INTRODUCTION TO FUSION SYSTEMS

Markus Linckelmann

Revised and corrected version March 2010

Abstract. Fusion systems were introduced by L. Puig in the early 1990’s as a common framework for fusion systems of finite groups or p-blocks of finite groups. Benson [4] suggested that every fusion system should give rise to a p-complete topological space, generalising the notion of a classifying space of a finite group. A criterion for the existence and uniqueness of such spaces was given by Broto, Levi and Oliver, who developed in [6] the homotopy theory of a class of topological spaces, called p-local finite groups, which use fusion systems as their underlying algebraic structure. The present notes, while not touching upon the subject of p-local finite groups directly, are intended to provide a detailed introduction to fusion systems as needed in the structure theory of finite groups, p-blocks and p-local finite groups. As motivation we give a brief review of some aspects, including classical theorems of Burnside and Frobenius, of the interplay between the local and global structure of finite groups in Section 1. We describe the concept of an abstract fusion system in Section 2. Section 3 carries over to fusion systems the notions of normalisers and centralisers of subgroups in a finite group. Using terminology introduced in Section 4, we give in Section 5 a proof of Alperin’s fusion theorem for fusion systems. In Section 6 we show that one can take quotients of fusion systems by certain subgroups, and the Sections 7 and 8 develop the analogues for fusion systems of normal subgroups and simple groups. The last Section generalises a control of fusion result from block theory to arbitrary fusion systems. Most of the material in this introduction to fusion system has appeared in print elsewhere; we give references as we go along.

Contents

§1 Local structure of finite groups 2 §2 Fusion systems 4 §3 Normalisers and centralisers 8 §4 Centric subgroups 14 §5 Alperin’s fusion theorem 17 §6 Quotients of fusion systems 20 §7 Normal fusion systems 22 §8 Simple fusion systems 24 §9 Normal subsystems and control of fusion 26

Typeset by AMS-TEX

§1 Local structure of finite groups

Throughout these notes we denote by p a prime number. The local structure of a finite group G at the prime p can be understood as the structure of a Sylow-p-subgroup P together with some information about the way in which P is embedded into G. The following definition makes this precise, and encodes the relevant information in a category, called fusion system:

Definition 1.1. Let G be a finite group and let P be a Sylow-p-subgroup of G. The fusion system of G on P is the category denoted by F_{P} (G), with the set of subgroups of P as objects and group homomorphisms induced by conjugation in G as morphisms; more precisely, for any two subgroups Q, R of P, we set

Hom_{F_{P} (G)}(Q, R) = Hom_{G}(Q, R) ,

the set of group homomorphisms φ : Q → R for which there is an element x ∈ G such that φ(u) = ^{x}u = xux^{−1} for all u ∈ Q. The composition of morphisms in F_{P} (G) is the usual composition of group homomorphisms.

With the notation of 1.1, if H is a subgroup of G containing P then P is a Sylow-p-subgroup of H and F_{P} (H) ⊆ F_{P} (G). We will say that H controls G-fusion in P if F_{P} (H) = F_{P} (G). If P is abelian, the following theorem of Burnside states that N_{G}(P) controls G-fusion in P:

Theorem 1.2. (Burnside) Let G be a finite group and let P be a Sylow-p-subgroup of G. If P is abelian then F_{P} (G) = F_{P} (N_{G}(P)).

Proof. Let Q be a subgroup of P and let φ : Q → P be a morphism in F_{P} (G). That is, there is an element x ∈ G such that ^{x}Q ⊆ P and φ(u) = xux^{−1} for all u ∈ Q. Since P is abelian, both P and ^{x}P are Sylow-p-subgroups of C_{G}(^{x}Q). Thus there is c ∈ C_{G}(^{x}Q) such that P = ^{cx}P. Then cx ∈ N_{G}(P), and we still have ^{cx}u = c(xux^{−1})c^{−1} = xux^{−1} = φ(u) for all u ∈ Q, because c commutes with all elements of the form xux^{−1} ∈ ^{x}Q, where u ∈ Q. Thus φ is induced by conjugation with an element in N_{G}(P), which proves the Theorem. □

Example 1.3. Let G = S_{4} be the symmetric group on four letters and let P = ⟨x, t⟩ be the subgroup generated by the cycle x = (1, 2, 3, 4) and the involution t = (1, 2)(3, 4). Then P is a Sylow-2-subgroup of S_{4}, and clearly P is a dihedral group of order 8. The group P has a Klein four subgroup V = {(1), (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)} such that Aut_{G}(V) contains a cyclic group of order 3, having conjugation by (1, 2, 3) as generator. This automorphism cannot be extended to an automorphism of P because the automorphism group of a dihedral group is easily seen to be a 2-group again. Thus N_{G}(P) does not control fusion in this case.

A finite group G is called p-nilpotent if P has a normal complement K; that is, K is a normal subgroup of G such that G ≅ K ⋊ P. Note that then K = O_{p′}(G), the largest normal subgroup of G of order prime to p. The following theorem of Frobenius illustrates that being p-nilpotent is a property which can be read off the p-local structure of G:

Theorem 1.4. (Frobenius) Let G be a finite group and let P be a Sylow-p-subgroup of G. The following are equivalent:

(i) G is p-nilpotent. (ii) N_{G}(Q) is p-nilpotent for any non-trivial subgroup Q of P.

(iii) We have F_{P} (G) = F_{P} (P). (iv) For any Q ⊆ P the group Aut_{G}(Q) = N_{G}(Q)/C_{G}(Q) is a p-group.

Sketch of proof. Suppose that G is p-nilpotent; say G = K ⋊ P, where K = O_{p′}(G) and P is a Sylow-p-subgroup of P. Since P ≅ G/K is a p-group, K is equal to the set of all p′-elements in G. Thus, for any subgroup H, the intersection H ∩ K is the set of all p′-elements in H, and hence H is p-nilpotent with H ∩ K = O_{p′}(H). In particular, N_{G}(Q) is p-nilpotent for any subgroup Q of P. Thus (i) ⇒ (ii). Let x ∈ G, u ∈ P such that ^{x}u ∈ P. Write x = yz with y ∈ K and z ∈ P. Then [y,^{z}u] ∈ P ∩ K = {1}, hence ^{x}u = ^{z}u, which proves that any morphism in F_{P} (G) is induced by conjugation with an element in P. Thus (i) ⇒ (iii). The implication (iii) ⇒ (iv) is trivial. Let Q be a subgroup of P. Since Q and O_{p′}(N_{G}(Q)) are normal subgroups of N_{G}(Q) of coprime orders, they centralise each other and hence O_{p′}(N_{G}(Q)) = O_{p′}(C_{G}(Q)). Thus if N_{G}(Q) is p-nilpotent then N_{G}(Q)/C_{G}(Q) is a p-group. This shows the implication (ii) ⇒ (iv). In order to show that (iv) implies (i) we proceed by induction over the order of G. The hypothesis (iv) passes down to subgroups of G, and hence we may assume that every proper subgroup of G is p-nilpotent. Let Q be a non-trivial subgroup of P. If N_{G}(Q) ≠ G then N_{G}(Q) is p-nilpotent by induction. If G = N_{G}(Q) then G/Q is p-nilpotent by induction. Let L be the subgroup of G containing Q such that L/Q = O_{p′}(G/Q). Then Q is a normal Sylow-p-subgroup of L, hence L = Q ⋊ K for some p′-subgroup K of L. The assumption that N_{G}(Q)/C_{G}(Q) is a p-group implies that K centralises Q, hence L = Q × K. Then K is a normal p-complement in G = N_{G}(Q). We next show that F_{P} (G) = F_{P} (P). One can do this directly, but this verification is also a trivial consequence of Alperin’s fusion theorem 5.2 below. In particular, Z = Z(P) has the property that if x ∈ G such that ^{x}Z ⊆ P then ^{x}Z = Z. Set N = N_{G}(Z). Since N is p-nilpotent, the quotient N/[N, N] has a non-trivial p-group as quotient. By a theorem of Grün this implies that G/[G, G] has a non-trivial p-group as quotient. Thus the smallest normal subgroup O^{p}(G) of G with G/O^{p}(G) a p-group is a proper subgroup of G, hence p-nilpotent, and a normal p-complement of O^{p}(G) is also one for G itself. This completes the proof of the implication (iv) ⇒ (i). □

Glauberman and Thompson sharpened this for odd p as follows. The Thompson subgroup J(P) of a finite p-group P is the subgroup of P generated by the set of abelian subgroups of P of maximal order.

Theorem 1.5. (Glauberman-Thompson, [11, Ch. 8, Theorem 3.1]) Let p be an odd prime, let G be a finite group and let P be a Sylow-p-subgroup of G. Then G is p-nilpotent if and only if N_{G}(Z(J(P))) is p-nilpotent.

Glauberman’s ZJ-theorem in [9] is a sufficient criterion for F_{P} (G) to be controlled by the normaliser of the center of the Thompson subgroup J(P).

Theorem 1.6. (Glauberman, [9]) Let p be an odd prime. If no subquotient of G is isomorphic to Qd(p) = (C_{p} × C_{p}) ⋊ SL_{2}(p) then F_{P} (G) = F_{P} (N_{G}(Z(J(P)))).

The semi-direct product Qd(p) is understood with the natural action of SL_{2}(p) on C_{p} × C_{p}. One easily checks that the Theorem does not hold for the group G = Qd(p). The relevance of the following theorem, due to R. Solomon, lies in the fact that had there been a finite group G with a 2-local structure as stated below, then G could have been chosen simple, and thus would have given rise to a new finite simple group.

Theorem 1.7. (R. Solomon, [25]) There is no finite group G having a Sylow-2-subgroup P of Spin_{7}(3) such that F_{P} (Spin_{7}(3)) ⊆ F_{P} (G) and such that all involutions in P are G-conjugate.

Alperin and Broué showed in [2] that, extending ideas of Brauer, every p-block of a finite group admits a local structure which can be described in terms of Brauer pairs and which has formal properties very similar to those of the categories F_{P} (G) above; see [13] (this volume) for details on fusion systems of blocks.

2 Fusion systems

Puig’s abstract notion of fusion systems on p-groups, which we are going to describe in this section, captures the common properties of fusion systems of finite groups and p-blocks. If P, Q, R are subgroups of a finite group G, we denote as before by Hom_{P} (Q, R) the set of group homomorphisms φ : Q → R for which there is y ∈ P satisfying φ(u) = yuy^{−1} for all u ∈ Q; we write Aut_{P} (Q) = Hom_{P} (Q, Q). Thus Aut_{P} (Q) is canonically isomorphic to N_{P} (Q)/C_{P} (Q); in particular Aut_{Q}(Q) ≅ Q/Z(Q) is the group of inner automorphisms of Q.

Definition 2.1. A category on a finite p-group P is a category F whose objects are the subgroups of P and whose morphism sets Hom_{F}(Q, R) consist, for any two subgroups Q, R of P, of injective group homomorphisms with the following properties:

(i) if Q is contained in R then the inclusion Q ⊆ R is a morphism in F; (ii) for any φ ∈ Hom_{F}(Q, R), the induced isomorphism Q ≅ φ(Q) and its inverse are morphisms in F; (iii) composition of morphisms in F is the usual composition of group homomorphisms.

Definition 2.2. Let F be a category on a finite p-group P. A subgroup Q of P is called fully F-centralised if |C_{P} (R)| ≤ |C_{P} (Q)| for any subgroup R of P such that R ≅ Q in F, and Q is called fully F-normalised if |N_{P} (R)| ≤ |N_{P} (Q)| for any subgroup R of P such that R ≅ Q in F.

If G is a finite group with Sylow-p-subgroup P, we will see in 2.9 below that a subgroup Q of P is fully F_{P} (G)-centralised if and only if C_{P} (Q) is a Sylow-p-subgroup of C_{G}(Q); similarly, Q is fully F_{P} (G)-normalised if and only if N_{P} (Q) is a Sylow-p-subgroup of N_{G}(Q). The following definition is due to Broto, Levi and Oliver [6].

Definition 2.3. Let F be a category on a finite p-group P, and let Q be a subgroup of P. For any morphism φ : Q → P in F, we set

N_{φ} = {y ∈ N_{P} (Q)| there is z ∈ N_{P} (φ(Q)) such that φ(^{y}u) = ^{z}φ(u) for all u ∈ Q} .

In other words, N_{φ} is the inverse image in N_{P} (Q) of the group

Aut_{P} (Q) ∩ (φ^{−1} ∘ Aut_{P} (φ(Q)) ∘ φ)

Note that in particular QC_{P} (Q) ⊆ N_{φ} ⊆ N_{P} (Q).

Definition 2.4. A fusion system on a finite p-group P is a category F on P such that Hom_{P} (Q, R) ⊆ Hom_{F}(Q, R) for any two subgroups Q, R of P, and such that the following two properties hold:

(I-S) Aut_{P} (P) is a Sylow-p-subgroup of Aut_{F}(P). (II-S) every morphism φ : Q → P in F such that φ(Q) is fully F-normalised extends to a morphism ψ : N_{φ} → P (that is, ψ|_{Q} = φ).

This terminology is, of course, coherent with the previous section - we will prove in 2.10 that fusion systems of finite groups as defined in §1 are indeed fusion systems in the sense of 2.4. As mentioned before, it follows from work of Alperin and Broué [2], that every p-block of a finite group gives rise to a fusion system on any of its defect groups; see [13] for details. There are examples, due to Levi and Oliver [18], Ruiz and Viruel [23], of fusion systems which cannot be fusion systems of any finite group, and it has been shown by Kessar [12], Kessar and Stancu [16], that these “exotic” fusion systems cannot even occur as fusion systems of blocks. It is not known at present whether every fusion system of a block of some finite group occurs as fusion system of some (other) finite group. As far as terminology goes, Broto, Levi and Oliver [6] use the term saturated fusion system for what is called fusion systems in 2.4, and Puig calls fusion systems (full) Frobenius systems. The “extension axiom” (II-S) relates the role of N_{φ} as object of F to its image N_{φ}/Q in Aut_{F}(Q). Definition 2.4 is equivalent to the definition given in Broto, Levi, Oliver [6] which uses a priori stronger axioms; the following two Propositions, due to Stancu [26], show that these two versions are equivalent.

Proposition 2.5. ([26]) Let F be a fusion system on a finite p-group P. A subgroup Q of P is fully F-normalised if and only if Q is fully F-centralised and Aut_{P} (Q) is a Sylow-p-subgroup of Aut_{F}(Q).

