Markus Linckelmann - University of...

INTRODUCTION TO FUSION SYSTEMS

Markus Linckelmann

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 [3]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 [5]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 detailedintroduction 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, includingclassical 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 andcentralisers 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 Sections7 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 fusionsystems. 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 9§4 Centric subgroups 16§5 Alperin’s fusion theorem 19§6 Quotients of fusion systems 24§7 Normal fusion systems 26§8 Simple fusion systems 28§9 Normal subsystems and control of fusion 31

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2 MARKUS LINCKELMANN

§1 Local structure of finite groups

Throughout these notes we denote by p a prime number. The local structure of afinite group G at the prime p can be understood as the structure of a Sylow-p-subgroupP 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 acategory, called fusion system:

Definition 1.1. Let G be a finite group and let P be a Sylow-p-subgroup of G. Thefusion system of G on P is the category denoted by FP (G), with the set of subgroupsof 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

HomFP (G)(Q,R) = HomG(Q,R) ,

the set of group homomorphisms ϕ : Q → R for which there is an element x ∈ G suchthat ϕ(u) = xu = xux−1 for all u ∈ Q. The composition of morphisms in FP (G) is theusual 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 FP (H) ⊆ FP (G). We will say that H controls G-fusion in P ifFP (H) = FP (G). If P is abelian, the following theorem of Burnside states that NG(P )controls G-fusion in P :

Theorem 1.2. (Burnside) Let G be a finite group and let P be a Sylow-p-subgroup ofG. If P is abelian then FP (G) = FP (NG(P )).

Proof. Let Q be a subgroup of P and let ϕ : Q→ P be a morphism in FP (G). That is,there is an element x ∈ G such that xQ ⊆ P and ϕ(u) = xux−1 for all u ∈ Q. Since Pis abelian, both P and xP are Sylow-p-subgroups of CG(xQ). Thus there is c ∈ CG(xQ)such that P = cxP . Then cx ∈ NG(P ), and we still have cxu = c(xux−1)c−1 = xux−1 =ϕ(u) for all u ∈ Q, because c commutes with all elements of the form xux−1 ∈ xQ, whereu ∈ Q. Thus ϕ is induced by conjugation with an element in NG(P ), which proves theTheorem.

Example 1.3. Let G = S4 be the symmetric group on four letters and let P = 〈x, t〉 bethe 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 S4, and clearly P is a dihedral group of order 8. Thegroup P has a Klein four subgroup V = (1), (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3) suchthat AutG(V ) is cyclic of order 3, having conjugation by (1, 2, 3) as generator. Thisautomorphism cannot be extended to an automorphism of P because the automorphismgroup of a dihedral group is easily seen to be a 2-group again. Thus NG(P ) does notcontrol fusion in this case.

INTRODUCTION TO FUSION SYSTEMS 3

A finite group G is called p-nilpotent if P has a normal complement K; that is, Kis a normal subgroup of G such that G ∼= K ⋊ P . Note that then K = Op′(G), thelargest normal subgroup of G of order prime to p. The following theorem of Frobeniusillustrates that being p-nilpotent is a property which can be read off the p-local structureof G:

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

(i) G is p-nilpotent.

(ii) NG(Q) is p-nilpotent for any non-trivial subgroup Q of P .

(iii) We have FP (G) = FP (P ).

(iv) For any Q ⊆ P the group AutG(Q) = NG(Q)/CG(Q) is a p-group.

Sketch of proof. Suppose that G is p-nilpotent; say G = K ⋊P , where K = Op′(G) andP is a Sylow-p-subgroup of P . Since P ∼= G/K is a p-group, K is equal to the set ofall p′-elements in G. Thus, for any subgroup H, the intersection H ∩ K is the set ofall p′-elements in H, and hence H is p-nilpotent with H ∩K = Op′(H). In particular,NG(Q) is p-nilpotent for any subgroup Q of P . Thus (i) ⇒ (ii). Let x ∈ G, u ∈ Psuch that xu ∈ P . Write x = yz with y ∈ K and z ∈ P . Then [y, zu] ∈ P ∩K = 1,hence xu = zu, which proves that any morphism in FP (G) is induced by conjugationwith an element in P . Thus (i) ⇒ (iii). The implication (iii) ⇒ (iv) is trivial. LetQ be a subgroup of P . Since Q and Op′(NG(Q)) are normal subgroups of NG(Q) ofcoprime orders, they centralise each other and hence Op′(NG(Q)) = Op′(CG(Q)). Thusif NG(Q) is p-nilpotent then NG(Q)/CG(Q) is a p-group. This shows the implication(ii) ⇒ (iv). In order to show that (iv) implies (i) we proceed by induction over theorder of G. The hypothesis (iv) passes down to subgroups of G, and hence we mayassume that every proper subgroup of G is p-nilpotent. Let Q be a non-trivial subgroupof P . If NG(Q) 6= G then NG(Q) is p-nilpotent by induction. If G = NG(Q) thenG/Q is p-nilpotent by induction. Let L be the subgroup of G containing Q such thatL/Q = Op′(G/Q). Then Q is a normal Sylow-p-subgroup of L, hence L = Q ⋊ K forsome p′-subgroup K of L. The assumption that NG(Q)/CG(Q) is a p-group implies thatK centralises Q, hence L = Q×K. Then K is a normal p-complement in G = NG(Q).We next show that FP (G) = FP (P ). One can do this directly, but this verification isalso 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 xZ ⊆ P then xZ = Z. Set N = NG(Z). SinceN is p-nilpotent, the quotient N/[N,N ] has a non-trivial p-group as quotient. By atheorem of Grun this implies that G/[G,G] has a non-trivial p-group as quotient. Thusthe smallest normal subgroup Op(G) of G with G/Op(G) a p-group is a proper subgroupof G, hence p-nilpotent, and a normal p-complement of Op(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 abeliansubgroups of P of maximal order.

Theorem 1.5. (Glauberman-Thompson, [10, Ch. 8, Theorem 3.1]) Let p be an oddprime, 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 NG(Z(J(P ))) is p-nilpotent.

Glauberman’s ZJ-theorem in [8] is a sufficient criterion for FP (G) to be controlledby the normaliser of the center of the Thompson subgroup J(P ).

Theorem 1.6. (Glauberman, [8]) Let p be an odd prime. If no subquotient of G isisomorphic to Qd(p) = (Cp × Cp) ⋊ SL2(p) then FP (G) = FP (NG(Z(J(P )))).

The semi-direct product Qd(p) is understood with the natural action of SL2(p) onCp × Cp. 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 hadthere been a finite group G with a 2-local structure as stated below, then G could havebeen chosen simple, and thus would have given rise to a new finite simple group.

Theorem 1.7. (R. Solomon, [24]) There is no finite group G having a Sylow-2-subgroupP of Spin7(3) such that FP (Spin7(3)) ⊆ FP (G) and such that all involutions in P areG-conjugate.

Alperin and Broue showed in [2] that, extending ideas of Brauer, every p-block of afinite group admits a local structure which can be described in terms of Brauer pairsand which has formal properties very similar to those of the categories FP (G) above;see [12] (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 describein 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 HomP (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 AutP (Q) = HomP (Q,Q). Thus AutP (Q) is canonicallyisomorphic to NP (Q)/CP (Q); in particular AutQ(Q) ∼= Q/Z(Q) is the group of innerautomorphisms of Q.

Definition 2.1. A category on a finite p-group P is a category F whose objects are thesubgroups of P and whose morphism sets HomF (Q,R) consist, for any two subgroupsQ, 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 ∈ HomF (Q,R), the induced isomorphism Q ∼= ϕ(Q) and its inverse aremorphisms 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 calledfully F-centralised if |CP (R)| ≤ |CP (Q)| for any subgroup R of P such that R ∼= Q inF , and Q is called fully F-normalised if |NP (R)| ≤ |NP (Q)| for any subgroup R of Psuch that R ∼= Q in F .

IfG is a finite group with Sylow-p-subgroup P , we will see in 2.9 below that a subgroupQ of P is fully FP (G)-centralised if and only if CP (Q) is a Sylow-p-subgroup of CG(Q);similarly, Q is fully FP (G)-normalised if and only if NP (Q) is a Sylow-p-subgroup ofNG(Q). The following definition is due to Broto, Levi and Oliver [5].

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

Nϕ = y ∈ NP (Q)| there is z ∈ NP (ϕ(Q)) such that ϕ(yu) = zϕ(u) for all u ∈ Q .

In other words, Nϕ is the inverse image in NP (Q) of the group

AutP (Q) ∩ (ϕ−1 AutP (ϕ(Q)) ϕ)

Note that in particular QCP (Q) ⊆ Nϕ ⊆ NP (Q).

Definition 2.4. A fusion system on a finite p-group P is a category F on P suchthat HomP (Q,R) ⊆ HomF (Q,R) for any two subgroups Q, R of P , and such that thefollowing two properties hold:

(I-S) AutP (P ) is a Sylow-p-subgroup of AutF (P ).

(II-S) every morphism ϕ : Q→ P in F such that ϕ(Q) is fully F -normalised extends toa morphism ψ : Nϕ → P (that is, ψ|Q = ϕ).

This terminology is, of course, coherent with the previous section - we will prove in2.10 that fusion systems of finite groups as defined in §1 are indeed fusion systems in thesense of 2.4. As mentioned before, it follows from work of Alperin and Broue [2], thatevery p-block of a finite group gives rise to a fusion system on any of its defect groups;see [12] for details. There are examples, due to Levi and Oliver [17], Ruiz and Viruel [22],of fusion systems which cannot be fusion systems of any finite group, and it has beenshown by Kessar [11], Kessar and Stancu [15], that these “exotic” fusion systems cannoteven occur as fusion systems of blocks. It is not known at present whether every fusionsystem of a block of some finite group occurs as fusion system of some (other) finitegroup. As far as terminology goes, Broto, Levi and Oliver [5] use the term saturatedfusion 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 Nϕ as object of F toits image Nϕ/Q in AutF (Q). Definition 2.4 is equivalent to the definition given in Broto,Levi, Oliver [5] which uses a priori stronger axioms; the following two Propositions, dueto Stancu [25], show that these two versions are equivalent.

Proposition 2.5. ([25]) Let F be a fusion system on a finite p-group P . A subgroupQ of P is fully F-normalised if and only if Q is fully F-centralised and AutP (Q) is aSylow-p-subgroup of AutF (Q).

