Memoirs of the American Mathematical Society, ISSN 0065-9266 ; No

EMOIRS M of the American Mathematical Society

Number 986

The Generalized Fitting Subsystem of a Fusion System Michael Aschbacher

January 2011 • Volume 209 • Number 986 (end of volume) • ISSN 0065-9266

American Mathematical Society

Number 986

The Generalized Fitting Subsystem of a Fusion System

Michael Aschbacher

January 2011 • Volume 209 • Number 986 (end of volume) • ISSN 0065-9266

Library of Congress Cataloging-in-Publication Data Aschbacher, Michael, 1944- The generalized ﬁtting subsystem of a fusion system / Michael Aschbacher. p. cm. — (Memoirs of the American Mathematical Society, ISSN 0065-9266 ; no. 986) “January 2011, Volume 209, number 986 (end of volume).” Includes bibliographical references. ISBN 978-0-8218-5303-0 (alk. paper) 1. Sylow subgroups. 2. Algebraic topology. I. Title. QA174.2.A83 2011 512.2—dc22 2010038097

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Contents

Introduction 1 Chapter 1. Background 7 Chapter 2. Direct products 11

Chapter 3. E1 ∧E2 17 Chapter 4. The product of strongly closed subgroups 23 Chapter 5. Pairs of commuting strongly closed subgroups 25 Chapter 6. Centralizers 33 Chapter 7. Characteristic and subnormal subsystems 39

Chapter 8. T F0 49 Chapter 9. Components 61

Chapter 10. Balance 67 Chapter 11. The fundamental group of F c 71 Chapter 12. Factorizing morphisms 77 Chapter 13. Composition series 83 Chapter 14. Constrained systems 87 Chapter 15. Solvable fusion systems 91 Chapter 16. Fusion systems in simple groups 95 Chapter 17. An example 105 Bibliography 109

iii

Abstract

The notion of a fusion system was first defined and explored by Puig, in the context of modular representation theory. Later, Broto, Levi, and Oliver extended the theory and used it as a tool in homotopy theory. We seek to build a local theory of fusion systems, analogous to the local theory of finite groups, involving normal subsystems and factor systems. Among other results, we define the notion of a simple system, the generalized Fitting subsystem of a fusion system, and prove the L-balance theorem of Gorenstein and Walter for fusion systems. We define a notion of composition series and composition factors, and prove a Jordon-Hölder theorem for fusion systems.

Received by the editor December 19, 2007. Article electronically published on July 21, 2010; S 0065-9266(2010)00621-5. 2010 Mathematics Subject Classiﬁcation. Primary 20D20, 55R35. This work was partially supported by NSF-0504852.

c 2010 American Mathematical Society

Introduction

Fusion systems were defined and first studied by L. Puig, although Puig calls these objects Frobenius categories rather than saturated fusion systems; see in particular [P1]and[P2]. Our introduction to the subject was from [BLO], and we adopt the notation and terminology found there. Puig developed his theory of Frobenius categories primarily as a tool in the modular representation theory of finite groups. The work of Broto, Levi, and Oliver in [BLO], and later work by these authors and other homotopy theorists, was motivated by the study of the p-local homotopy theory of classifying spaces of finite groups and compact Lie groups. In particular, Broto, Levi, and Oliver define new objects called p-local finite groups, consisting of a saturated fusion system, together with an associated linking system, which possesses a p-completed classifying space. Our motivation is a bit different. We seek to translate results from the local theory of finite groups into the setting of saturated fusion systems, and use those results to prove theorems about fusion systems. Then we seek to use such theorems to in turn prove theorems about finite groups. We believe some theorems about finite groups have easier proofs in the category of fusion systems. In particular we hope to simplify portions of the proof of the classification of the finite simple groups using this approach. Thus this memoir continues our program (begun in [A1]) to translate results from the local theory of finite groups into the setting of saturated fusion systems. The reader is directed to [BLO] for basic notation and terminology involving fusion systems, and to [FGT] for notation and terminology involving finite groups. However in section 1 we review some basic material on fusion systems. The definition of a fusion system appears in Definition 1.1 in [BLO], and the definition of a saturated fusion system appears in Definition 1.2 in [BLO]. Both definitions can also be found in section 1 of this memoir. Let p be a prime and S a finite p-subgroup. Roughly speaking, a fusion system on S is a category F whose objects are the subgroups of S, and such that the set homF (P, Q) of morphisms from a subgroup P to a subgroup Q of S is a set of injective group homomorphisms from P into Q satisfying some weak axioms. If G is a finite group and S ∈ Sylp(G), F then S(G) is the fusion system on S such that homFS (G)(P, Q) consists of the g conjugation maps cg : P → Q for g ∈ G with P ≤ Q. Again, roughly speaking, F is saturated if it satisfies some axioms which are easily verified for FS (G)using Sylow’s Theorem. Let F be a saturated fusion system on S.In[A1] we defined the notion of a normal subsystem of F. The definition is repeated in section 1 of this memoir. Puig in [P1] and Linckelmann in [L], also define their own notions of “normal subsystem”. In each case, their normal subsystems are the same as our F-invariant subsystems, so their notions of normality are weaker than ours. However there is a

2 MICHAEL ASCHBACHER subtlety here: To Linckelmann, the term “subsystem” means saturated subsystem, so his notion is closer to ours than that of Puig, differing only in the condition (N1). (cf. section 1) Our definition is chosen to insure that the following property ∗ is satisfied: If G is a finite group with F (G)=Op(G), and F = FS (G), then the map H →FS∩H (H) is a bijection between the normal subgroups of G and the normal subsystems of F. In [A1] we also defined the factor system F/T for each subgroup T of S strongly closed in S with respect to F. The definition is the same as that of Puig in [P1]. In this memoir we begin to establish properties of normal subsystems and factor systems of saturated fusion systems. Here are some of our main results. Let p be a prime and F a saturated fusion system on a finite p-group S.In Example 6.4 of [A1], we saw that the intersection E1 ∩E2 of normal subsystems Ei of F need not be normal in F. However this is not a serious problem, since it develops that E1 ∩E2 is not quite the right object to consider. Rather in section 3 we prove:

Theorem 1. Let Ei be a normal subsystem of F on a subgroup Ti of S,for i =1, 2. Then there exists a normal subsystem E1 ∧E2 of F on T1 ∩ T2 contained in E1 ∩E2. Moreover, E1 ∧E2 is the largest normal subsystem of F normal in E1 and E2. The next result probably already appears somewhere in the literature in the special case where F = FS(G) is the system of a finite group G on a Sylow p- subgroup S of G. I have not been able to find a reference to such a theorem, but then I haven’t looked seriously. It would be a bit surprising if the result were not known for finite groups.

Theorem 2. Assume Ti, i =1, 2, are strongly closed in S with respect to F. Then T1T2 is strongly closed in S with respect to F.

If H1 and H2 are normal subgroups of a group G,thenH1H2 G.The analogue of this result may hold for saturated fusion systems. Here we content ourselves with a proof only in a very special case; this case suﬃces for our most immediate applications.

Theorem 3. Assume Ei F on Ti for i =1, 2, and that [T1,T2]=1.Then there exists a normal subsystem E1E2 of F on T1T2.FurtherifT1 ∩ T2 ≤ Z(Ei) for i =1, 2,thenE1E2 is a central product of E1 and E2.

In section 2 we discuss the direct product F1 ×F2 of fusion systems F1 and F2. Some of the results in this section appear already in section 1 of [BLO]; see in particular 1.5 in [BLO]. A central product F1 ×Z F2 is a factor system (F1 ×F1)/Z, for some Z ≤ Z(F1 ×F2) such that Z ∩Fi =1fori =1, 2. Theorem 3 bears some resemblance to earlier theorems about ﬁnite groups due to Gorenstein-Harris in [GH], and Goldschmidt in [Go2]. Namely in each of these papers, the authors prove the existence of certain normal subgroups of a group G under the hypothesis that for S ∈ Syl2(G), there are subgroups Ti of S for i =1, 2, such that [T1,T2]=1andTi is strongly closed in S with respect to G. Let E F be a system on T . In 6.7 we show the set of subgroups Y of CS(T ) such that E≤CF (Y ) has a largest member CS(E), and CS(E) is strongly closed in S with respect to F. Then in section 6 we deﬁne the centralizer in F of E to be a certain fusion system CF (E)onCS(E), and we prove:

INTRODUCTION 3

Theorem 4. E F E F ∈ E fc If then CF ( ) ,andforX CF ( ) , AutCF (E)(X) p = O (AutCF (T )(X))AutCS (E)(X). In section 7 we find that there is a characteristic subsystem Op(F)ofF on the subgroup [S, Op(F)] defined by p p fc [S, O (F)] = [U, O (AutF (U))] : U ∈F . The group [S, Op(F)] is the F-hyperfocal subgroup of S defined in Chapter 13 of [F]; this subgroup also appears in [BCGLO2]. Moreover in Theorem 13.6 of [P2], Puig shows that there exists a saturated F-invariant subsystem on the hyperfocal subgroup (which Puig calls the hyperfocal subcategory). Similarly in section 4 of [BCGLO2], it is shown that there exists a unique saturated subsystem of “p-power index” on the hyperfocal subgroup. In section 8 we show: Theorem 5. Let E F on T ,andT ≤ R ≤ S. Then there exists a unique saturated fusion subsystem RE of F on R such that Op(RE)=Op(E).Inparticular F = SOp(F). Part of the proof of this result was suggested by the proof of Theorem 4.6 in [BCGLO2]. p If F = FS (G) for some finite group G with S ∈ Sylp(G), then by 7.7, O (F)= p FS∩Op(G)(O (G)). Define F to be simple if F has no proper nontrivial normal subsystems. Define F to be quasisimple if F = Op(F)andF/Z(F) is simple. Define the components of F to be the subnormal quasisimple subsystems of F. Recall Op(F) is the largest subgroup of S normal in F. Define E(F) to be the normal subsystem of F generated by the set Comp(F) ∗ of components of F (which exists by Theorem 1), and set F (F)=E(F)Op(F). We call F ∗(F)thegeneralized Fitting subsystem of F. Of course all of these notions are similar to the analogous notions for groups. In section 9 we prove: Theorem 6. (1) E(F) is a characteristic subsystem of F. (2) E(F) is the central product of the components of F. (3) Op(F) centralizes E(F). ∗ ∗ (4) CF (F (F)) = Z(F (F)). In section 10 we prove a version of the Gorenstein-Walter theorem on so called L-balance [GW]:

Theorem 7. For each fully normalized subgroup X of S, E(NF (X)) ≤ E(F). It is worth noting that the proof of L-balance for a group G requires that the components of G/Op (G) satisfy the Schreier conjecture, or when p =2,aweak version of the Schreier conjecture due to Glauberman. Our proof of Theorem 7 requires no deep results. The theorem does not quite imply L-balance for groups, since there is not a nice one to one correspondence between quasisimple groups and quasisimple fusion systems. The proof can be translated into the language of groups, but even there at some point one seems to need some result like Theorem A of Goldschmidt in [Go2], which is only proved for p = 2 without the classiﬁcation. Still, something is going on here, which suggests that in studying fusion systems, one may be lead to new theorems or better proofs of old theorems about ﬁnite

4 MICHAEL ASCHBACHER groups. Indeed examples show that some steps in the Classification of the finite simple groups have easier proofs in the category of saturated 2-fusion systems, and that often it is possible to translate such results back to the category of finite groups, to obtain simplifications of the corresponding parts of the proof of the Classification. In section 5 of [BCGLO2], the authors of that paper study what they call “subsystems of F of index prime to p”. In particular [BCGLO2] parameter- izes such subsystems via the subgroups of a certain group Γp (F), and prove ∼ c p c c Γp (F) = π1(F )/O (π1(F )), where π1(F ) is the fundamental group of the centric subcategory F c of F. In our next theorem, we show the normal subsystems of F on S are the subsys- c ∼ tems corresponding to the normal subgroups of Γp (F), and that π1(F ) = Γp (F). More generally, in section 11 we state and give alternate proofs for some of the results from section 5 of [BCGLO2] in the language of normal subsystems, using our theory of such subsystems. To begin, we need to recall some definitions. From 3.11, there exists a smallest p p normal subsystem O (F)ofF on S.ForP ≤ S, define B(P )=O (AutF (P )). By parts (1) and (3) of 5.2 in [A1], B is a constricted F-invariant map on S.Set B = E(B). By 5.5 in [A1], B is an F-invariant subsystem of F. In Definition 3.3 p of [BCGLO2], the subsystem B is denoted by O∗ (F). In Definition 3.1 of [BCGLO2], the authors define a subsystem of F prime to p to be a saturated subsystem E on S containing B. Then on page 44 of [BCGLO2], they (essentially) define 0 c AutF (S)=α ∈ AutF (S):α|P ∈ homB(P, S)forsomeP ∈F . 0 As B is F-invariant, AutF (S) AutF (S), so we can define 0 Γp (F)=AutF (S)/AutF (S).

The fundamental group π1(C) of a small category C is a standard notion from the theory of symplicial sets; see section 11 for the deﬁnition in the special case of the small category F c. We can at last state our next theorem:

c ∼ Theorem 8. (1) π1(F ) = Γp (F). 0 (2) The map E → AutE (S)/AutF (S) is a bijection between the set of normal subsystems of F on S and the set of normal subgroups of Γp (F). p 0 c (3) F = O (F) iff AutF (S)=AutF (S) iff π1(F )=1. (4) F is simple iff the following hold: (a) For each normal subsystem D of F on a subgroup D of S, we have D = S. 0 (b) AutF (S)=AutF (S). Theorem 8 is proved in section 11. A homomorphic image of F is a fusion system which is the image of F under a surjective morphism of fusion systems. Recall that a subgroup T of S is strongly closed in S with respect to F if for each P ≤ T and φ ∈ homF (P, S), Pφ ≤ T . Given a strongly closed subgroup T of S, in section 8 of [A1] we defined a factor system F/T of F on S/T ,andshowed that F/T is a saturated fusion system. In Definition 12.13 we define a functor Θ=ΘF,T from F to F/T and prove:

INTRODUCTION 5

Theorem 9. (1) For each subgroup T of S strongly closed in S with respect to F, ΘF,T : F→F/T is a surjective morphism of fusion systems. (2) The map T →F/T is a bijection between the set of strongly closed subgroups of S and the set of isomorphism classes of homomorphic images of F.

In [P1], Puig defines the same factor system and proves the map ΘF,T is a surjective morphism of fusion systems, although using a different approach. Given our notions of normal subsystem and factor system, it becomes possible in Definitions 13.1 and 13.3 to define the notion of a composition series λ for F and its family F (λ) of factors. From 13.5, all factors of λ aresimplesystems.Thenin section 13 we prove: Theorem 10 (Jordon-Hölder Theorem for fusion systems). If λ and μ are composition series for F then λ and μ have the same length and F (λ)=F (μ). Given Theorem 10, we can define the composition factors of F to be the family F (λ) of factors for any composition series λ of F. A saturated fusion system is said to be exotic if it is not the fusion system FS(G) of any finite group G with Sylow group S. Exotic fusion systems are known to exist; indeed we describe one in section 17. In the remainder of the paper we study solvable fusion systems and the composition factors of fusion systems of groups. We find that some of those factors are exotic, so already even in the study of the fusion systems of finite groups, it is necessary to pass to more general systems. The fusion system of order p is the system FG(G), where G is the group of order p. Define F to be solvable if all composition factors of F are of order p. Fusion systems of p-solvable groups are solvable, but by Theorem 11 below, while each solvable system is the fusion system of a finite group, there are many non-p- solvable groups with solvable fusion systems. There is an alternate definition of solvability for fusion systems due to Puig in section 19 of [P2]. There Puig defines F to be solvable if a certain series of invariant subsystems descends to the identity. Puig’s series is analogous to the derived series in a group, so his definition of solvability is as natural as the one given above. However it turns out that F is solvable in Puig’s sense iff F = FS(G) for some p-solvable group with S ∈ Sylp(G), so the two notions of solvability of fusion systems define different classes of fusion systems. The fusion system F is constrained if F has a normal centric subgroup. A ∗ model for F is a finite group G such that S ∈ Sylp(G), F (G)=Op(G), and F = FS (G). Write G(F) for the set of models of F.IfG(F) = ∅ then F is constrained. Conversely by the fundamental result Proposition C in [BCGLO1], if F is constrained then G(F) = ∅ and all models for F are isomorphic. Section 15 contains various conditions on a saturated fusion system F which are equivalent to the solvability of F. Most of these equivalences are easy to prove. However in section 15 we also prove the following result, whose proof seems to require the classification of the finite simple groups: Theorem 11. The saturated fusion system F is solvable iff F is constrained, and for G ∈G(F), all composition factors of G of order divisible by p are of order p or p-Goldschmidt.

AgroupG with Sylow p-subgroup S is p-Goldschmidt if FS (G)=FS(NG(S)). Using the classiﬁcation of ﬁnite simple groups and a result of Flores and Foote and

6 MICHAEL ASCHBACHER

Foote in [FF]and[F], one can enumerate the simple p-Goldschmidt groups; the list appears in Theorem 15.6. By Theorem 10, if E is a composition factor of the system of some finite group, then E is a composition factor of FS (G) for some finite simple group G and 1 = S ∈ Sylp(G). If FS (G) is simple, then E = FS (G), but (as Theorem 11 shows) even though G is simple, FS (G) need not be simple. Suppose F = FS(G) for some nonabelian finite simple group G and S ∈ Sylp(G), but F is not simple. Then there is a proper nontrivial normal subsystem E of F on some subgroup T of S. One possibility is that T is proper in S. Thus one important problem is to determine the triples (G, S, T )with1= T

~~CHAPTER 1~~

~~Background~~

For background involving fusion systems, for the most part we refer the reader to [BLO]or[A1]. However in this section we recall some notation, terminology, and definitions involving fusion systems used frequently in this paper. Let C be a category and A and B objects in C.WritehomC(A, B)forthe set of morphisms in C from A to B,andAutC(A) for the set of automorphisms of A in C. Given an isomorphism α : A → B in C,writeα∗ for the isomorphism ∗ ∗ −1 α : AutC(A) → AutC(B) defined by α : φ → α φα. Let S be a group. A fusion category on S is a category F whose objects are the subgroups of S,andsuchthatforP, Q ≤ S,homF (P, Q)isasetofinjective group homomorphisms from P to Q.Forx, y ∈ S, xy = y−1xy is the conjugate of y x by y,andcy : S → S defined by cy : x → x is conjugation by y. A fusion system on S is a fusion category F on S such that: s (1) for each s ∈ S and P, Q ≤ S with P ≤ Q, cs : P → Q is in homF (P, Q), and −1 (2) for each φ ∈ homF (P, Q), φ : P → Pφ and φ : Pφ → P are in homF (P, Pφ) and homF (Pφ,P), respectively. Usually S will be a finite p-group for some prime p. In the remainder of the section assume p is a prime and F is a fusion system on a finite p-group S.WriteP ∈F to indicate P is an object in F;thatisP is a subgroup of S. Given P ∈F,let F P = {Pφ : φ ∈ homF (P, S)} be the set of F-conjugates of P . Define P to be fully centralized, fully normalized F f if for all Q ∈ P , |CS(P )|≥|CS(Q)|, |NS (P )|≥|NS (Q)|, respectively. Write F for the set of fully normalized subgroups of S. Our fusion system F is saturated if: f (I) For all P ∈F , P is fully centralized and AutS(P ) ∈ Sylp(AutF (P )), and (II) whenever P ∈Fand φ ∈ homF (P, S) such that Pφ is fully centralized, then each α ∈ Nφ extends to a member of homF (Nφ,S), where { ∈ ∗ ∈ } Nφ = g NS(P ):cg AutS(Pφ) , using the ∗-notation defined earlier. In the remainder of the section assume the fusion system F is saturated. Recall from [A1] that a subsystem E of F on a subgroup T of S is F-invariant if (I1) T is strongly closed in S with respect to F,and ∗ (I2) for each P ≤ Q ≤ T , φ ∈ homE (P, Q), and α ∈ homF (Q, S), φα ∈ homE (Pα,T). Further E is normal in F (written E F)ifE is F-invariant, E is saturated, and E satisfies condition (N1) in F:

~~8 MICHAEL ASCHBACHER~~

ˆ ˆ (N1) Each φ ∈ AutE (T ) extends to φ ∈ AutF (TCS(T )) such that [φ, CS(T )] ≤ Z(T ). For T ≤ S strongly closed in S with respect to F, we can define the factor system F/T as in 8.6 in [A1]. F For P ≤ S,setVP = PCS(P ). Define P to be centric if for each Q ∈ P , c fc f c CS(Q) ≤ Q.WriteF for the set of centric subgroups of S,andsetF = F ∩F . The normalizer in F of P is the fusion system N = NF (P )onNS (P ) such that for Q, R ≤ NS (P ), homN (P, Q) consists of those φ ∈ homF (R, Q) which extend f to some φˆ ∈ homF (PR,PQ) acting on P .ByatheoremofPuig,ifP ∈F ,then NF (P ) is a saturated fusion system. We say P is normal in F and write P F if F = NF (P ). We say that F is constrained if there is a centric subgroup of S which is normal in F. If F is constrained then the set G(F)ofmodelsofF consists of the finite ∗ groups G such that S ∈ Sylp(G), F = FS (G), and F (G)=Op(G). By a result from [BCGLO1], constrained saturated fusion systems possess models, which are determined up to isomorphism. F F f Given T strongly closed in S with respect to ,let T be the set of subgroups F F fc ∈Ff ≤ U of T fully normalized in ,and T the set of U T such that CT (U) U. ∈Ff In 4.1 of [A1], given U T , we defined a constrained saturated system D D (U)= F,T (U)=NNF (V )(U), UCT (U) and given a subsystem E of F on T , we also defined the subsystem E E (U)= F,E (U)=NNE (UCT (U))(U). Then we obtain the model

~~G(U)=GF,T (U) ∈G(D(U)), and when E F, also E(U) D(U)byTheorem2in[A1], and by Theorem 1 in [A1]wehaveauniquemodel~~

H(U)=HF,E (U) ∈G(E(U)), with H(U) G(U). See 5.1 in [A1] for the deﬁnition of an F-invariant map on T , and 5.2.3 in [A1] for the deﬁnition of a constricted F-invariant map. A strongly F normalized chain in T is a chain

~~C =(U0 ··· Un = T ) f of subgroups of T such that for each 0 ≤ i~~

~~1. BACKGROUND 9~~

~~∈Ff strongly normalized. As U1α T , we conclude from 1.1.2 in [A1]thatwemay choose α to extend to γ ∈ homF (NS(U1),S). As NS (U) ≤ NS(U1),~~

~~NS(U)β ≤ NS (U1)β ≤ NS(U1β)=NS (U1α), ∈Ff ∈Ff C so as U T , also Uα T and U1α = NT (Uα). Thus (Uα)isstrongly normalized.~~

~~CHAPTER 2~~

~~Direct products~~

Let p be a prime, and Fi be a fusion system on a ﬁnite p-group Si for i =1, 2. Set S = S1 × S2 and let πi : S → Si be the projection of S on Si.ForPi,Qi ≤ Si ∈ × ∈ × × and φi homFi (Pi,Qi), deﬁne φ1 φ2 hom(P1 P2,Q1 Q2)by

~~φ1 × φ2 :(x1,x2) → (x1φ1,x2φ2).~~

Define the category F = F1 ×F2 to have objects the subgroups of S,andfor P, Q ≤ S, define homF (P, Q) to consist of the maps φ such that φ =(φ1 × φ2)|P , ≤ ∈ Pφ Q,andφi homFi (Pπi,Qπi). Call the map φi the ith projection of φ. Observe φi is uniquely determined by the equality φπi = πiφi. Often we write Pi for Pπi. Write Fî for the subcategory of F whose objects are the subgroups of S and with hom (P, Q)={φ ∈ homF (P, Q):φ − =1}. Fî 3 i

(2.1). F1 ×F2 is a fusion system on S1 × S2. Proof. ≤ ∈ Let P, Q S.Forφi homFi (Pi,Qi), φi is a monomorphism of groups, so φ1 ×φ2 : P1 ×P2 → Q1 ×Q2 is a monomorphism. Thus φ =(φ1 ×φ2)|P : P → Q1×Q2 is a monomorphism, so if Pφ ≤ Q then φ : P → Q is a monomorphism. If Pφ ≤ R ≤ Q then Piφi ≤ Ri ≤ Qi,soasFi is a fusion system, φi regarded as a map from Pi to Ri is in Fi, and hence φ regardedasamapfromP to R is in F.If φ is an isomorphism then also φi : Pi → Qi is an isomorphism, so as Fi is a fusion −1 → F −1 → F system, φi : Qi Pi is an i-map, and hence φ : Q P is an -map. Finally ∈ × ∈ for s =(s1,s2) S, cs = cs1 cs2 ,socs homF (S, S), completing the proof.

~~(2.2). F = Fˆ1, Fˆ2 .~~

~~Proof. For φ ∈ homF (P, Q), φ =(φ × 1)(1 × φ ) with φ × 1 ∈ hom (P, S) 1 2 1 Fˆ1 and 1 × φ ∈ hom (P (φ × 1),Q). 2 Fˆ2 1~~

(2.3). Assume E is a fusion system on S and Fi is a subsystem of E for i =1, 2. Assume ≤ ∈ ˆ ∈ (a) For each P, Q Si and φ homFi (P, Q), φ extends to φ homE (PS3−i, ˆ QS3−i) with φ =1on S3−i. E ˆ ∈ ≤ (b) = φ : φ homFi (P, Q),P,Q Si,i =1, 2 . That is each morphism in E is a composition of restrictions of such morphisms φˆ. Then E = F1 ×F2. Proof. ≤ ∈ ˆ × Let i = 1 or 2, P, Q Si,andφ homFi (P, Q). Then φ = φ 1 ˆ or 1 × φ,soφ ∈ homF (PS3−i,QS3−i), and then E≤Fby (b). Thus by 2.2, it remains to show Fˆi ⊆E.

~~12 MICHAEL ASCHBACHER~~

Let U, V ≤ S and ψ ∈ homFˆ (U, V ). Then ψ =(ψ1 × 1)|U , with ψ1 ∈ × ∈ 1 E homF1 (U1,V1). But ψ1 1 homE (U1S2,V1S2)by(a),soas is a fusion system, ψ ∈ homE (U, S). Then as Uψ ≤ V , also ψ ∈ homE (U, V ), completing the proof. (2.4). F F Assume for i =1, 2 that i = Si (Gi) for some ﬁnite group Gi with Si ∈ Sylp(Gi).LetG = G1 × G2.ThenF1 ×F2 = FS (G),soF1 ×F2 is saturated.

Proof. Let E = FS(G); we check that the hypothesis of 2.3 are satisfied by E ≤ → ∈ .ForP, Q Si,homFi (P, Q) consists of the maps φi = cgi|P : P Q with gi ˆ ∈ NG (P, Q). Now φi = cg | homE (PS3−i,QS3−i)is1onS3−i, so condition i i PS3−i ∈ ∈ ˆ ˆ (a) of 2.3 is satisfied. Further for g = g1g2 G with gi Gi, cg = cg1 cg2 = φ1φ2, so condition (b) is also satisfied. (2.5). ≤ ≤ × For U S, NF (U) NF (U1U2)=NF1 (U1) NF2 (U2). Proof. ≤ × ≤ ∈ First NS(U) NS1 (U1) NS2 (U2). Further for P NS(U), φ homF (P, S)isinNF (U)iffφ extends to ϕ ∈ homF (UP,S) acting on U.Inthat × ∈ event, ϕ is the restriction of ϕ1 ϕ2 with ϕi homFi (UiPi,Si) acting on Ui, so ϕ1 × ϕ2 ∈ homF (U1U2P1P2,S) extends φ and acts on U1U2.ThusNF (U) ≤ NF (U1U2). Further specializing to the case U = U1U2, the remarks above show × NF (U)=NF1 (U1) NF2 (U2). (2.6). Let U ≤ S.Then

(1) CS(U)=CS1 (U1)CS2 (U2). ∈Fc ∈Fc (2) If U then Ui i . ∈Fc × ∈Fc (3) If Ui i for i =1, 2,thenU1 U2 . fc (4) Assume Fi is saturated for i =1, 2,andU ∈F . Then there exists ∈ ∈Ffc × ∈Ffc φi homFi (NSi (Ui),Si) with Uiφi i and U(φ1 φ2) . Proof. Part (1) is a straightforward calculation. Suppose U ∈Fc.Thenfor ∈ ≤ ≤ each φ homF (U, S), CS(Uφ) Uφ.But(Uφ)i = Uiφi,soby(1),CSi (Uiφi) Uφ∩ Si ≤ Uiφi, establishing (2). ∈Fc × ∈Fc Assume Ui i for i =1, 2, and set W = U1 U2.ThenW iﬀ for each ∈ ≤ × ∈ φ homF (W, S), CS(Wφ) Wφ.Butφ = φ1 φ2 with φi homFi (Ui,Si), and × CS(Wφ)=CS1 (U1φ1) CS(U2φ2)by(1).Now(3)follows. Finally assume the hypothesis of (4), and using 1.1.2 in [A1], pick φi ∈ ∈Ff × ≤ × homFi (NSi (Ui),Si) with Uiφi i .Letφ = φ1 φ2.ThenNS (U)φ NS1 (U1)φ1 ≤ ∈Ffc ∈Ffc ∈Fc ∈Ffc NS2 (U2)φ2 S,soasU , also Uφ .By(2),Ui i ,soUiφ i .

(2.7). If F1 and F2 are saturated then F1 ×F2 is saturated. Proof. By Theorem A in [BCGLO1] it suffices to show: c (a) F = AutF (U):U ∈F ,and (b) The saturation axioms are satisfied at each member of F c. As Fi is saturated, (a) is satisfied in Fi. Then by 2.6.3, c c Fˆ = Aut (U × S − ):U ∈F ≤AutF (U):U ∈F , i Fî i 3 i i i so(a)issatisfiedbyF by 2.2. c c Thus it remains to verify (b). Let U ∈F and set W = U1U2.AsU ∈F, f U is fully centralized. Suppose U ∈F and pick φi and φ = φ1 × φ2 as in ∈Ffc F ∈G 2.6.4. Then Uiφi i ,soas i is saturated there is Gi (NFi (Uiφi)). Let

~~2. DIRECT PRODUCTS 13~~

× × E Qi = NSi (Uiφi), Q = Q1 Q2,andG = G1 G2. By 2.5, = NF (Wφ)= × F NF1 (U1φ) NF2 (U2φ), so by 2.4, NF (Wφ)= Q(G). By 2.5, NF (Uφ)=NE (Uφ), F so NF (Uφ)= NQ(Uφ)(NG(Uφ)), and hence NF (Uφ) is saturated. Thus NF (U) is saturated, so AutS(U) ∈ Sylp(AutF (U)), and hence U satisﬁes axiom I for saturation. Next suppose φ ∈ homF (U, S). Then ∗ Nφ = {x ∈ NS (U):cxφ ∈ AutS(Uφ)} ≤{ ∈ × ∗ ∈ } × (x1,x2) NS1 (U1) NS2 (U2):cxi φi AutSi (Uiφi) = Nφ1 Nφ2 . ∈Fc ∈ × By 2.6.2, Ui i ,soφi extends to ϕi homFi (Nφi ,Si). Let ϕ =(ϕ1 ϕ2)|U . Then ϕ extends φ to Nφ, verifying axiom II, and completing the proof. Deﬁnition 2.8.

~~Write U = U(F1, F2) for the set of subgroups D of Z(F1) × Z(F2) such that D ∩ Z(Fi)=1fori =1, 2. For D ∈U, deﬁne the central product of F1 and F2 with respect to D to be~~

~~F1 ×D F2 =(F1 ×F2)/D.~~

(2.9). Let D ∈U(F1, F2).Then (1) D ≤ Z(F1 ×F2)=Z(F1) × Z(F2). + + (2) Let θ : F→F = F1 ×D F2 be the natural map θ : s → s = sD and φθ : x+ → (xφ)+.Thenθ is a surjective morphism of fusion systems. F →F+ F F (3) θ : i i = iθ is an isomorphism, when i is regarded as a subsystem of F. + (4) If F1 and F2 are saturated, then so is F . F F ∈ (5) Suppose i = Si (Gi) for some ﬁnite group Gi with Si Sylp(Gi),and + set G = G1 × G2.ThenD ≤ Z(G),soG = G/D is a central product of G1 and + + G2,andF = FS+ (G ). E + F + E (6) Assume is a fusion system on S and i is a saturated subsystem of + on Si for i =1, 2. Assume: +fc (i) For i =1, 2,eachP ∈F ,andeachφ ∈ AutF + (P ), φ extends to i i ˆ + ˆ + φ ∈ AutF + (PS − ) with φ =1on S − . i 3 i 3 i ˆ +fc (ii) E = φ : φ ∈ AutF + (P ),P∈F ,i=1, 2 . i i Then E = F1 ×D F2.

Proof. A straightforward calculation shows Z(F)=Z(F1) × Z(F2), so as D ≤ Z(F1) × Z(F2), (1) holds. Part (2) follows from 8.3 in [A1], and (3) follows from the fact that Si ∩ ker(θ)=Si ∩ D =1. If F1 and F2 are saturated, then so is F by 2.7. Then (4) follows from 8.5 in [A1]. Assume the hypothesis of (5). By 2.4, F = FS(G). Then as D ≤ Z(F), + D ≤ Z(G), so G is a central product of G1 and G2 as D ∩ Gi = 1. Finally 8.7 in [A1] completes the proof of (5). Assume the hypothesis of (6). We repeat the proof of 2.3. As in that proof, E≤F+ Fˆ+ ⊆E F + by conditions (i) and (ii), and it remains to show i ,since = ˆ+ ˆ+ + + +fc F , F by 2.2 and (2). As F is saturated, F = AutF + (P ):P ∈F by 1 2 i i i i ˆ+ ˆ +fc ˆ+ A.10 in [BLO]. Thus F = φ : φ ∈ AutF + (P ):P ∈F ,soF ≤Eby (i) and i i i i (ii), completing the proof of (6).

~~14 MICHAEL ASCHBACHER~~

One can extend the notion of the direct product of fusion systems to the notion of a direct product Fi i∈I of a family (Fi : i ∈ I) of an arbitrary finite number of fusion systems, either by a recursive construction, or by an obvious generalization of the definitions for families of size 2. In particular from the former point of view, by induction on |I|, the direct product of saturated fusion systems is saturated. Similarly one can extend Definition 2.8 to construct the central product ( Fi)/Z ∈ i I ≤ F ∩ F of a family of fusion systems with respect to Z i Z( i) such that Z Z( i)=1 for each i ∈ I.

(2.10). Let (Fi : i ∈ I) and (F˜i : i ∈ I) be families of fusion systems, and F → F˜ (βi : i i) a family of morphisms. Then there exists a unique morphism β = βi : Fi → F˜i extending βi for each i. Indeed i i i xβ =( xi)β = xiβi i i and φβ =( φi)β = φiβi i i ∈ F for x S = i Si and φi an i-map. Proof. Proceeding by induction on |I| = n,wemaytaken =2.Itiswell known that β : S = S1 × S2 → S˜ = S˜1 × S˜2 is the unique group homomorphism ≤ ∈ ∈ extending β1 and β2.LetPi,Qi Si and φi homFi (Pi,Qi). Then φiβi hom (P β ,Q β ), so φ β × φ β ∈ hom (P β × P β ,Q β × Q β ). Then if F˜i i i i i 1 1 2 2 F˜ 1 1 2 2 1 1 2 2 φ ∈ homF (P, Q) with Pπi = Pi and Qπi = Qi,andwithφ =(φ1 × φ2)|P , define φβ =(φ1β1 × φ2β2)|Pβ. It is straightforward to check that β is a morphism, and using 2.2, β is unique. (2.11). F ∈ F F F + F Let ( i : i I) be a family of fusion systems, = i i, = /Z a central product, and θ : F→F+ the natural map. Assume α ∈ Aut(F +) {F + F ∈ } is an isomorphism permuting i = iθ : i I . Then there exists a unique αˆ ∈ Aut(F) such that αθˆ = θα. Moreover Fiαˆ = Fiσ,whereσ ∈ Sym(I) is defined F + F + by i α = iσ. Proof. F + F + ∈ F →F+ By hypothesis i α = iσ for some σ Sym(I). Define θi : i i F F →F −1 to be the restriction of θ to i, and defineα î : i iσ byα î = θiαθiσ .Thenˆαi ∈ F is an isomorphism for each i I, so by 2.10,α ˆ = i αî is an automorphism of . F F + →F+ By constructionα îθ = θαi on i,whereαi : i iσ is the restriction of α to F +. Then by 2.10, i θα = θ · αi = θαi = αîθ =( αî) · θ =ˆαθ, i i i i andα ˆ is unique subject to this property by 2.10. In the last part of the final lemma of this section, it is convenient to prove a result about quasisimple systems, a notion not introduced until section 7.

2. DIRECT PRODUCTS 15 (2.12). F ≤ ≤ F p F Let ( i :1 i p) be a family of fusion systems, = i=1 i, F + = F/Z a central product, and α an automorphism of F + of order p such F + F + ≤ ≤ that i α = i+1 for each 1 i p, where the indices are read modulo p.Let θ : F→F+ be the natural map. Then (1) Let αˆ ∈ Aut(F) such that αθˆ = θα.LetD = CF (ˆα) be the fusion system on C (ˆα) such that for P, Q ≤ C (ˆα), homD(P, Q)={φ ∈ homF (P, Q):φαˆ = φ}. S ∼ S Then D = F1δ = F1, CS(ˆα) is the full diagonal subgroup D of S = S1 ×···×Sp p−1 determined by αˆ, δ : S1 → S is the map δ : x → (x, xα,ˆ ··· ,xαˆ ),andfor ≤ ∈ ··· p−1 P, Q S1 and φ homF1 (P, Q), φδ =(φ, φα,ˆ ,φαˆ )|Pδ. + p (2) If F1 is quasisimple then E = O (CF + (α)) = CF (ˆα)θ is quasisimple, with + + ∼ + E /Z(E ) = F1/Z(F1),andCF + (α)=E CZ(F +)(α). Proof. We ﬁrst prove (1), where appealing to 2.11 and replacing F +, α by F, αˆ, we may assume Z =1andα =ˆα. Visibly F1δ ≤ CF (α)andδ : F1 →F1δ is an isomorphism. Further it is well known that D = CS(α). Finally if P, Q ≤ S1 and ∈ ×···× ∈ i−1 i−1 φ homD(Pδ,Qδ)thenφ =(φ1 φp)|Pδ for some φi homFi (Pα ,Qα ). As α centralizes φ, φiα = φi+1,so(1)holds. Now assume F1 is quasisimple and remove the assumption that Z = 1. Adopt the notation in section 8 of [A1] in discussing the factor system F +.Asˆαθ = θα, α =ˆα+, so from section 8 in [A1], + CS+ (α)=C+(ˆα)={s :[s, αˆ] ∈ Z}. + + As α is free on S and acts on Z(F) ≥ Z, it follows that CS+ (α) ≤ CS(ˆα) Z(F ). + ∼ Similarly CF + (α) ≤ CF (ˆα)θZ(F ). By (1), CF (ˆα) = F1 is quasisimple, so CF (ˆα)θ p is also quasisimple, and hence CF (ˆα)θ = O (CF + (α)), so (2) holds.