Proof. Assume that Q is fully F-normalised. We first show that then Q is also fully F-centralised. Let φ : R → Q be an isomorphism in F such that R is fully F-centralised. By the extension axiom (II-S) in 2.4 there is a morphism ψ : RC_{P} (R) → P in F such that ψ|_{R} = φ. Hence ψ maps C_{P} (R) to C_{P} (Q), which implies that |C_{P} (R)| ≤ |C_{P} (Q)|, hence equality since R is fully Fcentralised. Thus Q is fully F-centralised. Choose Q to be of maximal order such that Q is fully F-normalised but Aut_{P} (Q) is not a Sylow-p-subgroup of Aut_{F}(Q). Then Q is a proper subgroup of P by the Sylow axiom 2.4.(I-S). Choose a p-subgroup S of Aut_{F}(Q) such that Aut_{P} (Q) is a proper normal subgroup of S. Let φ ∈ S − Aut_{P} (Q). Since φ normalises Aut_{P} (Q), for every y ∈ N_{P} (Q) there is z ∈ N_{P} (Q) such that φ(^{y}u) = ^{z}φ(u) for all u ∈ Q. In other words, N_{φ} = N_{P} (Q). Since Q is fully F-normalised, it follows from 2.4.(II-S) that there is an automorphism ψ of N_{P} (Q) in F such that ψ|_{Q} = φ. Since φ has p-power order, by decomposing ψ into its p-part and its p′-part we may in fact assume that ψ has p-power order. Let τ : N_{P} (Q) → P be a morphism in F such that τ(N_{P} (Q)) is fully F-normalised. Now τψτ^{−1} is a p-element in Aut_{F}(τ(N_{P} (Q))), thus conjugate to an element in Aut_{P} (τ(N_{P} (Q))). Therefore we may choose τ in such a way that there is y ∈ N_{P} (τ(N_{P} (Q))) satisfying τψτ^{−1}(v) = ^{y}v for any v ∈ τ(N_{P} (Q)). Since ψ|_{Q} = φ, the automorphism τψτ^{−1} of τ(N_{P} (Q)) stabilises τ(Q). Thus y ∈ N_{P} (τ(Q)). Since Q is fully Fnormalised we have N_{P} (τ(Q)) ⊆ τ(N_{P} (Q)), hence ψ(u) = ^{τ^{−1}(y)}u for all u ∈ N_{P} (Q). But then in particular φ ∈ Aut_{P} (Q), contradicting our initial choice of φ. Thus Aut_{P} (Q) is a Sylow-p-subgroup in Aut_{F}(Q). The converse is easy since |N_{P} (Q)| = |Aut_{P} (Q)| · |C_{P} (Q)|. This proves the result. □

Lemma 2.6. Let F be a fusion system on a finite p-group P let Q, R be subgroups of P and let φ : Q → R be an isomorphism in F such that R is fully F-normalised. Then there is an isomorphism ψ : Q → R in F such that N_{ψ} = N_{P} (Q), or equivalently, such that ψ can be extended to a morphism from N_{P} (Q) to P in F.

Proof. The group φ ∘ Aut_{P} (Q) ∘ φ^{−1} is a p-subgroup of Aut_{F}(R). Since R is fully F-normalised, Aut_{P} (R) is a Sylow-p-subgroup of Aut_{F}(R) by 2.5. Thus there is β ∈ Aut_{F}(R) such that β ∘ φ ∘ Aut_{P} (Q) ∘ φ^{−1} ∘ β^{−1} ⊆ Aut_{P} (R). Set ψ = β ∘ φ. The above inclusion means precisely that for any y ∈ N_{P} (Q) there is z ∈ N_{P} (R) such that ψ ∘ c_{y} ∘ ψ^{−1} = c_{z}, where c_{y} and c_{z} are the automorphisms of Q and R induced by conjugation with y and z, respectively. Equivalently, for any y ∈ N_{P} (Q) there is z ∈ N_{P} (R) such that ψ ∘ c_{y} = c_{z} ∘ ψ, which in turn means that N_{ψ} = N_{P} (Q). The extension axiom (II-S) implies that ψ can be extended to a morphism from N_{P} (Q) to P in F. □

Proposition 2.7. ([26]) Let F be a fusion system on a finite p-group P. Let Q be a subgroup of P. Every morphism φ : Q → P such that φ(Q) is fully F-centralised extends to a morphism ψ : N_{φ} → P in F (that is, ψ|_{Q} = φ).

Proof. Let φ : Q → P be a morphism in F such that φ(Q) is fully F-centralised. Let ρ : φ(Q) → P be a morphism in F such that R = ρ(φ(Q)) is fully F-normalised. By 2.6 we may choose ρ in such a way that that N_{ρ} = N_{P} (φ(Q)). In particular, ρ extends to a morphism σ : N_{P} (φ(Q)) → P. But then N_{φ} ⊆ N_{ρ∘φ}, hence ρ ∘ φ extends to a morphism τ : N_{φ} → P. Thus τ(N_{φ}) ⊆ σ(N_{P} (φ(Q))), and hence we get a morphism σ^{−1}|_{τ(N_{φ})} ∘ τ : N_{φ} → P which extends φ as required. The result follows. □

Definition 2.8. Let F, F′ be a fusion systems on finite p-groups P, P′, respectively. A morphism of fusion systems from F to F′ is a pair (α, Φ) consisting of a group homomorphism α : P → P′ and a covariant functor Φ : F → F′ with the following properties:

(i) for any subgroup Q of P we have α(Q) = Φ(Q); (ii) for any morphism φ : Q → R in F we have Φ(φ) ∘ α = α ∘ φ.

Note that Φ, if it exists, is determined by α. Thus the set of morphisms from F to F′ can be viewed as a subset of the set of group homomorphisms from P to P′. In particular, the fusion systems F, F′ are isomorphic if there is a group isomorphism α : P ≅ P′ such that Hom_{F′}(α(Q), α(R)) = α ∘ Hom_{F}(Q, R) ∘ α^{−1}|_{α(Q)} for all subgroups Q, R of P.

If one takes the view that understanding a fusion system F on a given finite p-group P amounts to understanding what morphisms beyond those induced by P itself are morphisms in F, one ends up looking at a category obtained from “dividing out by inner automorphisms”:

Definition 2.9. Let F be a fusion system on a finite p-group P. The orbit category of F is the category F̄ having the subgroups of P as objects and, for any two subgroups Q, R of P we have HomF̄(Q, R) = Aut_{R}(R)\Hom_{F}(Q, R), with composition of morphisms induced by that in F.

This makes sense: with the notation above, the group Aut_{R}(R) acts on Hom_{F}(Q, R) by composition of group homomorphisms, and if φ, φ′ ∈ Hom_{F}(Q, R) are in the same Aut_{R}(R)-orbit and ψ, ψ′ ∈ Hom_{F}(R, S) in the same Aut_{S}(S)-orbit, a trivial verification shows that ψ ∘ φ, ψ′ ∘ φ′ ∈ Hom_{F}(Q, S) are in the same Aut_{S}(S)-orbit as well, for any subgroups Q, R, S of P.

In order to show that fusion systems of finite groups are indeed fusion systems in the sense of 2.4 we collect a few technicalities in the following Lemma.

Lemma 2.10. Let G be a finite group, let P be a Sylow-p-subgroup of G and set F = F_{P} (G). Then F is a category on P, and for any subgroup Q of P the following hold:

(i) Q is fully F-centralised if and only if C_{P} (Q) is a Sylow-p-subgroup of C_{G}(Q). (ii) Q is fully F-normalised if and only if N_{P} (Q) is a Sylow-p-subgroup of N_{G}(Q).

Proof. Any inclusion morphism is in F because it can be viewed as conjugation by the unit element. If φ : Q → R is a morphism in F then there is x ∈ G such that φ(u) = ^{x}u for all u ∈ Q. Then the induced isomorphism Q ≅ ^{x}Q is still given by conjugation with x, hence also a morphism in F. The composition of morphisms in F is the usual composition of group homomorphisms by definition 1.1. Thus F is a category on P in the sense of 2.1. Let Q be a subgroup of P. Let S be a Sylow-p-subgroup of C_{G}(Q) containing C_{P} (Q). Then there is x in G such that ^{x}(QS) ⊆ P. Then conjugation by x is an isomorphism φ : Q ≅ ^{x}Q in F, and ^{x}S ⊆ C_{P} (^{x}Q). It follows that |C_{P} (Q)| ≤ |S| ≤ |C_{P} (^{x}Q)|. Thus Q is fully F-centralised if and only if |C_{P} (Q)| = |S|, or if and only if C_{P} (Q) = S. This proves (i). The same argument with normalisers instead of centralisers proves (ii). □

Theorem 2.11. Let G be a finite group and let P be a Sylow-p-subgroup of G. The category F_{P} (G) is a fusion system.

Proof. Set F = F_{P} (G). Clearly F is a category on P, and we have F_{P} (P) ⊆ F. Thus we only have to check the two axioms (I-S) and (II-S) from 2.4. We have Aut_{F}(P) ≅ N_{G}(P)/C_{G}(P), and the image of P in this quotient group is a Sylow-p-subgroup, which implies that Aut_{P} (P) ≅ P/Z(P) is a Sylow-p-subgroup of Aut_{F}(P). This proves (I-S). Let now Q be a subgroup of P and let φ : Q → P be a morphism in F. Set R = φ(Q) and suppose that R is fully F-normalised. Let x ∈ G such that φ(u) = ^{x}u for all u ∈ Q. We have

N_{φ} = {y ∈ N_{P} (Q) | ∃ z ∈ N_{P} (R) : φ(^{y}u) = ^{z}φ(u) (∀u ∈ Q)} .

Since φ is given by conjugation with x, this translates to

N_{φ} = {y ∈ N_{P} (Q) | ∃ z ∈ N_{P} (R) : ^{xy}u = ^{zx}u (∀u ∈ Q)} .

The equation^{xy}u = ^{zx}u for all u ∈ Q means that x^{−1}z^{−1}xy centralises Q, hence z^{−1}xyx^{−1} centralises xQ = R, and so we can write xyx^{−1} = zc for some c ∈ C_{G}(R). This shows that we have

xN_{φ} ⊆ N_{P} (R)C_{G}(R) .

Since R is fully F-normalised, by 2.10 the group N_{P} (R) is a Sylow-p-subgroup of N_{G}(R), hence of N_{P} (R)C_{G}(R). Thus there is an element d ∈ C_{G}(R) such that ^{dx}N_{φ} ⊆ N_{P} (R). Define ψ : N_{φ} → P by ψ(y) = ^{dx}y for all y ∈ N_{φ}. We claim that ψ extends φ. Indeed, if u ∈ Q then ψ(u) = ^{dx}u = dφ(u) = φ(u) because d centralises R = φ(Q). This proves (II-S). □

Theorem 2.12. Let G be a finite group, let P be a Sylow-p-subgroup of G and let K be a normal p′-subgroup of G. Set Ḡ = G/K and denote by P̄ the image of P in Ḡ. The canonical group homomorphism α : G → Ḡ induces an isomorphism of fusion systems F_{P} (G) ≅ FP̄(Ḡ).

Proof. Since K is a p′-group the map α induces an isomorphism P ≅ P̄. Thus, for any two subgroups Q, R of P the map Hom_{F}(Q, R) → HomF̄(Q̄, R̄) induced by α is injective, where Q̄, R̄ are the canonical images of Q, R in P̄. In order to show that it is surjective, let ψ : Q̄ → R̄ be a morphism in F̄. Let S be the inverse image in P of ψ(Q̄). Then there is x ∈ G such that conjugation by x induces ψ; in other words, ^{x}Q ⊆ KS and hence both ^{x}Q and S are Sylow-p-subgroups of KS. Thus there is y ∈ K such that ^{yx}Q = S. Then φ : Q → R defined by φ(u) = ^{yx}u for u ∈ Q induces the morphism ψ as required. □

We note an elementary group theoretic consequence for future reference:

Corollary 2.13. Let G be a finite group, let K be a normal p′-subgroup of G and let α : G → G/K be the canonical surjection. Let T be a Sylow-p-subgroup of G/K and let P be a Sylow-p-subgroup of G such that α(P) = T. Let y ∈ G/K. Then there is x ∈ G such that α(x) = y and such that α(P ∩^{x}P) = T ∩^{y}T.

Proof. Set R = T ∩ ^{y}T. Then ^{y^{−1}}R ⊆ T. Thus conjugation by y^{−1} is a morphism R → ^{y^{−1}}R in F_{T} (G/K). By 2.12 this lifts to a morphism Q → ^{x^{−1}}Q for some x ∈ G satisfying α(x) = y, where Q is the inverse image of R in P. Then Q ⊆ P ∩^{x}P, hence α(Q) ⊆ α(P ∩^{x}P) ⊆ α(P) ∩ α(^{x}P) = T ∩^{y}T = α(Q) and hence equality. □

The following theorem by Levi and Oliver, using Solomon’s work mentioned in 1.7 above, implies that there are exotic fusion systems which are not of the form F_{P} (G) for any finite group G with Sylow-p-subgroup P:

Theorem 2.14. (Levi, Oliver [18]) Let P be a Sylow-2-subgroup of Spin_{7}(3). There is a fusion system F on P such that F_{P} (Spin_{7}(3)) ⊆ F and such that all involutions in P are F-conjugate.

Ruiz and Viruel classified in [23] all possible fusion systems on extraspecial p-groups of order p^{3} and, using the classification of finite simple groups, showed that some of them are exotic.

§3 Normalisers and centralisers

The concept of K-normalisers in fusion systems as well as the main result of this section, Theorem 3.6, are due to Puig [22]. Our presentation follows partly [6, Appendix]. We start with some particular cases of K-normalisers. As in group theory, one can define normalisers and centralisers in fusion systems:

Definition 3.1. Let F be a fusion system on a finite p-group P and let Q be a subgroup of P.

(i) The normaliser of Q in F is the category N_{F}(Q) on N_{P} (Q) having as morphisms all group homomorphisms φ : R → S, for R, S subgroups of N_{P} (Q), for which there exists a morphism ψ : QR → QS in F satisfying ψ(Q) = Q and ψ|_{R} = φ.

(ii) The centraliser of Q in F is the category C_{F}(Q) on C_{P} (Q) having as morphisms all group homomorphisms φ : R → S, for R, S subgroups of C_{P} (Q), for which there exists a morphism ψ : QR → QS in F satisfying ψ|_{Q} = Id_{Q} and ψ|_{R} = φ.

(iii) We denote by QC_{F}(Q) the subcategory of N_{F}(Q) on QC_{P} (Q) having as morphisms all group homomorphisms φ : R → S, for R, S subgroups of QC_{P} (Q), for which there exists a morphism ψ : QR → QS and v ∈ Q such that ψ|_{Q} = c_{v} and ψ|_{R} = φ, where c_{v} is the automorphism of Q given by conjugation with v.

(iv) We denote by N_{P} (Q)C_{F}(Q) the subcategory of N_{F}(Q) on N_{P} (Q) having as morphisms all group homomorphisms φ : R → S, for R, S subgroups of N_{P} (Q), for which there exists a morphism ψ : QR → QS and v ∈ N_{P} (Q) such that ψ|_{Q} = c_{v} and ψ|_{R} = φ.

Theorem 3.2. Let F be a fusion system on a finite p-group P and let Q be a subgroup of P. If Q is fully F-normalised then N_{F}(Q) is a fusion system on N_{P} (Q), and if Q is fully F-centralised then C_{F}(Q) is a fusion system on C_{P} (Q), QC_{F}(Q) is a fusion system on QC_{P} (Q) and N_{P} (Q)C_{F}(Q) is a fusion system on N_{P} (Q).

Note that we have inclusions of categories C_{F}(Q) ⊆ QC_{F}(Q) ⊆ N_{P} (Q)C_{F}(Q) ⊆ N_{F}(Q). Definition 3.1 and Theorem 3.2 are part of the more general concept:

Definition 3.3. ([22]) Let P be a finite p-group, let Q be a subgroup of P and let K be a subgroup of Aut(Q). The K-normaliser of Q in P is the subgroup

N_{P}^{K}(Q) = {y ∈ N_{P} (Q) | ∃α ∈ K : α(u) = yuy^{−1} ∀ u ∈ Q} .

We set Aut_{P}^{K}(Q) = K ∩ Aut_{P} (Q) and Aut_{F}^{K}(Q) = K ∩ Aut_{F}(Q).