Proof. Assume that Q is fully F -normalised. We first show that then Q is also fullyF -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 : RCP (R) → P in F suchthat ψ|R = ϕ. Hence ψ maps CP (R) to CP (Q), which implies that |CP (R)| ≤ |CP (Q)|,hence equality since R is fully F -centralised. Thus Q is fully F -centralised. Choose Qto be of maximal order such that Q is fully F -normalised but AutP (Q) is not a Sylow-p-subgroup of AutF (Q). Then Q is a proper subgroup of P by the Sylow axiom 2.4.(I-S).Choose a p-subgroup S of AutF (Q) such that AutP (Q) is a proper normal subgroupof S. Let ϕ ∈ S − AutP (Q). Since ϕ normalises AutP (Q), for every y ∈ NP (Q) thereis z ∈ NP (Q) such that ϕ(yu) = zϕ(u) for all u ∈ Q. In other words, Nϕ = NP (Q).Since Q is fully F -normalised, it follows from 2.4.(II-S) that there is an automorphismψ of NP (Q) in F such that ψ|Q = ϕ. Since ϕ has p-power order, by decomposing ψinto its p-part and its p′-part we may in fact assume that ψ has p-power order. Let τ :NP (Q) → P be a morphism in F such that τ(NP (Q)) is fully F -normalised. Now τψτ−1

is a p-element in AutF (τ(NP (Q))), thus conjugate to an element in AutP (τ(NP (Q))).Therefore we may choose τ in such a way that there is y ∈ NP (τ(NP (Q))) satisfyingτψτ−1(v) = yv for any v ∈ τ(NP (Q)). Since ψ|Q = ϕ, the automorphism τψτ−1 ofτ(NP (Q)) stabilises τ(Q). Thus y ∈ NP (τ(Q)). Since Q is fully F -normalised we have

NP (τ(Q)) ⊆ τ(NP (Q)), hence ψ(u) = τ−1(y)u for all u ∈ NP (Q). But then in particularϕ ∈ AutP (Q), contradicting our initial choice of ϕ. Thus AutP (Q) is a Sylow-p-subgroupin AutF (Q). The converse is easy since |NP (Q)| = |AutP (Q)| · |CP (Q)|. This proves theresult.

Lemma 2.6. Let F be a fusion system on a finite p-group P let Q, R be subgroups ofP and let ϕ : Q→ R be an isomorphism in F such that R is fully F-normalised. Thenthere is an isomorphism ψ : Q → R in F such that Nψ = NP (Q), or equivalently, suchthat ψ can be extended to a morphism from NP (Q) to P in F .

Proof. The group ϕ AutP (Q) ϕ−1 is a p-subgroup of AutF (R). Since R is fullyF -normalised, AutP (R) is a Sylow-p-subgroup of AutF (R) by 2.5. Thus there is β ∈AutF (R) such that β ϕ AutP (Q) ϕ−1 β−1 ⊆ AutP (R). Set ψ = β ϕ. Theabove inclusion means precisely that for any y ∈ NP (Q) there is z ∈ NP (R) suchthat ψ cy ψ

−1 = cz, where cy and cz are the automorphisms of Q and R inducedby conjugation with y and z, respectively. Equivalently, for any y ∈ NP (Q) there isz ∈ NP (R) such that ψ cy = cz ψ, which in turn means that Nψ = NP (Q). The


extension axiom (II-S) implies that ψ can be extended to a morphism from NP (Q) toP in F .

Proposition 2.7. ([25]) Let F be a fusion system on a finite p-group P . Let Q be asubgroup of P . Every morphism ϕ : Q→ P such that ϕ(Q) is fully F-centralised extendsto a morphism ψ : Nϕ → 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. Letρ : ϕ(Q) → P be a morphism in F such that R = ρ(ϕ(Q)) is fully F -normalised. By2.6 we may choose ρ in such a way that that Nρ = NP (ϕ(Q)). In particular, ρ extendsto a morphism σ : NP (ϕ(Q)) → P . But then Nϕ ⊆ Nρϕ, hence ρ ϕ extends to amorphism τ : Nϕ → P . Thus τ(Nϕ) ⊆ σ(NP (ϕ(Q))), and hence we get a morphismσ−1|τ(Nϕ) τ : Nϕ → 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 grouphomomorphism α : P → P ′ and a covariant functor Φ : F → F ′ with the followingproperties:

(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 Fto 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 isomorphismα : P ∼= P ′ such that HomF ′(α(Q), α(R)) = αHomF (Q,R)α−1|α(Q) for all subgroupsQ, R of P .

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

Definition 2.9. Let F be a fusion system on a finite p-group P . The orbit category ofF is the category F having the subgroups of P as objects and, for any two subgroups Q,R of P we have HomF (Q,R) = AutR(R)\HomF (Q,R), with composition of morphismsinduced by that in F .

This makes sense: with the notation above, the group AutR(R) acts on HomF (Q,R)by composition of group homomorphisms, and if ϕ, ϕ′ ∈ HomF (Q,R) are in the sameAutR(R)-orbit and ψ, ψ′ ∈ HomF (R, S) in the same AutS(S)-orbit, a trivial verificationshows that ψ ϕ, ψ′ ϕ′ ∈ HomF (Q, S) are in the same AutS(S)-orbit as well, for anysubgroups Q, R, S of P .

In order to show that fusion systems of finite groups are indeed fusion systems in thesense 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 =FP (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 CP (Q) is a Sylow-p-subgroup of CG(Q).

(ii) Q is fully F-normalised if and only if NP (Q) is a Sylow-p-subgroup of NG(Q).

Proof. Any inclusion morphism is in F because it can be viewed as conjugation bythe unit element. If ϕ : Q → R is a morphism in F then there is x ∈ G such thatϕ(u) = xu for all u ∈ Q. Then the induced isomorphism Q ∼= xQ is still given byconjugation with x, hence also a morphism in F . The composition of morphisms inF is the usual composition of group homomorphisms by definition 1.1. Thus F is acategory on P in the sense of 2.1. Let Q be a subgroup of P . Let S be a Sylow-p-subgroup of CG(Q) containing CP (Q). Then there is x in G such that x(QS) ⊆ P .Then conjugation by x is an isomorphism ϕ : Q ∼= xQ in F , and xS ⊆ CP (xQ). Itfollows that |CP (Q)| ≤ |S| ≤ |CP (xQ)|. Thus Q is fully F -centralised if and only if|CP (Q)| = |S|, or if and only if CP (Q) = S. This proves (i). The same argument withnormalisers instead of centralisers proves (ii).

Theorem 2.11. Let G be a finite group and let P be a Sylow-p-subgroup of G. Thecategory FP (G) is a fusion system.

Proof. Set F = FP (G). Clearly F is a category on P , and we have FP (P ) ⊆ F . Thuswe only have to check the two axioms (I-S) and (II-S) from 2.4. We have AutF (P ) ∼=NG(P )/CG(P ), and the image of P in this quotient group is a Sylow-p-subgroup, whichimplies that AutP (P ) ∼= P/Z(P ) is a Sylow-p-subgroup of AutF (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) = xu for all u ∈ Q.We have

Nϕ = y ∈ NP (Q) | ∃ z ∈ NP (R) : ϕ(yu) = zϕ(u) (∀u ∈ Q) .

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

Nϕ = y ∈ NP (Q) | ∃ z ∈ NP (R) : xyu = zxu (∀u ∈ Q) .

The equation xyu = zxu for all u ∈ Q means that x−1z−1xy centralises Q, hencez−1xyx−1 centralises xQ = R, and so we can write xyx−1 = zc for some c ∈ CG(R).This shows that we have

xNϕ ⊆ NP (R)CG(R) .

Since R is fully F -normalised, by 2.10 the group NP (R) is a Sylow-p-subgroup of NG(R),hence of NP (R)CG(R). Thus there is an element d ∈ CG(R) such that dxNϕ ⊆ NP (R).Define ψ : Nϕ → P by ψ(y) = dxy for all y ∈ Nϕ. We claim that ψ extends ϕ. Indeed,if u ∈ Q then ψ(u) = dxu = 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 bea normal p′-subgroup of G. Set G = G/K and denote by P the image of P in G. Thecanonical group homomorphism α : G → G induces an isomorphism of fusion systemsFP (G) ∼= FP (G).

Proof. Since K is a p′-group the map α induces an isomorphism P ∼= P . Thus, forany two subgroups Q, R of P the map HomF (Q,R) → HomF (Q, R) induced by α isinjective, where Q, R are the canonical images of Q, R in P . In order to show that it issurjective, let ψ : Q→ R be a morphism in F . Let S be the inverse image in P of ψ(Q).Then there is x ∈ G such that conjugation by x induces ψ; in other words, xQ ⊆ KSand hence both xQ and S are Sylow-p-subgroups of KS. Thus there is y ∈ K such thatyxQ = S. Then ϕ : Q→ R defined by ϕ(u) = yxu for u ∈ Q induces the morphism ψ asrequired.

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 letα : G→ G/K be the canonical surjection. Let T be a Sylow-p-subgroup of G/K and letP be a Sylow-p-subgroup of G such that α(P ) = T . Let y ∈ G/K. Then there is x ∈ Gsuch that α(x) = y and such that α(P ∩ xP ) = T ∩ yT .

Proof. Set R = T ∩ yT . Then y−1

R ⊆ T . Thus conjugation by y−1 is a morphism

R → y−1

R in FT (G/K). By 2.12 this lifts to a morphism Q → x−1

Q for some x ∈ Gsatisfying α(x) = y, where Q is the inverse image of R in P . Then Q ⊆ P ∩ xP , henceα(Q) ⊆ α(P ∩ xP ) ⊆ α(P ) ∩ α(xP ) = T ∩ yT = α(Q) and hence equality.

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

Theorem 2.14. (Levi, Oliver [17]) Let P be a Sylow-2-subgroup of Spin7(3). There isa fusion system F on P such that FP (Spin7(3)) ⊆ F and such that all involutions in Pare F-conjugate.

Ruiz and Viruel classified in [22] all possible fusion systems on extraspecial p-groupsof order p3 and, using the classification of finite simple groups, showed that some ofthem are exotic.

§3 Normalisers and centralisers

The concept of K-normalisers in fusion systems as well as the main result of thissection, Theorem 3.6, are due to Puig [21]. Our presentation follows partly [5, Appendix].


We start with some particular cases of K-normalisers. As in group theory, one can definenormalisers 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 subgroupof P .

(i) The normaliser of Q in F is the category NF (Q) on NP (Q) having as morphisms allgroup homomorphisms ϕ : R → S, for R, S subgroups of NP (Q), for which there existsa morphism ψ : QR→ QS in F satisfying ψ(Q) = Q and ψ|R = ϕ.

(ii) The centraliser of Q in F is the category CF (Q) on CP (Q) having as morphisms allgroup homomorphisms ϕ : R→ S, for R, S subgroups of CP (Q), for which there existsa morphism ψ : QR→ QS in F satisfying ψ|Q = IdQ and ψ|R = ϕ.

(iii) We denote by QCF (Q) the subcategory of NF (Q) on QCP (Q) having as morphismsall group homomorphisms ϕ : R → S, for R, S subgroups of QCP (Q), for which thereexists a morphism ψ : QR→ QS and v ∈ Q such that ψ|Q = cv and ψ|R = ϕ, where cvis the automorphism of Q given by conjugation with v.

(iv) We denote by NP (Q)CF(Q) the subcategory of NF (Q) on NP (Q) having as mor-phisms all group homomorphisms ϕ : R → S, for R, S subgroups of NP (Q), for whichthere exists a morphism ψ : QR→ QS and v ∈ NP (Q) such that ψ|Q = cv and ψ|R = ϕ.

Theorem 3.2. Let F be a fusion system on a finite p-group P and let Q be a subgroupof P . If Q is fully F-normalised then NF (Q) is a fusion system on NP (Q), and if Qis fully F-centralised then CF (Q) is a fusion system on CP (Q), QCF (Q) is a fusionsystem on QCP (Q) and NP (Q)CF(Q) is a fusion system on NP (Q).

