choice of (M, γ), and completes the proof. □

Lemma 7.6. Let T be a subgroup of S, such that T is weakly closed in F and properly contains Y. Let Γ^{+} be the union of Γ with the set of subgroups P of S such that T ≤ P. Then γ extends to an automorphism of L_{Γ^{+}}(M).

Proof. Set Φ = Γ^{+}, K = L_{Φ}(M), L_{T} = N_{L}(T), and let Γ_{T} be the set of all Q ∈ Γ such that T ≤ Q. Then L_{T} = L_{Γ_{T}} (N_{M}(T)) since T ≤ S_{w} for all w ∈ D(L) ∩ W(N_{M}(T)).

Let H be the reduced core of N_{M}(T) with respect to S, set Y_{T} = O_{p}(H), and set D_{T} = Z(Y_{T}). Also, set H = N_{L}(Y_{T}), let γ_{T} be the restriction of γ to L_{T}, and let η be the restriction of γ to H. Recall that Y_{T} is weakly closed in N_{F}(T) by 6.5. Then, since T is weakly closed in F, it follows that also Y_{T} is weakly closed in F. If Y_{T} ∈ Γ then H = H by 2.3(c), and η is an automorphism of H. On the other hand, suppose that Y_{T} ∉ Γ, and let R be the set of all R ∈ Γ containing Y_{T}. Since J(Q, V) ∈ Γ for all Q ∈ Γ, by 6.6(c), it follows that J(R, V) ≰ Y_{T} for any R ∈ R. Setting V_{T} = Ω_{1}(D_{T}), 6.6(c) yields also J(R, V_{T}) ≠ Y_{T} for R ∈ R. Similarly, J(S, D) ≰ Y_{T}, and then J(S, D_{T}) ≠ Y_{T}. Thus, the hypothesis of 6.11 is satisfied with (H, R) in place of (M, Γ), and we conclude that η extends to an automorphism of H. Thus, in any case, η either is, or extends to, an automorphism of H, and then 6.8(b) implies that γ_{T} extends to an automorphism β_{T} of N_{M}(T).

Since T is weakly closed in F, it is vacuously true that any pair of distinct F-conjugates of T generates a member of Γ. Then 5.15(a) implies that γ extends to an automorphism β of K, such that β restricts to γ on L and to β_{T} on N_{M}(T). This yields the lemma. □

Definition 7.7. Let \overline{S} be the image of S in G, and let Q be the set of all non-identity subgroups U of \overline{S} such that N\overline{S}(U) ∈ Syl_{p}(N_{G}(U)) and such that N_{G}(U)/U has a strongly p-embedded subgroup. We say that G has an essential splitting if there exists a subgroup H of G having the following properties.

(1) N_{G}(\overline{S}) ≤ H.

(2) O_{p}(H) ≠ 1 and O_{p}(H) is weakly closed in F\overline{S}(G).

(3) For each U ∈ Q, either O^{p′}(N_{G}(U)) ≤ H or O^{p′}(N_{G}(U)) centralizes C_{W} (U).

Proposition 7.8. G has no essential splitting.

Proof. Suppose false, and let H be a subgroup of G satisfying the conditions (1) through (3) in definition 7.7. Set \overline{B} = O_{p}(H), and let B be the preimage of \overline{B} in S. Then B = S ∩ C_{M}(D)B is a Sylow p-subgroup of C_{M}(D)B, and it follows that B is weakly closed in F := F_{S}(M).

Let Σ be the union of Γ with the set of overgroups of B in S. By 5.17 there is an F-filtration F = (Σ_{i}, R_{i})_{i=0}^{N} of Σ with R_{0} = B, and with the property that R_{i} is fully normalized in F, and of maximal order subject to R_{i} ∉ Σ_{i−1} (1 ≤ i ≤ N). Thus, Σ_{0} is the set of overgroups of B in S, and Σ_{N} = Σ.

Set K = L_{Σ}(M) and let K_{i} be the restriction of K to Σ_{i}, as in 2.9.1. Thus K_{0} = N_{M}(B). Let σ be an extension of γ to an automorphism of K, as given by 7.6, and let σ_{i} be the restriction of σ to K_{i}. Then σ_{N} = σ, while σ_{0} is a rigid automorphism of the group N_{M}(B). Since H contains the image of S in G, we have S ≤ N_{M}(B), and then 1.10(c) yields σ_{0} = c_{z} (conjugation by z) for some z ∈ Z(S). If also σ is given on all of K by z-conjugation, then γ is given by z-conjugation on L, and γ extends to the automorphism c_{z} of M. Since (M, γ) is a counterexample to 7.1, we conclude that the largest index m such that σ_{m} = c_{z} is smaller than N.

Set R = R_{m+1}, set L = N_{M}(R), and let X be the subgroup of L generated by C_{L}(Z(S)) together with the set of all N_{L}(P) as P varies over the set of proper overgroups of R in N_{S}(R). Each such P is in Σ_{m} by the construction of F. The restriction of σ_{L} to L (see 3.8) acts on X, and σ_{L} ∘ c_{z^{−1}} centralizes a set of generators for X, so σ_{L} acts as c_{z} on X. If X = L then 5.15(a) implies that σ_{m+1} is given by c_{z} on all of K_{m+1}, contrary to the choice of m. Thus X ≠ L, and hence X/R is strongly p-embedded in L/R. Moreover, we now have [Z(S), O^{p′}(L)] ≠ 1.

For any subgroup E of L, let \overline{E} be the image of E in G. Then \overline{L} ≠ \overline{X} since C_{L}(D) ≤ X. Then \overline{X}/\overline{R} is strongly p-embedded in \overline{L}/\overline{R}, and then condition (3) in definition 7.7 says that either O^{p′}(\overline{L}) ≤ H or [C_{W} (S), O^{p′}(\overline{L})] = 1. Suppose O^{p′}(\overline{L}) ≤ H. As B ≤ R_{i} for all i, where B is weakly closed in F, it follows that L ≤ N_{M}(B), and hence σ_{L} = c_{z}. Again, 5.15(a) implies that σ_{m+1} = c_{z} on K_{m+1}, contradicting the choice of m. Thus, O^{p′}(\overline{L}) centralizes C_{W} (S), and then O^{p′}(\overline{L}) centralizes Z(S) by 6.2(c). Then also O^{p′}(L) centralizes Z(S).

Set C = L_{Σ}(C_{M}(Z(S))). Then C is the set of all f ∈ K such that [Z(S), f] = 1, and thus C is σ-invariant. Since |C| < |L|, 6.12 applies and yields an extension of σ to an automorphism σ^{∗} of C_{M}(Z(S)). Then σ^{∗} is the identity automorphism, by 1.10(c), and thus σ_{L} is the identity automorphism of L. Then σ_{L} = c_{z}, and we again have a contradiction via 5.15(a). □

In the remaining arguments, whenever X is a subgroup of M we write \overline{X} for the image C_{M}(D)X/C_{M}(D) of X in G.

Lemma 7.9. Assume that G is one of the classical groups that appear in 7.4.2. Then either G has an essential splitting in the sense of 7.7, or G = SL_{2}(q) (q a power of p).

Proof. In 7.4.2 it is given that W is a direct sum of copies of the natural G-module U, or that G = SL_{n}(q) (n ≥ 4) and W is a direct sum of copies of U and its dual U^{∗}. Then C_{G}(C_{W} (S)) is equal to either C_{G}(C_{U}(S)) or, in the exceptional case, to C_{G}(C_{U}(S)) ∩ C_{G}(C_{U^{∗}}(S)). Since the definition of essential splitting depends only on G, and on the subgroup C_{G}(C_{W} (S)) of G, we may therefore assume, for the sake of simplicity, that W = U or, exceptionally, that W = U ⊕ U^{∗}. In either case, write W_{0} for the subspace U of W.

Set G_{0} = [G, G] if p = 2 and G = O_{2n}^{ϵ}(q), and otherwise set G_{0} = G. Let T be the Sylow p-subgroup \overline{S} of G, set T_{0} = T ∩ G_{0}, let P be the set of minimal parabolic subgroups L of G_{0} over T_{0} such that C_{W} (T_{0}) is not L-invariant, and set H = ⟨P⟩T. Let Q be the set of all subgroups Q of T such that N_{T} (Q) ∈ Syl_{p}(N_{G}(Q)) and such that N_{G}(Q)/Q has a strongly p-embedded subgroup. Set K = C_{G}(C_{W} (T)). It will then suffice to show:

(a) O_{p}(H) is weakly closed in F_{T} (G), and if G ≠ SL_{2}(q) then O_{p}(H) ≠ 1.

(b) There exists a parabolic subgroup K^{∗} of G_{0} such that O^{p′}(K^{∗})T ≤ K ≤ K^{∗}T.

(c) For each Q ∈ Q, either N_{G}(Q) ≤ H or O^{p′}(N_{G}(Q)) centralizes C_{U}(T). Indeed, only (a) and (c) are needed, but (b) will play a role in obtaining these points.