In other words, N_{P}^{K}(Q) consists of all elements in N_{P} (Q) which induce, by conjugation, an automorphism of Q belonging to the automorphism subgroup K. Note that C_{P} (Q) ⊆ N_{P}^{K}(Q), and if K contains all inner automorphisms of Q then also Q ⊆ N_{P}^{K}(Q). We have Aut_{P}^{K}(Q) ≅ N_{P}^{K}(Q)/C_{P} (Q). There are various special cases of this construction we will encounter most frequently: if K = {Id_{Q}} then N_{P}^{K}(Q) = C_{P} (Q) and Aut_{P}^{K}(Q) = Aut_{F}^{K}(Q) = {Id_{Q}}; if K = Aut_{Q}(Q) then N_{P}^{K}(Q) = QC_{P} (Q) and Aut_{P}^{K}(Q) = Aut_{F}^{K}(Q) = Aut_{Q}(Q); finally, if K = Aut(Q) then N_{P}^{K}(Q) = N_{P} (Q), Aut_{P}^{K}(Q) = Aut_{P} (Q) and Aut_{F}^{K}(Q) = Aut_{F}(Q). For φ : Q → P any injective group homomorphism the group φK = φ ∘ K ∘ φ^{−1} is a subgroup of Aut(φ(Q)), and it makes thus sense to consider N_{P}^{^{φ}K}(φ(Q)).

Definition 3.4. Let F be a category on a finite p-group P, let Q be a subgroup of P and let K be a subgroup of Aut(Q). We say that Q is fully K-normalised in F if |N_{P}^{K}(Q)| ≥ |N_{P}^{^{φ}K}(φ(Q))| for any morphism φ : Q → P in F.

Thus Q is fully Aut(Q)-normalised in F if and only if Q is fully normalised in the sense of 2.2, and Q is fully {Id_{Q}}-normalised if and only if Q is fully F-centralised. Note that being fully Aut_{Q}(Q)-normalised is also equivalent to being fully centralised.

Definition 3.5 Let F be a fusion system on a finite p-group P, let Q be a subgroup of P and let K be a subgroup of Aut(Q). The K-normaliser of Q in F is the subcategory N_{F}^{K}(Q) of F on N_{P}^{K}(Q) having morphism sets

Hom_{N_{F}^{K}(Q)}(R, S) = {φ ∈ Hom_{F}(R, S) | ∃ ψ ∈ Hom_{F}(QR, QS) : ψ|_{R} = φ, ψ|_{Q} ∈ K} .

for any two subgroups R, S in N_{P}^{K}(Q).

If K = {Id_{Q}} then N_{F}^{K}(Q) = C_{F}(Q) and if K = Aut(Q) then N_{F}^{K}(Q) = N_{F}(Q). Other cases of interest, as mentioned above, include K = Aut_{Q}(Q) which yields N_{F}^{K}(Q) = QC_{F}(Q) and K = Aut_{P} (Q) which yields N_{F}^{K}(Q) = N_{P} (Q)C_{F}(Q). Thus Theorem 3.2 is a consequence of the following more general result.

Theorem 3.6. (Puig [22]) Let F be a fusion system on a finite p-group P, let Q be a subgroup of P and let K be a subgroup of Aut(Q). Suppose that Q is fully K-normalised in F. Then N_{F}^{K}(Q) is a fusion system on N_{P}^{K}(Q).

Lemma 3.7. Let F be a category on a finite p-group P such that F contains F_{P} (P), let Q be a subgroup of P and K a subgroup of Aut(Q).

(i) If Q is fully F-centralised and if Aut_{P}^{K}(Q) is a Sylow-p-subgroup of Aut_{F}^{K}(Q) then Q is fully K-normalised in F, and for any morphism φ : Q → P in F such that φ(Q) is fully φ ∘ K ∘ φnormalised in F, the group φ(Q) is fully F-centralised and Aut_{P}^{φ∘K∘φ^{−1}}(φ(Q)) is a Sylow-p-subgroup of Aut_{F}^{φ∘K∘φ^{−1}}(φ(Q)). (ii) If Q is fully F-centralised and K ⊆ Aut_{P} (Q) then Q is fully K-normalised in F.

Proof. Statement (i) follows from |N_{P}^{K}(Q)| = |Aut_{P}^{K}(Q)|·|C_{P} (Q)|. If K ⊆ Aut_{P} (Q) then Aut_{P}^{K}(Q) = Aut_{F}^{K}(Q), and hence (ii) is a special case of (i). □

Proposition 3.8. Let F be a fusion system on a finite p-group P, let Q be a subgroup of P and let K be a subgroup of Aut(Q).

(i) The group Q is fully K-normalised in F if and only if Q is fully F-centralised and Aut_{P}^{K}(Q) is a Sylow-p-subgroup of Aut_{F}^{K}(Q). (ii) Let φ : Q → R be an isomorphism in F and set L = φ ∘ K ∘ φ^{−1}. Suppose that R is fully L-normalised in F. Then there are morphisms τ : Q · N_{P}^{K}(Q) → P in F and κ ∈ K such that τ|_{Q} = φ ∘ κ. (iii) Let ψ : QN_{P}^{K}(Q) → P be a morphism in F. If Q is fully K-normalised in F then ψ(Q) is fully ψ ∘ K ∘ ψ^{−1}-normalised in F. (iv) If H is a normal subgroup of K and if Q is fully K-normalised in F then Q is fully H-normalised in F.

Proof. (i) Suppose that Q is fully K-normalised in F. By 2.6 there is an isomorphism φ : Q → R in F such that R is fully F-normalised and such that φ extends to a morphism ψ : N_{P} (Q) → P in F. Set L = φ ∘ K ∘ φ^{−1}. Thus ψ maps N_{P}^{K}(Q) to N_{P}^{L}(R). Since Q is fully K-normalised we have |N_{P}^{K}(Q)| ≥ |N_{P}^{L}(R)| and hence ψ induces in fact an isomorphism N_{P}^{K}(Q) ≅ N_{P}^{L}(R). Any such isomorphism restricts to an isomorphism C_{P} (Q) ≅ C_{R}(Q). Since R is fully F-normalised, R is fully F-centralised by 2.5 and thus Q is fully F-centralised. The group φ ∘ Aut_{P}^{K}(Q) ∘ φ^{−1} is a p-subgroup of the subgroup Aut_{F}^{L}(R) of Aut_{F}(R). By 2.5 the group Aut_{P} (R) is a Sylow-p-subgroup of Aut_{F}(R). Thus some conjugate of Aut_{P} (R) intersected with Aut_{F}^{L}(R) will be a Sylow-p-subgroup of Aut_{F}^{L}(R), and this Sylow-p-subgroup can, of course, be chosen to contain the p-subgroup φ ∘ Aut_{F}^{K}(Q) ∘ φ^{−1}. Thus there is β ∈ Aut_{F}(R) such that

φ ∘ Aut_{F}^{K}(Q) ∘ φ^{−1} ⊆ β ∘ Aut_{P} (R) ∘ β^{−1} ∩ Aut_{F}^{L}(R)

and such that the right side of this inclusion is a Sylow-p-subgroup of Aut_{F}^{L}(R). Then Aut_{P} (R) ∩ Aut_{F}^{β^{−1}∘L∘β}(R) = Aut_{P}^{β^{−1}∘L∘β}(R) is a Sylow-p-subgroup of Aut_{F}^{β^{−1}∘L∘β}(R). Since Q is fully Knormalised we have |Aut_{P}^{K}(Q)| ≥ |Aut_{P}^{β^{−1}∘L∘β}(R)| which implies that Aut_{P}^{K}(Q) is a Sylow-p-subgroup of Aut_{F}^{K}(Q). The converse follows from 3.7.(i). (ii) Since R is fully K-normalised in F it is in particular fully F-centralised. Set N = Q · N_{P}^{K}(Q). Then φ ∘ N ∘ φ^{−1} is a p-subgroup of Aut_{F}^{L}(R). By (i), Aut_{P}^{L}(R) is a Sylow-p-subgroup of Aut_{F}^{L}(R). Thus there is λ ∈ Aut_{F}^{L}(R) such that λ ∘ φ ∘ N ∘ φ^{−1} ∘ λ^{−1} ⊆ Aut_{P} (R). This means that N ⊆ N_{λ∘φ}, and hence λ ∘ φ extends to a morphism τ : Q · N_{P}^{K}(Q) → P. Now λ ∘ φ = φ ∘ (φ^{−1} ∘ λ ∘ φ) and κ = φ^{−1} ∘ λ ∘ φ ∈ Aut_{P}^{K}(Q) is as required.

(iii) The map ψ sends N_{P}^{K}(Q) to N_{P}^{ψ∘K∘ψ^{−1}}(ψ(Q)). In particular, |N_{P}^{K}(Q)| ≤ |N_{P}^{ψ∘K∘ψ^{−1}}(ψ(Q))|, whence the result. (iv) If Q is fully K-normalised in F then Q is fully F-centralised and Aut_{P}^{K}(Q) is a Sylow-p-subgroup of Aut_{F}^{K}(Q), by (i). If also H is normal in K then Aut_{F}^{H}(Q) is normal in Aut_{F}^{K}(Q), and hence Aut_{P}^{H}(Q) = Aut_{P}^{K}(Q) ∩ Aut_{F}^{H}(Q) is a Sylow-p-subgroup in Aut_{F}^{H}(Q). Thus, again by (i), Q is fully H-normalised in F. □

Proof of Theorem 3.6. Clearly N_{F}^{K}(Q) is a category on N_{P}^{K}(Q) in the sense of 2.1, and N_{F}^{K}(Q) contains F_{N_{P}^{K}(Q)}(N_{P}^{K}(Q)). For any subgroup R of N_{P}^{K}(Q) and any subgroup I of Aut(R) we set

K ∗ I = {α ∈ Aut(QR) | α|_{Q} ∈ K, α|_{R} ∈ I} .

Then

3.6.1. N_{N_{P}^{K}(Q)}^{I}(R) = N_{P}^{K∗I}(QR)

is the subgroup of all y ∈ N_{P} (Q) ∩ N_{P} (R) such that conjugation by y induces on Q an automorphism in K and on R an automorphism in I. With this notation,

3.6.2. the restriction map Aut_{F}^{K∗I}(QR) → Aut_{N_{F}^{K}(Q)}^{I}(R) is surjective.

Indeed, any β ∈ Aut_{N_{F}^{K}(Q)}(R) extends to some α ∈ Aut_{F}(QR) with α|_{Q} ∈ K, and since β = α|_{R}, the surjectivity follows. We observe next that

3.6.3. for any subgroup R of N_{P}^{K}(Q) and any subgroup I of Aut(R) there is a morphism φ : QR → QN_{P}^{K}(Q) in F such that φ|_{Q} ∈ K and such that φ(QR) = Qφ(R) is fully φ∘(K ∗I)∘φ^{−1}-normalised.

To see this, let ρ : QR → P be a morphism in F such that ρ(QR) is fully ρ ∘ (K ∗ I) ∘ ρ^{−1}- normalised. Set σ = ρ|_{Q}. Since Q is fully K-normalised in F, by 3.8.(ii) applied to σ^{−1} and σ(Q) there is a morphism τ : σ(Q)N_{P}^{σ∘K∘σ^{−1}}(σ(Q)) → P and κ ∈ σ ∘ K ∘ σ^{−1} such that τ|_{σ(Q)} = σ^{−1} ∘ κ.

Set φ = τ ∘ρ. Then φ|_{Q} = τ ∘ρ|_{Q} = τ ∘σ = σ^{−1} ∘κ∘σ ∈ K. Note that ρ(QR)N_{P}^{ρ∘(K∗I)∘ρ^{−1}}(ρ(QR)) ⊆ σ(Q)N_{P}^{σ∘K∘σ^{−1}}(σ(Q)) and that ρ(QR) is fully ρ ∘ (K ∗ I) ∘ ρ^{−1}-normalised in F. It follows from 3.8.(iii) applied to the appropriate restriction of τ and ρ(QR) that RQ is fully K ∗ I-normalised in F. This proves 3.6.3. We show next that

3.6.4. if a subgroup R of N_{P}^{K}(Q) is fully I-normalised in N_{F}^{K}(Q) for some subgroup I of Aut(R) then QR is fully K ∗ I-normalised in F.

Indeed, by 3.6.3 there exists a morphism φ : QR → QN_{P}^{K}(Q) in F such that φ|_{Q} ∈ K and such that Qφ(Q) is fully φ ∘ (K ∗ I) ∘ φ^{−1}-normalised. We have N_{P}^{K∗I}(QR) = N_{N_{P}^{K}(Q)}^{I}(R) by 3.6.1

and |N_{N_{P}^{K}(Q)}^{I}(R)| ≥ |N_{N_{P}^{K}(Q)}^{φ∘I∘φ^{−1}}(φ(R))| because R is fully I-normalised. Now N_{N_{P}^{K}(Q)}^{φ∘I∘φ^{−1}}(φ(R)) =

N_{P}^{K∗(φ∘I∘φ^{−1})}(φ(R)) by 3.6.1 and since φ|_{Q} ∈ K we have K ∗ (φ ∘ I ∘ φ^{−1}) = φ ∘ (K ∗ I) ∘ φ^{−1}, from which we get that the last group is equal to N_{P}^{φ∘(K∗I)∘φ^{−1}}(φ(R)). In particular, |N_{P}^{K∗I}(QR)| ≥ |N_{P}^{φ∘(K∗I)∘φ^{−1}}(φ(R))|, which proves the statement 3.6.4. We use this now to prove the stronger version of the Sylow axiom as in 2.5.

3.6.5. if a subgroup R of N_{P}^{K}(Q) is fully N_{F}^{K}(Q)-normalised then R is fully N_{F}^{K}(Q)-centralised and Aut_{N_{P}^{K}(Q)}(R) is a Sylow-p-subgroup of Aut_{N_{F}^{K}(Q)}(R).

In order to show 3.6.5 it suffices, by 3.7.(i), to show this for some subgroup R′ of N_{P}^{K}(Q) isomorphic to R in N_{F}^{K}(Q). Set A = Aut(R). By 3.6.3 there is a morphism φ : QR → QN_{P}^{K}(Q) such that φ|_{Q} ∈ K and such that Qφ(R) is fully φ ∘ (K ∗ A) ∘ φ^{−1}-normalised in F. By 3.8.(iv), Qφ(R) is then also fully φ ∘ (K ∗ Id) ∘ φ^{−1}-normalised in F, where Id is the trivial automorphism group of R. Since φ|_{Q} ∈ K we have φ ∘ (K ∗ Id) ∘ φ^{−1} = K ∗ Id, where now Id denotes abusively also the trivial automorphism group of φ(R). Note that N_{P}^{K∗Id}(Qφ(R)) = C_{N_{P}^{K}(Q)}(φ(R)). Thus, if ψ : QR → QN_{P}^{K}(Q) is any other morphism in F such that ψ|_{Q} ∈ K then, using that Qφ(R) is fully K ∗ Id-normalised in F, we get

|C_{N_{P}^{K}(Q)}(ψ(R))| = |N_{P}^{K∗Id}(Qψ(R))| ≤ |N_{P}^{K∗Id}(Qφ(R))| = |C_{N_{P}^{K}(Q)}(φ(R))|

This shows that φ(R) is fully N_{F}^{K}(Q)-centralised. Set B = φ ∘ A ∘ φ^{−1} = Aut(φ(R)). Since Qφ(R) is fully K ∗ B-normalised in F it follows from 3.8.(i) that Aut_{P}^{K∗B}(Qφ(R)) is a Sylow-p-subgroup of Aut_{F}^{K∗B}(Qφ(R)). The restriction map Aut_{F}^{K∗B}(Qφ(R)) → Aut_{N_{F}^{K}(Q)}(φ(R)) from 3.6.2 is surjective, and it maps Aut_{P}^{K∗B}(Qφ(R)) to Aut_{N_{P}^{K}(Q)}(φ(R)), which is hence a Sylow-p-subgroup of Aut_{N_{F}^{K}(Q)}(φ(R)). This proves 3.6.5.