Note that we have inclusions of categories CF (Q) ⊆ QCF (Q) ⊆ NP (Q)CF (Q) ⊆NF (Q). Definition 3.1 and Theorem 3.2 are part of the more general concept:

Definition 3.3. ([21]) Let P be a finite p-group, let Q be a subgroup of P and let Kbe a subgroup of Aut(Q). The K-normaliser of Q in P is the subgroup

NKP (Q) = y ∈ NP (Q) | ∃α ∈ K : α(u) = yuy−1 ∀ u ∈ Q .

We set AutKP (Q) = K ∩ AutP (Q) and AutKF (Q) = K ∩ AutF (Q).

In other words, NKP (Q) consists of all elements in NP (Q) which induce, by conju-

gation, an automorphism of Q belonging to the automorphism subgroup K. Note thatCP (Q) ⊆ NK

P (Q), and ifK contains all inner automorphisms ofQ then alsoQ ⊆ NKP (Q).

We have AutKP (Q) ∼= NKP (Q)CP (Q)/CP (Q). There are various special cases of this con-

struction we will encounter most frequently: if K = IdQ then NKP (Q) = CP (Q)

and AutKP (Q) = AutKF (Q) = IdQ; if K = AutQ(Q) then NKP (Q) = QCP (Q) and

AutKP (Q) = AutKF (Q) = AutQ(Q); finally, if K = Aut(Q) then NKP (Q) = NP (Q),

AutKP (Q) = AutP (Q) and AutKF (Q) = AutF (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 makesthus sense to consider N

ϕKP (ϕ(Q)).

Definition 3.4. Let F be a fusion system on a finite p-group P , let Q be a subgroupof P and let K be a subgroup of Aut(Q). We say that Q is fully K-normalised in F if|NK

P (Q)| ≥ |NϕKP (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 thesense of 2.2, and Q is fully IdQ-normalised if and only if Q is fully F -centralised. Notethat being fully AutQ(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 ofP and let K be a subgroup of Aut(Q). The K-normaliser of Q in F is the subcategoryNK

F (Q) of F on NKP (Q) having morphism sets

HomNKF

(Q)(R, S) = ϕ ∈ HomF (R, S) | ∃ ψ ∈ HomF (QR,QS) : ψ|R = ϕ, ψ|Q ∈ K .

for any two subgroups R, S in NKP (Q).

If K = IdQ then NKF (Q) = CF (Q) and if K = Aut(Q) then NK

F (Q) = NF (Q).Other cases of interest, as mentioned above, include K = AutQ(Q) which yieldsNK

F (Q) = QCF (Q) and K = AutP (Q) which yields NKF (Q) = NP (Q)CF(Q). Thus

Theorem 3.2 is a consequence of the following more general result.

Theorem 3.6. (Puig [21]) Let F be a fusion system on a finite p-group P , let Q be asubgroup of P and let K be a subgroup of Aut(Q). Suppose that Q is fully K-normalisedin F . Then NK

F (Q) is a fusion system on NKP (Q).

The proof requires the following Proposition (part (i) generalises 2.5).

Proposition 3.7. Let F be a fusion system on a finite p-group P , let Q be a subgroupof 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

AutKP (Q) is a Sylow-p-subgroup of AutKF (Q).

(ii) Let ϕ : Q → R be an isomorphism in F and set L = ϕ K ϕ−1. Suppose that Ris fully L-normalised in F . Then there are morphisms τ : Q · NK

P (Q) → P in F andκ ∈ K such that τ |Q = ϕ κ.

(iii) Let ψ : QNKP (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 Q is fully F-normalised and K ⊆ AutP (Q) then Q is fully K-normalised in F .

Proof. (i) Suppose that Q is fully K-normalised in F . By 2.6 there is an isomorphismϕ : Q→ R in F such that R is fully F -normalised and such that ϕ extends to a morphism


ψ : NP (Q) → P in F . Set L = ϕ K ϕ−1. Thus ψ maps NKP (Q) to NL

P (R). SinceQ is fully K-normalised we have |NK

P (Q)| ≥ |NLP (R)| and hence ψ induces in fact an

isomorphism NKP (Q) ∼= NL

P (R). Any such isomorphism restricts to an isommorphismCP (Q) ∼= CR(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 AutKP (Q) ϕ−1 is a p-subgroup of the

subgroup AutLF (R) of AutF (R). By 2.5 the group AutP (R) is a Sylow-p-subgroup of

AutF (R). Thus some conjugate of AutP (R) intersected with AutLF (R) will be a Sylow-

p-subgroup of AutLF (R), and this Sylow-p-subgroup can, of course, be chosen to contain

the p-subgroup ϕ AutKF (Q) ϕ−1. Thus there is β ∈ AutF (R) such that

ϕ AutKF (Q) ϕ−1 ⊆ β AutP (R) β−1 ∩ AutLF (R)

and such that the right side of this inclusion is a Sylow-p-subgroup of AutLF (R). Then

AutP (R) ∩ Autβ−1Lβ

F (R) = Autβ−1LβP (R) is a Sylow-p-subgroup of Autβ

−1LβF (R).

Since Q is fully K-normalised we have |AutKP (Q)| ≥ |Autβ−1LβP (R)| which implies that

AutKP (Q) is a Sylow-p-subgroup of AutKF (Q).

Suppose conversely that Q is fully F -centralised and that AutKP (Q) is a Sylow-

p-subgroup of AutKF (Q). Let ϕ : Q → R be an isomorphism in F and set L =

ϕ K ϕ−1. Then |CP (R)| ≤ |CP (Q)| and since AutKF (Q) ∼= AutLF (R) we get

that |AutLP (R)| ≤ |AutKP (Q)|, which together implies |NLP (R)| = |CP (R)||AutLP (R)| ≤

|CP (Q)||AutKP (Q)| = |NKP (Q)|, showing that Q is fully K-normalised in F .

(ii) Since R is fully K-normalised in F it is in particular fully F -centralised. Set N =

Q · NKP (Q). Then ϕ N ϕ−1 is a p-subgroup of AutLF (R). By (i), AutLP (R) is a

Sylow-p-subgroup of AutLF (R). Thus there is λ ∈ AutLF (R) such that λ ϕ N ϕ−1 λ−1 ⊆ AutP (R). This means that N ⊆ Nλϕ, and hence λ ϕ extends to a morphism

τ : Q ·NKP (Q) → P . Now λ ϕ = ϕ (ϕ−1 λ ϕ) and κ = ϕ−1 λ ϕ ∈ AutKP (Q) is as

required.

(iii) The map ψ sends NKP (Q) to NψKψ−1

P (ψ(Q)). In particular, |NKP (Q)| ≤

|NψKψ−1

P (ψ(Q))|, whence the result.

(iv) If K ⊆ AutP (Q) then |NKP (Q)| = |CP (Q)| · |K|. Thus if Q is fully F -normalised, it

is fully F -centralised, whence the statement.

Proof of Theorem 3.6. Clearly NKF (Q) is a category on NK

P (Q) in the sense of 2.1. Forany subgroup R of NK

P (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 INK

P (Q)(R) = NK∗I

P (QR)

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


3.6.2. the restriction map AutK∗IF (QR) → AutINK

F(Q)(R) is surjective.

Indeed, any β ∈ AutNKF

(Q)(R) extends to some α ∈ AutF (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 NKP (Q) and any subgroup I of Aut(R) there is a morphism

ϕ : QR → QNKP (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 (I ∗K) ρ−1-normalised. Set σ = ρ|Q. Since Q is fully K-normalised in F , by 3.7.(ii) applied to

σ−1 and σ(Q) there is a morphism τ : σ(Q)NσKσ−1

P (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 κσ ∈ K.

Note that ρ(QR)Nρ(K∗I)ρ−1

P (ρ(QR)) ⊆ σ(Q)NσKσ−1

P (σ(Q)) and that ρ(QR) is fullyρ (K ∗ I) ρ−1-normalised in F . It follows from 3.7.(iii) applied to the appropriaterestriction of τ and ρ(QR) that RQ is fully K ∗ I-normalised in F . This proves 3.6.3.

We prove now the Sylow axiom I-S from 2.4. Set R = NKP (Q). We have to show that the

inner automorphism group I = AutR(R) is a Sylow-p-subgroup of AutNKF

(Q)(R). Set

A = Aut(R). By 3.6.2, the restriction map AutK∗AF (QR) → AutNK

F(Q)(R) is surjective,

and it maps AutK∗IF (QR) onto I = AutR(R). It suffices to show that AutK∗I

F (QR)

contains a Sylow-p-subgroup of AutK∗AF (QR). Applying 3.6.3 with R = NK

P (Q) showsthat QR is fully ϕ (K ∗ A) ϕ−1-normalised for some ϕ : QR → QNK

P (Q) in Fwhose restriction to Q is in K, and hence ϕ (K ∗ A) ϕ−1 = K ∗ A. Thus the group

AutK∗AP (QR) is a Sylow-p-subgroup of AutK∗A

F (QR) by 3.7.(i). But since R = NKP (Q)

we get AutK∗AP (QR) = AutK∗I

P (QR), which is clearly contained in AutK∗IF (QR). This

proves that the Sylow axiom I-S holds. Before we proceed with the proof of the extensionaxiom, we show that

3.6.4. if a subgroup R of NKP (Q) is fully I-normalised in NK

F (Q) for some subgroup Iof Aut(R) then QR is fully K ∗ I-normalised in F .

Indeed, by 3.6.3 there exists a morphism ϕ : QR → QNKP (Q) in F such that

ϕ|Q ∈ K and such that Qϕ(Q) is fully ϕ (K ∗ I) ϕ−1-normalised. We have

NK∗IP (QR) = N I

NKP (Q)

(R) by 3.6.1 and |N INK

P (Q)(R)| ≥ |NϕIϕ−1

NKP (Q)

(ϕ(R))| because R

is fully I-normalised. Now NϕIϕ−1

NKP (Q)

(ϕ(R)) = NK∗(ϕIϕ−1)P (ϕ(R)) by 3.6.1 and since

ϕ|Q ∈ K we haveK∗(ϕIϕ−1) = ϕ(K∗I)ϕ−1, from which we get that the last group

is equal to Nϕ(K∗I)ϕ−1

P (ϕ(R)). In particular, |NK∗IP (QR)| ≥ |N

ϕ(K∗I)ϕ−1

P (ϕ(R))|,which proves the statement 3.6.4.

For the proof of the extension axiom II-S, let R be a subgroup of NKP (Q) and let

ϕ : R → NKP (Q) be a morphism in NK

F (Q) such that ϕ(R) is fully NKF (Q)-normalised.

We consider the group Nϕ as defined in 2.3 for the morphism ϕ in the category NKF (Q);

that is, Nϕ is the subgroup of all y ∈ NNKP (Q)(R) for which there exists an element


z ∈ NNKP (Q)(ϕ(R)) satisfying ϕ(yuy−1) = zϕ(u)z−1 for all u ∈ R. Set I = AutNϕ

(R).