~~CHAPTER 3~~

~~E1 ∧E2~~

In this section F is a saturated fusion system over the ﬁnite p-group S,andEi, i =1, 2, are normal subsystems of F on Ti.SetT = T1 ∩ T2. By 3.6.1 in [A1], E1 ∩E2 is F-invariant on T , so in particular T is strongly closed in S with respect to F. Notation 3.1. ∈Ffc Let U T ,andfori =1, 2, set TU,i = UCTi (U)andTU,1,2 = TU,1TU,2. Deﬁne D D 1,2(U)=NF (V ), where V = VTU,1,2 .Thus 1,2(U) is a fusion system on NS (U). (3.2). ∈Ffc Let U T and V = VTU,1,2 .Then (1) U = T ∩ TU,i = T ∩ TU,1,2 = T ∩ V for i =1, 2. (2) NS(U)=NS (TU,i)=NS(TU,1,2)=NS (V ) for i =1, 2. (3) T ∈Ffc for i =1, 2,andV ∈Ffc. U,i Ti (4) D1,2(U) is saturated and constrained.

(5) AutF (U)=AutD1,2(U)(U). ≤ ≤ ≤ ≤ ∩ (6) TU,i VTU,i VU for i =1, 2,andTU,1,2 V VU .FurtherTU,i = VU Ti. Proof. ≤ ∈Ffc First TU,i = UCTi (U)andCT (U) U as U T .Thus ∩ ∩ T TU,i = U(T CTi (U)) = UCT (U)=U.

Similarly U = T ∩ TU,1,2 = T ∩ V , establishing (1). Then (1) implies (2), and (3) ∈Ff ∈Ffc follows from (2) and the choice of U T .AsV by (3), part (4) follows from 1.2.1 in [A1]. The proof of (5) is the same as that of 4.3.3 in [A1]. ≤ ≤ By deﬁnition, TU,i TU,iCS(TU,i)=VTU,i ,andTU,i = UCTi (U) UCS (U)= ≤ ≤ ≤ ≤ VU . Similarly CS(TU,i) CS(U) VU ,andTU,1,2 V VU .AsTU,i = UCTi (U), arguing as in (1), we conclude TU,i = Ti ∩ VU .Thatis(6)holds. Notation 3.3. ∈Ffc Let U T ,andfori =1, 2, form the groups Gi = Gi(U)=GF,Ti (U)and

Hi = Hi(U)=HF,Ei (U) of 7.1 and part (3) of Theorem 2 of [A1], with respect E F D D E to the normal subsystem i of . Similarly let i(U)= F,Ti (U)and i(U)= E F,Ei (U) be the fusion systems of 4.1 in [A1], deﬁned with respect to the normal subsystem Ei of F. Finally let D(U)=DF,T (U) be the fusion system of 4.1 in [A1] deﬁned with respect to the strongly closed subgroup T of S,andchoose G(U)=GF,T (U) ∈G(D(U)) as in 7.1 in [A1]. (3.4). ∈Ffc Let U T . Then for i =1, 2: (1) Hi acts on TU,3−i.

~~(2) Hi acts on V , VTU,i ,andTU,i. (3) Ei(U) ≤D1,2(U).~~

~~18 MICHAEL ASCHBACHER~~

Proof. Let H = Hi, X = TU,i,andY = TU,3−i. Then working in G = Gi, [H, Y ] ≤ CH (U)U = K.FurtherX ∈ Sylp(K)andX G as VX G ∗ and X = Ti ∩ VX . Therefore X = Op(K)=F (K). Next [X, Y ] ≤ X ∩ Y = ∩ ∈Ffc U(CT1 (U) T2)=UCT (U)=U as U T .ThusCY (U) centralizes X/U and U, ∗ so as F (K)=X,[K, CY (U)] ≤ CK (X/U) ∩ CK (U) ≤ X.ThusK acts on XY . Then as [H, Y ] ≤ K, H acts on Op(KY )=XY and hence also on XY ∩ T3−i = Y . This establishes (1). Next XCS(X)=VX G and V ≤ YVX ,soH acts on VX and V = CYVX (XY ), establishing (2). Finally (2) implies (3). Notation 3.5. ∈Ffc D D Let U T . By 3.2.4, = 1,2(U) is saturated and constrained, so by 2.5.1 in D [A1], there exists a model G = G1,2(U)for . Notice Xi = NTi (U) is strongly closed in NS (U) with respect to D. By 3.4.3, Ei(U) ≤D, and arguing as in the proof of 4.4 in [A1], Ei(U)isD-invariant. As Ei is saturated, so is Ei(U) by 4.4.2 in [A1]. ∈ For h NHi (Xi), h acts on Y = TU,3−i by 3.4.1, and h centralizes CS(Xi)/Z(Xi). ˆ Thus φ = c | extends φ = c | to YVX and centralizes CS(Xi)/Z(Xi). In h YVXi h Xi i ˆ particular φ ∈D, so condition N(D, Ei(U)) of [A1] is satisfied. Thus Ei(U) D, E F so by Theorem 1 in [A1], i(U)= Xi (Hi) for a unique normal subgroup Hi of G ∈ ∼ with Xi Sylp(Hi). By 2.5.2 in [A1], Hi = Hi via an isomorphism which is the identity on Xi,sowewriteHi for Hi. By 3.4.2, Hi acts on TU,j , VTU,i ,andV ,so Hi also acts on these groups. Set H = H(U)=H1 ∩ H2.ThusH is a normal subgroup of G,andas Ti ∈ Sylp(Hi), T = T1 ∩ T2 is Sylow in H. Next by 3.2.5, AutF (U)=AutD(U), so as AutD(U)=AutG(U)andH G, AutH (U) AutF (U). F fc Define A = A1,2 on T by A(U)=AutH(U)(U). Then extend A to a constricted F-invariant map A = A1,2 on T as in 5.2.3 in [A1]. Then (as in 5.1 in [A1]) define

~~E1 ∧E2 = E(A1,2).~~

Write E for E1 ∧E2. (3.6). E ∈Ff ∈ ∈Ffc ∈ (1) = A(Uϕ):U T ,ϕ AutF (T ) = A(Uϕ):U T ,ϕ AutF (T ) . (2) E≤E1 ∩E2. Proof. From 3.5, E = E(A)andA is a constricted F-invariant map. Therefore (1) follows from parts (5) and (6) of 5.5 in [A1]. ∈Ffc ∈GE Let U T . By construction in 3.5, Hi(U) ( i(U)), so AutHi(U)(U)= ≤ AutEi(U)(U) AutEi (U). Thus ≤ ≤ A(U)=AutH(U)(U) AutHi(U)(U) AutEi (U), as H(U)=H1(U) ∩ H2(U). Then for ϕ ∈ AutF (T ), ∗ ≤ ∗ A(Uϕ)=A(U)ϕ AutEi (U)ϕ = AutEi (Uϕ), as Ei is F-invariant. Therefore by (1), E≤ ∈Ffc ∈ ≤E AutEi (Uϕ):U T ,ϕ AutF (T ) i, establishing (2).

~~3. E1 ∧E2 19~~

~~(3.7). ∈Ffc Let U T .Then D ≤D (1) (U)=ND1,2(U)(VU ) 1,2(U).~~

~~(2) G(U)=NG1,2(U)(VU ). (3) H(U) G(U). (4) If L is a normal subsystem of F on T normal in E1 and E2 then L(U)= HF,L(U) G(U) and L(U) ≤ H(U).~~

Proof. By 3.2.6, V ≤ VU and for i =1, 2, TU,i = VU ∩ Ti,soTU,1,2 = (VU ∩ T1)(VU ∩ T2) G(U). Thus (1) holds. Let G = G1,2(U)andH = H(U). From (1), G(U)andNG(VU )) are in G(D(U)), so we may choose our member G(U) of G(D(U)) to be NG(VU ), and hence (2) holds. Recall from 3.5 that H G.Also[H, CS(U)] ≤ CH (U)=Z(U), so H acts on VU , and hence H ≤ NG(VU )=G(U). Thus (3) holds. Assume the setup of (4). From section 1, L = L(U)=HF,L(U) G(U). L E L By hypothesis, i.Thenas is a system on T , NL(U) NEi (U)(cf. 8.23.2) so L(U) ≤Ei(U) by 4.2.2 in [A1]. Thus again from section 1, there is p G p amodelLi for L(U)normalinHi.NowL0 = O (L)=T = O (Li)and L = L0X and Li = L0Xi,whereX, Xi are Hall p -subgroups of NL(T ),NLi (T ), respectively. Also Xi acts on NL(T ), so we may choose Xi to act on X.For ∈ ∈ −1 xi Xi there is x X with cx = cxi on T . Therefore y = x xi is a p -element centralizing T .SetQ = Op(G)andQ0 = Op(L0). Then [X, Q] ≤ L ∩ Q = Q0 and as Li is subnormal in G,also[X1,Q] ≤ Q0,so[y, Q] ≤ Q0. But now y is a p -element in CG(Q/Q0) ∩ CG(Q0), so xi = x ∈ L, and hence L = Li ≤ Hi.Thus L ≤ H1 ∩ H2 = H(U), proving (4).

(3.8). ∈Ffc ∈D f ≤ ≤ ∈ Let U T , V = VTU,1,2 , P (U) ,withU P T ,andβ ∈Ffc homF (NS (P ),S) with Pβ T .SetPi = PTU,i and Wi = VVTU,i for i =1, 2. Then ∩ ∩ ∩ ∩ (1) P = Pi T = PV T = PWi T , U = T Wi,andNHi(U)(Pi)=NHi(U)(P ) for i =1, 2. ˇ → (2) There is an isomorphism β : NG1,2(U)(P ) NG1,2(Pβ)(Vβ) extending β : NS(P ) ∩ NS (U) → NS (Pβ) ∩ NS(Uβ). ˇ (3) NHi(U)(Pi)β = NHi(Pβ)(TU,iβ). ˇ (4) NH(U)(P )β = NH(Pβ)(Uβ). ∗ (5) AutH(Pβ)(Uβ)=AutNH(U)(P )(U)β . Proof. ∩ ≤ ≤ First Pi = PCTi (U), so Pi T = PCT (U)=P as CT (U) U P . ∩ ∩ ∩ ∩ ≤ Similarly P = PV T = PWi T and U = Wi T .AsP = Pi T , NHi(U)(Pi) NHi(U)(P ), and the opposite inclusion follows as Pi = PTU,i with TU,i Hi(U). Thus (1) holds. Part (2) follows as in the proof of parts (8) and (9) of 7.2 in [A1], using 3.2. By (1), P = Pi ∩ T so NS (Pi) ≤ NS (P ). Thus by 7.2 and 7.18 in [A1], applied to Pi,TU,i, Ei in the role of P, U, E, it follows that ˇ → (*) β|NS (Pi) extends to βi : NGi(TU,i )(Pi) NGi(Piβ)(VTU,i β), and ˇ (**) NHi(U)(Pi)βi = NHi(Piβ)(TU,iβ).

~~Next the maps in N F (PiV )andN F (PTVT )arethoseonPWi N (VTU,i ) N (V ) U,i acting on U,sothetwosystemsareequal.Thus~~

~~(!) their models NGi(TU,i )(PiV )andNG1,2(U)(P )arealsoequal.~~

~~20 MICHAEL ASCHBACHER~~

~~Similarly~~

(!!) NGi(Piβ)(Wiβ)=NG1,2(Pβ)(Wiβ). As ∩ NGi(TU,i )(PiV )=NGi(TU,i )(Pi) NGi(TU,i )(V ) and ∩ NGi(Piβ)(Wiβ)=NGi(Piβ)(VTU,i β) NGi(Piβ)(Vβ), ˇ we conclude from (*) that NGi(TU,i )(PiV )βi = NGi(Piβ)(Wiβ). Then it follows ˇ from (!) and (!!) that NG1,2(U)(PVTU,i )βi = NG1,2(Pβ)(Wiβ). Then using 2.3 and ˇ ˇ ∈ ∩ ≤ 2.4.2 in [A1], β = czi βi on NG1,2(U)(PVTU,i ), for some zi Z(NS (P ) NS (U))

~~NG1,2(U)(NHi(U)(Pi)). Therefore (3) follows from (**) and the observation in 3.5 ≤ ≤ that Hi(U) NG1,2(U)(VTU,i ), so that NHi(U)(Pi) NG1,2(U)(PVTU,i ).~~

Claim K = NH(Pβ)(Uβ)=NH(Pβ)(TU,1,2β). For let Ki = NHi(Pβ)(Uβ). Then ∩ ≤ ≤ [CT3−i (U)β,Ti Ki] CT (Uβ) Uβ, ∗ so as Ti ∩ Ki ∈ Sylp(Ki)andF (Ki)=Op(Ki), Ki acts on TU,3−iβ.ThusK = K1 ∩ K2 acts on TU,1,2β = TU,1βTU,2β, establishing the claim. By (1), ∩ ∩ NH1(U)(P1) NH2(U)(P2)=NH1(U)(P ) NH2(U)(P )=NH(U)(P ), and similarly ∩ ∩ ∩ NH1(Pβ)(TU,1β) NH2(Pβ)(TU,2β)=H1(Pβ) H2(Pβ) NG1,2(Pβ)(TU,1βTU,2β)

~~= NH(Pβ)(TU,1,2β)=NH(Pβ)(Uβ), by the claim. Now (4) follows from (3) and these two observations. Finally (4) implies (5).~~

(3.9). Assume F = FS(G) for some ﬁnite group G with S ∈ Sylp(G),and E F ∈ for i =1, 2, i = Ti (Hi) for some normal subgroup Hi of G with Ti Sylp(Hi). Set H = H1 ∩ H2.ThenT ∈ Sylp(H) and E = E1 ∧E2 = FT (H).Inparticular E F. Proof. ∈Ffc Let U T , D = NS(U), Yi = TU,i, Y = Y1Y2, V = VY ,and D = D1,2(U). Since D = NF (V )andF = FS(G), D = FD(NG(V )). Similarly E E F i(U)= F,E (U)=N E (U)= (NH (Yi)). i N i (Yi) NTi (U) i

~~Further from 3.4.2, NH (Yi)=NH (V ), so Ei(U)=F (Hi(U)), where Hi(U)= i i NTi (U)~~

~~NHi (V ). From 3.5, ∩ ∩ H(U)=H1(U) H2(U)=NH1 (V ) NH2 (V )=NH (V ).~~

Thus A(U)=AutNH (V )(U). Next [NH (U),Y] ≤ UCH (U), so YCH (U) NH (U). Also Z(U)isSylow in CH (U), so Y is Sylow in YCH (U). Hence by a Frattini argument, NH (U)= CH (U)I,whereI = NH (Y ). Also [CS(Y ),I] ≤ CI (Y ), so I acts on CI (Y )CS(Y )Y = CI (Y )V .FurtherCT (Y ) ≤ CT (U)=Z(U), so V = CS(Y )Y is Sylow in VCI (Y ). Then by another Frattini argument, I = CI (Y )NI (V ). Then as U ≤ Y , ≤ AutH (U)=AutNH (U)(U)=AutI (U)=AutNI (V )(U) AutNH (V )(U)=A(U). Let E = FT (H). Then AutE (U)=AutH (U)=A(U), so (in the notation of section 5 in [A1]) AutE,c = A.ThusE = E(AutE,c)=E(A)=E, completing the proof.

~~3. E1 ∧E2 21~~

(3.10). Assume F = FS(G) for some finite group G with S ∈ Sylp(G) and ∗ F (G)=Op(G).Then E F (1) For i =1, 2, i = Ti (Hi) for a unique normal subgroup Hi of G with Ti ∈ Sylp(Hi). (2) Set H = H1 ∩ H2.ThenT ∈ Sylp(H) and E = E1 ∧E2 = FT (H). (3) E F. Proof. Part (1) is a consequence of Theorem 1 in [A1]. Then (2) and (3) follow from (1) and 3.9. We are now in a position to prove Theorem 1. We first prove E = E1 ∧E2 F. To do so, we verify the conditions of part (4) of Theorem 3 in [A1] for the groups H(U) we have constructed in 3.5, and then appeal to Theorem 3 in [A1] to conclude that E F.ObservethatE≤E1 ∩E2 by 3.6.2. ∈Ffc Let U T . By 3.7.3, H(U) G(U). By 3.6.1, condition (ii) of part (4) of Theorem 3 in [A1] is satisfied. Finally condition (i) of part (4) of Theorem 3 in [A1] is satisfied by 3.8.5. Thus E F. Finally suppose L F is a system on T normal in E1 and E2. By 3.7.4, L(U) ≤ H(U), so AutL(U)=AutL(U)(U) ≤ AutH(U)(U)=A1,2(U), and hence L = E(AutL) ≤ E(A1,2)=E.FurtherE Ei for i =1, 2 (cf. paragraph two of the proof of 7.2). Thus we have established Theorem 1. Define N (F,T) to be the set of all normal subsystems of F on T .IfN (F,T) is nonempty, define the normal subsystem of F generated by T to be: T F = E. E∈N(F,T ) (3.11). If N (F,T) is nonempty then T F is the smallest normal subsystem of F over T . Proof. This follows from Theorem 1.

~~CHAPTER 4~~

~~The product of strongly closed subgroups~~

In this section F is a saturated fusion system over the ﬁnite p-group S,andTi, i =1, 2, are nontrivial subgroups of S which are strongly closed in S with respect to F.SetT = T1T2. We prove Theorem 2 in this section. The proof involves a series of reductions. Assume the Theorem is false, and choose Y ≤ T and α ∈ homF (T,S) such that Yα T . Subject to this constraint choose Y with n = |T : Y | minimal. As F is saturated it follows from Alperin’s Fusion Theorem (cf. A.10 in [BLO]) that: fc (4.1). F = AutF (R):R ∈F . In particular by 4.1, we can choose Y and α so that Y ≤ R ∈Ffc and α ∈ AutF (R). Subject to this constraint, choose R maximal. Let Q = NS (R), Qi = ∩ ∈Ffc D NTi (R), and Ri = Ti R.AsR , NF (R)= (R) is saturated and constrained by 1.2.1 in [A1], so we may choose G ∈G(D(R)). Observe that as Ti is strongly closed in S with respect to F, Ri G.Furtherα = cg|Y for some g ∈ G.

(4.2). (1) If Q, g ≤H ≤ G then R = Op(H). (2) R = Op(G). Proof. Visibly (1) implies (2). Suppose H is a counterexample to (1), and fc set P = Op(H) and let β ∈ homF (NS(P ),S) with Pβ ∈F .AsR ≤ H ≤ G and R ≤ Op(G), R ≤ P .ThenasH is a counterexample to (1), R

~~(4.3). (1) For i =1, 2, Ti R. (2) For i =1, 2, Qi R. (3) R = Q. (4) Y NG(Q) ≤T .~~

Proof. Suppose T1 ≤ R.Nowfory ∈ Y , y = y1y2 with yi ∈ Ti.Then ∈ ≤ −1 ∈ g ∈ y1 T1 R,soy2 = y1 y R.AsRi G, yi Ri for i =1, 2, so g yα = y ∈ R1R2 ≤ T .ThusYα≤ T , contrary to the choice of α.

~~Therefore (1) holds. In particular R~~

~~24 MICHAEL ASCHBACHER~~

~~(4.4). (1) Y = T ∩ R. (2) Y Q. Proof. If Y~~

~~(4.5). There exist H ∈Hwith g ∈ H and R = Op(H). Let M be the unique maximal overgroup of Q in H and K = Op(H). Set K1 =[Q1,K].~~

~~(4.6). (1) K = K1. (2) K centralizes R/R1.~~

Proof. As Q1 Q, K1 =[K, Q1] is invariant under K and Q,soK1 KQ = H.ThusK1Q is an overgroup of Q in H, so either K1Q ≤ M or H = K1Q. In the latter case (1) holds. In the former as K1Q1 H, H = K1NH (P )by a Frattini argument, where P = Q ∩ K1Q1.ThusasK1 ≤ M, NH (P ) M,so H = NH (P ) by the uniqueness of M. Therefore P ≤ Op(H), so P ≤ R by 4.5. But now Q1 ≤ P ≤ R, contrary to 4.3.2. Thus (1) is established. Next [R, Q1] ≤ R ∩ T1 = R1.ThenasR and R1 are normal in H, (2) follows from (1). We are now in a position to obtain a contradiction, and hence establish Theorem 2. By 4.4.1, R1 ≤ Y , so by 4.6.2, K acts on Y .FurtherQ acts on Y by 4.4.2, so Y KQ = H, contrary to the choice of H. This completes the proof of Theorem 2.