We note at the outset that |G/G_{0}| ≤ 2 and that p = 2 if |G/G_{0}| ≠ 1.

Set K^{∗} = N_{G_{0}}(C_{W} (T_{0})). If W is irreducible then C_{W} (T_{0}) is a 1-dimensional subspace of W, and K^{∗} is a maximal parabolic subgroup of G_{0} over T_{0}. On the other hand, if W is reducible, so that G = SL_{n}(q) with n ≥ 4, then K^{∗} is the parabolic subgroup of Lie corank 2 obtained as the intersection of the two maximal parabolics L_{1} and L_{2} such that L_{i}/O_{p}(L_{i}) ≅ GL_{n−1}(q). Then O^{p′}(K^{∗}) centralizes C_{U}(T_{0}) and we obtain (b). Further:

(1) Either there is a unique minimal parabolic subgroup X of G_{0} over T_{0} not contained in K^{∗}, or there are two such (to be denoted X_{1} and X_{2}). In the latter case, X_{1}X_{2} is a group, and O^{p′}(X_{1}X_{2})/O_{p}(X_{1}X_{2}) ≅ SL_{2}(q) × SL_{2}(q).

Set H_{0} = H ∩ G_{0}. Then (1) shows that either G = G_{0} and H is a parabolic subgroup of G; or else G ≠ G_{0}, H = H_{0}T, and H_{0} is a minimal parabolic subgroup of G_{0}. Set P = O_{p}(H) and set P_{0} = P ∩ G_{0}. By 1.12, P_{0} is weakly closed in F_{T_{0}}(G_{0}). As H = N_{G}(P_{0}) = N_{G}(P), P is weakly closed in F := F_{T} (G). If H = G then G_{0} is itself a minimal parabolic subgroup of G_{0}, since the exceptional case where H_{0} is not a minimal parabolic occurs only when the Lie rank of G_{0} is greater than that of H_{0}. In the list of groups under consideration from 7.4.2, only SL_{2}(q) has Lie rank equal to 1, so (a) holds.

Let Q ∈ Q, set N = N_{G}(Q) and N_{0} = N ∩ G_{0}. Also, set E = O^{p′}(N), E_{0} = E ∩ G_{0}, and Q_{0} = Q ∩ G_{0}. In proving (c) we may assume that [C_{W} (T), E] ≠ 0.

As N/Q has a strongly p-embedded subgroup we have Q = O_{p}(N). Suppose that N ≤ G_{0}. Since N_{T} (Q) ∈ Syl_{p}(N_{G}(Q) by the definition of Q, a theorem of Borel and Tits [Theorem 3.1.3 in GLS] implies that N_{G_{0}}(Q) is a parabolic subgroup of G_{0} over T_{0}. Thus N/Q is a group of Lie type, as is E/Q. The only groups of Lie type having a strongly p-embedded subgroup are those of Lie rank 1, so N is a minimal parabolic subgroup of G_{0}. Since [C_{U}(T), E] ≠ 0, we have [C_{U}(T_{0}), E] ≠ 0, and thus N is either the unique minimal parabolic over T_{0} which does not normalize C_{U}(T_{0}), or N is one of the two such minimal parabolics given by (1). Thus N ≤ H, and we have (c) in this case. We have thus reduced the proof of (c) to the case where G ≠ G_{0}, so p = 2 and G = O_{2n}^{ϵ}(q) for some sign ϵ.

Suppose next that Q ≤ G_{0}. Then O_{2}(N_{0}) ≤ O_{2}(N) = Q, so the Borel-Tits theorem implies that N_{0} is a parabolic subgroup of G_{0} over T_{0}. Let K be an overgroup of Q in N such that K/Q is strongly embedded in N/Q, and set K_{0} = K ∩ G_{0}. Then |K_{0}/Q| is divisible by 2 since N_{T_{0}}(Q) ≤ K, and hence K_{0}/Q is strongly embedded in N_{0}/Q. As in the preceding paragraph, it follows that N_{0} is a minimal parabolic over T_{0}, and N_{0} is then the unique minimal parabolic over T_{0} which does not normalize C_{W} (T_{0}). Then N_{0} ≤ H, and since N = N_{0}T where T ≤ H, (c) holds in this case. We may therefore assume that Q ≰ G_{0}.

Set R = O_{p}(N_{G_{0}}(Q_{0})). Then R is E-invariant, as is N_{R}(Q_{0}). Then N_{QR}(Q) ≤ Q as Q = O_{p}(N), and thus R = Q_{0}. By the Borel-Tits theorem N_{G_{0}}(Q_{0}) is a parabolic subgroup of G_{0} over T_{0}.

Suppose Q_{0} = 1. Then |Q| = 2 and O_{2}(C_{G_{0}}(Q)) = 1. Then [8.7 in AsSe] implies that E_{0} ≅ Sp_{2n−2}(q). But E_{0} ≅ E/Q in this case, so E_{0} has a strongly embedded subgroup. This yields n = 2, whereas 7.4.2(2) excludes O_{4}^{ϵ}(q). Thus \overline{Q}_{0} ≠ 1, and N_{G_{0}}(Q_{0}) is a proper parabolic subgroup of G_{0}.

Set L = O^{2′}(N_{G_{0}}(Q_{0})), and set L̃ = L/Q_{0}. The Levi decomposition for N_{G}(Q_{0}) yields a direct product decomposition

L̃ = L̃_{1} × ⋯ × L̃_{k}

where each L̃_{i} is a (possibly disconnected) group of Lie type, and where each L̃_{i} is Q̃-invariant. We choose such a decomposition so as to maximize k. Then

Ẽ_{0} = CẼ_{1}(Q̃) × ⋯ × CẼ_{k}(Q̃),

and each CẼ_{i}(Q̃) has even order. But Ẽ_{0} ≅ E/Q, so Ẽ_{0} has a strongly embedded subgroup. We conclude that k = 1.

Suppose that N_{G_{0}}(Q_{0}) is disconnected. Then L̃ is a direct product of two factors (each of them a group of Lie type) which are interchanged by Q̃, and Ẽ is isomorphic to each of the factors. Then Ẽ ≅ SL_{2}(q) as Ẽ has a strongly embedded subgroup, and N_{G_{0}}(Q_{0}) is a product of two minimal parabolic subgroups, with root system of type A_{1} × A_{1} acted on non-trivially by T/T_{0}. But there is a unique minimal parabolic subgroup X over T_{0} which does not normalize C_{U}(T_{0}), and hence N_{G_{0}}(Q_{0}) normalizes C_{U}(T_{0}). Then [C_{U}(T_{0}), E_{0}] = 1 and [C_{U}(T), E] = 1, contrary to the choice of Q. We conclude that N_{G_{0}}(Q_{0}) is a connected parabolic, and moreover, X = N_{G_{0}}(Q_{0}).

From the discussion in [section 2.7 of GLS] on the relation between classical groups and groups defined as groups of Lie type, it now follows that N_{G_{0}}(Q_{0}) is contained in the stabilizer of a singular j-space U for some j > 1, and that L/Q_{0} acts as SL_{j}(q) on U. Let t be an 𝔽_{q}-transvection in T on U. Then t centralizes every singular T_{0}-invariant subspace of U, so [L, t] ≤ O_{2}(L) = Q_{0}. Since E_{0} ≤ L, Q_{0}⟨t⟩ is then an E-invariant subgroup of T. But Q_{0} is the unique largest E_{0}-invariant subgroup of T_{0}, so Q is the unique largest E_{0}-invariant subgroup of T, and thus Q = Q_{0}⟨t⟩. Then Q is L-invariant, and so L = E_{0}. Again, as E_{0}/Q_{0} has a strongly embedded subgroup, we conclude that L/Q_{0} has Lie rank 1, and thus E_{0} = O^{2′}(X). As H = XT we obtain E = E_{0}T ≤ H. Further, we now have H = N_{G}(Q_{0}), and so N ≤ H. This completes the proof of (c). □

Remark. The group Sym(8) is well known to be isomorphic to O_{6}^{+}(2), by means of a quadratic form on the natural irreducible Sym(8)-module preserved by Sym(8). Thus, the preceding lemma applies to the case where G = Sym(8) and W is the natural irreducible module.

Lemma 7.10. Suppose that p = 2, and suppose that G = Sym(n) is a symmetric group, with n ≥ 5, and that W is isomorphic to the natural irreducible G-module over 𝔽_{2}. Then n = 8, and W is a natural module for G.

Proof. We assume throughout that n ≠ 8. Let Ω = {1, 2, ⋯, n} be the standard G-set, and let G_{0} = Alt(n) be the subgroup of index 2 in G. Let Ṽ be the natural permutation module for G, identified with the set 2^{Ω} of subsets of Ω (where addition is given by symmetric difference of subsets). Let W̃ be the submodule of Ṽ consisting of subsets of Ω of even order, and let Z̃ be the 1-dimensional submodule {∅, Ω} of Ṽ. By definition, W ≅ W̃ if n is odd, and W ≅ W̃/Z if n is even.