Note that conjugation by any y ∈ Nϕ leaves Q invariant and induces an automorphismof Q belonging to K. Then Nϕ = N I

NKP (Q)

(R) = NK∗IP (QR). We have ϕ I ϕ−1 ⊆

AutNKP (Q)(ϕ(Q)) ⊆ AutNK

F(Q)(ϕ(R)), and hence ϕ(R) is fully ϕ I ϕ−1-normalised in

NKF (Q) by 3.7.(iv). Thus Qϕ(R) is fully K ∗ (ϕ I ϕ−1)-normalised by 3.6.4. Since ϕ

is a morphism in NKF (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 ∗ (ϕI ϕ−1))ψ = K ∗I because ψ|Q ∈ K and because ψ−1|R ϕ = IdR.Applying 3.7.(ii) to ψ and QR yields the existence of a morphism τ : QRNK∗I

P (QR) → Pand κ ∈ K ∗ I such that τ |QR = ψ κ. Since κ|R ∈ I = AutR(R) there is y ∈ R suchthat κ|R = cy , conjugation by y. Then τ |Nϕ

c−1y : Nϕ → P is a morphism in NK

F (Q),because it extends to the morphism τ |QNϕ

cy : QNϕ → P whose restriction to Q is

τ |Q c−1y |Q = ψ|Q κ|Q c−1

y |Q ∈ K. Moreover, the morphism τ |Nϕ c−1

y restricted to R

is equal to ψ|R κ|R c−1y |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 thequestions one would like to be able to address is that of control of fusion by normalisersof p-subgroups, a question which we can now rephrase as follows: when do we have anequality F = NF (Q) for some normal subgroup Q of P . The most basic case, Burnside’stheorem 1.2, remains true for arbitrary fusion systems:

Theorem 3.8. Let F be a fusion system on an abelian finite p-group P . Then F =NF (P ).

Proof. Let Q be a subgroup of P and let ϕ : Q → P be a morphism in F . Since P isabelian, every subgroup of P is fully F -normalised, and hence ϕ extends to a morphismψ : QCP (Q) → P , by 2.4.(II-S). Again, since P is abelian, we have QCP (Q) = P , andthus ψ ∈ AutF (P ).

We quote further results without proof, referring to the original papers. It has beenshown by Stancu that the conclusion of 3.8 remains true for some other classes of finitep-groups:

Theorem 3.9. ([26]) Let P be a metacyclic finite p-group for some odd prime p. Thenfor every fusion system F on P we have F = NF (P ).

Theorem 3.10. ([26]) Let P be a finite p-group of the form P = E × A, where Ais elementary abelian and where E is extra-special of exponent p2 if p is odd and notisomorphic to D8 if p = 2. Then for every fusion system F on P we have F = NF (P ).

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


Theorem 3.11. Let F be a fusion system on a finite p-group P . The following areequivalent:

(i) For any Q ⊆ P the group AutF (Q) is a p-group.

(ii) F = FP (P ).

(iii) For any non-trivial fully F-normalised subgroup Q of P we have NF (Q) =FNP (Q)(NP (Q)).

The proof of 3.11 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 3.12. ([13, Theorem A]) Let p be an odd prime and let F be a fusion systemon a finite p-group P . We have F = FP (P ) if and only if NF (Z(J(P ))) = FP (P ).

Gilotti and Serena [7] gave a general criterion when a fusion system of a finite groupis of the form NF (Q), and Stancu generalised this to arbitrary fusion systems. Beforewe state this, we need the following terminology:

Definition 3.13. Let F be a fusion system on a finite p-group P and let Q be asubgroup 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 inF we have ϕ(R ∩Q) ⊆ Q.

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

Theorem 3.14. ([26, 6.11]) Let F be a fusion system on a finite p-group P , and let Qbe a subgroup of P . The following are equivalent:

(i) F = NF (Q).

(ii) The subgroup Q is strongly F-closed and there is a central series Q = Qn > Qn−1 >· · · > Q1 > 1 with Qi 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 centralseries of weakly F -closed subgroups, and hence 3.14 applies:


Corollary 3.15. Let F be a fusion system on a finite p-group P and let Q be an abeliansubgroup of P . We have F = NF (Q) if and only if Q is strongly F-closed.

We give an alternative proof of 3.15 following 9.2 below. By [13], Glauberman’s ZJ-theorem 1.6 carries over as well, but it requires some extra effort to define a notion ofQd(p)-free fusion systems, replacing the hypothesis on G having no subquotient isomor-phic to Qd(p) in 1.6; see 4.7 and 4.8 below for more details. When specialised to fusionsystems of finite groups, normalisers and centralisers in the fusion system correspond tothe fusion systems of the relevant normaliser and centralisers:

Proposition 3.16. Let G be a finite group, let P be a Sylow-p-subgroup and set F =FP (G). Let Q be a subgroup of P .

(i) If Q is fully F-normalised then NFP (G)(Q) = FNP (Q)(NG(Q)).

(ii) If Q is fully F-centralised then CFP (G)(Q) = FCP (Q)(CG(Q)).

(iii) If Q is normal in G then F = NF (Q).

Proof. The statements make sense: if Q is fully F -normalised then NP (Q) is a Sylow-p-subgroup of NG(Q), and if Q is fully F -centralised then CP (Q) is a Sylow-p-subgroupof CG(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 iscalled F -centric if for any isomorphism ϕ : Q→ R in F we have CP (R) = Z(R).

Note that every F -centric subgroup Q of P is trivially fully F -centralised, because allcentralisers CP (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 Fc thefull subcategory of F having as objects all F -centric subgroups of P . We denote by Fc

the canonical image of Fc in the orbit category F of F .

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

Proof. Let ϕ : QCP (Q) → R be an isomorphism in F . Let ψ : ϕ(Q) → Q be the inverseof ϕ restricted to ϕ(Q). Since Q is fully F -centralised, ψ extends, by 2.6, to a morphismτ : ϕ(Q)CP (ϕ(Q)) → P whose image must be contained in QCP (Q). Since ϕ(Q) ⊆ Rwe have RCP (R) ⊆ ϕ(Q)CP (ϕ(Q)), hence |RCP (R)| ≤ |QCP (Q)| = |R|, which forcesCP (R) ⊆ R or CP (R) = Z(R). Thus QCP (Q) is F -centric.


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

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

Even though elementary, the crucial observation is that thanks to 4.4, taking centers ofcentric subgroups is a contravariant functor from Fc to the category of finitely generatedZ(p)-modules, where Z(p) = a

b| a, b ∈ Z, p 6 |b.

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

Z : Fc −→ mod(Z(p))

sending any F-centric subgroup Q of P to Z(Q) and sending the class of any morphismϕ : 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 uniquemap Z(R) → Z(Q) as claimed. This map does not depend on the choice of ϕ in its classmodulo inner automorphisms of R because any inner automorphism of R is the identityon Z(R). Thus Z is well-defined.

The following result appears in work of Kulshammer and Puig [16] for fusion systemsof p-blocks and in Broto, Castellana, Grodal, Levi, Oliver [4, §4] in general:

Theorem 4.6. Let F be a fusion system on a finite p-group P . For any F-centricfully normalised subgroup Q of P there is, up to isomorphism, a unique finite groupL = LF

Q having NP (Q) as Sylow-p-subgroup such that Q E L, CL(Q) = Z(Q) and

NF (Q) = FNP (Q)(L).

In other words, if F is a fusion system on a finite p-group P such that F = NF (Q)for some F -centric normal subgroup Q of P then F is automatically the fusion sys-tem of some finite group. The proof of 4.6 uses some of the cohomological machin-ery developed in [5], showing that the center functor from 4.5 applied to the fusionsystem NF (Q) is acyclic. The three properties of the group L = LF

Q imply that

AutF (Q) ∼= NL(Q)/CL(Q) = L/Z(Q), and hence L fits into a short exact sequenceof finite groups

1 −→ Z(Q) −→ L −→ AutF (Q) −→ 1


whose restriction to the Sylow-p-subgroup NP (Q) of L is of the form

1 −→ Z(Q) −→ NP (Q) −→ AutP (Q) −→ 1 .

This characterises the group L up to isomorphism because the restriction map on coho-molgy H2(AutF (Q);Z(Q)) → H2(AutP (Q);Z(Q)) is well-known to be injective.

Definition 4.7. ([13, 1.1]) A fusion system F on a finite p-group P is called Qd(p)-freeif Qd(p) is not involved in any of the groups LF

Q, with Q running over the set of F -centricfully F -normalised subgroups of P .

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

Theorem 4.8. ([13, Theorem B]) Let p be an odd prime and let F be a fusion systemon a finite p-group P . If F is Qd(p)-free then F = NF (Z(J(P ))).

While all morphisms in the fusion system F are monomorphisms, this is no longertrue 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 thecategory Fc is an epimorphism.

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

Lemma 4.10. Let F be a fusion system on a finite p-group P . Let Q, R be F-centricsubgroups of P such that Q ⊆ R, and let ϕ ∈ AutF (R). We have ϕ|Q = IdQ if and onlyif ϕ ∈ AutZ(Q)(R).

Proof. Assume that ϕ|Q = IdQ. We proceed by induction over [R : Q]. Considerfirst the case where Q is normal in R. Let u ∈ Q and v ∈ R. Then vu ∈ Q, hencevu = ϕ(vu) = ϕ(v)u, and thus v−1ϕ(v) ∈ CR(Q) = Z(Q), or equivalently, ϕ(v) = vz forsome z ∈ Z(Q). If ϕ has order prime to p in Aut(R) this forces ϕ = IdR. Thereforewe 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 ∈ AutP (R). Since ϕ restricts to IdQ andsince Q is F -centric this implies that ϕ ∈ AutZ(Q)(R). This proves 4.10 if Q is normal inR. In general, if ϕ|Q = IdQ then ϕ(NR(Q)) = NR(Q). Thus ϕ|NR(Q) ∈ AutZ(Q)(NR(Q))by the previous paragraph. Hence there is z ∈ Z(Q) such that cz ϕ|NR(Q) = IdNR(Q),where cz is the automorphism of R given by conjugation with z. By induction we getcz ϕ ∈ AutZ(NR(Q))(Q). As all involved groups are F -centric we have Z(NR(Q)) ⊆Z(Q), and thus ϕ ∈ AutZ(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-centricsubgroups of P such that Q ⊆ R, and let ϕ, ϕ′ ∈ HomF (R,P ) such that ϕ|Q = ϕ′|Q.Then ϕ(R) = ϕ′(R).

Proof. Let v ∈ NR(Q). For every u ∈ Q we have ϕ(vu) = ϕ′(vu), hence ϕ(v)−1ϕ′(v) ∈CP (ϕ(Q)) = Z(ϕ(Q)). It follows that ϕ(NR(Q)) = ϕ′(NR(Q)). By 4.10, ϕ|NR(Q) andϕ′|NR(Q) differ by conjugation with an element in Z(Q), and we may therefore assumethat their restrictions to NR(Q) actually coincide. The equality ϕ(R) = ϕ′(R) followsby induction.

Proof of Theorem 4.9. Let Q, R, S be F -centric subgroups of P , let ϕ ∈ HomF (Q,R)and let ψ, ψ′ ∈ HomF (R, S). Assume that the images of ψϕ and ψ′ϕ in HomFc(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), henceψ−1 ψ′ ∈ AutZ(ϕ(Q))(R) by 4.10. Thus the images of ψ, ψ′ in the orbit category areequal.

We conclude this section with a remark on what the notion of being centric means inthe 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 setF = FP (G). A subgroup Q of P is F-centric if and only if CG(Q) = Z(Q)×Op′(CG(Q)).