For the proof of the extension axiom II-S, let R be a subgroup of N_{P}^{K}(Q) and let φ : R → N_{P}^{K}(Q) be a morphism in N_{F}^{K}(Q) such that φ(R) is fully N_{F}^{K}(Q)-normalised. Note that then φ(R) is in particular fully N_{F}^{K}(Q)-centralised, by 3.6.5. We consider the group N_{φ} as defined in 2.3 for the morphism φ in the category N_{F}^{K}(Q); that is, N_{φ} is the subgroup of all y ∈ N_{N_{P}^{K}(Q)}(R) for which there exists an element z ∈ N_{N_{P}^{K}(Q)}(φ(R)) satisfying φ(yuy^{−1}) = zφ(u)z^{−1} for all u ∈ R. Set I = Aut_{N_{φ}}(R). Note that conjugation by any y ∈ N_{φ} leaves Q invariant and induces an automorphism of Q belonging to K. Then N_{φ} = N_{N_{P}^{K}(Q)}^{I}(R) = N_{P}^{K∗I}(QR). We have φ ∘ I ∘ φ^{−1} ⊆ Aut_{N_{P}^{K}(Q)}(φ(Q)) ⊆ Aut_{N_{F}^{K}(Q)}(φ(R)), and hence φ(R) is fully φ ∘ I ∘ φ^{−1}-normalised in N_{F}^{K}(Q) by 3.7.(ii). Thus Qφ(R) is fully K ∗ (φ ∘ I ∘ φ^{−1})-normalised by 3.6.4. Since φ is a morphism in N_{F}^{K}(Q) there exists a morphism ψ : QR → P in F such that ψ|_{Q} ∈ K and ψ|_{R} = φ. So ψ(QR) = Qφ(R) is fully K ∗ (φ ∘ I ∘ φ^{−1})-normalised by the above. Now ψ^{−1} ∘ (K ∗ (φ ∘ I ∘ φ^{−1})) ∘ ψ = K ∗ I because ψ|_{Q} ∈ K and because ψ^{−1}|_{R} ∘ φ = Id_{R}. Applying 3.8.(ii) to ψ and QR yields the existence of a morphism τ : QR · N_{P}^{K∗I}(QR) → P and κ ∈ K ∗ I such that τ|_{QR} = ψ ∘ κ. Since κ|_{R} ∈ I = Aut_{R}(R) there is y ∈ R such that κ|_{R} = c_{y}, conjugation by y. Then τ|_{N_{φ}} ∘ c_{y}^{−1} : N_{φ} → P is a morphism in N_{F}^{K}(Q), because it extends to the morphism τ|_{QN_{φ}} ∘ c_{y} : QN_{φ} → P whose restriction to Q is τ|_{Q} ∘ c_{y}^{−1}|_{Q} = ψ|_{Q} ∘ κ|_{Q} ∘ c_{y}^{−1}|_{Q} ∈ K. Moreover, the morphism τ|_{N_{φ}} ∘ c_{y}^{−1} restricted to R is equal to ψ|_{R} ∘ κ|_{R} ∘ c_{y}^{−1}|_{R} = ψ|_{R} = φ, and hence this morphism extends φ as required in the extension axiom. This completes the proof of 3.6. □

As in finite group theory, given a fusion system F on a finite p-group P, one of the questions one would like to be able to address is that of control of fusion by normalisers of p-subgroups, a question which we can now rephrase as follows: when do we have an equality F = N_{F}(Q) for some normal subgroup Q of P. The most basic case, Burnside’s theorem 1.2, remains true for arbitrary fusion systems:

Theorem 3.9. Let F be a fusion system on an abelian finite p-group P. Then F = N_{F}(P).

Proof. Let Q be a subgroup of P and let φ : Q → P be a morphism in F. Since P is abelian, every subgroup of P is fully F-normalised, and hence φ extends to a morphism ψ : QC_{P} (Q) → P, by 2.4.(II-S). Again, since P is abelian, we have QC_{P} (Q) = P, and thus ψ ∈ Aut_{F}(P). □

We quote further results without proof, referring to the original papers. It has been shown by Stancu that the conclusion of 3.9 remains true for some other classes of finite p-groups:

Theorem 3.10. ([27]) Let P be a metacyclic finite p-group for some odd prime p. Then for every fusion system F on P we have F = N_{F}(P).

Theorem 3.11. ([27]) Let P be a finite p-group of the form P = E × A, where A is elementary abelian and where E is extra-special of exponent p^{2} if p is odd and not isomorphic to D_{8} if p = 2. Then for every fusion system F on P we have F = N_{F}(P).

The parts of Frobenius’s theorem 1.4 related to the fusion system of a finite group remain true as well:

Theorem 3.12. Let F be a fusion system on a finite p-group P. The following are equivalent:

(i) For any Q ⊆ P the group Aut_{F}(Q) is a p-group. (ii) F = F_{P} (P). (iii) For any non-trivial fully F-normalised subgroup Q of P we have N_{F}(Q) = F_{N_{P} (Q)}(N_{P} (Q)).

The proof of 3.12 is a trivial adaptation of the ideas of the proof of 1.4. The p-nilpotency criterion 1.5 of Glauberman and Thompson can also be generalised:

Theorem 3.13. ([14, Theorem A]) Let p be an odd prime and let F be a fusion system on a finite p-group P. We have F = F_{P} (P) if and only if N_{F}(Z(J(P))) = F_{P} (P).

Gilotti and Serena [8] gave a general criterion when a fusion system of a finite group is of the form N_{F}(Q), and Stancu generalised this to arbitrary fusion systems. Before we state this, we need the following terminology:

Definition 3.14. Let F be a fusion system on a finite p-group P and let Q be a subgroup of P. (iv) Q is weakly F-closed if for every morphism φ : Q → P in F we have φ(Q) = Q.

(v) Q is strongly F-closed, if for any subgroup R of P and any morphism φ : R → P in F we have φ(R ∩ Q) ⊆ Q.

If Q is strongly F-closed then Q is weakly F-closed. One easily checks that if Q is strongly F-closed then for any subgroup R of P and any morphism φ : R → P in F we have in fact φ(R ∩ Q) = φ(R) ∩ Q. Indeed, the left side is contained in the right side by the above definition, and the other inclusion is obtained by applying this inclusion to φ(R) and the morphism φ^{−1} viewed as morphism from φ(R) to P. If F = N_{F}(Q) for some subgroup Q of P, then clearly Q is strongly F-closed. The converse of this statement is not true, in general. More precisely:

Theorem 3.15. ([27, 6.11]) Let F be a fusion system on a finite p-group P, and let Q be a subgroup of P. The following are equivalent:

(i) F = N_{F}(Q). (ii) The subgroup Q is strongly F-closed and there is a central series Q = Q_{n} > Q_{n−1} > ⋯ > Q_{1} > {1} with Q_{i} weakly F-closed for 1 ≤ i ≤ n − 1.

If Q is an abelian and strongly F-closed subgroup of P then Q > {1} is a central series of weakly F-closed subgroups, and hence 3.15 applies:

Corollary 3.16. Let F be a fusion system on a finite p-group P and let Q be an abelian subgroup of P. We have F = N_{F}(Q) if and only if Q is strongly F-closed.

We give an alternative proof of 3.16 following 9.2 below. By [14], Glauberman’s ZJ-theorem 1.6 carries over as well, but it requires some extra effort to define a notion of Qd(p)-free fusion systems, replacing the hypothesis on G having no subquotient isomorphic to Qd(p) in 1.6; see 4.7 and 4.8 below for more details. When specialised to fusion systems of finite groups, normalisers and centralisers in the fusion system correspond to the fusion systems of the relevant normaliser and centralisers:

Proposition 3.17. Let G be a finite group, let P be a Sylow-p-subgroup and set F = F_{P} (G). Let Q be a subgroup of P.

(i) If Q is fully F-normalised then N_{F_{P} (G)}(Q) = F_{N_{P} (Q)}(N_{G}(Q)). (ii) If Q is fully F-centralised then C_{F_{P} (G)}(Q) = F_{C_{P} (Q)}(C_{G}(Q)). (iii) If Q is normal in G then F = N_{F}(Q).

Proof. The statements make sense: if Q is fully F-normalised then N_{P} (Q) is a Sylow-p-subgroup of N_{G}(Q), and if Q is fully F-centralised then C_{P} (Q) is a Sylow-p-subgroup of C_{G}(Q), by 2.8. The rest is a trivial verification. □

§4 Centric subgroups

Definition 4.1. Let F be a fusion system on a finite p-group P. A subgroup Q of P is called F-centric if for any isomorphism φ : Q → R in F we have C_{P} (R) = Z(R).

Note that every F-centric subgroup Q of P is trivially fully F-centralised, because all centralisers C_{P} (R) of subgroups R isomorphic to Q in F have the same order |Z(Q)|.

Definition 4.2. Let F be a fusion system on a finite p-group P. We denote by F^{c} the full subcategory of F having as objects all F-centric subgroups of P. We denote by F̄^{c} the canonical image of F^{c} in the orbit category F̄ of F.

Proposition 4.3. Let F be a fusion system on a finite p-group P and let Q be a fully F-centralised subgroup of P. Then QC_{P} (Q) is F-centric.

Proof. Let φ : QC_{P} (Q) → R be an isomorphism in F. Set Q_{1} = φ(Q). Then |C_{P} (Q_{1})| ≤ |C_{P} (Q)| since Q is fully F-centralised. Since φ(C_{P} (Q)) ⊆ C_{P} (Q_{1}) we also have the converse inequality |C_{P} (Q)| ≤ |C_{P} (Q_{1})|, hence R = Q_{1}C_{P} (Q_{1}). In particular, C_{P} (R) ⊆ C_{P} (Q_{1}) ⊆ R, whence the result. □

Proposition 4.4. Let F be a fusion system on a finite p-group P and let Q, R be subgroups of P such that Q ⊆ R. If Q is F-centric then R is F-centric and we have Z(R) ⊆ Z(Q).

Proof. Let ψ : R → P be a morphism in F. If Q is F-centric then C_{P} (ψ(Q)) = Z(ψ(Q)), hence C_{P} (ψ(R)) ⊆ C_{P} (ψ(Q)) ⊆ ψ(R) and hence C_{P} (ψ(R)) = Z(ψ(R)). In particular, Z(R) = C_{P} (R) ⊆ C_{P} (Q) = Z(Q). □

Even though elementary, the crucial observation is that thanks to 4.4, taking centers of centric subgroups is a contravariant functor from F̄^{c} to the category of finitely generated ℤ_{(p)}-modules, where ℤ_{(p)} = \{\frac{a}{b} | a, b ∈ ℤ, p |̸b\}.

Theorem 4.5. Let F be a fusion system on a finite p-group P There is a unique functor

Z : F̄^{c} ⟶ mod(ℤ_{(p)})

sending any F-centric subgroup Q of P to Z(Q) and sending the class of any morphism φ : Q → R between F-centric subgroups Q, R in F to the unique morphism Z(φ) : Z(R) → Z(Q) which sends z ∈ Z(R) to the unique element y ∈ Z(Q) satisfying φ(y) = z.

Proof. If φ : Q → R is a morphism in F with Q, R F-centric, then the subgroup φ(Q) of R is F-centric, and hence Z(R) ⊆ Z(φ(Q)) = φ(Z(Q)). Thus φ induces a unique map Z(R) → Z(Q) as claimed. This map does not depend on the choice of φ in its class modulo inner automorphisms of R because any inner automorphism of R is the identity on Z(R). Thus Z is well-defined. □

The following result appears in work of Külshammer and Puig [17] for fusion systems of p-blocks and in Broto, Castellana, Grodal, Levi, Oliver [5, §4] in general:

Theorem 4.6. Let F be a fusion system on a finite p-group P. For any F-centric fully normalised subgroup Q of P there is, up to isomorphism, a unique finite group L = L_{Q}^{F} having N_{P} (Q) as Sylow-p-subgroup such that Q ⊴ L, C_{L}(Q) = Z(Q) and N_{F}(Q) = F_{N_{P} (Q)}(L).

In other words, if F is a fusion system on a finite p-group P such that F = N_{F}(Q) for some F-centric normal subgroup Q of P then F is automatically the fusion system of some finite group. The proof of 4.6 uses some of the cohomological machinery developed in [6], showing that the center functor from 4.5 applied to the fusion system N_{F}(Q) is acyclic. The three properties of the group L = L_{Q}^{F} imply that Aut_{F}(Q) ≅ N_{L}(Q)/C_{L}(Q) = L/Z(Q), and hence L fits into a short exact sequence of finite groups

1 ⟶ Z(Q) ⟶ L ⟶ Aut_{F}(Q) ⟶ 1

whose restriction to the Sylow-p-subgroup N_{P} (Q) of L is of the form

1 ⟶ Z(Q) ⟶ N_{P} (Q) ⟶ Aut_{P} (Q) ⟶ 1 .

This characterises the group L up to isomorphism because the restriction map on cohomolgy H^{2}(Aut_{F}(Q); Z(Q)) → H^{2}(Aut_{P} (Q); Z(Q)) is well-known to be injective.

Definition 4.7. ([14, 1.1]) A fusion system F on a finite p-group P is called Qd(p)-free if Qd(p) is not involved in any of the groups L_{Q}^{F}, with Q running over the set of F-centric fully F-normalised subgroups of P.

Glauberman’s ZJ-theorem admits the following version for fusion systems (which for fusion systems of blocks has also been noted by G. R. Robinson):

Theorem 4.8. ([14, Theorem B]) Let p be an odd prime and let F be a fusion system on a finite p-group P. If F is Qd(p)-free then F = N_{F}(Z(J(P))).

While all morphisms in the fusion system F are monomorphisms, this is no longer true in the orbit category. However, we have the following:

Theorem 4.9. Let F be a fusion system on a finite p-group P. Every morphism in the category F̄^{c} is an epimorphism.

The technicalities for the proof of 4.9 are given in the following two well-known lemmas.

Lemma 4.10. Let F be a fusion system on a finite p-group P. Let Q, R be F-centric subgroups of P such that Q ⊆ R, and let φ ∈ Aut_{F}(R). We have φ|_{Q} = Id_{Q} if and only if φ ∈ Aut_{Z(Q)}(R).

Proof. Assume that φ|_{Q} = Id_{Q}. We proceed by induction over [R : Q]. Consider first the case where Q is normal in R. Let u ∈ Q and v ∈ R. Then ^{v}u ∈ Q, hence ^{v}u = φ(^{v}u) = ^{φ(v)}u, and thus v^{−1}φ(v) ∈ C_{R}(Q) = Z(Q), or equivalently, φ(v) = vz for some z ∈ Z(Q). If φ has order prime to p in Aut(R) this forces φ = Id_{R}. Therefore we may assume that the order of φ is a power of p. Upon replacing R by a fully F-normalised F-conjugate we may assume that φ ∈ Aut_{P} (R). Since φ restricts to Id_{Q} and since Q is F-centric this implies that φ ∈ Aut_{Z(Q)}(R). This proves 4.10 if Q is normal in R. In general, if φ|_{Q} = Id_{Q} then φ(N_{R}(Q)) = N_{R}(Q). Thus φ|_{N_{R}(Q)} ∈ Aut_{Z(Q)}(N_{R}(Q)) by the previous paragraph. Hence there is z ∈ Z(Q) such that c_{z} ∘φ|_{N_{R}(Q)} = Id_{N_{R}(Q)}, where c_{z} is the automorphism of R given by conjugation with z. By induction we get c_{z} ∘ φ ∈ Aut_{Z(N_{R}(Q))}(Q). As all involved groups are F-centric we have Z(N_{R}(Q)) ⊆ Z(Q), and thus φ ∈ Aut_{Z(Q)}(R) as claimed. The converse is trivial. □

Lemma 4.11. Let F be a fusion system on a finite p-group P, let Q, R be F-centric subgroups of P such that Q ⊆ R, and let φ, φ′ ∈ Hom_{F}(R, P) such that φ|_{Q} = φ′|_{Q}. Then φ(R) = φ′(R).