Recall the notation V = Ω_{1}(D). We now write V = V_{0} × V_{1}, where V_{1} ≤ C_{V} (G), and where V_{1} is a G-submodule of V chosen to be as small as possible subject to V = V_{0}C_{V} (G). Thus, V_{0} is indecomposable for G. We claim that V_{0} is isomorphic to a Gsubmodule of Ṽ or of Ṽ/Z. Indeed, by an elementary calculation [exercise 3 on page 74 of FGT] we have |H^{1}(G_{0}, W)| = 1 if n is odd, |H^{1}(G_{0}, W)| = 2 if n is even, and Ṽ indecomposable for G_{0} if n is even. The claim follows in a straightforward way from this exercise, and from the observations made in the preceding paragraph. In particular, transpositions in G are transvections on V_{0}, and hence also on V.

Set K = C_{M}(C_{W} (S)) and let \overline{K} be the image of K in G. Then K ≤ C_{M}(C_{D}(S)) by 6.2(c). By 3.8 there is an automorphism γ_{K} of L_{Γ}(K) given by restricting γ. Since |K| < |M|, and since best offenders on D are also best offenders on C_{D}(O_{2}(K)) by 6.6(c), we may apply 6.12 with (K, S, O_{2}(K), Γ) in the role of (M, S, Y, Γ). Thus γ_{K} extends to an automorphism of K. As [Z(S), K] = 1, 1.10(c) yields:

(1) The identity map on K is the unique extension of γ_{K} to an automorphism of K. Let T be the set of subgroups T of S such that Y ≤ T, and such that \overline{T} is generated by a transposition. For any subgroup P of S set

P_{T} = ⟨T ∈ T | T ≤ P ⟩Y.

Set R = S_{T}, set R_{0} = R ∩ M_{0} where M_{0} is the preimage of Alt(n) in M, and set H = N_{M}(R). Then H = N_{M}(R_{0}), and \overline{H} ≅ 2^{m} ⋊ Sym(m) where m is the greatest integer in n/2.

If the set \{\overline{R}, \overline{R}_{0}\} contains all of the quadratic best offenders on D contained in \overline{S}, then every member of Γ contains R_{0}, and then L = L_{Γ}(M) = H, contrary to 7.2. Thus there exists a quadratic best offender \overline{A} ≤ \overline{S} with \overline{A} ∉ \{\overline{R}, \overline{R}_{0}\}. Since n ≠ 8 by assumption, 7.4.3 implies that |W/C_{W} (\overline{A})| = |\overline{A}| and that \overline{A} contains a transposition t. Notice that

(*)  |W/C_{W} (\overline{A})| ≤ |V/C_{V} (\overline{A})| ≤ |D/C_{D}(\overline{A})|.

Since \overline{A} is a best offender on D, hence also on V by 6.6(c), we conclude that the inequalities in (*) are equalities, and hence that D = C_{D}(\overline{A})V. Then C_{D}(\overline{A})C_{V} (t) has index 2 in D, and thus |D/C_{D}(t)| = 2. Since G is generated by n − 1 transpositions, we conclude that |D/C_{D}(G)| ≤ 2^{n−1}, and hence

(2) D = C_{D}(G)V. Further, since |D/C_{D}(t)| = 2, the preimage in S of any subgroup of ⟨T ⟩ generated by transpositions is in Γ. Thus

(3) T ⊆ Γ, and Q_{T} ∈ Γ for any Q ∈ Γ such that N_{M}(Q) ≰ H.

Notice that (3) yields O_{2}(H) ∈ Γ, so H = N_{L}(O_{2}(H)), and then γ restricts to an automorphism of H. By 1.10(c), γ |_{H}= c_{z} for some z ∈ Z(S), and where of course c_{z} is also an automorphism of M. Replacing γ with γ ∘ c_{z}^{−1}, we may assume:

(4) γ restricts to the identity automorphism on H.

Suppose that n = 2m is even, let Q ∈ Γ with N_{M}(Q) ≰ H, and let Q_{0} be a fully normalized F-conjugate of Q_{T}. As H is m-transitive on T , Q_{0} is in fact an H-conjugate of Q_{T}. Set X = N_{M}(Q_{0}). Then \overline{X} ≅ (2^{k} ⋊ Sym(k)) × Sym(n − 2k), where k is the number of transpositions in \overline{Q}_{0}. Since n is even it follows from [lemma 2.8 in BHS] that \overline{X} is generated by its subgroups \overline{X} ∩ \overline{H} and \overline{X} ∩ \overline{K}. Since C_{X}(D) ≤ K, we then have X = ⟨X ∩ H, X ∩ K⟩. Let γ_{X} be the restriction of γ to X. Then (1) and (5) imply that γ_{X} induces the identity map on X. Here N_{M}(Q_{T}) = X^{h} for some h ∈ H. We have (h^{−1}, x, h) ∈ D(L_{Γ}(M)) via Q_{T} for each x ∈ X, so γ restricts to the identity on N_{M}(Q_{T}) by (5). Since N_{M}(Q) ≤ N_{M}(Q_{T}) we conclude that γ restricts to the identity map on N_{M}(Q). Thus γ is the identity automorphism of L in the case that m is even, and we may therefore assume that n = 2m + 1 is odd.

Let G_{1} be a subgroup of G such that \overline{S} ≤ G_{1}, and with G_{1} ≅ Sym(2m). Let M_{1} be the pre-image of G_{1} in M and set L_{1} = L_{Γ}(M_{1}). Then L_{1} is γ-invariant by 3.8. Let γ_{1} be the restriction of γ to L_{1}. Then M_{1} = ⟨H, K⟩ (again by [2.8 in BHS]). The hypothesis of 7.1 holds with M_{1} in place of M, so the minimality of |M| implies that γ_{1} extends to an automorphism β_{1} of M_{1}. Then (1) and (5) imply that β_{1} is the identity automorphism.

Now let G_{2} be a subgroup of G such that G_{2} ≅ Sym(m) × Sym(m + 1), and such that \overline{S} ∩ G_{2} ∈ Syl_{2}(G_{2}). Let M_{2} be the preimage of G_{2} in M, set S_{2} = S ∩ M_{2}, and let Γ_{2} be the set of all subgroups Q of S_{2} with Q ∈ Γ. One observes that R ≤ M_{2}, so Γ_{2} is non-empty. Set L_{2} = L_{Γ_{2}}(M_{2}). Then L_{2} is γ-invariant by 3.8. Let γ_{2} be the restriction of γ to L_{2}. As |M_{2}| < |M|, 6.12 applies with (M_{2}, S_{2}, Γ_{2}) in the role of (M, S, Γ), yielding an extension of γ_{2} to an automorphism β_{2} of M_{2}. Then β_{2} = c_{u} for some u ∈ Z(S_{2}). As n is odd, and as remarked above, we have H^{1}(G, W) = 0. Thus V may be identified with W ×C_{V} (G). As G is generated by n−1 transpositions, it follows that |D/C_{D}(G)| = 2^{n−1} and that D = W × C_{D}(G). Thus, we may take u ∈ W.

Recall that Ω denotes the standard G-set. We may then take G_{1} to be the stabilizer in G of n, and we may take G_{2} to be the stabilizer in G of the partition (Δ_{1}, Δ_{2}) of Ω, where Δ_{1} = {1, ⋯, m}. Identify W with the set of even-order subsets of Ω. As β_{1} is the identity map on M_{1}, c_{u} centralizes M_{1} ∩ M_{2}, and it follows that c_{u} is either the identity map on M_{2} or that c_{u} is given on M_{2} by taking u = Ω − {n}. In either case, the automorphism c_{u} of M induces β_{i} on M_{i} (i = 1, 2). Replacing γ with γ ∘ c_{u}^{−1}, we may assume that both β_{1} and β_{2} are identity maps.

We now argue as we did in the case where n is even, taking an arbitrary Q ∈ Γ, taking Q_{0} to be a fully normalized F-conjugate of Q_{T}, and setting X = N_{M}(Q_{0}). As before, we have \overline{X} ≅ (2^{k} ⋊Sym(k))×Sym(n−2k), and now \overline{X} is generated by its subgroups \overline{X} ∩G_{1} and \overline{X} ∩ G_{2}. Thus X = ⟨X ∩ M_{1}, X ∩ M_{2}⟩. As each β_{i} is an identity map, the restriction γ_{X} of γ to X is the identity map on X. Since N_{M}(Q) ≤ N_{M}(Γ_{T}), and N_{M}(Q_{T}) is an H-conjugate of X, it follows as in the case when n is even that γ restricts to the identity map on N_{M}(Q). Thus, γ is the identity automorphism of L, and we have obtained a contradiction to the assumed non-existence of an extension of γ to an automorphism of M. □

Proof of Proposition 7.1 By 7.5, G has a unique J-component, and 7.4.2 then yields the possibilities for the structure of G and for the action of G on W. Recall that (M, γ) is a counterexample to 7.1. By 7.10, if G is a symmetric group Sym(n), and W its natural irreducible module, then n = 8. Since Sym(8) ≅ O_{6}^{+}(2) via a quadratic form on the natural irreducible module for Sym(8) (see the remark following 7.8), 7.8 and 7.9 imply that G is a symmetric group Sym(n) with n ≠ 8 and with W the natural module - or else that G = SL_{2}(q). Thus, we have only the case G = SL_{2}(q) and (by 7.4.2) W the natural SL_{2}(q)-module left to consider. But in this last case, G has a strongly p-embedded subgroup N_{G}(\overline{S}), and hence C_{M}(D)N_{M}(S) is a proper subgroup of M containing L_{Γ}(M). This contradicts 7.2, and thus the proof of 7.1 is complete.