Proof. By 2.8, Q is F -centric if and only if Z(Q) is a Sylow-p-subgroup of CG(Q).Since Z(Q) is a central subgroup of CG(Q) this happens if and only if CG(Q) splits asdescribed, by some standard group theory.

Using 4.12 it is easy to directly construct the groups LFQ in the case where F = FP (G):

just take LFQ = NG(Q)/Op′(CG(Q)). Since CG(Q) = Z(Q)×Op′(Q) we get a short exact

sequence1 −→ Z(Q) −→ LF

Q −→ NG(Q)/CG(Q) ∼= AutF (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 com-pletely determined by automorphism groups AutF (Q) with Q running over a certainsubset of subgroups of P . Its original version, due to Alperin [1], is stated for fu-sion systems of finite groups and the class of so-called radical subgroups of P , refinedby Goldschmidt [9] who extrapolates Alperin’s methods to show that the potentiallysmaller class of essential subgroups suffices to determine F . Puig showed in [20] that


essential subgroups are essential - that is, no smaller class will determine F - and showedfurthermore which subsets of AutF (Q) are actually necessary to determine F , with Qrunning 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 containsa Sylow-p-subgroup P of G and P 6= 1 but H ∩ xP = 1 for any x ∈ G − H. Theexistence of a strongly p-embedded subgroup is equivalent to the poset Sp(G) of non-trivial p-subgroups of G being disconnected. Indeed, if H is a strongly p-embeddedsubgroup of G then Sp(G) has at least two connected components, namely that of aSylow-p-subgroup P contained in H and that of xP with x ∈ G − H. Conversely, ifSp(G) has a connected component different from Sp(G) then the stabliser H of such aconnected component is easily checked to be strongly p-embedded. We denote by Op(G)the unique maximal normal subgroup of G whose order is a power of p, and we denoteby Op′(G) the unique maximal normal subgroup of G whose order is prime to p. Notethat if Op(G) 6= 1 then Sp(G) is connected, and hence G has no strongly p-embeddedsubgroup. If F is a fusion system on a finite p-group P then for any subgroup Q in Pwe have AutQ(Q) ⊳AutF (Q), and hence AutQ(Q) ⊆ Op(AutF (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 Op(AutF (Q)) = AutQ(Q).

(ii) A subgroup Q of P is called F -essential if Q is F -centric and if AutF (Q)/AutQ(Q)has a strongly p-embedded subgroup.

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

Theorem 5.2. (Alperin’s fusion theorem) Let F be a fusion system on a finite p-groupP . Every isomorphism in F can be written as a composition of finitely many isomor-phisms of the form ϕ : R→ S in F for which there exists a subgroup Q containing bothR, S and an automorphism α ∈ AutF (Q) such that α|R = ϕ and either Q = P or Q isfully F-normalised essential.

The picture to have in mind is this: given any isomorphism ϕ : R → S in F , theabove theorem claims that there are finitely many subgroups R0, R1, .., Rn of P , withisomorphisms ϕi : Ri → Ri+1 for 0 ≤ i ≤ n − 1, and fully F -normalised essentialsubgroups Q1, Q2, .., Qn such that Ri, Ri+1 are contained in Qi, with automorphismsαi ∈ AutF (Qi) satisfying αi|Ri

= ϕi for 0 ≤ i ≤ n− 1, and such that R = R0, Rn = Swith ϕ = ϕn−1 · · ·ϕ1 ϕ0.

The crucial ingredient which makes this work is the extension axiom. We separatesome of the arguments, and for the sake of providing future reference, allow some re-dundancy 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 upthe proof of 5.2 in such a way that its first half proves this weaker version of Alperin’sfusion system (which is, for instance, also proved in [5, Appendix]).

Lemma 5.3. Let F be a category on a finite p-group P and let Q be a fully F-normalisedsubgroup of P . Let α ∈ AutF (Q). There is a unique subgroup R of NP (Q) containingQCP (Q) such that AutR(Q) = AutP (Q) ∩ (α−1 AutP (Q) α), and we have R = Nα.

Proof. The intersection AutP (Q) ∩ (α−1 AutP (Q) α) is a subgroup of AutP (Q) ∼=NP (Q)/CP (Q), hence equal to AutR(Q) for a unique subgroup R of NP (Q) containingQCP (Q). By the definition of R, this group consists of all elements y ∈ NP (Q) forwhich there is z ∈ NP (Q) such that cy = α−1 cz α, or equivalently, such thatα cy = cz α, where cy, cz are the automorphisms given by conjugation with y, z,respectivley. Explicitly, the last equation means that α(yu) = zα(u) for all u ∈ Q, whichis equivalent to saying that y belongs to the group Nα.

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 NP (Q) containing QCP (Q)such that Op(AutF (Q)) = AutR(Q), and then every automorphism α ∈ AutF (Q) ex-tends to an automorphism β ∈ AutF (R).

Proof. Since Q is fully F -normalised, AutP (Q) ∼= NP (Q)/CP (Q) is a Sylow-p-subgroupof AutF (Q). Thus there is indeed a unique subgroup R of NP (Q) containing CP (Q)such that AutR(Q) = Op(AutF (Q)). Since AutQ(Q) is a normal p-subgroup of AutF (Q)we also have Q ⊆ R. Let α ∈ AutF (Q). Since AutR(Q) = Op(AutF (Q)) we haveα AutR(Q) α−1 = AutR(Q). This means that R ⊆ Nα by 5.3, hence α extends to amorphism β : R → P . For any y ∈ R and any any u ∈ Q we have α(yu) = β(yuy−1) =β(y)α(u), or equivalently, α cy α

−1 = cβ(y), where cy, cβ(y) are the automorhisms ofQ given by conjugation with y, β(y). Since AutR(Q) is normal in AutF (Q) this impliescβ(y) ∈ AutR(Q) and hence β(y) ∈ R as CP (Q) ⊆ R. Thus β ∈ AutF (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 ∈ AutF (Q) such that AutQ(Q) is a proper subgroup ofAutP (Q) ∩ (α AutP (Q) α−1). Then α extends to a morphism ψ : R → P in F forsome subgroup R of NP (Q) which properly contains Q.

Proof. Let R be the unique subgroup of NP (Q) containing QCP (Q) such thatAutR(Q) = AutP (Q) ∩ (α AutP (Q) α−1). The hypotheses imply that QCP (Q) isa proper subgroup of R; in particular, Q is a proper subgroup of R. By 5.3 we haveR ⊆ Nα, 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 bewritten as a composition of finitely many isomorphisms of the form ψ : R → R′ for whichthere exists a subgroup Q of P containing R, R′ and an automorphism α ∈ AutF (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 anisomorphism in A then so is its inverse ψ−1 and any restriction of ψ to a subgroup Vof R induces an isomorphism ψ|V : V → V ′ in A, where V ′ = ψ(V ). We have to showthat 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 writtenas composition of isomorphisms as claimed by induction over [P : R]. For R = S = Pthere is nothing to prove, so we may assume that R is a proper subgroup of P . Theargument we are going to use constantly is as simple as this: whenever Nϕ contains Rproperly we can extend ϕ to Nϕ, and then ϕ is in A by induction. We use this argumentto first show that we can assume that S is fully F -normalised. Indeed, let τ : S → T bean isomorphism in F such that T is fully F -normalised. By 2.5 we can choose τ suchthat τ can be extended to NP (S). Thus, by induction, τ belongs to the class A, andhence, so does τ−1. Therefore, in order to show that ϕ is in A it suffices to show thatτ ϕ is in A. In other words, after replacing S by T and ϕ by τ ϕ, we may assumethat S is fully F -normalised. Then there is always some isomorphism ψ : R → S whichextends to NP (R), and hence which is in A by induction. Thus it suffices to show thatthe automorphism ϕψ−1 of S is in A. In other words, after replacing R by S and ϕ byϕ ψ−1 we are down to assuming that R = S is fully F -normalised and ϕ ∈ AutF (R).By induction and Lemma 5.4 we may assume that R is F -radical centric.

Suppose that R is not essential. Set X = AutP (R) and Y = ϕ−1 AutP (R)ϕ. SinceR is fully F -normalised, the groups X , Y are Sylow-p-subgroups of AutF (R). Since R isnot essential, the poset of non-trivial p-subgroups of AutF (R)/AutR(R) is connected. Inother words, there is a finite family X0 | 1 ≤ j ≤ m of Sylow-p-subgroups of AutF (R)such that X = X0, Xj ∩ Xj+1 6= AutR(R) for 0 ≤ j ≤ m − 1 and Xm = Y . For

0 ≤ j ≤ m let γj ∈ AutF (R) such that γj AutP (R) γ−1j = Xj ; since X0 = AutP (R)

we can take γ0 = IdR. We are going to prove, by induction over j, that the γj are in A.For j = 0 this is trivial. Suppose 0 < j ≤ m. We have Xj−1 ∩Xj 6= AutR(R), hence

AutP (R) ∩ (γ−1j−1 γj AutP (R) γ−1

j γj−1) 6= AutR(R) .

Thus, by Lemma 5.5 and induction the morphism γ−1j γj−1 is in A. By our induction

over j the morphism γj−1 is in A, hence so is γj . In particular, γm is in A. Since

Y = γm AutP (R) γ−1m = ϕ−1 AutP (R) ϕ

we get that ϕ γm normalises AutP (R), hence extends to NP (R). Thus ϕ γm is in A.But then ϕ is in A as well because γm and its inverse are so, which completes the proof.

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


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

Proof. Let R be an F -centric radical subgroup of P . Since F = NF (Q), the imageAutQ(R) of Q in AutF (R) is a normal p-subgroup of AutF (R). Since R is F -radicalthis implies that AutQ(R) ⊆ AutR(R) = Op(AutF (R)). Thus Q ⊆ RCP (R). Since R isalso F -centric we have RCP (R) = R, whence the result.

As an application of Alperin’s fusion theorem we get the following characterisationsof fusion systems (cf. [18, §8]):

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

(i) F = F ′.

(ii) For any fully F ′-centralised subgroup Z of order p of P we have HomF (Z, P ) =HomF ′(Z, P ) and CF (Z) = CF ′(Z).

Proof. Suppose that (ii) holds. Let Q be a non trivial subgroup of P and let ϕ ∈AutF (Q). Let Z be a subgroup of order p of Z(Q). Let ψ : Z → P be a morphism inF ′ such that ψ(Z) is fully F ′-centralised. Since Q ⊆ CP (Z), the morphism ψ extendsto 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) ∈ AutF ′(τ(Q)). Thus, after replacing Q byτ(Q), we may assume that Z is fully F ′-centralised. By the assumptions, the morphismϕ−1|ϕ(Z) : ϕ(Z) → Z belongs to F ′, and hence extends to a morphism κ : Q → P inF ′ (since Q = ϕ(Q) ⊆ CP (ϕ(Z))). Then κ ϕ : Q → P restricts to the identity on Z,hence κ ϕ is a morphism in CF (Z) = CF ′(Z). In particular, κ ϕ is a morphism inF ′. But then so is ϕ, because κ is in F ′. Alperin’s fusion theorem implies now (i). Theconverse is trivial.