~~CHAPTER 5~~

~~Pairs of commuting strongly closed subgroups~~

In this section F is a saturated fusion system over the ﬁnite p-group S,and Ti, i =1, 2, are subgroups of S strongly closed in S with respect to F, such that [T1,T2]=1.ByTheorem2,T = T1T2 is also strongly closed in S with respect to F.

~~(5.1). Let T1 ∩ T2 ≤ Ui ≤ Ti for i =1, 2.Then~~

~~(1) NT (U1U2)=NT1 (U1)NT2 (U2).~~

~~(2) CT (U1U2)=CT1 (U1)CT2 (U2). (3) Ti ∩ U1U2 = Ui for i =1, 2. (4) U U ∈Fc iﬀ U ∈Fc for i =1and 2. 1 2 T i Ti~~

Proof. Let U = U1U2.ThenU ∩ Ti = Ui(U3−i ∩ Ti) ≤ Ui(T1 ∩ T2)=Ui, establishing (3). By (3) and as [T1,T2]=1, ∩ ∩ NT (U1U2)=NT (U1) NT (U2)=T2NT1 (U1) T1NT2 (U2) ∩ ∩ = NT1 (U1)(T2 T1NT2 (U2)) = NT1 (U1)NT2 (U2)(T1 T2)=NT1 (U1)NT2 (U2), so (1) holds. A similar argument establishes (2). ∈Fc ∈ ≤ ≤ Next U T iff for each φ homF (U, S), CT (Uφ) Uφ iff CTi (Uiφ) Uiφ for i =1, 2by(2),iffU ∈Fc for i =1, 2. Thus (4) holds. i Ti Notation 5.2. Set P = F c ×Fc .Fori =1, 2, let Pi contains of those (U ,U ) ∈Psuch that T1 T2 1 2 fc fc U ∈F and V ∈ NF (U ) . i Ti U1U2 i (5.3). ∈P Let (U1,U2) and V = VU1U2 .Then ∩ (1) V = VU1 VU2 . ∩ ∩ (2) For i =1, 2, V Ti = Ui = VUi Ti. (3) NS(V )=NS (U1) ∩ NS(U2)=NS(U1U2). (4) T1 ∩ T2 ≤ U1 ∩ U2. ∈Fc (5) U1U2 T .

~~Proof. As U3−i ≤ CS(Ui), ∩ ∩ ∩ VU1 VU2 = U1CS(U1) U2CS(U2)=U1(CS(U1) U2CS(U2))~~

= U1U2(CS(U1) ∩ CS(U2)) = U2U2CS(U1U2)=V, so (1) holds. As U ∈Fc , T ∩ V = U , so (2) follows from (1). Then (1) and (2) i Ti i Ui i imply (3). As [T ,T ]=1,T ∩ T ≤ Z(T ), so (4) follows as U ∈Fc .Then(4) 1 2 1 2 i Ti and 5.1.4 imply (5).

~~26 MICHAEL ASCHBACHER~~

~~(5.4). ∈P Let (U1,U2) and V = VU1U2 .Then (1) The maps~~

NF (U2) F NF (U1) F (X1,U2) → (X1,U2) and (U1,X2) → (U1,X2) F ×{ } { }× F are bijections of the orbits of NF (U2) on U1 U2 and NF (U1) on U1 U2 F F × F with the orbits of on U1 U2 , respectively. ∈Ffc ∈Ff ∈Ff ∈ f (2) If for i =1or 2, Ui T ,thenU1U2 T iff V iff V NF (Ui) f iff U1U2 ∈ NF (Ui) . ∈Ffc ∩ ∈Ffc (3) If for i =1or 2, Ui T then Ui = Ti VUi and VUi . Proof. Part (1) is a standard result in group theory; the same proof works in the context of fusion systems. ∈Ffc ∈N f ∈ f ∈Ff Assume U1 T . By 5.3.3, V F (U1) iff U1U2 NF (U1) and V ∈Ff ∈Ff | |≥| | ∈VF iff U1U2 T .IfV then NS (V ) NS (W ) for all W ,soas NS(V ) ≤ NS (U1) by 5.3.3, certainly |NS(V ) ∩ NS(U1)|≥|NS(W ) ∩ NS(U1)| for all NF (U1) f W ∈ V .ThatisV ∈ NF (U1) . f f F Conversely assume V ∈ NF (U1) and let W ∈F ∩ V .Then|NS (V )|≤ | | ∈Ffc ∩ ∈Ff NS(W ) . Continue to assume U1 T . By 5.3.2, U1 = V T1,soasU1 T , there is α ∈ homF (NS(W ∩ T1),S)with(W ∩ T1)α = U1. By 5.3.3, NS(W ) ≤ NS(W ∩ T1)andNS (W ∩ T1)α ≤ NS (U1), so NS (W )α ≤ NS(Wα), and hence F F NF (U ) |NS(W )|≤|NS (Wα)|. Also Wα ∈ W = V ,soby(1),Wα ∈ V 1 .Thusas f f V ∈ NF (U1) , |NS(Wα)|≤|NS (V )|,so|NS (W )| = |NS(V )|, and hence V ∈F . Hence (2) holds. ∩ By 5.3.2, Ui = Ti VUi ,soNS (Ui)=NS (VUi ). Then (3) follows.

(5.5). ∈P1 Let (U1,U2) ,andsetV = VU1U2 .Then F fc (1) V and VU1 are in ,sowecanformG(V )=GF,S(V ) and G(VU1 )= ∩ GF,S(VU1 ), ND(V )(VU1 ) is a saturated constrained fusion system on X = NS (V ) N (V ), ND (V )=ND (V ),andN = N (V ) and M = N (V ) S U1 (V ) U1 (VU1 ) 1 G(V ) U1 1 G(VU1 ) G ˇ → are in (ND(V )(VU1 )),sothesetExt(1) of isomorphisms 1:N1 M1 which extend the identity map on X with 1ˇcV = cV is nonempty. 2 (2) Let α ∈ homF (NS(V ),S) such that (U1,U2)α =(X1,X2) ∈P.Then ∈Ffc ∈Ffc D →D U1U2α = X1X2 T , Vα = VX1X2 ,andα˙ : (V ) (Vα) is an isomorphism. Hence there is an isomorphism αˇ : G(V ) → G(Vα) extending α : ∗ NS(V ) → NS (Vα) such that αcˇ Vα = cV α . → ∈ (3) α˙ : ND(V )(VU2 ) ND(Vα)(VX2 ) is an isomorphism, N2 = NG(V )(VU2 ) G ∈G (ND(V )(VU2 ), N2αˇ = M2 = NG(Vα)(VX2 ) (ND(Vα)(VX2 ),andαˇ|N2 is in Ext(α). (4) If Ei F on Ti for i =1or 2, then Ei(Ui) D(V ),andthere E F exists a unique normal subgroup Hi(V ) of G(V ) with i(V )= NT (V )(Hi(V )) and ∈ i NTi (V ) Sylp(Hi(V )). Proof. ∈Ffc ∈Ff ∈Ffc ≤ By 5.4.3, VU1 , while by 5.4.2, V ,soV as CS(V ) ≤ V .ThenasV VU1 , we can apply parts (8) and (9) of 7.2 in [A1]toV,VU1 ,S,1 in the roles of U, P, T, β to conclude that (1) holds. ∈Ffc ∈ Next α exists as in (2) by 5.4.1. By 5.3.5 and 5.4.2, X1X2 T and VX1X2 F fc. Now we apply parts (6) and (7) of 7.2 in [A1]toV,S intheroleofU, T to ∗ conclude that (2) holds. Then ND(V )(VU2 )α = ND(Vα)(VU2 α)=ND(Vα)(VX2 ),

~~5. PAIRS OF COMMUTING STRONGLY CLOSED SUBGROUPS 27~~

∈G and N2αˇ = M2, with M2 (ND(Vα)(VX2 ) via the argument used to establish (1). −1 ∈G Thus applying α , N2 (ND(V )(VU2 )and(3)holds. Suppose that E1 F on T1. Then by Theorem 2 in [A1], there is a nor- E F mal subgroup H(U1)=HF,E1 (U)ofG(VU1 ) such that 1(U1)= Q1 (H(U1)) and ∈ ≤ NT1 (U1)=Q1 Sylp(H(U1)). Now [H(U1),VU1 ] U1,soH(U1)actsonV as ≤ ≤ E ≤D ˇ−1 ≤ U1 V VU1 .Thus 1(U1) (V ) and setting H1(V )=H(U1)1 , H1(V ) N1. By 5.3.2, U1 = T1 ∩ V G(V ). Then arguing as in the proof of 4.3 in [A1], E1(U1)isD(V )-invariant. From Theorem 2 in [A1], E1(U1) is saturated, and as ≤ D E E D [H(U1),VU1 ] U1,( (V ), 1(U1)) satisﬁes (N1). Therefore 1(U1) (V ), so H1(V ) G(V )byTheorem1in[A1]. Suppose on the otherhand that E2 F on T2. Then by Theorem 2 in [A1], there is H(X2) G(VX2 ) with properties as above. This time H(X2)actsonVα, −1 so E1(X2) ≤D(Vα), and setting H2(V )=H(X2)ˇα , H2(V ) ≤ N2 and E2(U2)= −∗ −∗ E2(X2)α ≤D(Vα)α = D(V ). Then arguing as above, E2(V ) D(V )and H2(V ) G(V ).

~~In the remainder of the section we assume: Hypothesis 5.6.~~

~~For i =1, 2, Ei F on Ti,and[T1,T2]=1.SetT = T1T2. Notation 5.7.~~

~~Suppose (Y1,Y2) ∈P. Then there exists α ∈ homF (NS (Y1),S)with(U1,U2)= ∈P1 ∈Ffc (Y1,Y2)α .SetU = U1U2 and V = VU1U2 . By 5.4.2 and 5.3.5, U T . Moreover~~

D(U)=DF,T (V )=NF (V )=DF,S(U)=D(V ), and hence also G(U)=GF,T (U)=GF,S(V )=G(V ). By 5.5.4, we can form Hi(U)=Hi(V ) G(V ). Set H(U)=H1(U)H2(U). Then H(U) G(U) ∈ and by 5.3.4 and 5.1.1, NT (U)=NT1 (U1)NT2 (U2) Sylp(H(U)). If in addition ∈Ffc ∈ → Y1Y2 T , then by 2.3 and 2.4 in [A1], there isα ˇ Ext(α), so thatα ˇ : G(Y1Y2) G(U) is an isomorphism extending α : NS (Y1Y2) → NS(U). In this case we set −1 −1 Hi(Y1Y2)=Hi(U)ˇα ,andH(Y1Y2)=H(U)ˇα .ThusHi(Y1Y2)andH(Y1Y2) are normal in G(Y1Y2). (5.8). ∈P ∈Ffc Let (U1,U2) with U1U2 T .Then ∈Efc ∈ (1) For i =1, 2, Ui i and NTi (Ui) Sylp(Hi(U1U2)).

(2) Hi(U1U2) acts on Q3−i = NT3−i (U3−i), and indeed on each overgroup of T1 ∩ T2 in Q3−i,fori =1, 2. (3) If T1 ∩ T2 ≤ Z(E1) ∩ Z(E2) then [H1(U1U2),H2(U1U2)] = 1 and T1 ∩ T2 ≤ Z(H(U1U2)). Proof. Let U = U1U2,andfori =1, 2, let Hi = Hi(U)andQi = NTi (U). i Fix i ∈{1, 2}. By 5.7 there is α ∈ homF (NS(Ui),S)with(U1,U2)α ∈P.Let

~~Wi = Uiα and W = W1W2. By 5.3.4 and 5.1, for j =1, 2, Qj = NTj (Uj),~~

NTj (W )=NTj (Wj), NT (U)=Q1Q2,andNT (W )=NT1 (W1)NT2 (W2). Then as ∈Ffc ∈Efc ∈ U T , NT (U)α = NT (W ), so Qj α = NTj (Wj). Hence as Wi i , also Ui E fc −1 −1 i , while by 5.7, NTi (Wi)isSylowinHi(W ), so Qi = NTi (Wi)α = NTi (Wi)ˇα −1 is Sylow in Hi(W )ˇα = Hi(U). Thus (1) holds.

~~28 MICHAEL ASCHBACHER~~

By (1), Qi ∈ Sylp(Hi), and by 5.7, Hi G(U). Now Q1 centralizes Q2, ≤ ≤ ∩ so [Q1,H2] CH2 (Q2) U2, and hence H2 acts on T1 Q1U2 = Q1. Indeed [Q1,H2] ≤ Q1 ∩ U2 = T1 ∩ T2.Thatis(2)holds. Assume T0 = T1 ∩ T2 ≤ Z(E1) ∩ Z(E2). Then (cf. 7.11.2) T0 is in the center ≤ ≤ ∩ ≤ ≤ p of Hi,soT0 Z(H). Further [Q1,H2] Q1 U2 T0 CU2 (H2), so O (H2) p centralizes Q1.ThenasQ2 centralizes Q1, H2 = O (H2)Q2 centralizes Q1.Thus ≤ ≤ ≤ p [H1,H2] CH1 (Q1) U1 CH1 (H2), so O (H2) centralizes H1.ThenH2 = p Q2O (H2) centralizes H1, completing the proof of (3).

Deﬁnition 5.9. ∈Ffc For each U T , we deﬁne normal subgroups Hi(U), i =1, 2andH(U)of ∈ ∈ G(U)=GF,T (U), such that NTi (U) Sylp(Hi(U)) and NT (U) Sylp(H(U)). First let U = T ∩ U and suppose U ∈Fc for i =1, 2. Using 5.4, we may i i i Ti ∈ ∈Ffc ≤ choose α homF (NS(U1U2),S) such that (U1U2)α T .AsNS (U) NS(U1U2), ∈Ffc ∈Ffc it follows that Uα T , so replacing U by Uα, we may assume U1U2 T .

In this case define Hi(U)=NHi(U1U2)(U)andH(U)=NH(U1U2)(U). Notice ≤ NTi (U) NTi (U1U2), which is Sylow in Hi(U1U2), so NTi (U)isSylowinHi(U). Similarly NT (U) ∈ Sylp(H(U)). ≤ ≤ ∩ ≤ Next observe CS(U) V = VU1U2 and [H(U),V] T V = U1U2 U by 5.1.2, so H(U)actsonVU = UCS(U). Also [U, V ] ≤ T ∩ V ≤ U,soV acts on U. ≤ ≤ Further [U, CS(U1U2)] CU (U1U2)=Z(U1U2) by 5.1.2. Thus AutCS (U1U2)(U) O2(AutF (U)), so AutV (U) ≤ O2(AutF (U)). Thus for φ ∈ AutF (U), V ≤ Nφ,so as F is saturated, φ ∈ NF (UV). This is AutF (U) ≤ NF (UV) ≤ NF (V ). Therefore V G(U). Thus we can take G(U)=NG(U1U2)(VU ), and we have H(U)and Hi(U)normalinG(U)inthiscase. In general our definition is recursive, with the recursion based on the parameter | | ˆ ∩ ˆ ˆ ˆ n = CT (U1U2):Z(U1U2) .LetUi = CTi (Ui) NS (U)andU = UU1U2.Then ˆ ≤ ˆ ≤ ˆ ∩ [Ui,U] Z(Ui), so Ui Op(G(U)) by an argument above. Then Ui = CTi (Ui) ˆ ≤ Op(G(U)) G(U). Also CS(U) CS(U), so VUˆ G(U). Arguing as above, ˆ ∈Ffc we may assume U T . Thus by 7.2.8 in [A1], we can take G(U)=NG(Uˆ)(VU ). Proceeding recursively, we define H (U)=N (U)andH(U)=N (U), i Hi(Uˆ) H(Uˆ) and observe that Hi(U)andH(U) are indeed normal in G(U), with Sylow groups

NTi (U)andNT (U). (5.10). ∈Ffc ∈D f ≤ ≤ ∈ Let U T and P (U) with U P T .Letβ homF (NS(P ), ∈Ffc S) with Pβ T .Then ˇ (1) NH(U)(P )β = NH(Pβ)(Uβ). ∗ (2) AutH(Pβ)(Uβ)=AutNH(U)(P )(U)β .

Proof. As usual let Ui = Ti ∩ U and Pi = Ti ∩ P . ∈Fc ∈P ≤ Suppose ﬁrst that U1U2 T ,sothat(U1,U2) by 5.1.4. As U1U2 P1P2, c fc also P P ∈F .Letα ∈ homF (N (U ),S) with X = U α ∈F . By 5.4, 1 2 T 1 S 1 1 1 1 T1 ∈Ffc ∈ we may choose α1 so that X =(U1U2)α1 T . Similarly we may choose β1 fc fc homF (N (P ),S)sothatP β ∈F and (P P )β ∈F . S 1 1 1 T1 1 2 1 T −1 Next by construction of H1(U) in 5.9, H1(U)=H1(U1α1)ˇα1 , G(U)= −1 G(Uα1)ˇα1 ,andH1(X)=H1(X1) is the normal subgroup of G(X1) with F (H (X )) = E (X ). The analogous statement also holds for P β . NT1 (X1) 1 1 1 1 1 1

~~5. PAIRS OF COMMUTING STRONGLY CLOSED SUBGROUPS 29~~

Thus by 7.18 in [A1] applied to E1, −1 ˇ −1 NH1(X)(P1α1)ˇα1 β1 = NH1(X1)(P1α1)ˇα1 β1 −1 = NH1(P1β1)(X1α1 β1)=NH1(P1β1)(U1β1). Then ˇ −1 ˇ NH1(U1U2)(P1)β1 =NH1(X)(P1α1)ˇα1 β1 =NH1(P1β1)(U1β1)=NH1((P1P2)β1)(U1β1). −1 ˇ Then applying β1 β to this equality, NH1(U1U2)(P1)β = NH1((P1P2)β)(U1β). By ˇ symmetry, NH2(U1U2)(P2)β = NH2((P1P2)β)(U2β), so ˇ ˇ ˇ NH(U1U2)(P1P2)β = NH1(U1U2)(P1)βNH2(U1U2)(P2)β

= NH1((P1P2)β)(U1β)NH2((P1P2)β)(U2β)=NH((P1P2)β)((U1U2)β), as Hi(U1U2)actsonP3−i and Hi((P1P2)β)actsonU3−iβ by 5.8.2. Therefore ˇ ∩ ∩ ˇ NH(U)(P )β =(NH(U1U2)(P1P2) N(U) N(P ))β ∩ ∩ = NH((P1P2)β)((U1U2)β) N(Uβ) N(Pβ)=NH(Pβ)(Uβ), so (1) holds in this case. Then (1) implies that (2) also holds in this case. Now assume we have a counterexample with t = |T : U| minimal, subject to this constraint with n = |CT (U1U2):Z(U1U2)| minimal, and subject to these constraint, with m = |P Uˆ : P | minimal, where Uˆi and Uˆ are deﬁned in 5.9. By the discussion above, n>1

Thus to establish the lemma in this case, it suﬃces to show that NH(Pβ)(Uβ)acts on Uβˆ ,asthenitalsoactsonPβˆ . ˆ K ∗ Let K = NHi(Pβ)(Uβ)) and J = (Uiβ) .AsF (Hi(Pβ)) = Op(Hi(Pβ)), ∗ ˆ ≤ ∩ p also F (J)=Op(J). But [Uiβ,Uβ] CTi (Uiβ) Uβ = Z(Uiβ), so O (J) cen- ∈Fc tralizes Uβ. However as Uβ T , Uβ contains a Sylow group of CK (Uβ), so as ∗ p ˆ ˆ F (J)=Op(J). O (J) = 1. That is Uiβ K, so indeed Uβ NH(Pβ)(Uβ).