Proof of the Main Theorem As pointed out at the beginning of this section, Proposition 7.1 provides the remaining step required for the proof of Proposition 6.10. Then 6.11, and 6.14 through 6.18 - which were proved under the assumption that Proposition 6.10 holds - yield a contradiction to the presumed non-existence or non-uniqueness of a centric linking system L whose fusion system F_{S}(L) is a given saturated fusion system.

APPENDIX

In [OV], Bob Oliver and Joana Ventura introduced a category T of “transporter systems” and isomorphisms of transporter systems. Part of the structure of any given transporter system consists of a functor ρ : T → F, where T is a category and where F is a fusion system on a finite p-group; and one says that the given transporter system is a “transporter system on (or over) F”. The category T has a full subcategory T^{c} of “centric linking systems” whose definition is far different from the one given in 2.9 here. The aim of this section is to show that transporter systems are the “same” as localities, that the two definitions of centric linking system are essentially equivalent, and to obtain the following result.

Theorem A. Let F be a saturated fusion system on a finite p-group S. Then there exists a centric linking system T over F (in the sense of [OV] or [BLO]), and T is unique up up to isomorphism of transporter systems.

In this way we will establish that our Main Theorem yields existence and uniqueness of “centric linking systems” in either sense of this term. In order to do this, we first review the definitions in [OV].

Let S be a finite p-group and let X be a collection of subgroups of S with S ∈ X. There is then a category T_{X}(S) whose set of objects is X, and whose morphism-sets are given by

Mor_{T_{X}(S)}(P, Q) = N_{S}(P, Q)

for P, Q ∈ X. Composition is given by multiplication in S.

Here is the definition of transporter system from [OV], but with the notions of left and right composition reversed from their original meanings, in order to maintain consistency with our policy of taking all categories in the right-handed sense.

Definition X.1. Let F be a fusion system over a finite p-group S. A transporter system associated to F is a non-empty finite category T , together with a pair of functors

T_{Ob(T )}(S) ⟶^{ϵ} T ⟶^{ρ} F

satisfying the following conditions. (A1) Ob(T ) ⊆ Ob(F), and Ob(T ) is closed under F-conjugacy and overgroups. Also,

ϵ is the identity on objects and ρ is the inclusion on objects. (A2) For each P, Q ∈ Ob(T ), the kernel

E(P) =^{def} Ker[ρ_{P} : Aut_{T} (P) → Aut_{F}(P)]

acts freely on Mor_{T} (P, Q) by left composition, and ρ_{P,Q} is the orbit map for this action. Also, E(Q) acts freely on Mor_{T} (P, Q) by right composition.

(B) For each P, Q ∈ Ob(T ), ϵ_{P,Q} : N_{S}(P, Q) → Mor_{T} (P, Q) is injective, and the

composite ρ_{P,Q} ∘ ϵ_{P,Q} sends g ∈ N_{S}(P, Q) to c_{g} ∈ Hom_{F}(P, Q). (C) For all ϕ ∈ Mor_{T} (P, Q) and all g ∈ P, the diagram

P ⟶^{ϕ} Q
ϵ_{P} (g)↓  ↓^{ϵ_{Q}(g′)}
P ⟶^{ϕ} Q

commutes in T , where g′ is the image of g under ρ(ϕ). (I) ϵ_{S}(S) ∈ Syl_{p}(Aut_{T} (S)). (II) Let ϕ ∈ Iso_{T} (P, Q), and let P ⊴ \overline{P} ≤ S and Q ⊴ \overline{Q} ≤ S be such that ϕ^{−1}∘ϵ_{P} (\overline{P})∘

ϕ ≤ ϵ_{Q}(\overline{Q}). Then there exists \overline{ϕ} ∈ Mor_{T} (\overline{P} , \overline{Q}) such that ϵ_{P,\overline{P}} (1)∘\overline{ϕ} = ϕ∘ϵ_{Q,\overline{Q}}(1).

If moreover F is saturated, Ob(T ) = F^{c}, and E(P) = Z(P) for all objects P, then T is a centric linking system.

Definition X.2. Let T = (T , ϵ, ρ) and T′ = (T′, ϵ′, ρ′) be transporter systems over a fusion system F on S, with Ob(T ) = Ob(T′). An isomorphism T → T′ (of transporter systems) consists of an invertible functor α : T → T′ (of categories) such that, in righthand notation, ϵ ∘ α = ϵ′ and α ∘ ρ′ = ρ.

Let L = (L, Δ, S) be a locality over S. Set T = Cat(L, Δ) (as defined in 2.8(1)) and let F be the fusion system F_{S}(L) on S, generated by the conjugation maps between objects. There is a functor

ϵ : T_{Δ}(S) → T

for which ϵ_{Ob} : Δ → Ob(T ) is the identity map, and where each ϵ_{P,Q} : N_{S}(P, Q) → Mor_{T} (P, Q) is an inclusion map. There is also a functor

ρ : T → F,

such that ρ_{Ob} : Δ → Ob(F) is the inclusion map of Δ into the set of all subgroups of S, and such that ρ_{P,Q}(ϕ) is the conjugation map c_{g} : P → Q, where g is the unique element of L such that ϕ = (g, P, Q). (See the discussion in 2.8(1)). We note that the functoriality of ρ depends on the condition (O2) in the definition 2.6 of “objective partial group”.

Proposition X.3.

(a) Let L = (L, Δ, S) be a locality. Then the diagram

(*)  T_{Δ}(S) ⟶^{ϵ} T ⟶^{ρ} F

of categories and functors is a transporter system, and if L is a centric linking system in the sense of 2.9 then T is a centric linking system in the sense of [BLO] or [OV].

(b) Let L = (L, Δ, S) and L′ = (L′, Δ, S) be localities having the same set Δ of objects, and let β : L → L′ be a rigid isomorphism. Define (T , ϵ, ρ) as above, and define (T′, ϵ′, ρ′) in the analogous way. There is then an isomorphism T → T′ of localities, given on objects by P ↦ P and on morphisms by (f, P, Q) ↦ (fβ, P, Q).

Proof. (a): By definition of T , Δ = Ob(T ), and then also Δ ⊆ Ob(F) since Δ is a set of subgroups of S. Since F is generated by the conjugation maps c_{f} : P → Q with f ∈ N_{L}(P, Q), with P, Q ∈ Δ, the condition (O2) in 2.5 implies that Δ is closed under F-conjugacy. Since Δ is overgroup closed by 2.9, we then have (A1).

Let P, Q ∈ Δ and define E(P) and E(Q) as in (A2). Since left and right cancellation holds in L, by 2.2, E(P) acts freely on Mor_{T} (P, Q) by left composition, and E(Q) acts freely by right composition. Let (f, P, Q) and (g, P, Q) ∈ Mor_{T} (P, Q) lie in the same fiber of the map ρ_{P,Q} : Mor_{T} (P, Q) → Hom_{F}(P, Q). Then the conjugation maps c_{f} and c_{g} from P to Q are equal. Set P′ = P^{f}(= P^{g}) and regard c_{g^{−1}} as a map from P′ to P. Then c_{fg^{−1}} = c_{f} ∘ c_{g^{−1}} is the identity map on P, so that fg^{−1} ∈ E(P). This shows that each fiber of ρ_{P,Q} is contained in an orbit of E(P). The reverse inclusion holds since

ρ_{P,Q}(hf) = ρ(hf) = ρ(h)ρ(f) = ρ(f),

for any h ∈ E(P). Thus (A2) holds.