Lemma 5.8. Let F , F ′ be fusion systems on a finite p-group P such that F ′ ⊆ F . Let

Qϕ

−→ Rψ

−→ S be a sequence of two composable morphisms in F such that Q, R, S areF-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 ψϕ. If ψ, ψϕ are in F ′, then so is ϕ = ψ−1|Im(ψϕ)ψϕ.Assume now that ϕ and ψ ϕ are morphisms in F ′. Up to replacing Q by ϕ(Q), wemay assume that ϕ is the inclusion Q ⊆ R. Let v ∈ NR(Q). Then, for any u ∈ Q, wehave ψ(vu) = ψ(v)u. Thus the morphism ψ|Q extends to a morphism τ : NR(Q) → P inF ′. By 4.11, we have τ(NR(Q)) = ψ(NR(Q)) and hence ψ−1 τ ∈ AutZ(Q)(NR(Q)) by4.10. Thus ψ|NR(Q) is a morphism in F ′. It follows inductively, that ψ is a morphismin 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) HomF (Q,P ) = HomF ′(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 bea minimal F -centric subgroup of P contained in R. Let ϕ ∈ HomF (R,P ). Thenϕ|Q ∈ HomF (Q,P ) = HomF ′(Q,P ). But then ϕ ∈ HomF ′(R,P ) by 5.8. Alperin’sfusion 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 = NF (Q) for some normalsubgroup Q of P , we can define a quotient category F/Q as follows:

Definition 6.1. (Puig [21]) Let F be a fusion system on a finite p-group P suchthat F = NF (Q) for some normal subgroup Q of P . We define the category F/Q onP/Q as follows: for any two subgroups R, S of P containing Q, a group homomorphismψ : R/Q→ S/Q is a morphism in the category F/Q if there exists a morphism ϕ : R→ Sin F satisfying ϕ(u)Q = ψ(uQ) for all u ∈ R.

Theorem 6.2. (Puig [21]) Let F be a fusion system on a finite p-group P such thatF = NF (Q) for some normal subgroup Q of P . The category F/Q is a fusion systemon P/Q.

Proof. Clearly F/Q is a category on P/Q. Since AutF/Q(P/Q) is a quotient of AutF (P ),the group AutP/Q(P/Q) is a Sylow-p-subgroup of AutF/Q(P/Q); thus the Sylow axiomI-S holds. It remains to show that the extension axiom II-S holds as well. Let R, Sbe subgroups of P containing Q, and let ψ : R/Q → S/Q be an isomorphism in F/Qsuch that S/Q is fully F -normalised. Since NP/Q(S/Q) = NP (S)/Q this implies thatR is fully F -normalised. Thus, by 2.6, there is a morphism ρ : NP (R) → P suchthat ρ(R) = S. Denote by σ : NP (R)/Q → P/Q the morphism induced by ρ. Inorder to show that ψ extends to Nψ it suffices to show that ψ σ−1|S/Q extends toNψσ−1|S/Q

, because the inverse image of Nψ is contained in NP (R). In other words,

we may assume that R = S is fully F -normalised and that ψ ∈ AutF/Q(R/Q). LetK be the kernel of the canonical surjective map AutF (R) → AutF/Q(R/Q). ThenX = K ∩ AutP (R) is a Sylow-p-subgroup of K. The Frattini argument implies thatAutF (R) = KNAutF (R)(X). Note that AutP (R) normalises X , so AutP (R) remains aSylow-p-subgroup of NAutF (R)(X). Thus we have a short exact sequence of finite groups

1 −→ K ∩NAutF (R)(X) −→ NAutF (R)(X)α

−→ AutF/Q(R/Q) −→ 1

The group L = (K ∩NAutF (R)(X))/X is a p′-group, and we have a short exact sequence

1 −→ L −→ NAutF (R)(X)/X −→ AutF/Q(R/Q) −→ 1


By 5.3, the image of Nψ in AutF/Q(R/Q) is the intersection of two Sylow-p-subgroups

AutP/Q(R/Q) ∩ (ψ−1 AutP/Q(R/Q) ψ). By 2.11 there is τ ∈ NAutF (R)(X) which

lifts ψ such that the canonical map α sends AutP (R) ∩ (τ−1 AutP (R) τ) onto thisintersection, or equivalently, the canonical map Nτ → Nψ is surjective. This means thatif δ : Nτ → P is a morphism in F which extends τ then its image γ modulo Q is amorphism 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 = CF (Z) for some subgroup Z of Z(P ). Inthat case one can be more precise about connections between F and F/Z. The followingtwo Theorems are well-known.

Theorem 6.3. Let F be a fusion system on a finite p-group P and let Z be a subgroupof Z(P ) such that F = CF (Z). Let Q be a subgroup of P containing Z. Then thefollowing hold.

(i) The canonical group homomorphism AutF (Q) → AutF/Z(Q/Z) is surjective, havingas 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 ∈ AutF (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 grouphomomorphism ζ : Q → Z. The map sending such a ϕ to ζ in turn is easily seento be an injective group homomorphism from ker(AutF (Q) → AutF/Z(Q/Z)) to theabelian p-group Hom(Q,Z), with group structure induced by that of Z. This proves(i). If Q/Z is F/Z-centric then CP/Z(R/Z) ⊆ R/Z for every subgroup R isomorphicto Q in F . Thus CP (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 AutF (Q) → AutF/Z(Q/Z), beinga p-group, must be contained in AutQ(Q), hence in that case we get an isomorphismAutF (Q)/AutQ(Q) ∼= AutF/Z(Q/Z)/AutQ/Z(Q/Z), and henceQ/Z is F/Z-radical.Thisproves (iii). Suppose now that Q is F -centric radical. We may assume that Q/Z isfully F/Z-centralised. The group Q/Z is F/Z-radical by (iii). The kernel K of thecanonical map AutF (Q) → AutF/Z(Q/Z) is a p-group, by (i). Since Q is F -radical thisimplies K ⊆ AutQ(Q). Let C be the inverse image in P of CP/Z(Q/Z). That is, theimage in P/Z of any element in C centralises Q/Z, and hence AutC(Q) ⊆ K. ThusAutC(Q) ⊆ AutQ(Q), which implies C ⊆ QCP (Q). Since Q is F -centric this impliesin fact that C ⊆ Q. Taking images in P/Z yields CP/Z(Q/Z) ⊆ Q/Z, or equivalently,CP/Z(Q/Z) = Z(Q/Z). Since Q/Z was chosen to be fully F/Z-centralised, this impliesthat 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 = CF (Z)and F ′ = CF ′(Z) for some subgroup Z of Z(P ). Suppose that F ⊆ F ′. Then F = F ′ ifand 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 AutF ′(Q) → AutF ′/Z(Q/Z) is contained inAutQ(Q). Thus K is also the kernel of the canonical map AutF (Q) → AutF/Q(Q/Z).Since AutF/Z(Q/Z) = AutF ′/Q′(Q/Z) it follows that AutF (Q) and AutF ′(Q) have thesame order. The assumption F ⊆ F ′ implies AutF (Q) = AutF ′(Q). The equalityF = 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. ([13, 3.4]) Let P be a finite p-group, let Q be a normal subgroup of Pand let F ,G be fusion systems on P such that F = PCF (Q) and such that G ⊆ F .Let R be a normal subgroup of P containing Q. We have G = NF (R) if and only ifG/Q = NF/Q(R/Q).

Proof. Suppose that G/Q = NF/Q(R/Q). In order to show the equality G = NF (R)we proceed by induction over the order of Q. If Q = 1 there is nothing to prove. IfQ 6= 1 then Z = Q ∩ Z(P ) 6= 1 because Q is normal in P . Since F = PCF (Q) wehave in fact F = CF (Z). Set F = F/Z and G = G/Z. Similarly, set P = P/Z,Q = Q/Z and R = R/Z. Then F , G are fusion systems on P satisfying F = PCF (Q)and G ⊆ F . The canonical isomorphism P /Q ∼= P/Q induces obviously isomorphismsof fusion systems G/Q ∼= G/Q and NF/Q(R/Q) ∼= NF/Q(R/Q). Thus, by induction, we

get that G = NF (R). Note that NF (R) = NF (R)/Z. Any G-centric radical subgroupof P contains R. Thus, by 6.3, any G-centric radical subgroup of P contains R. SinceG = NF (R) this implies that G = NG(R). Since also G ⊆ F it follows that G ⊆ NF (R).Thus, by 6.4 applied to G and NF (R) we get the equality G = NF (R). The converse istrivial.

When specialised to fusion systems of finite groups, the notion of quotients of fusionsystems 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 asubgroup of P which is normal in G. Set F = FP (G). Then F = NF (Q) and we haveF/Q = FP/Q(G/Q).

Proof. Trivial verification.


7 Normal fusion systems

The material of this and the following section is from [18] and is an attempt to imitatethe 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 ona subgroup P ′ of P . We say that F normalises F ′ if P ′ is strongly F -closed and if forevery isomorphism ϕ : Q→ Q′ in F and any two subgroups R, R′ of Q ∩ P ′ we have

ϕ HomF ′(R,R′) ϕ−1|ϕ(R) ⊆ HomF ′(ϕ(R), ϕ(R′)) .

We say that F ′ is normal in F and write F ′ E F if F ′ is contained in F and F normalisesF ′.

In other words, F normalises F ′ if for any isomorphism ϕ : Q → Q′ in F and anymorphism ψ : R → R′ in F ′ such that 〈R,R′〉 ⊆ Q, we have 〈ϕ(R), ϕ(R′)〉 ⊆ P ′ andthe induced morphism ϕ ψ ϕ−1 : ϕ(R) → ϕ(R′) is a morphism in F ′. Note that thisimplies that we have in fact an equality

ϕ HomF ′(R,R′) ϕ−1|ϕ(R) = HomF ′(ϕ(R), ϕ(R′)) .

Indeed, the left side is contained in the right side by the definition, and the otherinclusion 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 fusiontheorem this implies in particular that if R, S are subgroups of P ′ which are isomorphicin F then AutF ′(R) ∼= AutF ′(S).

The unique category on the trivial subgroup 1 of P is a fusion system which isnormal in any fusion system F on P . The obvious motivating example for the definitionof 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 Nbe a normal subgroup of G. We have FP∩N(N) E FP (G).

Proof. Trivial.

Proposition 7.3. Let F be a fusion system on a finite p-group P . Then FP (P ) isnormal in F if and only if F = NF (P ).

Proof. Suppose that FP (P ) E F . Then in particular for any morphism ϕ : R → Pin F and any u ∈ NP (R) there is v ∈ NP (ϕ(R)) such that ϕ(ur) = vϕ(r) for allr ∈ R. Whenever ϕ(R) is fully F -centralised, ϕ extends to a morphism ψ : NP (R) → Pin F . In particular, this holds if R, and hence ϕ(R), are F -centric. But then alsoNP (R) and ψ(NP (R)) are F -centric. Inductively, it follows that ϕ can be extendedto an automorphism of P belonging to F . Thus, by Alperin’s fusion theorem, we getF = NF (P ). The converse is easy.

In fact, Proposition 7.3 remains true with P replaced by any subgroup of P , by aresult of Stancu [26, 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 stronglyF-closed abelian subgroup of P then FQ(Q) is normal in F .