~~30 MICHAEL ASCHBACHER~~

~~ˆ Thus we may assume P is not normal in P ,soP~~

~~E = E1E2 = E(A).~~

Theorem 5.12. Assume Hypothesis 5.6. Then E1E2 is a normal subsystem of F on T = T1T2. Proof. ∈Ffc Let U T . By construction of the group H(U) in 5.9, H(U) G(U)andNT (U) ∈ Sylp(H(U)). Further using 5.5.6 in [A1], condition (ii) of part (4) of Theorem 3 in [A1] is satisfied by A and E as E = E(A). Also condition (i) of part (4) of Theorem 3 in [A1] is satisfied by 5.10.2. Thus the Theorem follows from Theorem 3 in [A1]. j (5.13). Assume T1 ∩T2 ≤ Z(Ei) for i =1, 2.LetQ be the set of pairs (U1,U2) c fc such that U ∈F , U U ∈F ,andU − = T − .Then j Tj 1 2 T 3 j 3 j ∗ j (1) E = A(U)ϕ : U = U1U2 with (U1,U2) ∈Q ,ϕ∈ AutF (T ),j=1, 2 . ∈Efc ∈ ˆ ∈ (2) For each P j and φ AutEj (P ), φ extends to φ AutE (PT3−j) with ˆ φ =1on T3−j . E ˆ ∈ ∈Efc (3) = φ : φ AutEj (P ):P j ,j =1, 2 . (4) E is a central product of E1 and E2. Proof. ∈Ffc Let U T . We first claim that A(U)=AutA(U˜)(U)forsome ˜ ˜ ˜ ˜ ∈Ec U = U1U2 containing U, with Ui i . Recall the definition of H(U) in 5.9. If ∈Ffc ˜ U1U2 T then by definition, H(U)=NH(U1U2)(U), so we may take Ui to be the × ˜ projection on Ti of the preimage of U in T1 T2,sinceNH(U1U2)(U)actsonUi. In general, the definition of H(U) in 5.9 was recursive on the parameter n defined there. Then proceeding by induction on n, we see that the claim holds. By the claim, E ∈Ec = A(U):U = U1U2,Ui i .

~~5. PAIRS OF COMMUTING STRONGLY CLOSED SUBGROUPS 31~~

Then by 3.6 and 5.5.3 in [A1], E ∗ ∈Ffc ∈ = A(U)ϕ : U = U1U2 T ,ϕ AutF (T ) . ∈Ffc Further for U = U1U2 T , A(U)=AutH (U)=AutH1(U)(U)AutH2(U)(U), and by 5.8.2, Aut (U)=AutAut (U), where W = Uj T3−j . Therefore (1) holds. Hj (U) Hj (W ) ∈Qj ∈ Let (U1,U2) , U = U1U2,andφ AutEj (Uj ). Then φ = ch|Uj for ˆ ˆ some h ∈ Hj (U), and from 5.8.3, φ = ch|U extends φ to AutE (U) with φ =1 fc fc on T − . That is (2) holds when P ∈F . In general, given P ∈E ,thereis 3 j Tj j fc α ∈ homF (N (P ),S) with Pα = U ∈F .SetU = U T − and H = H(U). As S j Tj j 3 j E is F-invariant, −∗ −∗ AutH (U)α = AutE (U)α ≤ AutE (PT3−j ), ∈ E ∗ −∗ and each φ AutEj (P ) extends to the -map φα α which is 1 on T3−j . Therefore ∈ ˆ E E ≤E E (2) holds, and for φ AutEj (P ), φ is an -map, so ,where is the second subsystem in (3). Next AutH (U)= AutH1(U)(U),AutH2(U)(U) , with { ˆ ∈ }⊆E AutHi(U)(U)= φ : φ AutEi (Ui) , and similarly for ϕ ∈ AutF (T ), ∗ { ˆ ∈ }⊆E AutHi(U)(U)ϕ = φ : φ AutEi (Uiϕ) . Therefore E⊆E, completing the proof of (3). Part (4) follows from (2), (3), and 2.9.6. We now establish Theorem 3. In particular we may assume Hypothesis 5.6 holds. By 5.12, E F.FurtherifT1 ∩ T2 ≤ Z(Ei)fori =1, 2, then by 5.13.4, E is a central product of E1 and E2.ThustheTheoremisproved.

~~CHAPTER 6~~

~~Centralizers~~

In this section F is a saturated fusion system over the ﬁnite p-group S,andE is a normal subsystem of F over T . Notation 6.1. ∈Ff D D E E For U T , deﬁne (U)= F,T (U)and (U)= F,E (U) as in 4.1 of [A1], and form the group G(U)=GF,T (U) and its normal subgroup H(U)=HF,E (U)of D F ∈ E 7.19.3, such that (U)= NS (U)(G(U)) with NS(U) Sylp(G(U)), and (U)= FN (U)(H(U)), with NT (U) ∈ Sylp(H(U)). Recall VT = TCS(T ). Set T I = CS(H(U)) and CS(E)= Iϕ. fc ∈F ϕ∈AutF (VT ) U T

~~Denote by X the set of subgroups X of CS(T ) such that E⊆CF (X).~~

~~(6.2). (1) CS(E) ≤ I ≤ CS(T ) ≤ VT . (2) AutF (VT ) acts on CS(E). (3) X = X ,where X { ≤ ⊆ ∈Ffc ∈ } = X CS(T ):AutE (Uϕ) CF (X) for all U T and all ϕ AutF (VT ) .~~

~~(4) Each subgroup of CS(E) is in X .~~

Proof. Set X = CS(E). By construction, X ≤ I ≤ CS(H(T )) ≤ CS(T ), so (1) holds. By (1), Iϕ ≤ VT for each ϕ ∈ AutF (VT ), and of course AutF (VT ) permutes {Iϕ : ϕ ∈ AutF (VT )} via right multiplication. Thus (2) holds. By Theorem 3 and 5.5.6 in [A1], E = E(AutE,c) ⊆ CF (X)foreachX ∈X ,so (3) holds. ∈Ffc ≤ ⊆ By construction, for each U T and Y X, AutH(U)(U) CF (Y ), and AutH(U)(U)=AutE (U). Then by (2), for each ϕ ∈ AutF (VT ), AutE (Uϕ) ⊆ CF (Y ). Thus (3) implies (4). (6.3). ∈X ∈Ffc ∈N f Let X and U T . Assume XU F (U) and set V = VXU, D D = NS (XU), Q = NT (U),and (U, X)=NNF (V )(XU).Then (1) U = T ∩ V . N (XU) (2) D(U, X)=N Aut(V ) (V ) is a saturated and constrained fusion system NF (U) on D.LetG = G(U, X)=NG(U)(V ) ∈G(D(U, X)). (3) E(U) D(U, X) and there exists H = H(U, X) G with Q ∈ Sylp(H) and E(U)=FQ(H). (4) FD(HD)=FD(H(U)D), and there exists an isomorphism ιX,U : HD → H(U)D which is the identity on D. (5) CD(H(U, X)) = CD(H(U)). ∈ F fc ≤ (6) There exists α homF (NS (Q),S) with Qα and Uα in T , NS(Q)ˇα f NS(Qα),and(XU)α ∈ NF (Uα) .

~~34 MICHAEL ASCHBACHER~~

(7) There is an isomorphism αˇ : G(U) → G(Uα) extending α : NS (U) → ∗ NS(Uα) such that αcˇ Uα = cU α ,wherecW : G(W ) → Aut(W ) is the conjugation map for W ∈{U, Uα}.FurtherH(U)ˇα = H(Uα). (8) NG(Qα)(Uα)=NG(U)(Q)ˇα and NH(Qα)(Uα)=NH(U)(Q)ˇα.

~~Proof. By deﬁnition of X , X ≤ CS(T ) ≤ CS(U), while CT (U) ≤ U as ∈Ffc U T .Thus~~

T ∩ V = T ∩ UXCS(UX)=U(T ∩ XCS(UX)) ≤ U(T ∩ CS(U)) = UCT (U)=U, proving (1). Argue as in the proof of 4.2 in [A1] to establish (2). Next [H(U),VU ] ≤ H(U) ∩ VU = U by (1), so as V ≤ VU , H(U)actsonXU and VU .ThusE(U)=FQ(H(U)) ≤D(U, X)=D. Arguing as in in 4.3.1 in [A1], E(U)isD-invariant, and as E is saturated, so is E(U) by 4.3.2 in [A1]. Also [NH(U)(Q),CS(Q)] ≤ CH(U)(Q)=Z(Q), so condition N(D, E(U)) of section 6 in [A1] is satisﬁed. Therefore E(U) D,sobyTheorem1in[A1], there is a model H for E(U)normalinG.Thus(3)holds. f Let W = UCD(U)andB = ND(U)(XU). As XU ∈ NF (U) , B is a saturated constrained fusion system on D by 1.2.1 in [A1], and as W ≤ VU , G1 = NG(U)(W ) is a model for B.AsV ≤ W , also B = ND(U,X)(W ), so G2 = NG(W ) is also amodelforB. Hence by 2.5 in [A1], the identity map 1B on B gives rise to an isomorphism β :(G1,D,W) → (G2,D,W)inExt(1B); in particular by 2.5.4 in [A1], β : G1 → G2 is an isomorphism which is the identity on D.Furtheras 1B : E(U) →E(U), β maps the unique model H(U)forE(U)normalinG1 to the unique module H normal in G2. Therefore (4) holds with ιX,U = β|H(U)D.Then (4) implies (5). Part (6) follows from parts (5) and (6) of 7.2 in [A1]. The ﬁrst statement in part (7) follows from 7.2.7 in [A1], and the second statement and part (8) follow from 7.18 in [A1].

(6.4). Let X ≤ CS(T ) and φ ∈ homF (X, S).Then (1) If φ extends to ϕ ∈ hom(XT,S) then X ∈X iﬀ Xφ ∈X. (2) If Xφ is fully centralized in F then X ∈X iﬀ Xφ ∈X. (3) If X ∈X then XF ∩Ff ∩X = ∅. F (4) If X ∈X and X ∈ X ,thenNT (X )=CT (X ). Proof. Assume the hypothesis of (1). As T is strongly closed in S, ϕ acts on T .LetP ≤ T and α ∈ homE (P, T). If X ∈X then α extends toα ˆ ∈ homF (PX,S) withα ˆ =1onX.Thenˆαϕ∗ is an extension of αϕ∗ withαϕ ˆ ∗ =1onXφ,so Xφ ∈X. Similarly if Xφ ∈X then X ∈X, so (1) holds. Then (1) implies (2). f Assume X ∈X. Then there is α ∈ homF (X, S) with Xα ∈F and hence Xα is fully centralized. Thus Xα ∈X by (2), so (3) holds. Let X ∈ XF .By(3)there F f is Y ∈ X ∩F ∩X. By 1.1 in [A1]thereisβ ∈ homF (NS (X ),S) with X β = Y . Then NT (X )β ≤ NT (Y )=CT (Y ), so (4) holds. (6.5). ∈X ∈Ffc Assume X , U T ,and F (*) for each Y ∈ X with |CT (Y )| > |U|, Y ∈X. Then X ≤ CS(H(U)). Proof. Assume otherwise and choose a counterexample with |T : U| minimal. Set Q = NT (U)andV = VXU.AsX ∈X, X ≤ CS(Q). Let β ∈

~~6. CENTRALIZERS 35~~

≤ homNF (U)(NS(UX),S) with UXβ fully normalized in NF (U). Then Q = Qβ NS(UX)β ≤ NS(UXβ), and as X ≤ CS(Q), also Xβ ≤ CS(Q). If U = T then Xβ ∈Xby 6.4.1, while if U = T then U1. f Let W = YQ and pick γ ∈ homF (W, S) with Wγ ∈F . By 1.1.2 in [A1], there is χ ∈ AutF (Wγ) such that γχ extends to ψ ∈ homF (NS(W ),NS(Wγ)). As T centralizes Y , NT (W )=NT (Q)so

NT (Q)ψ = NT (W )ψ ≤ NT (Wγ), andinparticularNT (Q)ψ ≤ T centralizes Yψ = Yγχ.As|NT (Q)| > |Q| = |P | (since P = T ), it follows from minimality of n that (2) Yγχ= Yψ∈X. As Wγ ∈Ff , it follows from 1.1.2 in [A1] that there exists θ ∈ Aut(Wγ) such that αγθ extends to μ ∈ homF (NS(XP),S). Then as X does not centralize NT (XP), Xμ = Xαγθ = Yγθ does not centralize NT (XP)μ ≤ NT (Wγ), which we record as: (3) Yγθ does not centralize NT (Wγ). f Let ρ ∈ homF (NS(TQγ),S) such that TQγρ ∈F .ThenNS (Wγ) ≤ NS(Qγ) ≤ NS(TQγ), so NS (Wγ)ρ ≤ NS (Wγρ) and hence as Wγ is fully normalized, so is fc Wγρ. Thus, replacing γ by γρ, we may assume TQγ ∈F .

~~36 MICHAEL ASCHBACHER~~

By (2), Yψ centralizes TQγ. Form the groups G = G(TQγ,Yψ)andH = H(TQγ,Yψ)of6.3.LetK = CG(H)andSK = S ∩ K,sothatSK ∈ Sylp(K). By (2) and minimality of n, hypothesis (*) of 6.5 is satisﬁed for the pair Yψ,TQγ f in the role of “X, U”. Thus Yψ ≤ SK by 6.5. As Wγ ∈F , θ and χ extend to θˆ andχ ˆ on VWγ. These maps act on Qγ and hence on TQγ, and then also on ˆ ∈ V = VWγTQγ = VYψTQγ .Thusθ|V = cg|V andχ ˆ|V = cf|V for some f,g G.Now ≤ −1 f −1 ≤ Yγχ = Yψ SK ,soasK G, Yγ = Yγχcf =(Yψ) SK ,andthen g Yγθ=(Yγ) ≤ SK , contrary to (3). This completes the proof.

~~(6.7). (1) X is the set of subgroups of CS(E). (2) CS(E) is strongly closed in S with respect to F.~~

Proof. We first prove (1). Let X ∈X. By 6.2.4, each subgroup of CS(E)isin X , so we must show X is contained in CS(E). As AutF (VT )actsonX , it suffices ≤ ∈Ffc to show each X I.LetU T ;itremainstoshowX centralizes H(U). But by 6.6, hypothesis (*) of 6.5 is satisfied, so X centralizes H(U) by 6.5, completing the proofof(1).

We next prove (2). By (1), we must show that if X ∈X then XF ⊆X.But this follows from 6.6. Notation 6.8. Write T for CS(E). By 6.7, T is strongly closed in S with respect to F,sowecan adopt the notation and appeal to the results involving strongly closed subgroups P F c ×Fc Pi appearing in earlier sections. Define = T T and define , i =1, 2, as in 5.2. X f X fc F f F fc ∈Xfc Write , for T , T ,etc.LetX , and pick representatives { ∈ }⊆N fc N c Ui : i I F (X)T for the orbits of NF (X)on F (X)T .By5.4,foreach ∈ ∈Ffc ∈XF ∈ f i I there exists U T and Xi such that UXi NF (U) and there exists αi ∈ isoF (UXi,XUi). Form D(U, Xi), G(U, Xi), and H(U, Xi) as in 6.3. ∈Xfc ∈Fc D D As X and Ui T , (U, Xi)=NF (VUXi ). Similarly form (X, Ui)= D →D NF (VXUi ). Then as in 5.5.3,α ˙ i : (U, Xi) (X, Ui) is an isomorphism, which induces an isomorphismα ˇi : G(U, Xi) → G(X, Ui)=G(D(X, Ui)). Set H(X, Ui)= ∈ F E H(U, Xi)ˇαi.ThenNT (Ui) Sylp(H(X, Ui)) and NT (Ui)(H(X, Ui)) = (Ui). Define

~~p C(X, Ui)=CG(X,Ui)(H(X, Ui)) and K(X, Ui)=O (C(X, Ui))NT (X). Next for X ∈Xfc, deﬁne~~

~~p A(X)=O (AutCF (T )(X))AutT (X).~~

As T is strongly closed in CS(X), AutF (X)=AutAutF (XT)(X)andAutCF (T )(X) AutAutF (XT)(X). Then as T is strongly closed in S, A(X) AutF (X). Hence we can extend A to a constricted F-invariant map on T as in 5.2.3 of [A1]. Deﬁne

C = CF (E)=E(A). (6.9). Let R = S ∩ K(X, T).Then E ∈ ∈ g ∈ ∈ (1) = AutE (Uiα):i I ,α AutF (XT) = AutE (Ui ):i I ,g NG(X,T )(R) . (2) If R = S ∩ K(X, Ui) for each i ∈ I ,thenR ≤ T .

~~6. CENTRALIZERS 37~~

Proof. As X ∈X, E≤CF (X). By 3.6.1 in [A1], E is NF (X)-invariant, and as E F, E is saturated. Finally [H(X, T),CS(T ) ∩ NS(X)] ≤ CH(X,T )(T )= Z(T ), so condition N(NF (X), E) of section 6 of [A1] is satisﬁed. Therefore E E c c NF (X). Then by 5.4 in [A1], = NF (X)T . Hence applying 5.5.6 in [A1], and recalling that {Ui : i ∈ I } is a set of representatives for the NF (X) orbits on c ∈ NF (X)T , the ﬁrst equality in (1) holds. Next for α NF (XT), α extends to VXT ,soα = g|XT for some g ∈ G = G(X, T). By a Frattini argument, G = NG(R)K(X, T), so as K(X, T) centralizes T , the second equality in (1) holds. 2 Assume the hypothesis of (2). From 6.8, there is (Xi,U) ∈P and αi ∈ isomF (UXi,XUi) inducing an isomorphismα ˇi : G(U, Xi) → G(X, Ui) with

H(X, Ui)=H(U, Xi)ˇαi. By 6.3.5, CNS (UXi)(H(U)) = CNS (UXi)(H(U, Xi)), so ∩ −1 as AutE (U)=AutH(U)(U), AutE (U) centralizes R =(S K(X, Ui))ˇαi ,and ∗ ∩ hence AutE (Ui)=AutE (U)αi centralizes R αˇi = S K(X, Ui)=R.Thenfor ∈ g ∗ g E g NG(R), AutE (Ui )=AutE (Ui)cg centralizes Rcg = R = R,so centralizes R by (1). That is R ≤ CS(E)=T . Recall the notion of a strongly normalized chain from section 1 and the chain C(U)=CE (U)forU ≤ T .

~~(6.10). Adopt Notation of 6.8 and let i ∈ I such that C(Ui)=(Ui = Ui,0 < ···~~

~~NG(X,Ui,j+1)(Ui,j ) which is the identity on NS (Ui,j ), such that NH(X,Ui,j )(Ui,j+1)θj~~

~~= NH(X,Ui,j+1)(Ui,j ). p p (2) O (C(X, Ui,j))θj = O (C(X, Ui,j+1)),soK(X, Ui,j)θj = K(X, Ui,j+1). (3) ωj = θj ···θn maps K(X, Ui,j) to K(X, T).~~

Proof. Part (1) follows from parts (6)-(8) of 6.3. Let U(j)=Ui,j and Yj = Y (X, U(j)) for Y ∈{G, H, C, K}.SetSj = S ∩ Gj,andQ = NT (X). ≤ ≤ ≤ Recall from 6.8 that for each j, Kj Q Cj ,soQ CSj (Hj). Then as Uj+1 ≤ ≤ Hj ,wehaveKj θj CGj (Uj+1)θj Gj+1. p ≤ ≤ ≤ Let Mj = O (Cj). Then Mj θj Kj θj CGj+1 (Uj+1), so [Mj θ, Hj+1] p ≤ CHj+1 (Uj+1)=Z(Uj+1),andthenasMj = O (Mj ), Mj θ Mj+1. Similarly −1 ∈ −1 ≤ Kj+1θj centralizes Uj+1 Sylp(Hj ), so Mj+1θj Mj . Thus (2) is established. Further (2) implies (3). (6.11). Adopt Notation of 6.8. Then (1) A(X)=AutK(X,T )(X). (2) NT (X) is Sylow in K(X, Ui) for each i ∈ I.