Condition (B) follows immediately from the definitions of the functors ϵ and ρ. The commutativity of the diagram in (C) is no more than the observation that if (f, P, Q) ∈ Mor_{T} (P, Q) and g ∈ P, then g^{f} is defined (via P^{f}), and g^{f} = g(ρ(f)) ∈ Q. Condition (I) is given by the hypothesis, in 2.9, that S ∈ Δ, so it only remains to establish (II). Here N_{L}(P) is isomorphic to Aut_{T} (P) via the map g ↦ (g, P, P) for P ∈ Δ. LetP, \overline{P} , Q, \overline{Q} ∈ Δ, with P ⊴ \overline{P} and Q ⊴ \overline{Q}, and let f ∈ L with P^{f} = Q. Then c_{f} induces an isomorphism N_{T} (P) → N_{T} (Q). If (\overline{P})c_{f} = \overline{Q} then the T -isomorphism (f, P, Q) extends to the T - isomorphism (f, \overline{P} , \overline{Q}). Thus (II) holds, and T is a transporter system.

Now suppose that L is a centric linking system in the sense of 2.9. That is, assume that Δ is the set of all F-centric subgroups, and that C_{L}(P) ≤ P for all P ∈ Δ. Then F is saturated, by 2.17(a), and E(P) = Z(P) for all P ∈ Δ, so T is a centric linking system in the sense of [OV]. Thus (a) holds.

(b): Let T , T′, and β : L → L′ be as given. Let α : T → T′ be the pair of maps, given on Ob(T ) = Δ by P ↦ P, and on morphisms by (f, P, Q) ↦ (fβ, P, Q). That α is then a functor is immediate from the fact that β is a homomorphism which sends each subgroup P of S to P. The invertibility of α is immediate from the invertibility of β, and it is trivially verified that ϵ ∘ α = ϵ′ (in right-hand notation). In order that α ∘ ρ′ be equal to ρ it is necessary and sufficient that each conjugation map c_{f} : P → Q with P, Q ∈ Δ be equal to the conjugation map c_{fβ}. Thus, let x ∈ P. Then x^{f} ∈ Q, so x^{f} ∈ S, and then

x^{f} = (x^{f})β = (xβ)^{fβ} = x^{fβ}.

Thus c_{f} = c_{fβ} as required, and (b) holds. □

Corollary X.4. Let F be a saturated fusion system on S, S a finite p-group. Then there exists a centric linking system (T , ϵ, ρ) whose fusion system is F.

Proof. The Main Theorem provides a centric linking system (L, Δ, S) in the sense of 2.9, with F = F_{S}(L), and then X.3(a) provides the required centric linking system in the sense of [OV]. □

For the remainder of this Appendix, let

T_{Δ}(S) ⟶^{ϵ} T ⟶^{ρ} F

be a transporter system. Set ι_{P,Q} = ϵ_{P,Q}(1), write ι_{P} for ϵ_{P} (1), and observe that ι_{P} is the identity element of Aut_{T} (P) by X.1(C). The morphisms ι_{P,Q} are called inclusion morphisms, and condition (B) implies that ρ sends inclusion morphisms in T to inclusion maps in F. Whenever P ≤ P′ ≤ S and Q ≤ Q′ ≤ S are in Ob(T ), and whenever

P ⟶^{ϕ} Q
ι_{P,P′} ↓  ↓^{ι_{Q,Q′}}
P′ ⟶^{ϕ′} Q′

is a commutative square in T , we say that ϕ is a restriction of ϕ′ (and sometimes write ϕ′ |_{P,Q}= ϕ); or we may say that ϕ′ is an extension of ϕ. Some of the results that follow can be found in [section 24 of P2]. Most notably, point (a) of Lemma X.8 (and which is the key point of this appendix) appears to be pre-figured in [Remark 24.12 in P2].

The following lemma collects what are for our purposes the key properties of T , established in [OV].

Lemma X.5. The following hold.

(a) All morphisms of T are both monomorphisms and epimorphisms in the categorical sense. That is, we have left and right cancellation for morphisms in T .

(b) For every morphism ϕ ∈ Mor_{T} (P, Q), and every P_{0}, Q_{0} ∈ Ob(T ) such that P_{0} ≤ P, Q_{0} ≤ Q, and ρ(ϕ) maps P_{0} into Q_{0}, there is a unique ϕ_{0} ∈ Mor_{T} (P_{0}, Q_{0}) such that ϕ_{0} = ϕ |_{P_{0},Q_{0}}. In particular, every morphism in T is the composite of an isomorphism followed by an inclusion morphism.

(c) Let ϕ and ϕ′ be T -homomorphisms P → Q, and let P_{0} and Q_{0} be objects of T with P_{0} ≤ P and Q_{0} ≤ Q. Suppose that ρ(ϕ) and ρ(ϕ′) map P_{0} into Q_{0}, and that ϕ |_{P_{0},Q_{0}}= ϕ′ |_{P_{0},Q_{0}}. Then ϕ = ϕ′.

(d) Let P, \overline{P} , Q, \overline{Q} be objects of T , with P ⊴ \overline{P} and with Q ⊴ \overline{Q}. If \overline{ϕ} ∈ Mor_{T} (\overline{P} , \overline{Q}) is an extension of ϕ ∈ Iso_{T} (P, Q) then the square

P ⟶^{ϕ} Q
ϵ_{P} (x)↓  ↓^{ϵ_{Q}(x(ρ(ϕ))}
P ⟶^{ϕ} Q

commutes for all x ∈ \overline{P}.

Proof. Three of these points are given by the following results in [OV]: (a) by 3.2(b) and 3.8, (b) by 3.2(c), and (d) by 3.3.

For the proof of (c): write ψ for ϕ |_{P_{0},Q_{0}}, and hence also for ϕ′ |_{P_{0},Q_{0}}. Then

ι_{P_{0},P} ∘ ϕ = ψ ∘ ι_{Q_{0},Q} = ι_{P_{0},P} ∘ ϕ′,

and (c) follows from left cancellation. □

Lemma X.6. Let ϕ_{0} : P_{0} → Q_{0}, ϕ : P → Q, and ϕ′ : P′ → Q′ be T -isomorphisms, and suppose that both ϕ and ϕ′ are extensions of ϕ_{0}. Then the following hold.

(a) P = P′ if and only if Q = Q′.

(b) There is a unique extension of ϕ_{0} to a T -isomorphism ϕ_{1} : P ∩ P′ → Q ∩ Q′, and each of ϕ and ϕ′ is an extension of ϕ_{1}.

Proof. (a) Suppose that P = P′. Let x ∈ N_{P} (P_{0}), and let y and y′ be the images of x under ρ(ϕ) and ρ(ϕ′), respectively. Then X.5(d), with ϕ_{0} in the role of ϕ, yields

ϕ_{0}^{−1} ∘ ϵ_{P_{0}}(x) ∘ ϕ = ϵ_{Q_{0}}(y) = ϵ_{Q_{0}}(y′).

As ϵ_{Q_{0}} is injective, by condition (B), we get y = y′, and thus ρ(ϕ) and ρ(ϕ′) agree on P_{1} := N_{P} (P_{0}). Let Q_{1} be the image of P_{1} under ρ(ϕ). By X.5(b) there is a restriction ϕ_{1} : P_{1} → Q_{1} of ϕ and a restriction ϕ′_{1} : P_{1} → Q_{1} of ϕ′, and X.2(c) implies that ϕ_{1} = ϕ′_{1}. Replacing ϕ_{0} by ϕ_{1} in (a), and applying induction on the index of P_{0} in P, we obtain Q = Q′ as desired. On the other hand, if Q = Q′ then we obtain P = P′ by working with ϕ_{0}^{−1}, ϕ^{−1}, and ϕ′^{−1}.

(b) Set P_{1} = P ∩ P′ and Q_{1} = Q ∩ Q′. Then ϕ and ϕ′ have restrictions ϕ_{1} and ϕ′_{1} to P_{1} which, in turn, restrict to ϕ_{0}. Then (a) implies that ϕ_{1} and ϕ′_{1} are T -isomorphisms P_{1} → Q_{1}, and X.5(c) yields ϕ_{1} = ϕ′_{1}. □

Define a relation ↑ on the set Mor(T ) of morphisms of T by ϕ ↑ ϕ′ if ϕ′ is an extension of ϕ. That is, ϕ ↑ ϕ′ if ϕ : P → Q and ϕ′ : P′ → Q′ with P ≤ P′, Q ≤ Q′, and with ι_{P,P′} ∘ ϕ′ = ϕ ∘ ι_{Q,Q′}. We may write also ϕ′ ↓ ϕ for ϕ ↑ ϕ′.

Lemma X.7. The following hold.

(a) The relation ↑ induces a partial order on Iso(T ).

(b) The relation ↑ respects composition of morphisms. That is, if ϕ ↑ ϕ′ and ψ ↑ ψ′, and the compositions ϕ ∘ ψ and ϕ′ ∘ ψ′ are defined, then (ϕ ∘ ψ) ↑ (ϕ′ ∘ ψ′).

Proof. The transitivity of the relation ↑ is easily verified. Suppose that both ϕ ↑ ϕ′ and ϕ ↓ ϕ′, where ϕ ∈ Iso_{T} (P, Q) and ϕ′ ∈ Iso_{T} (P′, Q′). Then P = P′, Q = Q′, ι_{P,P′} = ι_{P}, and ι_{Q,Q′} = ι_{Q}. Further, ι_{P} ϕ′ = ϕ ∘ ι_{Q} and then ϕ′ = ϕ since ι_{P} and ι_{Q} are identity morphisms in T . Thus (a) holds.