Proof. Since Q is abelian, the only morphisms in FQ(Q) are inclusions R ⊆ R′ ofsubgroups 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 ′ isnormal in F . Then for every subgroup Q of P the index [AutF (Q) : AutF ′(Q)] is primeto p.

Proof. Let Q be a subgroup of P , and let ϕ : Q → R be an isomorphism in F suchthat the subgroup R of P is fully F -normalised. Then AutP (R) is a Sylow-p-subgroupof AutF (R), and AutP (Q) ⊆ AutF ′(R). Since F ′ is normal in F , it follows that theSylow-p-subgroup ϕ−1 AutP (R) ϕ of AutF (Q) is contained in AutF ′(Q). Thus theindex of AutF ′(Q) in AutF (Q) is prime to p.

Remark 7.6. Proposition 7.5 is not true, in general, without the assumption that F ′ isnormal in F . Consider the case of a fusion system F on P such that there is a subgroupQ of P which is fully F -centralised but not fully F -normalised, and set F ′ = FP (P ).Then AutP (Q) = AutF ′(Q) is not a Sylow-p-subgroup of AutF (Q). The following is anexample for this situation.

Example 7.7. Let G = S8 be the symmetric group on eight letters, set E1 =〈(15)(26)(37)(48)〉, E2 = 〈(13)(24), (57)(68)〉, E4 = 〈(12), (34), (56), (78)〉 . ThenP = (E4 ⋊ E2) ⋊ E1 is a Sylow-2-subgroup of G. Set F = FP (G). The subgroupE4 of P is F -centric, hence Q = E4 ⋊ 〈(13)(24)(57)(68)〉 and R = E4 ⋊E1 are F -centricas well. Conjugating Q by (35)(46) yields R, hence Q ∼= R in F . Clearly Q is nor-mal in P ; in particular, Q is fully F -normalised. Conjugating (15)(26)(37)(48) ∈ R by(13)(24) ∈ E2 yields (17)(28)(35)(46). This is not an element in R since 7 does notbelong 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.

8 Simple fusion systems

Definition 8.1. (cf. [18, 4.1]) A fusion system F on a non trivial finite p-group P iscalled simple if F has no proper non trivial normal fusion subsystem.

The fusion system FP (G) of a finite simple group G with Sylow-p-subgroup P is notsimple in general. For instance, if G has an abelian Sylow-p-subgroup P of order atleast p2, then FP (G) cannot be simple by 8.4 below. However, if a simple fusion systemF on a finite p-group P is equal to FP (G) for some finite group G containing P asSylow-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 . Supposethat F = FP (G) for some finite group G having P as Sylow-p-subgroup. If Op′(G) = 1and if FP (H) 6= FP (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 suchthat F = FP (G), then G is simple.

Proof. Suppose that Op′(G) = 1 and that FP (H) 6= FP (G) for any proper subgroup Hof G containing P . Let N be a non-trivial normal subgroup of G. Then N∩P is a Sylow-p-subgroup of N , and FN∩P (N) is a normal fusion system in FP (G). As Op′(G) = 1,we have N ∩P 6= 1. As FP (G) is simple, this forces P ⊆ N and FP (N) = FP (G), henceN = G by the assumptions. Let now G be a finite group of minimal order such thatP is a Sylow-p-subgroup of G and such that F = FP (G). Then Op′(G) = 1, becausethe canonical map G → G/Op′(G) induces an isomorphism of fusion systems. By theminimality of G, we have FP (H) 6= FP (G) for any proper subgroup H of G containingP . Thus the second statement follows from the first.

Proposition 8.3. Let P be a finite p-group. Then FP (P ) is simple if and only if P iscyclic of order p.

Proof. By 7.4, for every subgroup Z of Z(P ) we have FZ(Z) E FP (P ), from which thestatement 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 = FP (P ).

Proof. If F is simple, then F = FP (P ) by 7.4, and hence |P | = p by 8.3. The converseis clear.

Proposition 8.5. (B. Oliver) Let F ,F ′ be fusion systems on a finite p-group P suchthat F ′ E F and such that AutF (P ) = AutF ′(P ). Then F ′ = F .

Proof. Suppose that F ′ 6= F . Let Q be a subgroup of maximal order such thatAutF ′(Q) 6= AutF (Q). By the assumptions, Q is a proper subgroup of P . SinceF ′ is normal in F we may assume that Q is fully F -normalised. Then AutP (Q)is a Sylow-p-subgroup of AutF (Q). Moreover, AutF ′(Q) is a normal subgroup ofAutF (Q) containing AutP (Q), and hence, by the Frattini argument, we have AutF (Q) =NAutF (Q)(AutP (Q))AutF ′(Q). By the extension axiom (II-S) every automorphism of Qin NAutF (Q)(AutP (Q)) extends to an automorphism of NP (Q) in F , hence in F ′ bythe maximality assumption on Q. This in turn implies that NAutF (Q)(AutP (Q)) ⊆AutF ′(Q), leading to the contradiction AutF (Q) = AutF ′(Q).

Corollary 8.6. Let F be a fusion system on a finite p-group P . Assume thatAutF (P ) = AutP (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 ′ E F .Then P ′ is strongly F -closed, hence P ′ = P by the assumptions. Since AutP (P ) ⊆AutF ′(P ) ⊆ AutF (P ), the assumptions imply further that AutF ′(P ) = AutF (P ). ThusF ′ = F by 8.5.

Corollary 8.7. Let F be a fusion system on a finite p-group P . Suppose that P isgenerated by the set of its subgroups of order p, that all subgroups of order p in P areF-conjugate and that AutF (P ) = AutP (P ). Then F is simple.

Proof. Let Q be a non-trivial strongly F -closed subgroup of P . Since all subgroups oforder 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 x2n

= 1 = t2 for some integer n ≥ 2 andtxt = x−1; that is, P is a dihedral 2-group of order 2n+1 ≥ 8.

Then P has three conjugacy classes of involutions, namely the classes of the elements

z = x2n−1

, t and xt. Besides the trivial fusion system FP = FP (P ), there are two othersystems, up to isomorphism. We denote by FI

P the fusion system on P generated byFP and an automorphism of order 3 of the Klein four group 〈z〉 × 〈t〉. Thus z and t areFIP -conjugate, while z and xt are not; hence there are now two FI

P -conjugacy classes ofinvolutions in P . We denote by FII

P the fusion system on P generated by FP and anautomorphism of order 3 on each of the Klein four groups 〈z〉× 〈t〉 and 〈z〉× 〈xt〉. Thusall involutions in P are FII

P -conjugate. Any fusion system on P is isomorphic to one ofFP , FI

P , FIIP and any of these systems appear as fusion systems FP (G) of some finite

group G having P as Sylow-2-subgroup. Any 2-block of a finite group having P as defectgroup has 1 or 2 or 3 isomorphism classes of simple modules, and then its fusion systemis isomorphic to FP or FI

P or FIIP , respectively. The only simple fusion system on P is

FIIP , and moreprecisely, if Q is the subgroup of index 2 in P generated by x2 and t, we

have FIIQ E FI

P . Here, for notational convenience, if Q is a Klein four group, we denote

by FIQ and by FII

Q the unique fusion system on Q generated by some automorphism oforder 3 of Q.

All three fusion systems arise from finite groups. If q is an odd prime power suchthat q ≡ ±1(mod8), then the group PSL2(q) has a dihedral Sylow-2-subgroup P ,and FP (PSL2(q)) = FII

P . In particular, FP (PSL2(q)) is simple in that case. Ifq ≡ ±3(mod8) then PSL2(q) has a Klein four group Q as Sylow-2-subgroup, and henceFP (PSL2(q)) cannot be simple by 8.4. In this case the inclusion FII

Q E FIP is realised

by the inclusion PSL2(q) E PGL2(q).

Theorem 8.9. ([18, 6.1]) Let q be an odd prime power such that q ≡ ±3(mod 8) andlet S be a Sylow-2-subgroup of Ω7(q). We have AutΩ7(q)(S) = AutS(S) and S hasno non-trivial proper strongly FΩ7(q)-closed subgroup. In particular, the fusion systemFS(Ω7(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 Spin7(q) over Fq. Then Spin7(q) has acentral involution z such that Spin7(q)/〈z〉

∼= Ω7(q), and hence S = P/〈z〉 is isomorphicto a Sylow-2-subgroup of Ω7(q). R. Solomon showed in [24] that if q ≡ ±3(mod8), nofinite group having P as Sylow-2-subgroup can have a fusion system which properlycontains FP (Spin7(q)), in which all involutions of P are conjugate and which has theproperty that CF (z)/〈z〉 ∼= FS(Ω7(q)). Levi and Oliver proved in [17, 2.1], that thereis actually for any odd prime power q a fusion system FSol(q) on P with the aboveproperties. We are going to call this the Solomon 2-fusion system FSol(q). Kessar showedin [11] that the fusion system FSol(3) cannot even occur as fusion system of a 2-block ofa finite group with P as defect group.

Theorem 8.10. ([18, 7.1]) The Solomon fusion system FSol(3) is simple.

We certainly expect that FSol(q) should be simple for any odd prime power q, butthis case is open at present. The simplicity of FSol(q) would follow from the material in[18] provided one could show that Aut(S) is a 2-group, where S is a Sylow-2-subgroupof Ω7(q). The following consequence of 5.7 is occasionally a useful criterion for thesimplicity of a fusion system:

Proposition 8.11. Let F , F ′ be fusion systems on a finite p-group P such that F ′ E F .If HomF (Z, P ) = HomF ′(Z, P ) and CF (Z)/Z is a simple fusion system on CP (Z)/Zfor any fully F ′-centralised subgroup Z of order p of P , then F ′ = F .

Proof. We have CF ′(Z) E CF (Z) and hence CF ′(Z)/Z E CF (Z)/Z. Thus, if CF (Z)/Zis simple for any fully F ′-centralised subgroup Z of order p of P , then CF ′(Z)/Z =CF (Z)/Z. Since p′-automorphisms lift uniquely through central p-extensions this impliesCF ′(Z) = CF (Z), hence F ′ = F by 5.7.

§9 Normal subsystems and control of fusion

The content of this section is from [19]. We show that if a normal subsystem F ′ ofa fusion system F is controlled by a single subgroup Q′ then F itself is controlled by asingle 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 ′ suchthat F ′ E F . Suppose that there is a subgroup Q′ of P ′ such that F ′ = NF ′(Q′). LetQ be the subgroup of P ′ generated by all subgroups of P ′ isomorphic to Q′ in F . ThenF = NF (Q).

The proof of Theorem 9.1 is based on one hand on ideas of G. R. Robinson, which haveled to [14, 1.5], and on the other hand on the notion of normal subsystems developed


in [18, §3]; see §7 above. It seems to be an open question as to whether a simple fusionsystem in the sense of 8.1 can have non trivial proper strongly closed subgroups. Thefollowing result, due to Stancu [26], can be obtained as a consequence of Theorem 9.1and provides a partial answer to this problem:

Corollary 9.2. ([26, Proposition 6.2]) Let p be a prime and let F be a fusion systemon a finite p-group P . Let Q be a subgroup of P . We have FQ(Q) E F if and only ifF = NF (Q).