Proof. Let V = VXT and form G = G(X, T), H = H(X, T), and K = ∈ K(X, T)asin6.8.Letα AutCF (T )(X)beap -element. Then α extends toα ˆ ∈ AutF (V ) centralizing T ,soˆα = gg for some p -element g ∈ CG(T ). p Now [g, H] ≤ CH (T )=Z(T ), so g ∈ O (CG(H)) ≤ K. Therefore A(X)= p ≤ O (AutCF (T )(X))AutT (X) AutK (X). As K centralizes T , the opposite inclusion is trivial, so (1) holds. By 1.1 we can choose our representatives Ui so that C(Ui)isastronglyNF (U)- normalized chain. Adopt the notation of 6.10 and its proof. By 6.10.3, S ∩ Kj = S ∩ Kn = Rn for each j. Asthisholdsforeachj ∈ I , it follows from 6.9.2 that Rn ≤ T . But of course NT (X) ≤ S ∩ Kn ≤ Rn,so(2)holds. (6.12). (1) A is a normal map on T .

~~38 MICHAEL ASCHBACHER~~

~~∈ E c p p (2) For each P CS( ) , O (A(P )) = O (AutCF (T )(P )) and A(P )=~~

AutCF (E)(P ). ∈ E fc p (3) For each P CS( ) , A(P )=O (AutCF (T )(P ))AutCS (E)(P ). c Proof. We first prove (2) and (3). Let P ∈C and pick μ ∈ homF (NS (P ),S) fc p with Pμ ∈X .ThenTμ = T as T is strongly closed in CS(P ). Now O (A(Pμ)) = p O (AutCF (T )(Pμ)), so as Tμ = T , p p −∗ p −∗ p O (A(P )) = O (A(Pμ))μ = O (AutCF (T )(Pμ))μ = O (AutCF (T )(P )), establishing the first part of (2). Moreover once (1) is established, 7.10.2 in [A1] fc gives the second part of (2). Further if P ∈C then NT (P )μ = NT (Pμ), so the first part of (2) implies (3). We next prove (1). Let X ∈Xfc; we must verify the three conditions (SA1)-(SA3) of 7.4 in [A1]. By 6.11, A(X)=AutK(X,T )(X)andNT (X) ∈ Sylp(K(X, T)), so AutT (X) ∈ Sylp(A(X)). Thus (SA1) holds. f Let P ∈D(X) with X ≤ P ≤ Q = NT (X). Suppose α ∈ AutA(P )(X) is a p -element. By (2), α extends toα ˆ ∈ AutF (PT) centralizing T .Thenˆα|X ∈ p ∈ ∈ α extends α|X ,soα|X O (AutCF (T )(X)). Also for y P , yα P and cy = cyα,so α ∈ NAut(X)(AutP (X)). That is AutA(P )(X) ≤ NA(X)(AutP (X)). Conversely, let β ∈ NA(X)(AutP (X)) be a p -element. Then β ∈ p ˆ ∈ O (AutCF (T )(X)), so β extends to β AutF (XT) centralizing T .Nowβ acts ˆ ≤ ˆ on AutP (X), so β acts on AutP (XT), and hence P Nβˆ,soβ lifts to a p -element ˜ β ∈ AutF (PT). Thus β ∈ AutA(P )(X), so AutA(P )(X)=NA(X)(AutP (X)), so that (SA2) holds. Let φ ∈ NA(Q)(X)beap -element and V = XCS(X). By (SA2), φ|X ∈ A(X) acts on AutQ(X). From 6.11.1, φ|X = cg|X for some g ∈ K(X, T). Now g|X ≤ ˆ acts on AutQ(X), so as CQ(X) X, g acts on Q.Letφ = cg|QCS (Q).Then ˆ [φ, CS(Q)] = [g, CS(Q)] ≤ CK(X,T )(Q)=Z(Q), so that (SA3) holds, completing the proof of (1). We are now in a position to prove Theorem 4. By 6.12.1, A is a normal map on T ,sobyTheorem3in[A1], C = CF (E)=E(A) F. Further by parts (2) ∈Cfc p and (3) of 6.12, for P , AutC(P )=A(P )=O (AutCF (T )(X)AutT (P ). This completes the proof of Theorem 4.

~~CHAPTER 7~~

~~Characteristic and subnormal subsystems~~

~~In this section F is a saturated fusion system over the finite p-group S. Definition 7.1. Define a subsystem E of F to be subnormal if there exists a subnormal series~~

~~E = Fn ··· F0 = F for E, consisting of subsystems of F such that for each 0~~

~~(7.3). Let α ∈ Aut(F) and E a subsystem of F on a subgroup T of S. (1) Eα is a subsystem of F on Tα.~~

~~40 MICHAEL ASCHBACHER~~

(2) If T is strongly closed in S with respect to F,thenTα is strongly closed in S with respect to F. (3) If E is F-invariant, then Eα is F-invariant. (4) If E F then Eα F. (5) If E F then Eα F. Proof. Straightforward. Define a subsystem E of F to be characteristic in F if E F and E is Aut(F)-invariant: that is for each α ∈ Aut(F), Eα = E. (7.4). If E and D are subsystems of F such that E D F, D is a system on D ≤ S,andAutF (D) ≤ Aut(E),thenE F.InparticularifE char D F, then E F. Proof. Let E, D be subsystems on T , D, respectively. Then E F,soE is saturated by 7.2.1. Let P ≤ T and γ ∈ homF (P, S). As D F, by 3.3 in [A1], there is ϕ ∈ AutF (D)andφ ∈ homD(Pϕ,S) such that γ = ϕφ on P .AsD F, AutF (D) ≤ Aut(D) (cf. 3.3 in [A1]), and if AutF (D) ≤ Aut(E)thenPϕ ≤ T . Moreover if E char D,thenAutF (D) ≤ Aut(E). Assume AutF (D) ≤ Aut(E). By the previous paragraph, Pϕ ≤ T ,soPγ = Pϕφ ≤ T as φ ∈ homD(Pϕ,S)andT is strongly closed in D with respect to D. Therefore T is strongly closed in S with respect to F.AsE D,thereexists μ ∈ AutD(T )andν ∈ homE (Pϕμ,S) with φ = μν.Thusγ = ϕφ = ϕμν, with ϕμ ∈ AutF (T ). That is (in the language of section 3 of [A1]), E is F-Frattini. Specialize to the case P = T .Asγ = ϕφ with ϕ ∈ AutF (D) ≤ Aut(E), and as φ ∈ AutD(T ) ≤ Aut(E), we have γ ∈ Aut(E). Therefore AutF (T ) ≤ Aut(E), so E is F-invariant by 3.3 in [A1]. Let V = TCD(T )CS(TCD(T ))), D(T )=NF (V ), and G ∈G(D(T )). As D F ∈Ff and T D, it follows from Theorem 2 in [A1] that there is H = H(T ) G with ND(TCD(T )) = FD(H)andD ∈ Sylp(H). Similarly as E D,weget K = K(T ) H with E(T )=NE (T )=FT (K)andT ∈ Sylp(K). ∗ Let X = CH (T )andY = CX (X/Z(T )). As F (X)=Op(X), Y = CD∩Y (T ). Let B be a Hall p-subgroup of K,sothatK = BT and [B,X] ≤ Z(T ). Then [B,CS(T )] ≤ Y = CD∩Y (T )and[B,CD∩Y (T )] ≤ CK (T )=Z(T ), so [B,CS(T )] ≤ Z(T ). Therefore (N1) is satisfied by E in F,soE F. Example 7.5. Recall from section 3 that SF is the smallest normal subsystem of F on S,and for T ≤ S, N (F,T) is the set of normal subsystems of F on T .ObserveSF is characteristic in F, using 7.3.4. More generally if: (i) T is strongly closed in S with respect to F,and (ii) Aut(F)actsonT ,and (iii) N (F,T) = ∅, t hen T F is characteristic in F. Definition 7.6. Define [S, Op(F)] = S ∩ Op(G(U)) : U ∈Ffc ,

~~7. CHARACTERISTIC AND SUBNORMAL SUBSYSTEMS 41~~

p where G(U)=GF,S(U) ∈G(D(U)). We show in the next lemma that [S, O (F)] F ∈Ffc is strongly closed in S with respect to . At that point, for U [S,Op(F)],wecan deﬁne p p H (U)=O (GF,[S,Op(F)](U))N[S,Op(F)](U). p p Set A (U)=AutHp(U)(U). Let A be the constricted F-invariant map on [S, Op(F)] determined by Ap(U) as in 5.2.3 in [A1], and set Op(F)=E(Ap).

(7.7). Let T =[S, Op(F)], T˜ = S ∩ Op(G(V )) : V ∈Ffrc ,andTˆ = S ∩ p ∈Ffc O (G(VU )) : U T .Then (1) T is strongly closed in S with respect to F. (2) T = T˜ = Tˆ. (3) Op(F) is characteristic in F on [S, Op(F)]. (4) Op(Op(F)) = Op(F). Proof. Assume for the moment that T is strongly closed in S with respect frc to F.LetV ∈F , U = T ∩ V , D = NS (V ), and β ∈ homF (NS(U),S) with ∈Ff p ∈Ffrc Uβ T .LetG = G(V )andH = O (G). As V , V = Op(G). By deﬁnition of T , T ∩ H ∈ Sylp(H), so U = Op(HT). Thus [CD(U),H] ≤ CH (U) ≤ f Z(U), so CD(U) ≤ V and CT ∩D(U) ≤ U.ThenasDβ ≤ NS(Vβ)andV ∈F , frc Dβ = NS(Vβ)andVβ ∈F . Hence as CDβ(Uβ) ≤ Vβ, CS(Uβ) ∩ NS (Vβ)= CDβ(Uβ) ≤ Vβ,soCS(Uβ) ≤ Vβ, and hence VUβ ≤ Vβ.ThenasCT ∩D(U) ≤ U, ≤ ∈Ffc ∗ ≤ we conclude that CT (Uβ) Uβ. Therefore Uβ T and NF (V )β NF (VUβ), ˇ ∩ p ≤ ∩ so by 7.2.8 in [A1], G(V )β = NG(VUβ)(Vβ), and hence (S O (G(V )))β S G(VUβ). F D D ∈Ffc F Next we claim that = ,where = NF (VW ):W T .As is frc saturated, F = AutF (V ):V ∈F by A.10 in [BLO]. We argue as in the proof of 5.5.3 in [A1]. Assume the claim is false and pick P ≤ S and α ∈ homF (P, S) such that α is not a D-map, and m = |S : P | is minimal subject to this constraint. frc Then m>1 and we pick Vi ∈F , αi ∈ AutF (Vi), and Pi as in the proof of 5.5.3 of [A1]. As in that proof, we may take α = α1|P and V1 = V .IfNS(U)=P then P = V = NS (U), so P = S, contradicting m>1. Hence P

fc p Now we prove (1). Let V ∈F , G = G(V )andH = O (G)NT (V ). As S fc p permutes F , T S,soasG = O (G)NS(V ), it follows that H G.By p deﬁnition of T , S ∩ O (G) ≤ T ,soQ = NT (V ) ∈ Sylp(H). In particular Q is fc strongly closed in NS (V ) with respect to G.ThenasF = AutF (V ):V ∈F , and as Q is strongly closed in NS (V ) with respect to G,(1)holds. frc Using the fact that F = AutF (V ):V ∈F and the claim, the same argument shows that T˜ and Tˆ are strongly closed. Let W = Op(G(V )). We claim frc S ∩ Op(G(V )) ≤ T˜.IfV = W then V ∈F , so the claim holds. Thus we may c fc assume V

42 MICHAEL ASCHBACHER closed, S ∩ Op(G(V )) ≤ T˜, establishing the claim. By the claim, T = T˜. A similar argument using paragraph one shows that T˜ = Tˆ, completing the proof of (2). f Next suppose V ≤ P ≤ T with P ∈D(V ) ,andβ ∈ homF (NS(P ),S) with fc p Pβ ∈F .ThenNH (P )=O (NH (P ))NQ(P ), so using 7.2.8 in [A1], ˇ p NH (P )β ≤ O (NG(Pβ)(Vβ))(NT (Pβ) ∩ NT (Vβ)) ≤ NHp(Pβ)(Vβ), and similarly we obtain the opposite inclusion. Then by Theorem 3 in [A1], E = Op(F)isnormalinF. fc p Visibly for α ∈ Aut(F)andU ∈F , Tα = T and A (U)α = AutHp(Uα)(Uα)= Ap(Uα), so E(Ap)α = E(Ap). Thus E is characteristic in F, establishing (3). p fc B F ∈F DF Finally let = O ( )andU T . Recall ,S (U)=NNF (VU )(U)and D ∈Ffc D D F,T (U)=N F (U), and as U , U = TU ,so F,S (U)= F,T (U)= N (VTU ) T DF,S(VU ). Therefore

G(VU )=GF,S(VU )=GF,S(U)=GF,T (U), p p so O (G(VU )) = O (GF,T (U)). p p Next DB,T (U)=NB(U)=EF,B(U)andO (GF,T (U))NT (U)=H (U) ∈ p p p G(EF,B(U)), so as GB,T (U) ∈G(DB,T (U)), GB,T (U)=H (U). Then as O (O (G)) = Op(G) for each ﬁnite group G, p p p p p O (GB,T (U)) = O (O (GF,T (U)) = O (GF,T (U)) = O (GF,S(U)). Let R =[T,Op(B)]. Then p fc p fc R = T ∩ O (GB,T (U)) : U ∈B = T ∩ O (GF,S(U)) : U ∈B p fc = S ∩ O (GF,S(U)) : U ∈B . F fc ⊆Bfc Also T ,so ≥ ∩ p ∈Ffc ∩ p ∈Ffc ˆ R S O (GF,S(U)) : U T = S O (GF,S(VU )) : U T = T = T, so R = T . Next

p p p p HF (U)=H (U)=O (GF,T (U))NT (U)=O (GB,T (U))NT (U) p p = O (GB,R(U))NR(U)=HB(U). p p p p B Then AF (U)=AutHF (U)=AutHB (U)=AB(U). Also for α a -map, α is also a F-map and Op(B) ≤B,so p p ∗ p ∗ p AB(Uα)=AB(U)α = AF (U)α = AF (Uα). Finally as R = T , Bf ⊆ Op(B)f , so using 5.5.4 in [A1], p f p f p p f p B = AF (P ):P ∈B = AB(P ):P ∈B ≤AB(P ):P ∈ O (B) = O (B), so B = Op(B), completing the proof of (4). The ﬁrst statement in part (2) of the next lemma is essentially Lemma 2.2 in [BCGLO2], attributed there to Puig.

~~p (7.8). (1) [S, O (F)] = R1 = R2,where p p fc R1 = [U, O (AutOp(F)(U))] : U ∈O (F) and p fc R2 = [U, O (AutF (U))] : U ∈F .~~

~~7. CHARACTERISTIC AND SUBNORMAL SUBSYSTEMS 43~~

p (2) Assume G is a ﬁnite group, S ∈ Sylp(G),andF = FS(G).Then[S, O (F)] p p p = S ∩ O (G) and O (F)=FS∩Op(G)(O (G)). Proof. Assume either p p (i) T =[S, O (F)], E = O (F), and R = R1 or R2,or p (ii) E = F = FS(G) for some ﬁnite group G and S ∈ Syl2(G), T = S ∩ O (G), and R = R2. Observe R ≤ T . In case (i) the objects of E are contained in T ,soR1 ≤ T , fc p p while R2 ≤ T as for U ∈F ,[U, O (AutF (U))] ≤ S ∩ O (GF,S) ≤ T . Similarly in p p case (ii), GF,S(U)=NG(U), and [U, O (NG(U))] ≤ U ∩ O (G) ≤ T . As R ≤ T ,toshowT = R it remains to show T ≤ R. Suppose not. Then as [T,T] ≤ Φ(T ), R[T,T]=Q

~~(7.9). Let R ≤ S. Then the following are equivalent: (1) R F. (2) FR(R) F.~~

~~Proof. Let E = FR(R). Then E is saturated.~~

Assume (1) holds. Then for all P ≤ S and φ ∈ homF (P, S), φ extends to φˆ ∈ homF (RP, S) acting on R. In particular if P ≤ R then Pφ ≤ Rφˆ = R,soR is ˆ ˆ strongly closed in S with respect to F. Also φ = φ|P with φ ∈ AutF (R), so (in the language of section 3 of [A1]) E is F-Frattini. Visibly AutF (R) ≤ Aut(E), so E is

~~44 MICHAEL ASCHBACHER~~

F-invariant by 3.3 in [A1]. As [CS(R),R] ≤ Z(R), condition (N1) is satisﬁed by E in F,soE is normal in F. That is (1) implies (2). Now assume (2). Then R is strongly closed in S with respect to F,soby1.6 in [BCGLO1] it remains to show that R is contained in each U ∈Ffrc. Assume U is a counterexample; then Q = NR(U) U.LetG ∈G(NF (U)), and set W = R ∩ U.AsU ∈Ffrc, U = F ∗(G). As R is strongly closed in S, W G. ∗ Also [Q, U] ≤ R ∩ U = W ,soQ ≤ H = CG(U/W). Then as U = F (G), ch CQ(W ) ≤ W .Letr ∈ Q and φ = cr|W .Thenforh ∈ H, crh|W = φ ∈ AutE (W ), h −1 h so crh|W = ct|W for some t ∈ Q.Thusr t ∈ CH (W ) ≤ Op(H) ≤ U,sor ∈ UQ. Thus UQ H,soQ ≤ Op(G)=U, contrary to the choice of U. Recall Op(F) is the largest subgroup of S normal in F. By 7.9, we can also regard Op(F) as the largest normal subsystem of F of the form FR(R)forsome R ≤ S. (7.10). (1) If R ≤ S with R F and α ∈ Aut(F),thenRα F. (2) Op(F) is a characteristic subsystem of F.

Proof. Part (1) is straightforward. Then as P = Op(F) is the largest subgroup of S normal in F, (1) implies that P is Aut(F)-invariant. Thus FP (P )α = FPα(Pα)=FP (P ), so (2) follows from 7.9 and our notational convention that we write Op(F)forFP (P ). Recall that the center of F is

~~Z(F)={z ∈ Op(F):zφ = z for all φ ∈ AutF (Op(F))}≤S.~~

We also write Z(F) for the fusion system FZ (Z) on the subgroup Z = Z(F)of S. (7.11). (1) Z(F) is a characteristic subsystem of F. fc (2) For each U ∈F , Z(F) ≤ Z(G(U)),whereG(U)=GF,S(U) ∈G(NF (U)).