Proof. Let v ∈ N_{R}(Q). For every u ∈ Q we have φ(^{v}u) = φ′(^{v}u), hence φ(v)^{−1}φ′(v) ∈ C_{P} (φ(Q)) = Z(φ(Q)). It follows that φ(N_{R}(Q)) = φ′(N_{R}(Q)). By 4.10, φ|_{N_{R}(Q)} and φ′|_{N_{R}(Q)} differ by conjugation with an element in Z(Q), and we may therefore assume that their restrictions to N_{R}(Q) actually coincide. The equality φ(R) = φ′(R) follows by induction. □

Proof of Theorem 4.9. Let Q, R, S be F-centric subgroups of P, let φ ∈ Hom_{F}(Q, R) and let ψ, ψ′ ∈ Hom_{F}(R, S). Assume that the images of ψ ∘ φ and ψ′ ∘ φ in HomF̄^{c}(Q, S) coincide. Up to replacing ψ′ by some S-conjugate, we may assume that ψ ∘ φ = ψ′ ∘ φ. Thus the restrictions to φ(Q) of ψ, ψ′ coincide. It follows from 4.11 that ψ(R) = ψ′(R). Thus ψ^{−1} ∘ ψ′ is an automorphism of R which restricts to the identity on φ(Q), hence ψ^{−1} ∘ ψ′ ∈ Aut_{Z(φ(Q))}(R) by 4.10. Thus the images of ψ, ψ′ in the orbit category are equal. □

We conclude this section with a remark on what the notion of being centric means in the case of fusion systems of finite groups:

Proposition 4.12. Let G be a finite group, let P be a Sylow-p-subgroup of G and set F = F_{P} (G). A subgroup Q of P is F-centric if and only if C_{G}(Q) = Z(Q) × O_{p′}(C_{G}(Q)).

Proof. By 2.8, Q is F-centric if and only if Z(Q) is a Sylow-p-subgroup of C_{G}(Q). Since Z(Q) is a central subgroup of C_{G}(Q) this happens if and only if C_{G}(Q) splits as described, by some standard group theory. □

Using 4.12 it is easy to directly construct the groups L_{Q}^{F} in the case where F = F_{P} (G): just take L_{Q}^{F} = N_{G}(Q)/O_{p′}(C_{G}(Q)). Since C_{G}(Q) = Z(Q) × O_{p′}(Q) we get a short exact sequence

1 ⟶ Z(Q) ⟶ L_{Q}^{F} ⟶ N_{G}(Q)/C_{G}(Q) ≅ Aut_{F}(Q) ⟶ 1

as above.

5 Alperin’s fusion theorem

Alperin’s fusion theorem states that a fusion system F on a finite p-group P is completely determined by automorphism groups Aut_{F}(Q) with Q running over a certain subset of subgroups of P. Its original version, due to Alperin [1], is stated for fusion systems of finite groups and the class of so-called radical subgroups of P, refined by Goldschmidt [10] who extrapolates Alperin’s methods to show that the potentially smaller class of essential subgroups suffices to determine F. Puig showed in [21] that essential subgroups are essential - that is, no smaller class will determine F - and showed furthermore which subsets of Aut_{F}(Q) are actually necessary to determine F, with Q running over essential subgroups. We need some terminology from finite group theory. A proper subgroup H of a finite group G is called strongly p-embedded if H contains a Sylow-p-subgroup P of G and P ≠ 1 but H ∩^{x}P = 1 for any x ∈ G − H. The existence of a strongly p-embedded subgroup is equivalent to the poset S_{p}(G) of non-trivial p-subgroups of G being disconnected. Indeed, if H is a strongly p-embedded subgroup of G then S_{p}(G) has at least two connected components, namely that of a Sylow-p-subgroup P contained in H and that of ^{x}P with x ∈ G − H. Conversely, if S_{p}(G) has a connected component different from S_{p}(G) then the stabliser H of such a connected component is easily checked to be strongly p-embedded. We denote by O_{p}(G) the unique maximal normal subgroup of G whose order is a power of p, and we denote by O_{p′}(G) the unique maximal normal subgroup of G whose order is prime to p. Note that if O_{p}(G) ≠ 1 then S_{p}(G) is connected, and hence G has no strongly p-embedded subgroup. If F is a fusion system on a finite p-group P then for any subgroup Q in P we have Aut_{Q}(Q) ◁ Aut_{F}(Q), and hence Aut_{Q}(Q) ⊆ O_{p}(Aut_{F}(Q)).

Definition 5.1. Let F be a fusion system on a finite p-group P.

(i) A subgroup Q of P is called F-radical if O_{p}(Aut_{F}(Q)) = Aut_{Q}(Q). (ii) A subgroup Q of P is called F-essential if Q is F-centric and if Aut_{F}(Q)/Aut_{Q}(Q) has a strongly p-embedded subgroup.

By the remarks preceding this definition, if Q is an F-essential subgroup of P then Q is F-centric radical. The converse need not be true, in general. Note also that an F-essential subgroup has to be a proper subgroup of P because Aut_{F}(P)/Aut_{P} (P) is a p′-group.

Theorem 5.2. (Alperin’s fusion theorem) Let F be a fusion system on a finite p-group P. Every isomorphism in F can be written as a composition of finitely many isomorphisms of the form φ : R → S in F for which there exists a subgroup Q containing both R, S and an automorphism α ∈ Aut_{F}(Q) such that α|_{R} = φ and either Q = P or Q is fully F-normalised essential.

The picture to have in mind is this: given any isomorphism φ : R → S in F, the above theorem claims that there are finitely many subgroups R_{0}, R_{1}, .., R_{n} of P, with isomorphisms φ_{i} : R_{i} → R_{i+1} for 0 ≤ i ≤ n − 1, and fully F-normalised essential subgroups Q_{1}, Q_{2}, .., Q_{n} such that R_{i}, R_{i+1} are contained in Q_{i}, with automorphisms α_{i} ∈ Aut_{F}(Q_{i}) satisfying α_{i}|_{R_{i}} = φ_{i} for 0 ≤ i ≤ n − 1, and such that R = R_{0}, R_{n} = S with φ = φ_{n−1} ∘ ⋯ φ_{1} ∘ φ_{0}.

The crucial ingredient which makes this work is the extension axiom. We separate some of the arguments, and for the sake of providing future reference, allow some redundancy in the following four Lemmas. In many applications it suffices to know that the class of automorphisms of F-centric radical subgroups determines F; we break up the proof of 5.2 in such a way that its first half proves this weaker version of Alperin’s fusion system (which is, for instance, also proved in [6, Appendix]).

Lemma 5.3. Let F be a category on a finite p-group P and let Q be a fully F-normalised subgroup of P. Let α ∈ Aut_{F}(Q). There is a unique subgroup R of N_{P} (Q) containing QC_{P} (Q) such that Aut_{R}(Q) = Aut_{P} (Q) ∩ (α^{−1} ∘ Aut_{P} (Q) ∘ α), and we have R = N_{α}.

Proof. The intersection Aut_{P} (Q) ∩ (α^{−1} ∘ Aut_{P} (Q) ∘ α) is a subgroup of Aut_{P} (Q) ≅ N_{P} (Q)/C_{P} (Q), hence equal to Aut_{R}(Q) for a unique subgroup R of N_{P} (Q) containing QC_{P} (Q). By the definition of R, this group consists of all elements y ∈ N_{P} (Q) for which there is z ∈ N_{P} (Q) such that c_{y} = α^{−1} ∘ c_{z} ∘ α, or equivalently, such that α ∘ c_{y} = c_{z} ∘ α, where c_{y}, c_{z} are the automorphisms given by conjugation with y, z, respectivley. Explicitly, the last equation means that α(^{y}u) = ^{z}α(u) for all u ∈ Q, which is equivalent to saying that y belongs to the group N_{α}. □

Lemma 5.4. Let F be a fusion system on a finite p-group P and let Q be a fully F-normalised subgroup of P. There is a unique subgroup R of N_{P} (Q) containing QC_{P} (Q) such that O_{p}(Aut_{F}(Q)) = Aut_{R}(Q), and then every automorphism α ∈ Aut_{F}(Q) extends to an automorphism β ∈ Aut_{F}(R).

Proof. Since Q is fully F-normalised, Aut_{P} (Q) ≅ N_{P} (Q)/C_{P} (Q) is a Sylow-p-subgroup of Aut_{F}(Q). Thus there is indeed a unique subgroup R of N_{P} (Q) containing C_{P} (Q) such that Aut_{R}(Q) = O_{p}(Aut_{F}(Q)). Since Aut_{Q}(Q) is a normal p-subgroup of Aut_{F}(Q) we also have Q ⊆ R. Let α ∈ Aut_{F}(Q). Since Aut_{R}(Q) = O_{p}(Aut_{F}(Q)) we have α ∘ Aut_{R}(Q) ∘ α^{−1} = Aut_{R}(Q). This means that R ⊆ N_{α} by 5.3, hence α extends to a morphism β : R → P. For any y ∈ R and any any u ∈ Q we have α(^{y}u) = β(yuy^{−1}) = ^{β(y)}α(u), or equivalently, α ∘ c_{y} ∘ α^{−1} = c_{β(y)}, where c_{y}, c_{β(y)} are the automorhisms of Q given by conjugation with y, β(y). Since Aut_{R}(Q) is normal in Aut_{F}(Q) this implies c_{β(y)} ∈ Aut_{R}(Q) and hence β(y) ∈ R as C_{P} (Q) ⊆ R. Thus β ∈ Aut_{F}(R). □

Lemma 5.5. Let F be a fusion system on a finite p-group P and let Q be a fully F-normalised subgroup of P. Let α ∈ Aut_{F}(Q) such that Aut_{Q}(Q) is a proper subgroup of Aut_{P} (Q) ∩ (α ∘ Aut_{P} (Q) ∘ α^{−1}). Then α extends to a morphism ψ : R → P in F for some subgroup R of N_{P} (Q) which properly contains Q.

Proof. Let R be the unique subgroup of N_{P} (Q) containing QC_{P} (Q) such that Aut_{R}(Q) = Aut_{P} (Q)∩ (α∘Aut_{P} (Q)∘α^{−1}). The hypotheses imply that QC_{P} (Q) is a proper subgroup of R; in particular, Q is a proper subgroup of R. By 5.3 we have R ⊆ N_{α}, and hence α extends to a morphism ψ : R → P as claimed. □

Proof of Theorem 5.2. We denote by A the class of all isomorphisms in F which can be written as a composition of finitely many isomorphisms of the form ψ : R → R′ for which there exists a subgroup Q of P containing R, R′ and an automorphism α ∈ Aut_{F}(Q) such that α|_{R} = ψ and such that either Q = P or Q is fully F-normalised essential. The class A is obviously closed under composition of isomorphisms, and if ψ : R → R′ is an isomorphism in A then so is its inverse ψ^{−1} and any restriction of ψ to a subgroup V of R induces an isomorphism ψ|_{V} : V → V′ in A, where V′ = ψ(V). We have to show that in fact every isomorphism in F is in the class A.

Let φ : R → S be an isomorphism in F. We are going to show that φ can be written as composition of isomorphisms as claimed by induction over [P : R]. For R = S = P there is nothing to prove, so we may assume that R is a proper subgroup of P. The argument we are going to use constantly is as simple as this: whenever N_{φ} contains R properly we can extend φ to N_{φ}, and then φ is in A by induction. We use this argument to first show that we can assume that S is fully F-normalised. Indeed, let τ : S → T be an isomorphism in F such that T is fully F-normalised. By 2.5 we can choose τ such that τ can be extended to N_{P} (S). Thus, by induction, τ belongs to the class A, and hence, so does τ^{−1}. Therefore, in order to show that φ is in A it suffices to show that τ ∘ φ is in A.

In other words, after replacing S by T and φ by τ ∘ φ, we may assume that S is fully F-normalised. Then there is always some isomorphism ψ : R → S which extends to N_{P} (R), and hence which is in A by induction. Thus it suffices to show that the automorphism φ∘ψ^{−1} of S is in A. In other words, after replacing R by S and φ by φ ∘ ψ^{−1} we are down to assuming that R = S is fully F-normalised and φ ∈ Aut_{F}(R). By induction and Lemma 5.4 we may assume that R is F-radical centric.

Suppose that R is not essential. Set X = Aut_{P} (R) and Y = φ^{−1} ∘ Aut_{P} (R) ∘ φ. Since R is fully F-normalised, the groups X, Y are Sylow-p-subgroups of Aut_{F}(R). Since R is not essential, the poset of non-trivial p-subgroups of Aut_{F}(R)/Aut_{R}(R) is connected. In other words, there is a finite family {X_{0} | 1 ≤ j ≤ m} of Sylow-p-subgroups of Aut_{F}(R) such that X = X_{0}, X_{j} ∩X_{j+1} ≠ Aut_{R}(R) for 0 ≤ j ≤ m−1 and X_{m} = Y. For 0 ≤ j ≤ m let γ_{j} ∈ Aut_{F}(R) such that γ_{j} ∘Aut_{P} (R)∘γ_{j}^{−1} = X_{j}; since X_{0} = Aut_{P} (R) we can take γ_{0} = Id_{R}. We are going to prove, by induction over j, that the γ_{j} are in A. For j = 0 this is trivial. Suppose 0 < j ≤ m. We have X_{j−1} ∩ X_{j} ≠ Aut_{R}(R), hence

Aut_{P} (R) ∩ (γ_{j−1}^{−1} ∘ γ_{j} ∘ Aut_{P} (R) ∘ γ_{j}^{−1} ∘ γ_{j−1}) ≠ Aut_{R}(R) .

Thus, by Lemma 5.5 and induction the morphism γ_{j}^{−1} ∘ γ_{j−1} is in A. By our induction over j the morphism γ_{j−1} is in A, hence so is γ_{j}. In particular, γ_{m} is in A. Since

Y = γ_{m} ∘ Aut_{P} (R) ∘ γ_{m}^{−1} = φ^{−1} ∘ Aut_{P} (R) ∘ φ

we get that φ ∘ γ_{m} normalises Aut_{P} (R), hence extends to N_{P} (R). Thus φ ∘ γ_{m} is in A. But then φ is in A as well because γ_{m} and its inverse are so, which completes the proof. □

Since F-essential subgroups are F-centric, Alperin’s fusion theorem implies in particular that a fusion system F is determined by its subcategory of F-centric radical subgroups. The following observation is useful to determine F-centric radical subgroups of P:

Proposition 5.6. Let F be a fusion system on a finite p-group P such that F = N_{F}(Q) for some normal subgroup Q of P. Then Q is contained in every F-centric radical subgroup of P.

Proof. Let R be an F-centric radical subgroup of P. Since F = N_{F}(Q), the image Aut_{Q}(R) of Q in Aut_{F}(R) is a normal p-subgroup of Aut_{F}(R). Since R is F-radical this implies that Aut_{Q}(R) ⊆ Aut_{R}(R) = O_{p}(Aut_{F}(R)). Thus Q ⊆ RC_{P} (R). Since R is also F-centric we have RC_{P} (R) = R, whence the result. □

As an application of Alperin’s fusion theorem we get the following characterisations of fusion systems (cf. [19, §8]):

Proposition 5.7. Let P be a finite p-group, and let F, F′ be fusion systems on P such that F′ ⊆ F. The following are equivalent.