Corollary 9.2 can be used to give an alternative proof of 3.15: if Q is an abelianstrongly F -closed subgroup of P , then by 7.4, FQ(Q) is normal in F , and hence F =NF (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 stronglyF-closed subgroup of P and let F ′ be a category on P ′ such that F ′ ⊆ F . Let Q be asubgroup 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 : RCP ′(R) →QCP (Q) in F . Since P ′ is strongly F -closed, this restricts to a morphism CP ′(R) →CP ′(Q) in F , in particular |CP ′(R) ≤ |CP ′(Q)|. Since Q, R are isomorphic in F ′ andsince R is fully F ′-centralised, this inequality is in fact an equality and so is Q is fullyF ′-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 amorphism ψ : NP (R) → NP (Q). Since P ′ is strongly F -closed, ψ induces a morphismNP ′(R) → NP ′(Q); in particular, |NP ′(R)| ≤ |NP ′(Q)|. Since Q, R are isomorphic inF ′ 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 fusionsystem on a subgroup P ′ of P such that F ′ E F and let Q be a subgroup of P ′. If Q isfully F-normalised then AutF ′(Q) ∩ AutP (Q) = AutP ′(Q).

Proof. If Q is fully F -normalised then AutP (Q) is a Sylow-p-subgroup of AutF (Q).Since AutF ′(Q) E AutF (Q) the intersection AutF ′(Q) ∩ AutP (Q) is then also a Sylow-p-subgroup of AutF ′(Q). By Proposition 9.3, Q is also fully F ′-normalised and henceAutP ′(Q) is a Sylow-p-subgroup of AutF ′(Q) as well. Clearly AutP ′(Q) ⊆ AutF ′(Q) ∩AutP (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 fusionsystem on a subgroup P ′ of P such that F ′ E F . Let ϕ : Q → Q′ be an isomorphismin F . Suppose that Q, Q′ are contained in P ′ and that Q′ is fully F ′-normalised. Thenthere is an automorphism τ ∈ AutF ′(Q′) such that the isomorphism τ ϕ : Q→ Q′ canbe extended to a morphism ψ : NP ′(Q) → P in F .

Proof. Since F ′ E F , the morphism ϕ induces an isomorphism AutF ′(Q) ∼= AutF (Q′)given by “conjugation” with ϕ. Since Q′ is fully F ′-normalised, the group AutP ′(Q′)is a Sylow-p-subgroup of AutF ′(Q′). Thus the image ϕ AutP ′(Q) ϕ−1 of AutP ′(Q)in AutF ′(Q′) is conjugate to a subgroup of AutP ′(Q′). In other words, there is τ ∈AutF ′(Q′) such that (τ ϕ) AutP ′(Q) (τ ϕ)−1 ⊆ AutP ′(Q′), or equivalently, theisomorphism AutF ′(Q) ∼= AutF ′(Q′) induced by “conjugation” with τϕmaps AutP ′(Q)to AutP ′(Q′).

Let now β : Q′ → R be an isomorphism in F such that R is fully F -normalised. ByProposition 9.3, R is then also fully F ′-normalised and by Corollary 9.4, AutF ′(R) ∩AutP (R) = AutP ′(R) is a Sylow-p-subgroup of AutF ′(R). Thus some AutF ′(R)-conjugate of β AutP ′(Q′) β−1 is contained in AutP ′(R). Thus, up to replacing β, wemay assume that β AutP ′(Q′) β−1 = AutP ′(R). Note that then NP ′(Q′) ∼= NP ′(R)because both Q′, R are fully F ′-normalised. Since R is in fact fully F -normalised,β extends to an isomorphism σ : NP ′(Q′) ∼= NP ′(R) in F . Also β τ ϕ mapsAutP ′(Q) into AutP ′(R), hence extends to a morphism ρ : NP ′(Q) → NP ′(R) in F .Then σ−1ρ : NP ′(Q) → NP ′(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 fusionsystem on a subgroup P ′ of P such that F ′ E 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 anautomorphism α ∈ AutF (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 fullyF ′-normalised. It follows from Proposition 9.5 that ψ can be chosen in such a way thatψϕ extends to a morphism α from NQ(P ′) = P ′ to P , and since P ′ is strongly F -closedit 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 fusionsystem on a subgroup P ′ of P such that F ′ E F . Let Q be a strongly F ′-closed subgroupof P ′. For any morphism ϕ : Q → P in F the subgroup ϕ(Q) is a strongly F ′-closedsubgroup of P ′ and there are automorphisms ψ ∈ AutF ′(ϕ(Q)) and α ∈ AutF (P ′) suchthat 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 automorphismα ∈ AutF (P ′) such that ψ ϕ = α|Q. Since F ′ E 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: ifϕ : Q → Q′ is an isomorphism in F such that Q′ is fully F -normalised then there


is an automorphism τ ∈ AutF (Q′) such that Nτϕ = NP (Q). In other words, for anysubgroup Q of P there is a morphism ϕ : Q → P such that ϕ(Q) is fully F -normalisedand such that ϕ extends to a morphism ψ : NP (Q) → P in F .

Proposition 9.8. Let F be a fusion system on a finite p-group P , let F ′ be a fusionsystem on a subgroup P ′ of P such that F ′ E F , and let Q be a subgroup of P ′. If Q isstrongly 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 anda morphism ϕ : R → P such that ϕ(R) is not contained in Q. Choose R maximal withthis property. Certainly ϕ(R) ⊆ P ′ because P ′ is strongly F -closed. Hence there is amorphism 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 ψϕ 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 ofϕ(R) in F ′ we may assume that ϕ extends to a morphism σ : NP ′(R) → NP ′(ϕ(R))in F . Restricting σ to NQ(R) yields a morphism (abusively still denoted by the sameletter) σ : NQ(R) → P ′. Since σ extends ϕ we still have that σ(NQ(R)) is not containedin Q. By the maximality of R with this property this forces R = Q. But then Q cannotbe weakly F -closed.

The next Proposition is an analogue of [14, 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 astrongly F-closed subgroup of P . For any subgroup Q of P there is an isomorphismQ ∼= 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 wemay assume that Q is fully F -normalised. Then there is an isomorphism ψ : Q∩P ′ → Tin F such that T is fully F -normalised, and by Proposition 9.5, we may choose ψ suchthat Nψ = NP (Q ∩ P ′). Then ψ extends to a morphism ϕ : NP (Q ∩ P ′) → P . ClearlyQ ⊆ NP (Q) ⊆ NP (Q ∩ P ′). Set R = ϕ(Q). Since ϕ maps NP (Q) to NP (R) and Q isfully F -normalised, R is fully F -normalised, too, and since P ′ is strongly F -closed wehave R∩P ′ = ϕ(Q)∩P ′ = ϕ(Q∩P ′) = ψ(Q∩P ′) = T , which is also fully F -normalised.

The following result generalises [14, 7.2] to arbitrary fusion systems and is the maintool 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 stronglyF-closed subgroup of P and let F ′ be a fusion system on P ′ contained in F . Let Q bea subgroup of P such that both Q, Q ∩ P ′ are fully F-normalised. Suppose that Q isF-radical centric. Then Q ∩ P ′ is F ′-centric. If in addition F ′ is normal in F , thenQ ∩ P ′ is F ′-radical centric and fully F ′-normalised.


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

Let Y be the subset of all ϕ ∈ AutF (Q) satisfying [Q,ϕ] ⊆ Q ∩ P ′, where [Q,ϕ] isas usual the subgroup generated by the set of elements of the form u−1ϕ(u), with urunning over Q.

One easily checks that Y is in fact a subgroup of AutF (Q). Since P ′ is stronglyF -closed, every automorphism of Q in F restricts to an automorphism of Q ∩ P ′ in F .Denote by ρ : AutF (Q) → AutF (Q∩P ′) the group homomorphism sending ϕ ∈ AutF (Q)to its restriction to Q ∩ P ′. Set X = Y ∩ ker(ρ); that is, X consists of all ϕ ∈ AutF (Q)such that [Q,ϕ] ⊆ Q ∩ P ′ and such that ϕ|Q∩P ′ = IdQ∩P ′ . Both X , Y are normalsubgroups of AutF (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 ϕn(u) = uτ(u)n for any positive integer n, and hence the order of ϕ is a powerof p. It follows that X is a p-group.

Clearly X contains all automorphisms of Q induced by conjugation with elementsin CP ′(Q ∩ P ′) ∩ NP (Q). As Q is F -radical centric, we have X ⊆ AutQ(Q), andhence CP ′(Q ∩ P ′) ∩ NP (Q) ⊆ Q. This forces CP ′(Q ∩ P ′) ⊆ Q, or equivalently,CP ′(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 Proposition9.3. It remains to show that Q∩P ′ is F ′-radical. Let S be the subgroup of NP ′(Q∩P ′)containing Q ∩ P ′ such that Op(AutF ′(Q ∩ P ′)) = AutS(Q ∩ P ′) (this makes senseas Q ∩ P ′ is fully F -normalised and hence fully F ′-normalised by Proposition 9.3).This is a characteristic p-subgroup of AutF ′(Q ∩ P ′), hence a normal p-subgroup ofAutF (Q ∩ P ′). Note that in particular Q normalises S; thus QS is a subgroup of P . IfQ ∩ P ′ is a proper subgroup of S then in particular S is not contained in Q and henceQ is a proper subgroup of QS, thus a proper subgroup of NQS(Q). But the image ofNQS(Q) in AutF (Q) is then a non-trivial normal p-subgroup of AutF (Q) because it isthe inverse image of the normal p-subgroup AutS(Q∩P ′) under the canonical restrictionmap ρ. This contradicts however the hypothesis that Q is F -radical.

Proof of Theorem 9.1. Let α ∈ AutF (P ′). Since F ′ is normal in F we have F ′ =NF ′(Q′) = NF ′(α(Q′)). Thus in fact F ′ = NF ′(Q), and in particular, given an F ′-centric radical subgroup S of P ′, we have Q ⊆ S. Futhermore, given ϕ ∈ AutF (S), byCorollary 9.6, there are α ∈ AutF (P ′) and ψ ∈ AutF ′(S) such that ψ ϕ = α|S. Notethat by the construction of Q we have α(Q) = Q. Also, since F ′ = NF ′(Q) we haveψ(Q) = Q and hence ϕ(Q) = Q. Thus ϕ belongs to NF (Q). Let R be an F -radicalcentric subgroup of P . After possibly replacing R by a subgroup isomorphic to R in Fwe 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. Thusany automorphism of R in F restricts to an automorphism of the F ′-centric radical


subgroup S = R ∩ P ′ of P ′, and hence belongs to NF (Q) by the first part of the proof.Alperin’s fusion theorem implies that F = NF (Q).

Proof of Corollary 9.2. Suppose that FQ(Q) E F . In particular Q is strongly F -closedin P . Then F = NF (Q) by Theorem 9.1. The converse is trivial.

Remark. The present paper is a slightly revised version of the article printed in “GroupRepresentation Theory” (eds. M. Geck, D. Testermann, J. Thevenaz), EPFL Press 2007.We added a statement (iv) in Theorem 6.3. In Theorem 6.4 we added the hypothesisF ⊆ F ′ (which is necessary) and rewrote the proofs of 6.4 and 6.5 accordingly.

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Markus Linckelmann

Department of Mathematical Sciences

Meston Building

Aberdeen, AB24 3UE

U.K.

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