Suppose that we are given ϕ ↑ ϕ′ and ψ ↑ ψ′, with ϕ ∘ ψ and ϕ′ ∘ ψ′ defined on objects P and P′ respectively. Set Q = P ϕ and R = Qψ, and set Q′ = P′ϕ′ and R′ = Q′ψ′. The following diagram, in which the vertical arrows are inclusion morphisms, adequately demonstrates that ϕ ∘ ψ ↑ ϕ′ ∘ ψ′.

P′ ⟶^{ϕ′} Q′ ⟶^{ψ′} R′
↑  ↑  ↑
P ⟶_{ϕ} Q ⟶_{ψ} R

This yields (b).

Let ≡ be the equivalence relation on Iso(T ) generated by the restriction of ↑ to isomorphisms. Let L be the set Iso(T )/ ≡ of equivalence classes. For ϕ ∈ Iso(T ) we write [ϕ] for the equivalence class containing ϕ.

Lemma X.8. Let f ∈ L.

(a) There is a unique maximal ϕ ∈ f with respect to ↑, and ϕ^{−1} is then maximal in [ϕ^{−1}].

(b) f ∩ Iso_{T} (P, Q) has cardinality at most 1 for any P, Q ∈ Ob(T )

Proof. Let ϕ : P → Q be maximal in f with respect to ↑. Suppose that there exists ϕ′ : P′ → Q′ in f such that ϕ is not an extension of ϕ′. Then ϕ′ may be chosen so that there exists ϕ_{0} : P_{0} → Q_{0} in f with ϕ_{0} ↑ ϕ and ϕ_{0} ↑ ϕ′. Among all such pairs (ϕ′, ϕ_{0}), choose one so that |P_{0}| is as large as possible. Then X.6(b) implies that P_{0} = P ∩ P′ and Q_{0} = Q ∩ Q′. It follows that N_{P′}(P_{0}) ≰ P, and so we may replace ϕ′ by the restriction of ϕ′ to N_{P′}(P_{0}) → N_{Q′}(Q_{0}). That is, we may assume that P_{0} ⊴ P′ and Q_{0} ⊴ Q′.

Let λ : Aut_{T} (P_{0}) → Aut_{T} (Q_{0}) be the isomorphism induced by conjugation by ϕ_{0}. Set P_{1} = N_{P} (P_{0}) and Q_{1} = N_{Q}(Q_{0}). Also, set P′′ = ⟨P_{1}, P′⟩ and Q′′ = ⟨Q_{1}, Q′⟩. Then X.5(d) implies that λ maps ϵ_{P_{0}}(P′′) onto ϵ_{P_{0}}(Q′′). By condition (II) in X.1, there is an extension of ϕ_{0} to a T -isomorphism P′′ → Q′′, and the maximality of P_{0} then yields P′′ ≤ P. Thus P′ ≤ P, and we have a contradiction. Thus f has a unique maximal element ϕ.

Set ψ = ϕ^{−1} and let ψ ↑ \overline{ψ}. Then \overline{ψ} : \overline{Q} → \overline{P} for some \overline{Q} containing Q and some \overline{P} containing P. Then ϕ ↑ \overline{ψ}^{−1}, so ϕ = \overline{ψ}^{−1} and ψ = \overline{ψ}. Thus ϕ^{−1} is maximal in its ≡-class, and (a) holds.

In order to prove (b), let ψ, ψ′ ∈ f ∩ Iso_{T} (P, Q). Then both ψ and ψ′ are restrictions of a single ϕ ∈ f, by (a). Now X.5(b) implies that ψ = ψ′. □

Define D to be the set of words w = (f_{1}, ⋯, f_{n}) ∈ W(L) such that there exists a sequence (ϕ_{1}, ⋯, ϕ_{n}) of T -isomorphisms with ϕ_{i} ∈ f_{i}, and a sequence (P_{0}, ⋯, P_{n}) of objects of T with ϕ_{i} : P_{i−1} → P_{i} for all i. We say also that w ∈ D via (P_{0}, ⋯, P_{n}), or via P_{0}. Define

Π : D → L

by Π(w) = f, where f is the unique maximal element of [ϕ_{1} ∘ ⋯ ∘ ϕ_{n}] given by X.8(a). That Π is well-defined follows from X.7(b) and an obvious induction on the length of w. Set 1 = [ι_{S}], and for any f ∈ L let f^{−1} be the equivalence class of ϕ^{−1}, where ϕ is the unique maximal member of f.

Proposition X.9. L, with the above structures, is a partial group. Moreover, the following hold.

(a) For any g ∈ S, the [ϵ_{S}(g)] is the set of all ϵ_{P,Q}(g) such that P^{g} = Q, and ϵ_{S}(g) is the maximal member of its class.

(b) [ι_{S}] is the set of all ι_{P}, P ∈ Ob(T ), and ι_{S} is the maximal member of its class.

(c) For any ϕ ∈ Iso(T ), [ϕ^{−1}] is the set of inverses of the members of [ϕ].

Proof. We first check that L is a partial group. Of course L is non-empty since T is non-empty. For any f ∈ L and any representative ϕ of f, f is a T -isomorphism between objects of T , so the word (f) of length 1 is in D. Now let w = (f_{1}, ⋯, f_{n}) be in D. Clearly, any prefix u = (f_{1}, ⋯ f_{k}) and any suffix v = (f_{k+1}, ⋯, f_{n}) of w is in D, so 2.1(1) holds for L. By definition Π(f) = f for f ∈ L, so 2.1(2) holds. Condition 2.1(3) is a straightforward consequence of associativity of composition of isomorphisms in T , and of the definition of Π.

That the inversion map f ↦ f^{−1} is an involutory bijection follows from X.8(a). Now let u = (f_{1}, ⋯, f_{n}) ∈ D via (P_{0}, ⋯, P_{n}), and set u^{−1} = (f_{n}^{−1}, ⋯, f_{1}^{−1}). Then u^{−1} ∈ D via (P_{n}, ⋯, P_{0}), so u^{−1} ∘ u ∈ D. One obtains a representative in the class Π(u^{−1} ∘ u) via a sequence of cancellations ϕ_{k}^{−1} ∘ ϕ of representatives ϕ_{k} ∈ f_{i}, so Π(u^{−1} ∘ u) is the equivalence class containing ι_{P_{0}}. Since ι_{P_{0}} ↑ ι_{S}, and since 1 = [ι_{S}] by definition, we get Π(u^{−1} ∘ u) = 1. Thus 2.1(4) holds in L, and L is a partial group.

We now prove (a). Let P ≤ P′ and Q ≤ Q′ in Ob(T ), and let g be an element of S such that P^{g} = Q and (P′)^{g} = Q′. The functoriality of ϵ yields

ϵ_{P,P′}(1) ∘ ϵ_{P′,Q′}(g) = ϵ_{P,Q′}(g) = ϵ_{P,Q}(g) ∘ ϵ_{Q,Q′}(1),

which means that ϵ_{P,Q}(g) ↑ ϵ_{P′,Q′}(g). In particular, we get ϵ_{P,Q}(g) ↑ ϵ_{S}(g). In order to complete the proof of (a), it now suffices to show that for any ϕ ∈ Iso_{T} (P, Q) with ϵ_{S}(g) ≡ ϕ, we have ϕ = ϵ_{P,Q}(g).

Suppose false, and let σ = (ϕ_{1}, ⋯, ϕ_{n}) be a sequence of T -isomorphisms with ϕ = ϕ_{1}, ϵ_{S}(g) = ϕ_{n}, and with either ϕ_{i} ↑ ϕ_{i+1} or ϕ_{i} ↓ ϕ_{i+1} for all i with 1 ≤ i < n. Among all (ϕ, P, Q) with ϕ ≠ ϵ_{P,Q}(g) and ϵ_{S}(g) ≡ ϕ, choose (ϕ, P, Q) so that the length of such a chain σ is as small as possible. Set ψ = ϕ_{2}. Then ψ = ϵ_{X,Y} (g), where X and Y are objects of T with X^{g} = Y. Suppose ϕ ↑ ψ. Applying the functor ρ to the commutative diagram

\begin{matrix}X & ⟶^{ϵ_{X,Y} (g)} & Y \\ ^{ι_{P,X}} ↓ & & ↓^{ι_{Q,Y}} \\ P & ⟶^{ϕ} & Q \end{matrix} ,

and applying condition (B) in X.1 to ρ(ϵ_{X,Y} (g)), we conclude that ρ(ϕ) is the restriction of c_{g} to the homomorphism ρ(ϕ) : P → Q. In particular, we get P^{g} = Q, so that also ϵ_{P,Q}(g) is a restriction of ϵ_{X,Y} (g). Then X.5(a) yields ϕ = ϵ_{P,Q}(g), and contrary to assumption. On the other hand, if ϕ ↓ ψ, then ϕ = ϵ_{P,Q}(g) by X.6, again contrary to assumption. This completes the proof of (a), and then (b) is the special case of (a) given by g = 1.