Proof. Let Z = Z(F) regarded as a subgroup of S,andZ = FZ (Z). For z ∈ Z and s ∈ S, zcs = z by deﬁnition of Z,soZ ≤ Z(S). Each F-map φ extends ˆ ˆ to φ on R = Op(F), so φ centralizes z by deﬁnition of Z. In particular AutF (Z)=1 and Z F,soZ F by 7.9. Visibly Z is Aut(F)-invariant, so (1) holds. Assume the setup of (2). Then Z ≤ Z(S) ≤ U,andthenforg ∈ G(U), cg|Z ∈ AutF (Z)=1,soZ ≤ Z(G(U)), completing the proof.

frc (7.12). (1) If U ∈F with G(U)=GF,S(U) p-closed, then U = S. frc (2) S = Op(F) iﬀ F = FS (G(S)) iﬀ F = {S}. (3) The following are equivalent: (a) F = Op(F). (b) F = FS (S). frc (c) For each U ∈F , AutF (U) is a p-group. (d) For each U ∈Ffrc, G(U) is a p-group. (e) Op(F) ≤ Z(F). (f) Op(F)=1. Proof. Let U ∈Ffrc and set G = G(U). As U ∈Ffrc, U = F ∗(G). Assume ∗ G is p-closed. Then U = F (G)=NS (U), and hence U = S.Thus(1)holds. If S = Op(F)thenF is constrained and F = FS (G(S)). If F = FS (G(S)), then S F,soNS (U) NF (U). Thus G(U)isp-closed, and hence U = S by

~~7. CHARACTERISTIC AND SUBNORMAL SUBSYSTEMS 45~~

(1), so F frc = {S}. Finally suppose F frc = {S}.AsF is saturated, frc F = AutF (U):U ∈F = AutF (S) , so S F, completing the proof of (2). Next by 7.9, F = Op(F)iffF = FS(S), in which case AutF (U)=AutS(U)isa p-group. Further G/Z(U)=AutF (U), so G is a p-group iff AutF (U)isap-group. Assume G is a p-group for each U ∈Ffrc.Thenby(1),F frc = {S},so p F = FS(G(S)) by (2). Further O (G(S)) = 1 and F = FS(S)asG(S)=S is a p-group, so (b) and (f) hold. Trivially (f) implies (e). Assume (e) holds and let T =[S, Op(F)]. Then S∩Op(G) ≤ T ≤ Z(F), so S∩Op(G) ≤ Z(G). Therefore as F ∗(G)=U, Op(G)=1. Thus (e) implies (d), completing the proof. (7.13). Assume F = Op(F).Then (1) If F = Op(F),thenF =1. (2) If E F and F = EZ(F) then F = E. p p Proof. If F = Op(F), then O (F) = 1 by 7.12.3, so (1) holds as F = O (F). Assume the hypothesis of (2), and let E be defined on T ≤ S and Z = Z(F). fc Let U ∈F and G = GF,S(U). As F = EZ, U = WZ,whereW = U ∩ T . ∈Ffc ∈Ffc Then NS (W )=NS(U)andCS(W )=CS(U), so as U , W T and U = VW .ThusG = GF,T (W ). From 5.9, G = HZ,whereH = HF,E (W ). Thus as Z ≤ Z(G) by 7.11, Op(G) ≤ H,soOp(G) ∩ S ≤ H ∩ S ≤ T . But by 7.7.2, [S, Op(F)] = Op(G(U)) ∩ S : U ∈Ffc ≤T ,andS =[S, Op(F)] as F = Op(F). Thus S = T . Finally as G = HZ, AutF (U)=AutG(U)=AutH (U)=AutE (U), fc fc so F = AutF (U):U ∈F = AutE (U):U ∈F = E. Recall F is quasisimple if F =1, F = Op(F), and F/Z(F)issimple.

(7.14). Assume F is quasisimple. Then F = Op(F)=Z(F). p Proof. As F =1and F = O (F), F = Op(F) by 7.13. Let Z = Z(F). Then F = NF (Z)=CF (Z), and Z ≤ Op(F) F. Therefore by 8.10.2 in [A1], Op(F)/Z F/Z.ButasF is quasisimple, F/Z is simple, so either F = Op(F) or Z = Op(F). We showed the former does not hold, so Op(F)=Z, completing the proof of the lemma. (7.15). Assume F/Z(F) is simple, and let E F. Then either E≤Z(F) or F = EZ(F), and in the latter case F = E if Z(F) ≤E. Proof. Let Z = Z(F)andE be a system on T ≤ S.AsZ(F)=Z, Z ≤ Z(S), so [T,Z]=1.ThusEZ F by Theorem 3, so replacing E by EZ, we may assume Z ≤E. Adopt the notation of section 8 of [A1]. By 8.10 in [A1], Eθ = E + F +, so as F + is simple, E + = F + or 1. In the latter case E≤Z(F),andintheformer E = F by 8.10 in [A1]. (7.16). Assume F = Op(F).ThenF is quasisimple iﬀ each proper normal subsystem of F is contained in Z(F). Proof. Let Z = Z(F)andF + = F/Z. Assume ﬁrst that F is quasisimple and let E F. By 7.15, either E≤Z(F)orF = EZ, and in the latter case F = E by 7.13.2. Conversely assume each proper normal subsystem of F is contained in Z.Then by 8.10 in [A1], F + is simple, so as F = Op(F), F is quasisimple.

~~46 MICHAEL ASCHBACHER~~

(7.17). Set Sp =[S, Op(F)].Then p p p f (1) O (F)=O (AutF (P ))AutSp (P ):P ∈ O (F) . p p fc (2) S = [U, O (AutF (U))] : U ∈F . (3) Assume E≤Fwith E saturated. Then Op(E) ≤ Op(F). Proof. By 7.6, A = Ap is a constricted F-invariant map on Sp and Op(F)= p F p ∈Op F f ∈Ffc E(A), so by 5.5.4 in [A1], O ( )= A (P ):P ( ) .ButforU Sp , p A(U)=AutHp(U)(U)=AutOp(AutF (U))(U)AutSp (U)sinceH (U)= p O (GF,Sp (U))NSp (U). Then as A is a constricted invariant map, A(P )= ∈ p F f AutOp(AutF (P ))(P )AutSp (P )forP O ( ) ,so(1)holds. Observe that (2) follows from 7.8.1. Assume the hypothesis of (3) with E asystemonR ≤ S. By 7.7.3, Op(E)is saturated, so replacing E by Op(E) and appealing to 7.7.4, we may assume E = p fc f O (E). Let U ∈E . Then there exists α ∈ homF (U, S) with Uα ∈F and ∈Ffc p ≤ VUα .ThenAutF (Uα)=AutAutF (VUα)(Uα), so [Uα,O (AutF (Uα))] p p [VUα,O (AutF (VUα))] ≤ S by (2). Then p p p p [U, O (AutE (U))]α =[Uα,O (AutEα∗ (Uα)] ≤ [Uα,O (AutF (Uα)] ≤ S , p p p so as S is strongly closed in S with respect to F,wehave[U, O (AutE (U))] ≤ S . Then Rp ≤ Sp by (2). f f Similarly for P ∈E ,thereisβ ∈ homF (P, S) with Pβ ∈F and p ∗ p (O (AutE (P )AutRp (P ))β = O (AutEβ∗ (Pβ)Aut(Rβ)p (Pβ) p p ≤ O (AutF (Pβ))AutSp (Pβ) ≤ O (F), as (Rβ)p ≤ Sp by the previous paragraph. Then since Op(F)isF-invariant, p p p p O (AutE (P ))AutRp (P ) ≤ O (F), so E = O (E) ≤ O (F) by (1). Hence (3) holds.

frc (7.18). Assume F0 F is a system on S0 ≤ S,andU ∈F .Then ∩ ∈Fc U0 = U S0 0 . Proof. F ∈Ff −Fc Assume otherwise. Conjugating in , we may assume U0 0 , ∩ so CS0 (U0) U0.ThenW = CS0 (U0) NS(U) U0.LetG = GF,S(U). Then [U, W ] ≤ U ∩ S0 = U0,so[G, W ] ≤ CG(U/U0) ∩ CG(U0), and hence W ≤ Op(G)= U.ButthenW ≤ U ∩ S0 = U0, a contradiction. (7.19). Assume B is a saturated subsystem of F on S with Op(B)=Op(F). Then B = F.

p p Proof. Let S0 =[S, O (F)], F0 = O (F), and for P ≤ S,setP0 = P ∩ S0. ∈Ffc ∈G If P0 0 then from 7.6, GF,S0 (P0)=HF,F0 (P0)NS(P0) (NF (VP0 )), where p ∈G HF,F0 (P0)=O (GF,S0 (P0))NS0 (P0) (NF0 (P0)). By symmetry, GB,S0 (P0)= ∈G ≤ HB,F0 (P0)NS (P0) with HB,F0 (P0) (NF0 (P0)). But NB(VP0 ) NF (VP0 ), so we ≤ may take GB,S0 (P0) GF,S0 (P0), and similarly HF,F0 (P0)=HB,F0 (P0). Then

~~GB,S0 (P0)=HB,F0 (P0)NS(P0)=HF,F0 (P0)NS(P0)=GF,S0 (P0). f Then for P ∈ NF (P0) ,~~

F B Aut (P )=AutNGF (P0)(P )=AutNGB (P0)(P )=Aut (P ). ,S0 ,S0 frc Next F = AutF (P ):P ∈F , so it suﬃces to show AutF (P )=AutB(P ) ∈Ffrc ∈Fc ∈ ∈ for P . By 7.18, P0 0 ,sothereexistsα homF (P, S) with U0 = P0α

~~7. CHARACTERISTIC AND SUBNORMAL SUBSYSTEMS 47~~

~~F fc ∈ f ∗ 0 and U = Pα NF (U0) . Thus by the previous paragraph, AutB(P )α = −∗ AutBα∗ (U)=AutF (U), so AutB(P )=AutF (U)α = AutF (P ), completing the proof.~~

~~CHAPTER 8~~

~~T F0~~

In this section F is a saturated fusion system over the ﬁnite p-group S, F0 is a normal subsystem of F on S0 ≤ S,andT is a subgroup of S strongly closed in S with respect to F, such that S0 ≤ T .GivenP ≤ S, we usually write P0 F ∩ ≤ T 0 { t ∈ ∈ t } for P S0.ForP0 S0,setP0 = P φ : t T,φ homF0 (P0,T) and F (T F )f = {P ≤ T : |N (P )|≥|N (Q )| for all Q ∈ P T 0 }.LetO, P be the 0 S0 0 T 0 T 0 0 0 setoforbitsofF0, S0 on subgroups of S0. Part of the treatment here (particularly Lemma 8.1) was suggested by the proof of Theorem 4.6 in [BCGLO2].

(8.1). Let P0 ≤ S0.Then F F O P 0 → s 0 S0 → sS0 (1) S is represented on and via s : P0 P0 , s : P0 P0 . (2) S acts on F f and each member of O contains a member of (T F )f . 0 0 S0 (3) The following are equivalent: (a) P ∈ (T F )f . 0 0 S0 F ∈Ff | | | 0 | (b) P0 0 and NT (P0):NS0 (P0) = NT (P0 ):S0 . F ∈Ff 0 S0 (c) P0 0 and NT (P0 ) acts on P0 . F ∈Ff 0 (d) P0 0 and NT (P0 )=NT (R0)S0. T F f (4) Let Q ∈ P 0 ∩ (T F ) . Then there exists ϕ ∈ homF (P ,Q ) such that 0 0 0 F0 0 0 0 NT (P0) ≤ Nϕ. Proof. F F ∈Ff ∈ Part (1) follows as 0 is -invariant. For Q0 0 and s S, s s F f NS0 (Q0)=NS0 (Q0) ,soS acts on 0 . F F R F f 0 0 Let be the set of orbits of S0 on 0 contained in P0 .ThenNS(P0 )acts on R, and by Proposition 1.16 in [BCGLO2], (|R|,p)=1.Thus F F 0 S0 ∈P 0 ∩Ff (*) NS (P0 )actsonQ0 for some Q0 0 0 . F F F 0 S0 0 | 0 | Next NT (P0 )actsonP0 iﬀ NT (P0 )=NT (P0)S0 iﬀ m = NT (P0 ):S0 = | | | | NT (P0)S0 : S0 = NT (P0):NS0 (P0) . Hence parts (b), (c), and (d) of (3) are | | equivalent. In particular by (*), NT (Q0):NS0 (Q0) = m. On the other hand F ≤ 0 | | | |≤ NT (P0) NT (P0 ), so NT (P0):NS0 (P0) = NT (P0)S0 : S0 m.Thusas Q ∈Ff and |N (Q ):N (Q )| = m, it follows that Q ∈ (T F )f and (a) is 0 0 T 0 S0 0 0 0 S0 equivalent to (b). Thus (3) is established, and as Q ∈ (T F )f , the proof of (2) is 0 0 S0 also complete. ∈ Assume the hypothesis of (4). For ϕ homF0 (P0,S0), let

~~ϕS0 = {ϕcx : x ∈ S0}. F F ≤ 0 0 ≤ S0 Let L = NT (P0). Then as L NT (P0 )andNT (P0 ) NT (Q0 )by(3),L acts on X { ∈ S0 ∈ } = ϕS0 : P0ϕ Q0 ,ϕ homF0 (P0,S0) ,~~

~~50 MICHAEL ASCHBACHER~~

→ ∗ ∈X { ∈ } via l : ϕS0 ϕcl S0.NowforX ,ΦX = ϕ X : P0ϕ = Q0 is nonempty, ∈ ∈X ∈ and ΦX = ϕNS0 (Q0) for any given ϕ ΦX .FurtherforX, Y , ϕ ΦX ,and ∈ ∈ |X | | | ψ ΦY , ψ = ϕα for some α AutF0 (Q0). Thus = AutF0 (Q0):AutS0 (Q0) ∈ ∈Ff F is prime to p,asAutS0 (Q0) Sylp(AutF0 (Q0)) since Q 0 and 0 is saturated. ∈X ∈ ∈ ∗ Therefore L acts on some X .Letϕ ΦX .Thenforl L, ϕcl = ϕcx for some ∗ ∗ ∗ x ∈ S0,soclϕ = clx−1 .Thenasclϕ ∈ AutF (Q0), clx−1 = clϕ ∈ AutT (Q0); that is L ≤ Nϕ, completing the proof of (4) and the lemma. Notation 8.2. Set (T F )fc =(T F )f ∩Fc.ForU ∈Ffc,writeA(U ) for the set of α ∈ 0 S0 0 S0 0 0 0 0 f homF (NS (U0),S) such that U0α ∈F.Forα ∈A(U0), set G(U0,α)= N H(U0α)NT (U0)α (where H(U0α)=HF,F0 (U0α)), and (U0,α) F −∗ = NT (U0)α(G(U0,α))α , regarded as a fusion system on NT (U0). (8.3). ∈Ffc Let U0 0 .Then (1) A(U0) = ∅. (2) For α ∈A(U0), NT (U0)α ∈ Sylp(G(U0,α)). (3) N (U0,α)=N (U0) is independent of the choice of α ∈A(U0). (4) N (U0) is a saturated constrained fusion system on NT (U0),andeachα ∈ A(U0) extends to an isomorphism αˇ : G(N (U0)) → G(U0,α),whereG(N (U0)) ∈ G(N (U0)).

Proof. Part (1) follows from 1.1.1 in [A1]. Let α ∈A(U0), H = H(U0α), and ∈Ff ≤ R = NT (U0)α.ThusG = G(U0,α)=HR.AsU0 0 and NS(U0)α NS(U0α), ∈ ∈ NS0 (U0)α = NS0 (U0α). Thus R Sylp(G)sinceNS0 (U0α) Sylp(H). Thus (2) holds. −1 Let β ∈A(U0)andsetγ = β α regardedasamapfromR1 = NT (U0)β to R. → ∗ N →N Set δ = γ|NS (U0β) : NS0 (U0β) NS0 (U0α). Then δ : F0 (U0β) F0 (U0α)is 0 ∗ an isomorphism of saturated constrained fusion systems, so by 2.3 in [A1], H1c1δ = Hc,whereH1 = H(U0β), c1 : G1 = H1R1 → Aut(U0β)andc : G → Aut(U0α) ∗ are the conjugation maps. Then as R1γ = R, G1c1γ = Gc. Hence by 2.4 in [A1], → F →F there is an isomorphismγ ˇ : G1 G extending γ.Then˙γ : R1 (G1) R(G)is ˙−1 ˙ −1 an isomorphism, so asγ ˙ = β α˙ , βγ˙ α˙ : N (U0,β) →N(U0,α)isanisomorphism which is the identity on NT (U0), and hence the identity map. This proves (3).

Next N (U0,α) is a constrained saturated fusion system, so asα ˙ : N (U0) → FR(G) is an isomorphism, (4) holds. Notation 8.4. fc f For U ∈ (T F ) and U ∈N(U ) with U ∩ S = U ,setN (U)=NN (U)and 0 0 S0 0 0 0 (U0) A(U)=AutN (U)(U). Set fc fc D = AutN (U), F : U ∈ (T F ) ,U∈N(U ) , (U) 0 0 0 S0 0 Df { ≤ ∈Df } Dfc regarded as a fusion system over T .Set 0 = P0 S0 : P0 and 0 = Df ∩Fc 0 0 .

(8.5). (1) F0 ⊆D. (2) Let P0 ≤ S0 and α ∈ homD(P0,T). Then there exists t ∈ T and φ ∈ t homF0 (P0,T) such that α = ctφ. (3) Df =(T F )f . 0 0 S0 D A ∈Dfc ∈N fc (4) = N (U)(U):U0 0 ,U (U0) .

~~8. T F0 51~~

∈Dfc ∈N f ∩ N (5) For U0 0 and U (U0) with U S0 = U0, (U) is a saturated ∈N fc N ∈ fusion system, and if U (U0) then (U) is constrained with NG(N (U0))(U) G(N (U)). ∈Dfc N ≤D (6) For U0 0 , A(U0)=AutD(U0).Further (U0) ,soforeach g g ∈ G(N (U0)) and D ≤ NT (U0) with D ≤ T , cg|D ∈ homD(D, T).

Proof. Part (1) follows from the definition of D. In particular as F0 = ∈Ffc ≤D D AutF0 (P0):P0 0 , we conclude from the definition of that fc fc fc (7) D = AutF (V ),AutN (U):V ∈F ,U ∈ (T F ) ,U∈N(U ) . 0 0 (U) 0 0 0 0 S0 0 ∈Ffc ≤ ≤ Assume the setup in (2). Then by (7), there exists Vi 0 ,1 i n, αi ∈B(Vi) with α = α1 ···αn, Pi ≤ Vi, Piαi = Pi+1,andP0 = P1,whereB(Vi)= f fc AutF (V )ifV ∈/ (T F ) ,andB(V )=AutN (V )ifV ∈ (T F ) .Intheformer 0 i i 0 S0 i (Vi) i i 0 S0 fc ∈ F ∈ F case αi = ϕi|Pi for some ϕi Aut 0 (Vi), while if Vi (T 0)S then αi = cti chi for ∈ ∈ ∈0 some ti NT (Vi)andhi H(Vi,βi). In particular chi = ϕi AutF0 (Vi). Thus in any case, αi = cti ϕi. F ∈ · ∗ ∗ F Finally given an 0-map φ and t T , φct = ct φct with φct an 0-map, so · α = cti ϕi = cti ϕ = ctϕ, i i for a suitable F0-map ϕ.Thus(2)holds. F ≤ D T 0 By (2), if P0 S0 then P0 = P0 , so (3) follows. B ∈Df ∈N fc Let = AutN (U)(U):U0 0 U (U0) . Suppose (4) fails and pick P ≤ T and α ∈ homD(P, T) such that α is not a B-map and m = |T : P | is fc minimal. As T ∈N(S0) , m>1. We can pick α = α1 ···αn, Vi,andPi as in the proofof(2).Asα = α1 ···αn, it suffices to show each αi is a B-map, so we can ∈Ffc ∈Df ∈ take n =1andα = α1.ThenV1 0 but V1 / 0 .Letβ isoF0 (V1,U0) with ∈Df ∈B ∈Df ∗ ∈B U0 0 .IfV1 = P then by minimality of m, β ,andasU0 0 , αβ ,so ∗ −∗ α =(αβ )β ∈B. Thus we may assume V1 = P , and it remains to show we can choose β so that β ∈B. Now apply the argument above and write β = β1 ···βk. As above we are done if Pi

∈Ffc N Pick U0 and U as in (5). By (3), U0 0 . By 8.3.4, (U0) is a saturated constrained fusion system and G(N (U0)) ∈G(N (U0)). Then (5) follows from the deﬁnition of N (U) and 1.2.1 in [A1]. ∈Dfc ∈ ∈ Let U0 0 and α AutD(U0). By (2), α = ctφ with t T and φ an F F F -map. Then U t = U φ−1 ∈ U 0 ,sot ∈ N (U 0 ). Then as U ∈ (T F )f 0 0 0 0 T 0 0 0 S0 by (3), we conclude from 8.1.3 that t = rs with r ∈ NT (U0)ands ∈ S0.Thus ∈ ∈ U0csφ = U0ctφ = U0α = U0,soϕ = csφ AutF0 (U0)andα = crϕ A(U), establishing the ﬁrst statement in (6). N N ∈N fc As (U0) is a saturated fusion system, (U0)= AutN (U0)(U):U (U0) by Alperin’s Fusion Theorem (cf. A.10 in [BLO]), so N (U0) ≤Dfrom 8.4. This completes the proof of (6).