(i) F = F′. (ii) For any fully F′-centralised subgroup Z of order p of P we have Hom_{F}(Z, P) = Hom_{F′}(Z, P) and C_{F}(Z) = C_{F′}(Z).

Proof. Suppose that (ii) holds. Let Q be a non trivial subgroup of P and let φ ∈ Aut_{F}(Q). Let Z be a subgroup of order p of Z(Q). Let ψ : Z → P be a morphism in F′ such that ψ(Z) is fully F′-centralised. Since Q ⊆ C_{P} (Z), the morphism ψ extends to a morphism τ : Q → P in F′. In order to show that φ is a morphism in F′, it suffices to show that τ ∘ φ ∘ τ^{−1}|_{τ(Q)} ∈ Aut_{F′}(τ(Q)). Thus, after replacing Q by τ(Q), we may assume that Z is fully F′-centralised. By the assumptions, the morphism φ^{−1}|_{φ(Z)} : φ(Z) → Z belongs to F′, and hence extends to a morphism κ : Q → P in F′ (since Q = φ(Q) ⊆ C_{P} (φ(Z))). Then κ ∘ φ : Q → P restricts to the identity on Z, hence κ ∘ φ is a morphism in C_{F}(Z) = C_{F′}(Z). In particular, κ ∘ φ is a morphism in F′. But then so is φ, because κ is in F′. Alperin’s fusion theorem implies now (i). The converse is trivial. □

Lemma 5.8. Let F, F′ be fusion systems on a finite p-group P such that F′ ⊆ F. Let Q ⟶^{φ} R ⟶^{ψ} S be a sequence of two composable morphisms in F such that Q, R, S are F-centric. If any two of the three morphisms φ, ψ, ψ ∘ φ are in F′, so is the third.

Proof. If φ, ψ are in F′, so is ψ ∘ φ. If ψ, ψ ∘ φ are in F′, then so is φ = ψ^{−1}|_{Im(ψ∘φ)} ∘ ψ ∘ φ. Assume now that φ and ψ ∘φ are morphisms in F′. Up to replacing Q by φ(Q), we may assume that φ is the inclusion Q ⊆ R. Let v ∈ N_{R}(Q). Then, for any u ∈ Q, we have ψ(^{v}u) = ^{ψ(v)}u. Thus the morphism ψ|_{Q} extends to a morphism τ : N_{R}(Q) → P in F′. By 4.11, we have τ(N_{R}(Q)) = ψ(N_{R}(Q)) and hence ψ^{−1} ∘ τ ∈ Aut_{Z(Q)}(N_{R}(Q)) by 4.10. Thus ψ|_{N_{R}(Q)} is a morphism in F′. It follows inductively, that ψ is a morphism in F′. □

Proposition 5.9. Let F, F′ be fusion systems on a finite p-group P such that F′ ⊆ F. The following are equivalent.

(i) F = F′. (ii) Hom_{F}(Q, P) = Hom_{F′}(Q, P) for every minimal F-centric subgroup Q of P.

Proof. Assume that (ii) holds. Let R be an F-centric subgroup of P, and let Q be a minimal F-centric subgroup of P contained in R. Let φ ∈ Hom_{F}(R, P). Then φ|_{Q} ∈ Hom_{F}(Q, P) = Hom_{F′}(Q, P). But then φ ∈ Hom_{F′}(R, P) by 5.8. Alperin’s fusion theorem implies (i). The converse is trivial. □

§6 Quotients of fusion systems

If F is a fusion system on a finite p-group P such that F = N_{F}(Q) for some normal subgroup Q of P, we can define a quotient category F/Q as follows:

Definition 6.1. (Puig [22]) Let F be a fusion system on a finite p-group P such that F = N_{F}(Q) for some normal subgroup Q of P. We define the category F/Q on P/Q as follows: for any two subgroups R, S of P containing Q, a group homomorphism ψ : R/Q → S/Q is a morphism in the category F/Q if there exists a morphism φ : R → S in F satisfying φ(u)Q = ψ(uQ) for all u ∈ R.

Theorem 6.2. (Puig [22]) Let F be a fusion system on a finite p-group P such that F = N_{F}(Q) for some normal subgroup Q of P. The category F/Q is a fusion system on P/Q.

Proof. Clearly F/Q is a category on P/Q. Since Aut_{F/Q}(P/Q) is a quotient of Aut_{F}(P), the group Aut_{P/Q}(P/Q) is a Sylow-p-subgroup of Aut_{F/Q}(P/Q); thus the Sylow axiom I-S holds. It remains to show that the extension axiom II-S holds as well. Let R, S be subgroups of P containing Q, and let ψ : R/Q → S/Q be an isomorphism in F/Q such that S/Q is fully F-normalised. Since N_{P/Q}(S/Q) = N_{P} (S)/Q this implies that S is fully F-normalised. Thus, by 2.6, there is a morphism ρ : N_{P} (R) → P such that ρ(R) = S. Denote by σ : N_{P} (R)/Q → P/Q the morphism induced by ρ. In order to show that ψ extends to N_{ψ} it suffices to show that ψ ∘ σ^{−1}|_{S/Q} extends to N_{ψ∘σ^{−1}|_{S/Q}}, because the inverse image of N_{ψ} is contained in N_{P} (R). In other words, we may assume that R = S is fully F-normalised and that ψ ∈ Aut_{F/Q}(R/Q). Let K be the kernel of the canonical surjective map Aut_{F}(R) → Aut_{F/Q}(R/Q). Then X = K ∩ Aut_{P} (R) is a Sylow-p-subgroup of K. The Frattini argument implies that Aut_{F}(R) = KN_{Aut_{F} (R)}(X). Note that Aut_{P} (R) normalises X, so Aut_{P} (R) remains a Sylow-p-subgroup of N_{Aut_{F} (R)}(X). Thus we have a short exact sequence of finite groups

1 ⟶ K ∩ N_{Aut_{F} (R)}(X) ⟶ N_{Aut_{F} (R)}(X) ⟶^{α} Aut_{F/Q}(R/Q) ⟶ 1

The group L = (K ∩ N_{Aut_{F} (R)}(X))/X is a p′-group, and we have a short exact sequence

1 ⟶ L ⟶ N_{Aut_{F} (R)}(X)/X ⟶ Aut_{F/Q}(R/Q) ⟶ 1

By 5.3, the image of N_{ψ} in Aut_{F/Q}(R/Q) is the intersection of two Sylow-p-subgroups Aut_{P/Q}(R/Q) ∩ (ψ^{−1} ∘ Aut_{P/Q}(R/Q) ∘ ψ). By 2.13 there is τ ∈ N_{Aut_{F} (R)}(X) which lifts ψ such that the canonical map α sends Aut_{P} (R) ∩ (τ^{−1} ∘ Aut_{P} (R) ∘ τ) onto this intersection, or equivalently, the canonical map N_{τ} → N_{ψ} is surjective. This means that if δ : N_{τ} → P is a morphism in F which extends τ then its image γ modulo Q is a morphism in F/Q which extends ψ. This proves that II-S holds for the category F/Q. □

Theorem 6.2 applies in particular when F = C_{F}(Z) for some subgroup Z of Z(P). In that case one can be more precise about connections between F and F/Z. The following two Theorems are well-known.

Theorem 6.3. Let F be a fusion system on a finite p-group P and let Z be a subgroup of Z(P) such that F = C_{F}(Z). Let Q be a subgroup of P containing Z. Then the following hold.

(i) The canonical group homomorphism Aut_{F}(Q) → Aut_{F/Z}(Q/Z) is surjective, having as kernel an abelian p-group. (ii) If Q/Z is F/Z-centric then Q is F-centric. (iii) If Q is F-radical then Q/Z is F/Z-radical. (iv) If Q is F-centric radical then Q/Z is F/Z-centric radical.

Proof. Let φ ∈ Aut_{F}(Q). Suppose that φ induces the identity on Q/Z. That is, φ(u) = uζ(u) for u ∈ Q and suitable ζ(u) ∈ Z. One checks that ζ is in fact a group homomorphism ζ : Q → Z. The map sending such a φ to ζ in turn is easily seen to be an injective group homomorphism from ker(Aut_{F}(Q) → Aut_{F/Z}(Q/Z)) to the abelian p-group Hom(Q, Z), with group structure induced by that of Z. This proves (i). If Q/Z is F/Z-centric then C_{P/Z}(R/Z) ⊆ R/Z for every subgroup R isomorphic to Q in F. Thus C_{P} (R) ⊆ R for any such R, hence Q is F-centric. This proves (ii). If Q is F-radical, the kernel of the canonical map Aut_{F}(Q) → Aut_{F/Z}(Q/Z), being a p-group, must be contained in Aut_{Q}(Q), hence in that case we get an isomorphism Aut_{F}(Q)/Aut_{Q}(Q) ≅ Aut_{F/Z}(Q/Z)/Aut_{Q/Z}(Q/Z), and hence Q/Z is F/Z-radical.This proves (iii). Suppose now that Q is F-centric radical. We may assume that Q/Z is fully F/Z-centralised. The group Q/Z is F/Zradical by (iii). The kernel K of the canonical map Aut_{F}(Q) → Aut_{F/Z}(Q/Z) is a p-group, by (i). Since Q is F-radical this implies K ⊆ Aut_{Q}(Q). Let C be the inverse image in P of C_{P/Z}(Q/Z). That is, the image in P/Z of any element in C centralises Q/Z, and hence Aut_{C}(Q) ⊆ K. Thus Aut_{C}(Q) ⊆ Aut_{Q}(Q), which implies C ⊆ QC_{P} (Q). Since Q is F-centric this implies in fact that C ⊆ Q. Taking images in P/Z yields C_{P/Z}(Q/Z) ⊆ Q/Z, or equivalently, C_{P/Z}(Q/Z) = Z(Q/Z). Since Q/Z was chosen to be fully F/Z-centralised, this implies that Q/Z is F/Z-centric, whence (iv). □

Theorem 6.4. Let F, F′ be fusion systems on a finite p-group P such that F = C_{F}(Z) and F′ = C_{F′}(Z) for some subgroup Z of Z(P). Suppose that F ⊆ F′. Then F = F′ if and only if F/Z = F′/Z.

Proof. Suppose that F/Z = F′/Z. Let Q be an F′-centric radical subgroup of P. Then, by 6.3, the kernel K of the canonical map Aut_{F′}(Q) → Aut_{F′/Z}(Q/Z) is contained in Aut_{Q}(Q). Thus K is also the kernel of the canonical map Aut_{F}(Q) → Aut_{F/Q}(Q/Z). Since Aut_{F/Z}(Q/Z) = Aut_{F′/Q′}(Q/Z) it follows that Aut_{F}(Q) and Aut_{F′}(Q) have the same order. The assumption F ⊆ F′ implies Aut_{F}(Q) = Aut_{F′}(Q). The equality F = F′ follows now from Alperin’s fusion theorem. The converse is trivial. □

The next Theorem is a slight generalisation of 6.4:

Theorem 6.5. ([14, 3.4]) Let P be a finite p-group, let Q be a normal subgroup of P and let F, G be fusion systems on P such that F = P C_{F}(Q) and such that G ⊆ F. Let R be a normal subgroup of P containing Q. We have G = N_{F}(R) if and only if G/Q = N_{F/Q}(R/Q).

Proof. Suppose that G/Q = N_{F/Q}(R/Q). In order to show the equality G = N_{F}(R) we proceed by induction over the order of Q. If Q = 1 there is nothing to prove. If Q ≠ 1 then Z = Q ∩ Z(P) ≠ 1 because Q is normal in P. Since F = P C_{F}(Q) we have in fact F = C_{F}(Z). Set F̄ = F/Z and Ḡ = G/Z. Similarly, set P̄ = P/Z, Q̄ = Q/Z and R̄ = R/Z. Then F̄, Ḡ are fusion systems on P̄ satisfying F̄ = P̄CF̄(Q̄) and Ḡ ⊆ F̄. The canonical isomorphism P̄/Q̄ ≅ P/Q induces obviously isomorphisms of fusion systems Ḡ/Q̄ ≅ G/Q and N_{F̄/Q̄}(R̄/Q̄) ≅ N_{F/Q}(R/Q). Thus, by induction, we get that Ḡ = NF̄(R̄). Note that NF̄(R̄) = N_{F}(R)/Z. Any Ḡ-centric radical subgroup of P̄ contains R̄. Thus, by 6.3, any G-centric radical subgroup of P contains R. Since Ḡ = NF̄(R̄) this implies that G = N_{G}(R). Since also G ⊆ F it follows that G ⊆ N_{F}(R). Thus, by 6.4 applied to G and N_{F}(R) we get the equality G = N_{F}(R). The converse is trivial. □

When specialised to fusion systems of finite groups, the notion of quotients of fusion systems coincides with what one expects:

Proposition 6.6. Let G be a finite group, let P be a Sylow-p-subgroup and let Q be a subgroup of P which is normal in G. Set F = F_{P} (G). Then F = N_{F}(Q) and we have F/Q = F_{P/Q}(G/Q).

Proof. Trivial verification. □

7 Normal fusion systems

The material of this and the following section is from [19] and is an attempt to imitate the definitions of normal subgroups and simple groups in the context of fusion systems.

Definition 7.1 Let F be a category on a finite p-group P, and let F′ be a category on a subgroup P′ of P. We say that F normalises F′ if P′ is strongly F-closed and if for every isomorphism φ : Q → Q′ in F and any two subgroups R, R′ of Q ∩ P′ we have

φ ∘ Hom_{F′}(R, R′) ∘ φ^{−1}|_{φ(R)} ⊆ Hom_{F′}(φ(R), φ(R′)) .

We say that F′ is normal in F and write F′ ⊴ F if F′ is contained in F and F normalises F′.

In other words, F normalises F′ if for any isomorphism φ : Q → Q′ in F and any morphism ψ : R → R′ in F′ such that ⟨R, R′⟩ ⊆ Q, we have ⟨φ(R), φ(R′)⟩ ⊆ P′ and the induced morphism φ ∘ ψ ∘ φ^{−1} : φ(R) → φ(R′) is a morphism in F′. Note that this implies that we have in fact an equality

φ ∘ Hom_{F′}(R, R′) ∘ φ^{−1}|_{φ(R)} = Hom_{F′}(φ(R), φ(R′)) .

Indeed, the left side is contained in the right side by the definition, and the other inclusion follows from applying this inclusion to φ^{−1}, φ(R), φ(R′) instead of φ, R, R′, respectively. Applied to R = R′ and S = φ(R) and making use of Alperin’s fusion theorem this implies in particular that if R, S are subgroups of P′ which are isomorphic in F then Aut_{F′}(R) ≅ Aut_{F′}(S).

The unique category on the trivial subgroup {1} of P is a fusion system which is normal in any fusion system F on P. The obvious motivating example for the definition of normal fusion systems is this:

Proposition 7.2. Let G be a finite group, let P be a Sylow-p-subgroup of G, and let N be a normal subgroup of G. We have F_{P ∩N}(N) ⊴ F_{P} (G).

Proof. Trivial. □

Proposition 7.3. Let F be a fusion system on a finite p-group P. Then F_{P} (P) is normal in F if and only if F = N_{F}(P).

Proof. Suppose that F_{P} (P) ⊴ F. Then in particular for any morphism φ : R → P in F and any u ∈ N_{P} (R) there is v ∈ N_{P} (φ(R)) such that φ(^{u}r) = ^{v}φ(r) for all r ∈ R. Whenever φ(R) is fully F-centralised, φ extends to a morphism ψ : N_{P} (R) → P in F. In particular, this holds if R, and hence φ(R), are F-centric. But then also N_{P} (R) and ψ(N_{P} (R)) are F-centric. Inductively, it follows that φ can be extended to an automorphism of P belonging to F. Thus, by Alperin’s fusion theorem, we get F = N_{F}(P). The converse is easy. □

In fact, Proposition 7.3 remains true with P replaced by any subgroup of P, by a result of Stancu [27, 6.2]. We include a proof in §9 below.