Let f = [ϕ] be an equivalence class, with ϕ maximal in f. One checks (by reversing pairs of arrows in the appropriate diagrams) that if ψ is a T -isomorphism, and ψ is a restriction of ϕ, then the T -isomorphism ψ^{−1} is a restriction of ϕ^{−1}. Point (c) follows from this observation. □

In view of X.9(a), there is no harm in writing g to denote the equivalence class [ϵ_{S}(g)], for g ∈ S.

Lemma X.10. Let ϕ : Z → W be a T -isomorphism, maximal in its ≡-class. Let X and Y be objects of T contained in Z, and let U and V be the images of X and Y, respectively, under ρ(ϕ). Suppose that there exist elements g and g′ in S such that the following diagram commutes.

(*)  \begin{matrix}X & ⟶^{ϕ|_{X,U}} & U \\ ^{ϵ_{X,Y} (g)}↓ & & ↓^{ϵ_{U,V} (g′)} \\ Y & ⟶^{ϕ|_{Y,V}} & V \end{matrix}

Then g ∈ Z, and g′ is the image of g under ρ(ϕ).

Proof. Let ϕ′ be the composition (in right-hand notation)

ϕ′ = ϵ_{Z^{g},Z}(g^{−1}) ∘ ϕ ∘ ϵ_{W,W^{g′}} (g′).

Thus, ϕ′ ∈ Iso_{T} (Z^{g}, W^{g′}), and the commutativity of (*) yields ϕ |_{Y} = ϕ′ |_{Y}. Thus, ϕ ≡ ϕ′, and the maximality of ϕ : Z → W implies that Z^{g} ≤ Z and W^{g′} ≤ W. That is, g ∈ N_{S}(Z) and g′ ∈ N_{S}(W). There is then a commutative diagram as follows.

Z ⟶^{ϕ} W
ϵ_{Z}(g)↓  ↓^{ϵ_{W} (g′)}
Z ⟶^{ϕ} W

Condition (I) in X.1 implies that there is an extension of ϕ to a T -isomorphism ⟨Z, g⟩ → ⟨W, g′⟩, and the maximality of ϕ then yields g ∈ Z and g′ ∈ W. Condition (C) in X.1 implies that g′ is the image under ρ(ϕ) of g. □

Set Δ = Ob(T ).

Corollary X.11. Let f ∈ L and let P ∈ Δ with the property that, for all x ∈ P, (f^{−1}, x, f) ∈ D and Π(f^{−1}, x, f) ∈ S. Let Q be the set of all such products Π(f^{−1}, x, f). Then Q ∈ Δ and there exists ψ ∈ f such that ψ ∈ Iso_{T} (P, Q).

Proof. As (f^{−1}, x, f) ∈ D there exist U, X, Y, V ∈ Δ and representatives ψ and \overline{ψ} of f such that

U ⟶^{\overline{ψ}^{−1}} X ⟶^{ϵ_{X,Y} (x)} Y ⟶^{ψ} V

is a chain of T -isomorphisms, and where the middle arrow in the diagram is indeed ϵ_{X,Y} (x) by Lemma X.9(a). As Π(f^{−1}, x, f) ∈ S there exists x′ ∈ S such that \overline{ψ}^{−1} ∘ ϵ_{X,Y} (x) ∘ ψ = ϵ_{U,V} (x′). Let ϕ : Z → W be the maximal element of f. Then X.10 implies that x ∈ Z, and x′ is the image of x under ρ(ϕ). In particular, we have P ≤ Z and Q ≤ W, and we may therefore take X = Y = P and U = V = Q, obtaining ψ ∈ Iso_{T} (P, Q). □

Lemma X.12. Let ψ : P → Q be a T -isomorphism, and let f = [ψ] be the equivalence class of ψ. Then P ≤ D(f), and P^{f} = Q in the partial group L.

Proof. For any g ∈ P, we have the composable sequence

Q ⟶^{ϕ^{−1}} P ⟶^{ϵ_{P} (g)} P ⟶^{ϕ} P

of T -isomorphisms, so (f^{−1}, g, f) is in D, and P ⊆ D(f). By X.1(C), ψ^{−1} ∘ ϵ_{P} (g) ∘ ψ = ϵ_{Q}(g′), where g′ ∈ Q. The class [ϵ_{Q}(g′)] is the same as [ϵ_{S}(g′)] by X.8(a); and we recall that we have introduced the convention to denote this class simply as g′. Thus g^{f} = g′, and so P^{f} ⊆ Q. The conjugation map g ↦ g^{f} is injective by 2.4(c), so P^{f} = Q, as required. □

Proposition X.13. (L, Δ, S) is a locality, and if (T , ϵ, ρ) is a centric linking system (in the sense of [BLO] and [OV]) then L is a centric linking system in the sense of definition 2.9.

Proof. First, L is a partial group, by X.9. In order to show that (L, Δ) is objective, let w = (f_{1}, ⋯, f_{n}) ∈ D. By definition, there exist representatives ψ_{i} of the classes f_{i}, and a sequence (P_{0}, ⋯, P_{n}) of objects of T , such that each ψ_{i} is a T -isomorphism P_{i−1} → P_{i}. Then P_{i−1}^{f_{i}} = P_{i} for all i, by X.12. Conversely, given w = (f_{1}, ⋯, f_{n}) ∈ W, and given (P_{0}, ⋯, P_{n}) ∈ W(Δ) with P_{i−1}^{f_{i}} = P_{i} for all i, it follows from X.11 that w ∈ D. Thus, (L, Δ) satisfies the condition (O1) of 2.6. The condition (O2) is given by X.11, so (L, Δ) is objective. That is, the condition (L1) for a locality holds. Also, since T is finite by X.1, L is finite.

The mapping Aut_{T} (S) → N_{L}(S) given by ψ ↦ [ψ] is a homomorphism, as follows from X.7. It is surjective by the definition of L, and injective by X.8(b). As ϵ_{S}(S) ∈ Syl_{p}(Aut_{T} (S)), by X.1(I), we conclude that S ∈ Syl_{p}(N_{L}(S)), and hence S is maximal in the poset of p-subgroups of L. That is, (L2) holds for L, and thus L is a locality.

Now suppose that (T , ϵ, ρ) is a centric linking system. That is, suppose that Δ = Ob(T ) is the set F^{c} of F-centric subgroups of S, and suppose for each object P that Z(P) = Ker(ρ_{P}). Let μ : Aut_{T} (P) → N_{L}(P) be the mapping ϕ ↦ [ϕ]. Then μ is a homomorphism by X.7(b). Let ϕ ∈ Ker(μ). Then [ϕ] = [ι_{S}], so ϕ ↑ ι_{S}, and then ϕ = ι_{P} by X.9(b). That is, ϕ is the identity element of Aut_{T} (P), and thus Ker(μ) = 1. Now let f ∈ N_{L} and let ψ ∈ f be the maximal element. Then ψ restricts to a T - automorphism ϕ of P by X.11, so μ is surjective, and hence an isomorphism. Since Z(P) = Ker(ρ_{P}) = C_{Aut_{T}} (P), we conclude that C_{L}(P) = Z(P), and hence L is a centric linking system in the sense of definition 2.9. □

Let ϕ : P → Q be a morphism in T (and not necessarily a T -isomorphism). Let Q_{0} be the image of P under the homomorphism ρ(ϕ). Then by X.5(b) there is a well-defined restriction ϕ_{0} = ϕ |_{P,Q_{0}} of ϕ to a T -isomorphism P → Q_{0}.

Lemma X.14. There is a functor η : T → Cat(L, Δ), such that η is the identity map on the set of objects, and such that

η_{P,Q} : Mor_{T} (P, Q) → Mor_{Cat(L,Δ)}(P, Q)

is the mapping ϕ ↦ (f, P, Q), where f is the ≡-class of the T -isomorphism ϕ_{0} : P → Q_{0}, and where Q_{0} is the image of P under ρ(ϕ).

Proof. Let ϕ : P → Q and ψ : Q → R be composable morphisms in T , let Q_{0} be the image of P under ρ(ϕ), and let R_{1} be the image of Q_{0} under ρ(ψ). The restrictions ϕ_{0}ϕ |_{P_{0},Q_{0}} and ψ_{1} = ψ |_{Q_{0},R_{1}} are then composable T -isomorphisms. Set θ = ϕ_{0} ∘ ψ_{1}. Then the product [ϕ_{0}][ψ_{1}] of ≡-classes is defined in L, and is equal to [θ], by X.7(b). Set f = [ϕ_{0}] and g = [ψ_{1}]. Thus [θ] = fg, so (f, P, Q) ∘ (g, Q, R) = (fg, Q, R) in Cat(L, Δ). This shows that η is a functor. □

Lemma X.15. There is a functor ξ : Cat(L, Δ) → T , such that ξ is the identity map on the set of objects, and such that

ξ_{P,Q} : Mor_{Cat(L,Δ)}(P, Q) → Mor_{T} (P, Q)

is the mapping (f, P, Q) ↦ ϕ |_{P,P^{f}} ∘ι_{P^{f} ,Q}, where ϕ is the maximal element in the ≡-class f. Moreover, ξ is invertible, and its inverse is η.