Proposition 7.4. Let F be a fusion system on a finite p-group P. If Q is a strongly F-closed abelian subgroup of P then F_{Q}(Q) is normal in F.

Proof. Since Q is abelian, the only morphisms in F_{Q}(Q) are inclusions R ⊆ R′ of subgroups R, R′ of Q. Since Q is strongly F-closed, the result follows. □

Proposition 7.5. Let F, F′ be fusion systems on a finite p-group P such that F′ is normal in F. Then for every subgroup Q of P the index [Aut_{F}(Q) : Aut_{F′}(Q)] is prime to p.

Proof. Let Q be a subgroup of P, and let φ : Q → R be an isomorphism in F such that the subgroup R of P is fully F-normalised. Then Aut_{P} (R) is a Sylow-p-subgroup of Aut_{F}(R), and Aut_{P} (Q) ⊆ Aut_{F′}(R). Since F′ is normal in F, it follows that the Sylow-p-subgroup φ^{−1}∘Aut_{P} (R)∘φ of Aut_{F}(Q) is contained in Aut_{F′}(Q). Thus the index of Aut_{F′}(Q) in Aut_{F}(Q) is prime to p. □

Remark 7.6. Proposition 7.5 is not true, in general, without the assumption that F′ is normal in F. Consider the case of a fusion system F on P such that there is a subgroup Q of P which is fully F-centralised but not fully F-normalised, and set F′ = F_{P} (P). Then Aut_{P} (Q) = Aut_{F′}(Q) is not a Sylow-p-subgroup of Aut_{F}(Q). The following is an example for this situation.

Example 7.7. Let G = S_{8} be the symmetric group on eight letters, set E_{1} = ⟨(15)(26)(37)(48)⟩, E_{2} = ⟨(13)(24), (57)(68)⟩, E_{4} = ⟨(12), (34), (56), (78)⟩ . Then P = (E_{4}⋊E_{2})⋊E_{1} is a Sylow-2-subgroup of G. Set F = F_{P} (G). The subgroup E_{4} of P is F-centric, hence Q = E_{4} ⋊ ⟨(13)(24)(57)(68)⟩ and R = E_{4} ⋊ E_{1} are F-centric as well. Conjugating Q by (35)(46) yields R, hence Q ≅ R in F. Clearly Q is normal in P; in particular, Q is fully F-normalised. Conjugating (15)(26)(37)(48) ∈ R by (13)(24) ∈ E_{2} yields (17)(28)(35)(46). This is not an element in R since 7 does not belong to the R-orbit of 1 (which is equal to {1, 2, 5, 6}). Thus R is not normal in P, and hence R is not fully F-normalised.

Remark 7.7. Aschbacher calls in [3] a subsystem F′ of F invariant if it satisfies the normality condition in 7.1, and F′ is called normal if it is invariant and has in addition the property that every φ ∈ Aut_{F′}(Q) extends to some ψ ∈ Aut_{F}(QC_{P} (Q)) such that ψ(z)z^{−1} ∈ Z(Q) for all z ∈ C_{P} (Q). It is then shown in [3, 6.3] that if F = F_{P} (G) for some finite group G and some Sylow-p-subgroup P and F′ = F_{Q}(N) for some normal subgroup N of G and Q = P ∩ N this additional condition is satisfied. The advantage of this stronger notion of normality is that it allows for carrying the analogy between normal subgroups and normal subsystems further; see [3].

8 Simple fusion systems

Definition 8.1. (cf. [19, 4.1]) A fusion system F on a non trivial finite p-group P is called simple if F has no proper non trivial normal fusion subsystem.

The fusion system F_{P} (G) of a finite simple group G with Sylow-p-subgroup P is not simple in general. For instance, if G has an abelian Sylow-p-subgroup P of order at least p^{2}, then F_{P} (G) cannot be simple by 8.4 below. However, if a simple fusion system F on a finite p-group P is equal to F_{P} (G) for some finite group G containing P as Sylow-p-subgroup, then G can be chosen to be simple:

Proposition 8.2. Let F be a simple fusion system on some finite p-group P. Suppose that F = F_{P} (G) for some finite group G having P as Sylow-p-subgroup. If O_{p′}(G) = 1 and if F_{P} (H) ≠ F_{P} (G) for any proper subgroup H of G containing P, then G is simple. In particular, if G has minimal order such that P is a Sylow-p-subgroup of G and such that F = F_{P} (G), then G is simple.

Proof. Suppose that O_{p′}(G) = 1 and that F_{P} (H) ≠ F_{P} (G) for any proper subgroup H of G containing P. Let N be a non-trivial normal subgroup of G. Then N ∩ P is a Sylow-p-subgroup of N, and F_{N∩P} (N) is a normal fusion system in F_{P} (G). As O_{p′}(G) = 1, we have N ∩ P ≠ 1. As F_{P} (G) is simple, this forces P ⊆ N and F_{P} (N) = F_{P} (G), hence N = G by the assumptions. Let now G be a finite group of minimal order such that P is a Sylow-p-subgroup of G and such that F = F_{P} (G). Then O_{p′}(G) = 1, because the canonical map G → G/O_{p′}(G) induces an isomorphism of fusion systems. By the minimality of G, we have F_{P} (H) ≠ F_{P} (G) for any proper subgroup H of G containing P. Thus the second statement follows from the first. □

Proposition 8.3. Let P be a finite p-group. Then F_{P} (P) is simple if and only if P is cyclic of order p.

Proof. By 7.4, for every subgroup Z of Z(P) we have F_{Z}(Z) ⊴ F_{P} (P), from which the statement follows. □

Proposition 8.4. Let P be a finite abelian p-group and let F be a fusion system on P. Then F is simple if and only if P has order p and F = F_{P} (P).

Proof. If F is simple, then F = F_{P} (P) by 7.4, and hence |P | = p by 8.3. The converse is clear. □

Proposition 8.5. (B. Oliver) Let F, F′ be fusion systems on a finite p-group P such that F′ ⊴ F and such that Aut_{F}(P) = Aut_{F′}(P). Then F′ = F.

Proof. Suppose that F′ ≠ F. Let Q be a subgroup of maximal order such that Aut_{F′}(Q) ≠ Aut_{F}(Q). By the assumptions, Q is a proper subgroup of P. Since F′ is normal in F we may assume that Q is fully F-normalised. Then Aut_{P} (Q) is a Sylow-p-subgroup of Aut_{F}(Q). Moreover, Aut_{F′}(Q) is a normal subgroup of Aut_{F}(Q) containing Aut_{P} (Q), and hence, by the Frattini argument, we have Aut_{F}(Q) = N_{Aut_{F} (Q)}(Aut_{P} (Q))Aut_{F′}(Q). By the extension axiom (II-S) every automorphism of Q in N_{Aut_{F} (Q)}(Aut_{P} (Q)) extends to an automorphism of N_{P} (Q) in F, hence in F′ by the maximality assumption on Q. This in turn implies that N_{Aut_{F} (Q)}(Aut_{P} (Q)) ⊆ Aut_{F′}(Q), leading to the contradiction Aut_{F}(Q) = Aut_{F′}(Q). □

Corollary 8.6. Let F be a fusion system on a finite p-group P. Assume that Aut_{F}(P) = Aut_{P} (P) and that P has no proper non trivial strongly F-closed subgroup. Then F is simple.

Proof. Let F′ be a fusion system on a non trivial subgroup P′ of P such that F′ ⊴ F. Then P′ is strongly F-closed, hence P′ = P by the assumptions. Since Aut_{P} (P) ⊆ Aut_{F′}(P) ⊆ Aut_{F}(P), the assumptions imply further that Aut_{F′}(P) = Aut_{F}(P). Thus F′ = F by 8.5. □

Corollary 8.7. Let F be a fusion system on a finite p-group P. Suppose that P is generated by the set of its subgroups of order p, that all subgroups of order p in P are F-conjugate and that Aut_{F}(P) = Aut_{P} (P). Then F is simple.

Proof. Let Q be a non-trivial strongly F-closed subgroup of P. Since all subgroups of order p of P are F-conjugate it follows that Q contains all subgroups of order p of P. But then Q = P by the assumptions on P, and hence F is simple by 8.6. □

Example 8.8. Let P = ⟨x⟩ ⋊ ⟨t⟩, such that x^{2^{n}} = 1 = t^{2} for some integer n ≥ 2 and txt = x^{−1}; that is, P is a dihedral 2-group of order 2^{n+1} ≥ 8.

Then P has three conjugacy classes of involutions, namely the classes of the elements z = x^{2^{n−1}}, t and xt. Besides the trivial fusion system F_{P} = F_{P} (P), there are two other systems, up to isomorphism. We denote by F_{P}^{I} the fusion system on P generated by F_{P} and an automorphism of order 3 of the Klein four group ⟨z⟩ × ⟨t⟩. Thus z and t are F_{P}^{I} -conjugate, while z and xt are not; hence there are now two F_{P}^{I} -conjugacy classes of involutions in P. We denote by F_{P}^{II} the fusion system on P generated by F_{P} and an automorphism of order 3 on each of the Klein four groups ⟨z⟩ × ⟨t⟩ and ⟨z⟩ × ⟨xt⟩. Thus all involutions in P are F_{P}^{II}-conjugate. Any fusion system on P is isomorphic to one of F_{P}, F_{P}^{I}, F_{P}^{II} and any of these systems appear as fusion systems F_{P} (G) of some finite group G having P as Sylow-2-subgroup. Any 2-block of a finite group having P as defect group has 1 or 2 or 3 isomorphism classes of simple modules, and then its fusion system is isomorphic to F_{P} or F_{P}^{I} or F_{P}^{II}, respectively. The only simple fusion system on P is F_{P}^{II}, and moreprecisely, if Q is the subgroup of index 2 in P generated by x^{2} and t, we have F_{Q}^{II} ⊴ F_{P}^{I}. Here, for notational convenience, if Q is a Klein four group, we denote by F_{Q}^{I} and by F_{Q}^{II} the unique fusion system on Q generated by some automorphism of order 3 of Q.

All three fusion systems arise from finite groups. If q is an odd prime power such that q ≡ ±1(mod8), then the group P SL_{2}(q) has a dihedral Sylow-2-subgroup P, and F_{P} (P SL_{2}(q)) = F_{P}^{II}. In particular, F_{P} (P SL_{2}(q)) is simple in that case. If q ≡ ±3(mod8) then P SL_{2}(q) has a Klein four group Q as Sylow-2-subgroup, and hence F_{P} (P SL_{2}(q)) cannot be simple by 8.4. In this case the inclusion F_{Q}^{II} ⊴ F_{P}^{I} is realised by the inclusion P SL_{2}(q) ⊴ P GL_{2}(q).

Theorem 8.9. ([19, 6.1]) Let q be an odd prime power such that q ≡ ±3(mod 8) and let S be a Sylow-2-subgroup of Ω_{7}(q). We have Aut_{Ω_{7}(q)}(S) = Aut_{S}(S) and S has no non-trivial proper strongly F_{Ω_{7}(q)}-closed subgroup. In particular, the fusion system F_{S}(Ω_{7}(q)) is simple.

Let q be an odd prime power such that q ≡ ±3(mod 8) and let P be a Sylow-2-subgroup of the 7-dimensional spinor group Spin_{7}(q) over 𝔽_{q}. Then Spin_{7}(q) has a central involution z such that Spin_{7}(q)/⟨z⟩ ≅ Ω_{7}(q), and hence S = P/⟨z⟩ is isomorphic to a Sylow-2-subgroup of Ω_{7}(q). R. Solomon showed in [25] that if q ≡ ±3(mod8), no finite group having P as Sylow-2-subgroup can have a fusion system which properly contains F_{P} (Spin_{7}(q)), in which all involutions of P are conjugate and which has the property that C_{F}(z)/⟨z⟩ ≅ F_{S}(Ω_{7}(q)). Levi and Oliver proved in [18, 2.1], that there is actually for any odd prime power q a fusion system F_{Sol(q)} on P with the above properties. We are going to call this the Solomon 2-fusion system F_{Sol(q)}. Kessar showed in [12] that the fusion system F_{Sol(3)} cannot even occur as fusion system of a 2-block of a finite group with P as defect group.

Theorem 8.10. ([19, 7.1]) The Solomon fusion system F_{Sol(3)} is simple.

We certainly expect that F_{Sol(q)} should be simple for any odd prime power q, but this case is open at present. The simplicity of F_{Sol(q)} would follow from the material in [19] provided one could show that Aut(S) is a 2-group, where S is a Sylow-2-subgroup of Ω_{7}(q). The following consequence of 5.7 is occasionally a useful criterion for the simplicity of a fusion system:

Proposition 8.11. Let F, F′ be fusion systems on a finite p-group P such that F′ ⊴ F. If Hom_{F}(Z, P) = Hom_{F′}(Z, P) and C_{F}(Z)/Z is a simple fusion system on C_{P} (Z)/Z for any fully F′-centralised subgroup Z of order p of P, then F′ = F.

Proof. We have C_{F′}(Z) ⊴ C_{F}(Z) and hence C_{F′}(Z)/Z ⊴ C_{F}(Z)/Z. Thus, if C_{F}(Z)/Z is simple for any fully F′-centralised subgroup Z of order p of P, then C_{F′}(Z)/Z = C_{F}(Z)/Z. Since p′- automorphisms lift uniquely through central p-extensions this implies C_{F′}(Z) = C_{F}(Z), hence F′ = F by 5.7. □

§9 Normal subsystems and control of fusion

The content of this section is from [20]. We show that if a normal subsystem F′ of a fusion system F is controlled by a single subgroup Q′ then F itself is controlled by a single subgroup Q, and we can take for Q the weak F-closure of Q′.

Theorem 9.1. Let p be a prime and let F be a fusion system on a finite p-group P. Let P′ be a strongly F-closed subgroup of P and let F′ be a fusion system on P′ such that F′ ⊴ F. Suppose that there is a subgroup Q′ of P′ such that F′ = N_{F′}(Q′). Let Q be the subgroup of P′ generated by all subgroups of P′ isomorphic to Q′ in F. Then F = N_{F}(Q).

The proof of Theorem 9.1 is based on one hand on ideas of G. R. Robinson, which have led to [15, 1.5], and on the other hand on the notion of normal subsystems developed in [19, §3]; see §7 above. It seems to be an open question as to whether a simple fusion system in the sense of 8.1 can have non trivial proper strongly closed subgroups. The following result, due to Stancu [27], can be obtained as a consequence of Theorem 9.1 and provides a partial answer to this problem:

Corollary 9.2. ([27, Proposition 6.2]) Let p be a prime and let F be a fusion system on a finite p-group P. Let Q be a subgroup of P. We have F_{Q}(Q) ⊴ F if and only if F = N_{F}(Q).

Corollary 9.2 can be used to give an alternative proof of 3.16: if Q is an abelian strongly F-closed subgroup of P, then by 7.4, F_{Q}(Q) is normal in F, and hence F = N_{F}(Q) by 9.2.

We divide the proofs in a series of statements.

Proposition 9.3. Let F be a fusion system on a finite p-group P. Let P′ be a strongly F-closed subgroup of P and let F′ be a category on P′ such that F′ ⊆ F. Let Q be a subgroup of P′.

(i) If Q is fully F-centralised then Q is fully F′-centralised.

(ii) If Q is fully F-normalised then Q is fully F′-normalised.

Proof. (i) Let τ : Q → R be an isomorphism in F′ such that R is fully F′-centralised. If Q is fully F-centralised, then τ^{−1} : R → Q extends to a morphism σ : RC_{P′}(R) → QC_{P} (Q) in F. Since P′ is strongly F-closed, this restricts to a morphism C_{P′}(R) → C_{P′}(Q) in F, in particular |C_{P′}(R) ≤ |C_{P′}(Q)|. Since Q, R are isomorphic in F′ and since R is fully F′-centralised, this inequality is in fact an equality and so is Q is fully F′-centralised, too.

(ii) Let now τ : Q → R be an isomorphism in F′ such that R is fully F′-normalised. Then there is an automorphism α of Q in F such that α ∘ τ^{−1} : R → Q extends to a morphism ψ : N_{P} (R) → N_{P} (Q). Since P′ is strongly F-closed, ψ induces a morphism N_{P′}(R) → N_{P′}(Q); in particular, |N_{P′}(R)| ≤ |N_{P′}(Q)|. Since Q, R are isomorphic in F′ and R is fully F′-normalised, this inequality is in fact an equality. □

Corollary 9.4. Let F be a fusion system on a finite p-group P, let F′ be a fusion system on a subgroup P′ of P such that F′ ⊴ F and let Q be a subgroup of P′. If Q is fully F-normalised then Aut_{F′}(Q) ∩ Aut_{P} (Q) = Aut_{P′}(Q).

Proof. If Q is fully F-normalised then Aut_{P} (Q) is a Sylow-p-subgroup of Aut_{F}(Q). Since Aut_{F′}(Q) ⊴ Aut_{F}(Q) the intersection Aut_{F′}(Q) ∩ Aut_{P} (Q) is then also a Sylow-p-subgroup of Aut_{F′}(Q). By Proposition 9.3, Q is also fully F′-normalised and hence Aut_{P′}(Q) is a Sylow-p-subgroup of Aut_{F′}(Q) as well. Clearly Aut_{P′}(Q) ⊆ Aut_{F′}(Q) ∩ Aut_{P} (Q), which implies the equality as stated. □

Proposition 9.5. Let F be a fusion system on a finite p-group P. Let F′ be a fusion system on a subgroup P′ of P such that F′ ⊴ F. Let φ : Q → Q′ be an isomorphism in F. Suppose that Q, Q′ are contained in P′ and that Q′ is fully F′-normalised. Then there is an automorphism τ ∈ Aut_{F′}(Q′) such that the isomorphism τ ∘ φ : Q → Q′ can be extended to a morphism ψ : N_{P′}(Q) → P in F.

Proof. Since F′ ⊴ F, the morphism φ induces an isomorphism Aut_{F′}(Q) ≅ Aut_{F}(Q′) given by “conjugation” with φ. Since Q′ is fully F′-normalised, the group Aut_{P′}(Q′) is a Sylow-p-subgroup of Aut_{F′}(Q′). Thus the image φ∘Aut_{P′}(Q)∘φ^{−1} of Aut_{P′}(Q) in Aut_{F′}(Q′) is conjugate to a subgroup of Aut_{P′}(Q′). In other words, there is τ ∈ Aut_{F′}(Q′) such that (τ ∘φ)∘Aut_{P′}(Q)∘(τ ∘φ)^{−1} ⊆ Aut_{P′}(Q′), or equivalently, the isomorphism Aut_{F′}(Q) ≅ Aut_{F′}(Q′) induced by “conjugation” with τ ∘ φ maps Aut_{P′}(Q) to Aut_{P′}(Q′).

Let now β : Q′ → R be an isomorphism in F such that R is fully F-normalised. By Proposition 9.3, R is then also fully F′-normalised and by Corollary 9.4, Aut_{F′}(R) ∩ Aut_{P} (R) = Aut_{P′}(R) is a Sylow-p-subgroup of Aut_{F′}(R). Thus some Aut_{F′}(R)-conjugate of β ∘ Aut_{P′}(Q′) ∘ β^{−1} is contained in Aut_{P′}(R). Thus, up to replacing β, we may assume that β ∘ Aut_{P′}(Q′) ∘ β^{−1} = Aut_{P′}(R). Note that then N_{P′}(Q′) ≅ N_{P′}(R) because both Q′, R are fully F′-normalised. Since R is in fact fully F-normalised, β extends to an isomorphism σ : N_{P′}(Q′) ≅ N_{P′}(R) in F. Also β ∘ τ ∘ φ maps Aut_{P′}(Q) into Aut_{P′}(R), hence extends to a morphism ρ : N_{P′}(Q) → N_{P′}(R) in F. Then σ^{−1}ρ : N_{P′}(Q) → N_{P′}(Q′) is a morphism in F which extends τ ∘ φ. □

Corollary 9.6. Let F be a fusion system on a finite p-group P. Let F′ be a fusion system on a subgroup P′ of P such that F′ ⊴ F. Let Q be a normal subgroup of P′. For any morphism φ : Q → P in F there is a morphism ψ : φ(Q) → P′ in F′ and an automorphism α ∈ Aut_{F}(P′) such that ψ ∘ φ = α|_{Q}.

Proof. Clearly there is a morphism ψ : φ(Q) → P′ in F′ such that ψ(φ(Q)) is fully F′-normalised. It follows from Proposition 9.5 that ψ can be chosen in such a way that ψ ∘ φ extends to a morphism α from N_{Q}(P′) = P′ to P, and since P′ is strongly F-closed it follows that α is in fact an automorphism of P′. □

Corollary 9.7. Let F be a fusion system on a finite p-group P. Let F′ be a fusion system on a subgroup P′ of P such that F′ ⊴ F. Let Q be a strongly F′-closed subgroup of P′. For any morphism φ : Q → P in F the subgroup φ(Q) is a strongly F′-closed subgroup of P′ and there are automorphisms ψ ∈ Aut_{F′}(φ(Q)) and α ∈ Aut_{F}(P′) such that ψ ∘ φ = α|_{Q}; in particular, α(Q) = φ(Q).

Proof. By Corollary 9.6 there is a morphism ψ : φ(Q) → P′ and an automorphism α ∈ Aut_{F}(P′) such that ψ ∘ φ = α|_{Q}. Since F′ ⊴ F clearly α(Q) is strongly F′-closed. Since ψ is a morphism in the category F′, this forces in particular φ(Q) = α(Q). □

Applied to the case P = P′ and F = F′, Proposition 9.5 yields again 2.5: if φ : Q → Q′ is an isomorphism in F such that Q′ is fully F-normalised then there is an automorphism τ ∈ Aut_{F}(Q′) such that N_{τ∘φ} = N_{P} (Q). In other words, for any subgroup Q of P there is a morphism φ : Q → P such that φ(Q) is fully F-normalised and such that φ extends to a morphism ψ : N_{P} (Q) → P in F.

Proposition 9.8. Let F be a fusion system on a finite p-group P, let F′ be a fusion system on a subgroup P′ of P such that F′ ⊴ F, and let Q be a subgroup of P′. If Q is strongly F′-closed and weakly F-closed then Q is strongly F-closed.

Proof. Suppose that Q is not strongly F-closed. Then there is a subgroup R of Q and a morphism φ : R → P such that φ(R) is not contained in Q. Choose R maximal with this property. Certainly φ(R) ⊆ P′ because P′ is strongly F-closed. Hence there is a morphism ψ : φ(R) → P′ in F′ such that ψ(φ(R)) is fully F′-normalised. Also ψ(φ(R)) is again not contained in Q because Q is strongly F′-closed and ψ is a morphism in F′. Thus, up to replacing φ by ψ ∘ φ we may assume that φ(R) is fully F′-normalised. Hence, by Proposition 9.5, up to replacing φ by φ composed with a suitable automorphism of φ(R) in F′ we may assume that φ extends to a morphism σ : N_{P′}(R) → N_{P′}(φ(R)) in F. Restricting σ to N_{Q}(R) yields a morphism (abusively still denoted by the same letter) σ : N_{Q}(R) → P′. Since σ extends φ we still have that σ(N_{Q}(R)) is not contained in Q. By the maximality of R with this property this forces R = Q. But then Q cannot be weakly F-closed. □

The next Proposition is an analogue of [15, 7.1] for arbitrary fusion systems.

Proposition 9.9. Let F be a fusion system on a finite p-group P and let P′ be a strongly F-closed subgroup of P. For any subgroup Q of P there is an isomorphism Q ≅ R in F such that both R and R ∩ P′ are fully F-normalised.

Proof. Since every subgroup of P is isomorphic to a fully F-normalised subgroup of P we may assume that Q is fully F-normalised. Then there is an isomorphism ψ : Q ∩ P′ → T in F such that T is fully F-normalised, and by Proposition 9.5, we may choose ψ such that N_{ψ} = N_{P} (Q ∩ P′). Then ψ extends to a morphism φ : N_{P} (Q ∩ P′) → P. Clearly Q ⊆ N_{P} (Q) ⊆ N_{P} (Q ∩ P′). Set R = φ(Q). Since φ maps N_{P} (Q) to N_{P} (R) and Q is fully F-normalised, R is fully F-normalised, too, and since P′ is strongly F-closed we have R ∩ P′ = φ(Q) ∩ P′ = φ(Q ∩ P′) = ψ(Q ∩ P′) = T, which is also fully F-normalised. □

The following result generalises [15, 7.2] to arbitrary fusion systems and is the main tool in the proof of Theorem 9.1.

Proposition 9.10. Let F be a fusion system on a finite p-group P, let P′ be a strongly F-closed subgroup of P and let F′ be a fusion system on P′ contained in F. Let Q be a subgroup of P such that both Q, Q∩P′ are fully F-normalised. Suppose that Q is F-radical centric. Then Q∩P′ is F′-centric. If in addition F′ is normal in F, then Q ∩ P′ is F′-radical centric and fully F′-normalised.

Proof. Any isomorphism ψ : S → Q∩P′ in F extends to a morphism C_{P} (S)S → C_{P} (Q∩P′)(Q∩P′) because Q ∩ P′ is fully F-centralised. Any such morphism sends C_{P′}(S) to C_{P′}(Q ∩ P′) because P′ is strongly F-closed. Thus we have |C_{P′}(S)| ≤ |C_{P′}(Q ∩ P′)|, and therefore, in order to show that Q ∩ P′ is F′-centric, it suffices to show that C_{P′}(Q ∩ P′) ⊆ Q ∩ P′.

Let Y be the subset of all φ ∈ Aut_{F}(Q) satisfying [Q, φ] ⊆ Q ∩ P′, where [Q, φ] is as usual the subgroup generated by the set of elements of the form u^{−1}φ(u), with u running over Q.

One easily checks that Y is in fact a subgroup of Aut_{F}(Q). Since P′ is strongly F-closed, every automorphism of Q in F restricts to an automorphism of Q ∩ P′ in F. Denote by ρ : Aut_{F}(Q) → Aut_{F}(Q ∩ P′) the group homomorphism sending φ ∈ Aut_{F}(Q) to its restriction to Q ∩ P′. Set X = Y ∩ ker(ρ); that is, X consists of all φ ∈ Aut_{F}(Q) such that [Q, φ] ⊆ Q ∩ P′ and such that φ|_{Q∩P′} = Id_{Q∩P′}. Both X, Y are normal subgroups of Aut_{F}(Q). Moreover, since any φ ∈ X restricts to the identity on Q ∩ P′, we have [[Q, φ], φ] = 1. Thus, if we write φ(u) = uτ(u) for u ∈ Q and some τ(u) ∈ Q ∩ P′, we get φ^{n}(u) = uτ(u)^{n} for any positive integer n, and hence the order of φ is a power of p. It follows that X is a p-group.

Clearly X contains all automorphisms of Q induced by conjugation with elements in C_{P′}(Q∩P′)∩ N_{P} (Q). As Q is F-radical centric, we have X ⊆ Aut_{Q}(Q), and hence C_{P′}(Q ∩ P′) ∩ N_{P} (Q) ⊆ Q. This forces C_{P′}(Q ∩ P′) ⊆ Q, or equivalently, C_{P′}(Q ∩ P′) = Z(Q ∩ P′). This proves that Q ∩ P′ is F′-centric.

Assume now that F′ is normal in F. Then Q ∩ P′ is fully F′-normalised by Proposition 9.3. It remains to show that Q ∩ P′ is F′-radical. Let S be the subgroup of N_{P′}(Q ∩ P′) containing Q ∩ P′ such that O_{p}(Aut_{F′}(Q ∩ P′)) = Aut_{S}(Q ∩ P′) (this makes sense as Q ∩ P′ is fully F-normalised and hence fully F′-normalised by Proposition 9.3). This is a characteristic p-subgroup of Aut_{F′}(Q ∩ P′), hence a normal p-subgroup of Aut_{F}(Q ∩ P′). Note that in particular Q normalises S; thus QS is a subgroup of P. If Q ∩ P′ is a proper subgroup of S then in particular S is not contained in Q and hence Q is a proper subgroup of QS, thus a proper subgroup of N_{QS}(Q). But the image of N_{QS}(Q) in Aut_{F}(Q) is then a non-trivial normal p-subgroup of Aut_{F}(Q) because it is the inverse image of the normal p-subgroup Aut_{S}(Q ∩ P′) under the canonical restriction map ρ. This contradicts however the hypothesis that Q is F-radical. □

Proof of Theorem 9.1. Let α ∈ Aut_{F}(P′). Since F′ is normal in F we have F′ = N_{F′}(Q′) = N_{F′}(α(Q′)). Thus in fact F′ = N_{F′}(Q), and in particular, given an F′-centric radical subgroup S of P′, we have Q ⊆ S. Futhermore, given φ ∈ Aut_{F}(S), by Corollary 9.6, there are α ∈ Aut_{F}(P′) and ψ ∈ Aut_{F′}(S) such that ψ ∘ φ = α|_{S}. Note that by the construction of Q we have α(Q) = Q. Also, since F′ = N_{F′}(Q) we have ψ(Q) = Q and hence φ(Q) = Q. Thus φ belongs to N_{F}(Q). Let R be an F-radical centric subgroup of P. After possibly replacing R by a subgroup isomorphic to R in F we may assume that R, R ∩ P′ are fully F-normalised by Proposition 9.9. Then R ∩ P′ is F′-centric radical by Proposition 9.10. In particular, Q ⊆ R by Proposition 5.6. Thus any automorphism of R in F restricts to an automorphism of the F′-centric radical subgroup S = R ∩ P′ of P′, and hence belongs to N_{F}(Q) by the first part of the proof. Alperin’s fusion theorem implies that F = N_{F}(Q). □

Proof of Corollary 9.2. Suppose that F_{Q}(Q) ⊴ F. In particular Q is strongly F-closed in P. Then F = N_{F}(Q) by Theorem 9.1. The converse is trivial. □

Remark. The present paper is a revised and corrected version of the article printed in “Group Representation Theory” (eds. M. Geck, D. Testermann, J. Thévenaz), EPFL Press 2007. We added a statement (iv) in Theorem 6.3. In Theorem 6.4 we added the hypothesis F ⊆ F′ (which is necessary) and rewrote the proofs of 6.4 and 6.5 accordingly. An omission in the proof of 3.6, pointed out by Susanne Danz, led to stating 3.4 for categories on p-groups, separating 3.7 into two statements 3.7, 3.8, and using those in the proof of 3.6 to show that the stronger BLO-version of the Sylow axiom holds for N_{F}^{K}(Q) in 3.6.5, needed in the proof of the extension axiom in N_{F}^{K}(Q) without referring to the previous 3.7, which was formulated for fusion systems only. We added the reference [3].

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