Proof. Let (f, P, Q) and (g, Q, R) be composable morphisms in Cat(L, Δ), and let ϕ ∈ f and ψ ∈ g be maximal. Then the composition ξ(f, P, Q) ∘ ξ(g, Q, R) is defined in T , as the following calculation shows.

ξ(f, P, Q) ∘ ξ(g, Q, R) = (ϕ |_{P,P^{f}} ∘ι_{P^{f} ,Q}) ∘ (ψ |_{Q,Q^{g}} ∘ι_{Q^{g},R})

= ϕ |_{P,P^{f}} ∘ψ |_{P^{f} ,P^{fg}} ∘ι_{P^{fg},Q^{g}} ∘ ι_{Q^{g},R} (*)

= ϕ |_{P,P^{f}} ∘ψ |_{P^{f} ,P^{fg}} ∘ι_{P^{fg},R}.

Set θ_{0} = ϕ |_{P,P^{f}} ∘ψ |_{P^{f} ,P^{fg}}. Then fg = [θ_{0}], by the definition of the product in L. Let θ be the maximal element of fg. Then θ_{0} = θ |_{P,P^{fg}}, and (*) then yields

ξ(f, P, Q) ∘ ξ(g, Q, R) = θ |_{P,P^{fg}} ∘ι_{P^{fg},R} = ξ(fg, P, R).

Thus, ξ is a functor.

Set P′ = P^{f}. By X.11 there exists γ ∈ f such that γ = ϕ |_{P,P′}. The functor ρ : T → F sends Mor_{T} (P, P′) to Hom_{F}(P, P′), so P^{f} is the image of P under ρ(γ). Then also P^{f} is the image of P under ρ(ϕ), since ρ(γ) is a restriction of the homomorphism ρ(ϕ) by X.1(C). We now note that

η(ξ(f, P, Q)) = η(ϕ |_{P,P^{f}} ∘ι_{P^{f} ,Q}).

By definition of η, η(ξ(f, P, Q)) is then (f′, P, Q), where f′ is the ≡-class of the T - isomorphism ϕ |_{P,P^{f}}. That is, f′ = f, and the composition ξ followed by η is the identity functor on Cat(L, Δ).

In the other order: consider ξ(η(θ)), where θ : A → B is an arbitrary T -morphism. Let B_{0} be the image of A under ρ(θ). Then η(θ) = (h, A, B) where h = [θ_{0}] and where θ_{0} : A → B_{0} is the restriction θ |_{A,B_{0}}. Applying ξ to (h, A, B) yields the T -morphism θ′ where

θ′ = θ^{∗} |_{A,B_{0}} ∘ι_{B_{0},B}

and where θ∗ is the maximal element in the ≡-class h. Maximality of θ^{∗} yields θ_{0} ↑ θ^{∗}, and then θ^{∗} |_{A,B_{0}}= θ_{0}. Now X.5(b) yields θ′ = θ, and thus η followed by ξ is the identity morphism on T , completing the proof. □

We are now able to prove Theorem A, and to thereby translate the Main Theorem into the language of [BLO] and [OV].

Theorem A. Let F be a saturated fusion system on the finite p-group S. Then, up to isomorphism of transporter systems, there exists a unique centric linking system (T , ϵ, ρ) over F.

Proof. Existence is given by X.4. Now let (T , ϵ, ρ) and (T′, ϵ′, ρ′) be centric linking systems over F in the sense of [OV]. Set (L, Δ, S) = (Iso(T )/ ≡, Ob(T ), S), and similarly define (L′, Δ′, S). Then Δ = Δ′ is the set of F-centric subgroups of S. By X.13 both L and L′ are F-centric linking systems in the sense of 2.9, so the Main Theorem yields a rigid isomorphism β : L → L′. Then X.3 yields an isomorphism β^{∗} : Cat(L, Δ) → Cat(L′, Δ) of categories. We now apply X.14 and X.15 to obtain a sequence

T ⟶^{η} Cat(L, Δ) ⟶^{β^{∗}} Cat(L′, Δ) ⟶^{ξ′} T′

of isomorphisms. Let α : T → T′ be the composite. It now remains to show that (in right-hand notation) ϵ ∘ α = ϵ′ and α ∘ ρ′ = ρ, in order to conclude that α fulfills the requirements of definition X.2 for an isomorphism of transporter systems.

Let δ : T_{Δ}(S) → Cat(L, Δ) be the functor which is the identity map on the set Δ of objects, and which sends x ∈ N_{S}(P, Q) to (x, P, Q). Let σ : Cat(L, Δ) → F be the functor which is the inclusion map Δ → Ob(F) on objects and which sends (f, P, Q) to c_{f} : P → Q. Define δ′ and σ′ with respect to Cat(L′, Δ) in the analogous way. We now check that the following diagram of categories and functors (in which “=” indicates the identity functor) commutes.

T_{Δ}(S)  T_{Δ}(S)  T_{Δ}(S)  T_{Δ}(S)
ϵ↓  ^{δ}↓  ↓_{δ′}  ↓_{ϵ′}
T ⟶^{η} Cat(L, Δ) ⟶^{β^{∗}} Cat(L′, Δ) ⟶^{ξ′} T′
ρ↓  ^{σ}↓  ↓_{σ′}  ↓^{ρ′}
F  F  F  F

Note that, by X.15, ξ′ is the inverse of a corresponding η′, so by symmetry it will suffice to check the two left-hand squares and the middle squares in this diagram for commutativity. Note also that all the arrows in the diagram act trivially on objects, so the problem is to check for commutivity when the arrows are applied to morphisms. We shall write all mappings to the right in the following calculations.

First: the two middle squares. Given x ∈ N_{S}(P, Q), one obtains

(xδ_{P,Q})β^{∗} = (x, P, Q)β^{∗} = (xβ, P, Q) = (x, P, Q) = xδ′_{P,Q}

since β is the identity on S. Also, given (f, P, Q) a morphism in Cat(L, Δ), one obtains

((f, P, Q)β^{∗})σ′ = (fβ, P, Q)σ′ = [c_{fβ} : P → Q] = [c_{f} : P → Q] = (f, P, Q)σ,

since c_{fβ} = c_{f} on any subgroup of S_{f}, again by the rigidity of β. Thus δ ∘ β^{∗} = δ′ and β^{∗} ∘ σ′ = σ.

Next, in order to show that ϵ ∘ η = δ, we need to verify that (xϵ_{P,Q})η = (x, P, Q) for x ∈ N_{S}(P, Q). By definition, η maps xϵ_{P,Q} to ([ϕ_{0}], P, Q), where ϕ_{0} is the restriction xϵ_{P,P^{x}} of xϵ_{P,Q}. Since x ∈ S, the maximal element of [ϕ_{0}] is xϵ_{S}, and [xϵ_{S}] is (by the convention established earlier) the element x of L. Thus ϵ ∘ η = δ.

Finally, let ϕ : P → Q be a T -morphism and set ϕ_{0} = ϕ_{P,P′} where P′ is the image of P under (ϕ)ρ. Applying η ∘ σ to ϕ we obtain c_{f} : P → Q where f = [ϕ_{0}]. Then P′ = P^{f} by X.12. For any x ∈ P, X.1(C) yields ϕ_{0}^{−1} ∘ xϵ_{P} ∘ ϕ_{0} = x′ϵ_{P′}, where x′ is the image of x under (ϕ_{0})ρ. Thus, conjugation c_{f} : P → P′ is, by the definition of the product Π in L, given by

x = [xϵ_{P}] ↦ [ϕ_{0}^{−1} ∘ xϵ_{P} ∘ ϕ_{0}] = [x′ϵ_{P′}] = x′,

and this shows that ((ϕ_{0})η)σ = (ϕ_{0})ρ. But ϕ = ϕ_{0} ∘ ι_{P′,Q}, where (ι_{P′,Q})η = (1, P′, Q), and where (1, P′, Q)σ is the inclusion map P′ ⊆ Q. Functoriality of η and σ then yields that (ϕ)(η ∘σ) is just (ϕ_{0})ρ followed by inclusion. Since also ρ sends inclusion morphisms to inclusion maps, the result is that (ϕ)(η ∘ σ) = (ϕ)ρ. This completes the proof that the big diagram commutes, and hence that α : T → T′ is an isomorphism of transporter systems. □

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Manhattan Kansas

   E-mail address: chermak@math.ksu.edu