~~(8.6). Let S0 ≤ P ≤ T .Then (1) AutD(T )=A(T ). D T (2) homD(P, T)=homN (S0)(P, T) and P = P .~~

~~52 MICHAEL ASCHBACHER~~

~~f (3) If P ∈N(S0) then N (P )=ND(P ),soA(P )=AutD(P ) and AutT (P ) ∈ Sylp(AutD(P )). (4) If α ∈ AutD(S0) and P ≤ Nα then α extends to a member of homD(P, T).~~

Proof. The proof of 5.5.2 in [A1] establishes (1). Similarly as S0 is strongly ∈ ∈ closed in T ,givenα homD(P, T), to show that α homN (S0)(P, T) it suffices (using the argument of 5.5.2 in [A1]) to take α = γP for some γ ∈ AutN (U)(U) ∈N fc and some U (S0) . By definition, α is induced in NN (S0)(U), so indeed ∈ α homN (S0)(P, T). Let β ∈ homD(P, T). By 8.4, N (S0)=FT (G(S0)) and G(S0)=HT,where ht H = H(S0). Thus β = cht|P for some h ∈ H and t ∈ T ,soP = Pβ ≤ T ,and h ht t hence P ≤ T .Then[P, h] ≤ T ∩ H = S0,soh acts on P .ThenPβ = P = P , so P D = P T .Thus(2)holds. ∈N f N N Assume P (S0) .Then (P )=NN (S0)(P ), so by (2), (P )=ND(P ), and hence AutD(P )=AutN (P )(P ). By 8.5.5, N (P )issaturated,soAutT (P ) ∈ Sylp(AutN (P )(P )). By definition, A(P )=AutN (P )(P ). Thus (3) holds. Assume α ∈ AutD(S0)andP ≤ Nα.By(2),α is induced in N (S0), so as N (S0) ⊆ is saturated, α extends to homN (S0)(P, T) homD(P, T), establishing (4).

~~(8.7). ≤ ∈ ∈Df ∩Fc Dfc Let P0 S0 and α isoD(P0,U0) with U0 0 0 = 0 .Thenα extends to a member of homD(Nα,T).~~

Proof. Assume false and choose a counterexample with m = |S0 : P0| minimal. By 8.6.4, m>1. ∈ ∈ t t By 8.5.2, α = ctα for some t T and α homF0 (P0,S0). Further Nα = Nα, t ∈ so replacing P, α by P ,α, we may assume α homF0 (P0,T). Suppose ﬁrst that P0

Nαγ ≤ NT (R0β) ∩ NT (U0β) ≤ (NT (R0) ∩ NT (U0))β, −1 −1 and hence Nαγδ ≤ NT (R0) ∩ NT (U0). Moreover γδ extends α, completing the proof in this case. ≤ ∩ So we may assume P Nα,andNα S0 = P0.Asm>1, P0

~~8. T F0 53~~

(8.8). Let P ∈Hand set P0 = P ∩ S0.Then ∈ ∈Df ≤ (1) There exists α isoD(P0,U0) with U0 0 and NT (P0) Nα. (2) α extends to αˆ ∈ homD(NT (P ),T). ∈ ∈N f (3) There exists χ homN (U0)(NT (P αˆ),NT (U0)) such that P αχˆ (U0) . (4) If P ∈Df then P αχˆ ∈Df . Proof. Part (1) follows from 8.5.3 and 8.1.4. Then (2) follows from 8.7. As f N (U0) is saturated, (3) follows from 1.1.2 in [A1]. Finally if P ∈D then as NT (P )ˆαχ ≤ NT (P αχˆ ), (4) follows. (8.9). Suppose P ∈Dc −H.Then ∈ (1) There exists α homD(P, T) such that CS0 (P0α) P0α. ∈ D ∩ (2) There exists P P such that OutS0 (P ) Op(OutD(P )) =1.

~~Proof. Assume (1) fails and choose a counterexample with m = |S0 : P0| ∈H ∈Fc minimal. As P/ , P0 / 0 ,soP0 = S0 and hence m>1. Thus P0~~

For U ∈H,writeA(U) for the set of those α ∈ homD(NT (U),T) such that U0α ∈ Df ∈N f −∗ ∈A ∈ 0 and Uα (U0) .SetA(U)=A(Uα)α for α (U). Suppose α, β −1 ∗ A(U), and set γ = β α : Uβ → Uα. Then by the proof of 8.3.3, γ : N (U0β) → N (U0α) is an isomorphism, so ∗ ∗ A(Uβ)γ = AutN (U0β)(Uβ)γ = AutN (U0α)(Uα)=A(Uα). Thus the deﬁnition of A(U) is independent of the choice of α ∈A(U). Hc H∩Dc fc ∈Hc ∈Dfc Let = .WriteH for the collection of U such that U0 0 fc and U ∈N(U0) . (8.11). Let P ∈H.Then (1) A(P ) = ∅. c c (2) P ∈H iﬀ for α ∈A(P ), Pα ∈N(P0α) . fc (3) If U ∈H then AutT (U) ∈ Sylp(A(U)). fc f (4) If U ∈H and U ≤ Q with Q ∈N(U) ,thenAutA(Q)(U)= NA(U)(AutQ(U)). Proof. Part (1) follows from parts (1)-(3) of 8.8. Let α ∈A(P )andset U0 = P0α, U = Pα,andN = N (U0). Then CT (P )α ≤ NT (P )α ≤ NT (U), so if

~~54 MICHAEL ASCHBACHER~~

c c c CT (P ) P then CT (U) U and hence U/∈N .ThusifU ∈N then P ∈H. c Conversely if P ∈H then as U = Pα, CT (Uφ) ≤ Uφ for all φ ∈ homN (U, T ), as N≤Dby 8.5.6, so U ∈Nc.Thus(2)holds. fc Suppose U ∈H . By deﬁnition in 8.4, A(U)=AutN (U)(U), and by 8.5.5, N (U) is a saturated fusion system on NT (U), so (3) follows. Indeed from 8.2 and N F ∈GN 8.3.4, = NT (U0)(G)forG ( ), and there is H = H(U0) G with ∈G N F H (NF0 (U0)) and G = HNT (U0). Then (U)= NT (U)(NH (U)NT (U)) and hence A(U)=AutH (U)AutT (U). Assume the hypothesis of (4). By parts (1)-(3) of 8.8 applied to S in the f role of T ,thereexistsα ∈ homF (N (Q),S) such that Q α ∈ (SF ) and Qα S 0 0 S0 F ∈Hfc is fully normalized in NS (Q0α)(H(Q0α)NS(Q0α). Hence also Qα .Let ∈{ } G(X)=GF,F0 (X)forX U0,Q0α . By 7.2 in [A1], α extends to an isomor- → phismα ˇ : NG(U0)(Q0) NG(Q0α)(VU0α). Then by 7.18 in [A1], NH(U0)(Q0)ˇα =

~~NH(Q0α)(U0α). Therefore~~

NH(U0)NT (U0)(Q0)ˇα = NH(Q0α)NT (Q0α)(U0α), so NG(N (U0))(Q0)ˇα = NG(N (Q0α))(U0α). Then ∩ ∩ NG(N (U))(Q)ˇα =(NG(N (U0))(Q0) N(U) N(Q))ˇα ∩ ∩ = NG(N (Q0α))(U0α) N(Uα) N(Qα)=NG(N (Qα))(Uα).

Next NA(U)(AutQ(U)) consists of those φ ∈ A(U) induced by g ∈ G = G(N (U)) acting on QCG(U). But CG(U)=CT (U) ≤ U ≤ Q, so these are the members of NG(Q). That is NA(U)(AutQ(U)) is the image of NG(N (U))(Q) under the conjugation map. Also AutA(Qα)(Uα) is the image of NG(N (Qα))(Uα) under the conjugation map, so from the last display, −∗ ∗ −∗ AutA(Q)(U)=AutA(Qα)(Uα)α = NA(U)(AutQ(U))α α = NA(U)(AutQ(U)), establishing (4).

Theorem 8.12. Let V ∈Hc.Then (1) Each α ∈ homD(V,T) extends to a member of homD(Nα,T). (2) A(V )=AutD(V ). The proof is almost exactly the same as that of Theorem 7.10 in [A1]. As- sume the Theorem is false, and let n be minimal subject to the condition that for c some V ∈H with n = |T : V |, either A(V ) = AutD(V ), or some member α of c homD(V,T)doesnotextendtoamemberofhomD(Nα,T). Pick V ∈H such that n = |T : V |, and pick α ∈ homD(V,T). Set U = Vα. The proof involves a series of reductions. (8.13). n>1.

~~Proof. Suppose n =1.ThenT = V = U = Nα, so 8.12.1 holds. Also 8.12.2 follows from 8.6.1. This contradicts the choice of n.~~

~~(8.14). If there exists V1 ≤ Nα such that V~~

~~8. T F0 55~~

∈ ∗ ∈ Let x V2 and write cx for conjugation by x on V1.Thenψ = cxα1 AutD(U1) and ∗ ∗ U1 V1 ∈ ψ|U = cxα1resU = cxresV α AutT (U) ∗ U1 V1 ∗ acts on AutU1 (U)asα1resU = resV α on AutV2 (V1). Thus ψ|U = ct|U for some t ∈ NT (U)∩NT (U1), since U is centric so the kernel of the map NT (U) → AutT (U) is Z(U) ≤ U1. Let γ ∈A(U1), N = N (U1γ), and G ∈G(N ). By minimality of n, AutD(U1γ) c = A(U1γ), and by 8.11.3, AutT (U1γ) ∈ Sylp(A(U1γ)). Further U ∈H,so c Uγ ∈N by 8.11.2, and hence CG(Uγ)=Z(Uγ), so AutZ(Uγ)(U1γ)isthekernelof → the conjugation map c : NA(U1γ)(Uγ) AutA(U1γ)(Uγ). Therefore as AutD(U1)= −∗ → A(U1γ)γ , AutZ(U)(U1)isthekernelofc : NAutD (U1)(U) AutAutD(U1)(U). Hence as ψ and ct are in AutD(U1) with ct = ψ on U, it follows that ψ ∈ ctAutZ(U)(U1) ⊆ AutT (U1). This establishes the claim. ≤ ∈ As V2 Nα1 and α1 extends to β on Nα1 , α2 = β|V2 homD(V2,T) extends α1. Continuing in this manner, α extends toα ˆ ∈ homD(Nα,T), completing the proofofthelemma. (8.15). Assume U ∈Hfc.Then: ∗ ∗ (1) There exists χ ∈ A(U) such that AutT (V )α χ ≤ AutT (U). (2) Let β = αχ.IfthereexistsV1 ≤ Nα such that V

~~56 MICHAEL ASCHBACHER~~

Proof. Let α ∈ AutD(U)andsetQ = NT (U). By 8.17 there exists χ ∈ A(U) ∈ ∈ and γ NAutD (Q)(U) such that γ extends αχ. By minimality of n, γ A(Q), so γ ∈ NA(Q)(U). By 8.11.4, AutA(Q)(U)=NA(U)(AutQ(U)), so β = γ|U ∈ A(U)acts −1 on AutQ(U). Then α = βχ ∈ A(U), establishing (1). Next by 8.5.5, A(U)=AutG(U)forG ∈G(N (U)). Thus α = ch|U for some ∗ ∗ ∗ h ∈ G.NowNα = {g ∈ Q : cgα ∈ AutQ(U)} and cgα = cg(ch|U ) = cgh ,so h h−1 Nα = {g ∈ Q : cgh ∈ AutQ(U)} = {g ∈ Q : g ∈ Q} = Q ∩ Q h−1 as ker(c)=CG(U)=Z(U) ≤ Q.Thusch : Nα = Q ∩ Q → Q extends α to Nα and is a D-map by 8.5.6. Thus (2) holds. We are now in a position to complete the proof of Theorem 8.12. We ﬁrst prove 8.12.2. Pick α so that U ∈Hfc.Thusα ∈A(V ), and by 8.18.1, A(U)=AutD(U), so by 8.10, −1 −∗ −∗ −1 A(V )=A(Uα )=A(U)α = AutD(U)α = AutD(Uα )=AutD(V ), estabishing 8.12.2. Next we prove 8.12.1 by applying Lemma 2.3 in [BCGLO1] (which we refer to in this paragraph as the Lemma) to D and K = {W ∈Hc : |T : W |≤n} in the roles of the system F and the set H of the Lemma. First by 8.18.2, condition (IIB)K of the Lemma holds. Second, by 8.17, condition (IIA)K of the Lemma holds. Therefore by part (b) of the Lemma, condition (II)K of the Lemma holds. c Finally as the members of H are fully centralized in D, condition (II)K is precisely 8.12.1. Thus the proof of Theorem 8.12 is complete. Deﬁnition 8.19

~~Denote the fusion system D deﬁned in 8.4 by T F0, and call this system the product of T with F0.~~

~~Theorem 8.20. Let F0 be a normal subsystem on S0 of the saturated fusion system F on S.LetS0 ≤ T ≤ S with T strongly closed in S with respect to F. Then the product system T F0 is saturated.~~

Proof. We verify the hypothesis of Theorem 2.2 in [BCGLO1]forD = T F0 in the role of F and Hc in the role of H. By 8.5.4, D is Hc-generated, as defined in [BCGLO1]. Further hypothesis (*) of Theorem 2.2 in [BCGLO1] is satisfied by 8.9.2. Visibly H, and hence also Hc is closed under conjugation, so it remains to show that D is Hc-saturated: That is the axioms for saturation hold for members of Hc. Axiom I is satisfied by 8.11.3 and 8.12.2, and axiom II is satisfied by 8.12.1. Thus the proof is complete.

(8.21). (1) F0 T F0. ∼ + + (2) T F0/F0 = FT + (T ),whereT = T/S0. (3) The map X → XF0 is a bijection between the set of subgroups of T containing S0 and the set of saturated subsystems of T F0 containing F0. (4) T F0 is the smallest saturated subsystem of F on T containing F0,andthe p p unique saturated subsystem B of F on T such that O (B)=O (F0). ∈Dfc N D p ≤ (5) For U0 0 , (U0)= T F0,S0 (U0) and O (AutD(U0)) AutF0 (U0). frc c (6) (T F0) ⊆H . Proof. ∈Dfc ∈GN Let U0 0 and G ( (U0)). By 8.3, there is an isomorphism fc ϕ ∈ homF (U ,W) with W ∈F extending to an isomorphismϕ ˇ : G → G(U ,ϕ)= 0 S0 0

~~8. T F0 57~~

∈ H(W )NT (U0)ϕ,whereH(W )=HF,F0 (W ). By 8.5.3 and construction in 8.2, U0 F f −1 −1 0 ,soH(U0ϕ)ˇϕ = H(U0)=H G(U0,ϕ)ˇϕ = G,andG = H(U0)NT (U0). In particular ED F (U )=F (H(U )), so H(U )=HD F (U ). , 0 0 NS0 (U0) 0 0 , 0 0 By 3.6.1 in [A1], F0 is D-invariant, and by 8.20, D is saturated. As H(S0)= ≤ HD,F0 (S0)and[CT (S0),H(S0)] [CS(S0),H(S0)] = Z(S0), condition (N1) in the definition of normality is satisfied. Thus (1) is established. N ≤D ≤ As (U0) D,S0 (U0), G G = GD,F0 (U0). By 8.5.6, AutD(U0)=A(U0)= N AutN (U0)(U0)=AutG(U0), so G = GCG (U0)=GCT (U0)=G.Thus (U0)= F F D NT (U0)(G)= NT (U0)(G )= D,S0 (U0), establishing the first statement in (5). As G = H(U0)NT (U0),

AutD(U0)=AutG(U0)=AutH(U0)(U0)AutT (U0), p ≤ ≤ so O (AutD(U0)) AutH (U0) AutF0 (U0), completing the proof of (5). Part (6) follows from 8.9.2. By construction in 8.4, D is a subsystem of F on T , and by 8.20, D is saturated. Let U ∈Dfc and set T p =[T,Op(D)]. Suppose first that U ∈Hfc. Then G(U)=NG(U), where G = G(U0), and as we saw above, G = HQ,where p p H = H(U0)andQ = NT (U0). Hence O (G(U)) ≤ H,so[U, O (AutD(U))] = p p p [U, O (G(U))] ≤ U∩H ≤ S0.ThenifT ≤ S0,wehave[U, O (ND(U))AutT p (U)] = p p frc [U, O (AutD(U))][U, T ] ≤ S0. In general, U ≤ V ∈D with G(U)=NG(V )(U) ∈Hc and AutD(U)=AutAutD(V )(U). By (6), V . Then by the first case, p p [U, O (AutD(U))] ≤ [V,O (AutD(V ))] ≤ S0. Thus by 7.8.1, p p fc T = [U, O (AutD(U))] : U ∈D ≤S0. p f p Let P ∈ O (D) .ThenP ≤ T ≤ S0. Hence there exists α ∈ homD(NT (P ),T) Df ∈Dfc p ≤ with both U = Pα and V = UCS0 (U)in 0 .ThenV 0 ,soO (AutD(V )) p ≤ AutF0 (V )by(5).ThenasAutD(U)=AutAutD(V )(U), O (AutD(U)) AutF0 (U). Hence p p −∗ ≤ −∗ ≤ O (AutD(P )) = O (AutD(U))α AutF0 (U)α AutF0 (P ). p p p f p But by 7.17.1, O (D)=O (AutD(P ))AutT p (P ):P ∈ O (D) ,soO (D) ≤F0. Then by 7.17.3 and 7.7.4, p p p p p O (D)=O (O (D)) ≤ O (F0) ≤ O (D), p p so O (D)=O (F0). On the other hand suppose B is a saturated subsystem of F on T containing F ∈Dfc ∗ ∈G D ∈G D 0, and let U0 and G ( B,T (U0)). As H(U0) ( F0,S0 (U0)), G(U0)= ∗ fc H(U0)NT (U0) ≤ G ,soN (U0) ≤NB(U0). Then for U ∈N(U0) , N (U)= ≤ ≤ ≤B D≤B NN (U0)(U) NB(U), so A(U)=AutN (U)(U) AutB(U) . Therefore by 8.5.4. Thus D is the smallest saturated subsystem of F on T containing F0. p p Assume further that O (B)=O (F0). Then applying 7.19 to the inclusion p p D≤B, and recalling that O (D)=O (F0), we conclude that D = B, completing the proof of (4). + By definition of D/F0 in 8.6 in [A1], D/F0 = N ,whereN = ND(S0)and + N is defined earlier in section 8 of [A1]. But N = FT (G), where G = G(S0), + + + so by 8.8 in [A1], N = FT + (G ), where G = G/H and H = H(S0). Then as + ∼ G = HT, G = G/H = T/S0, completing the proof of (2).

~~58 MICHAEL ASCHBACHER~~

Let X be the set of normal subgroups of T containing S0 and X ∈X.Then X is the set of overgroups of S0 in T strongly closed in T with respect to T ,and then as S is strongly closed in T with respect to D, X is the set of overgroups of S0 strongly closed in T with respect to D by 8.6.2. Therefore applying 8.20 to the saturated fusion system D in the role of F, XF0 is saturated. Further by (4), p p XF0 is the unique saturated subsystem B of T F0 on X with O (B)=O (F0). p p Also for any saturated subsystem E of D on X containing F0, O (F0) ≤ O (E) ≤ p p O (D)=O (F0) by 7.17 and (4), so XF0 is the unique saturated subsystem of D on X containing F0. Then (3) follows by induction of the subnormal length in T of overgroups X of S0 in T . (8.22). F = SOp(F). p Proof. Apply 8.21.4 to O (F)andS in the roles of F0 and T . Observe that Theorem 5 follows from parts (3) and (4) of 8.21, and from 8.22. (8.23). Let U ∈Ff .Then (1) NF (U), CF (U),andInnF (U) are saturated fusion systems. (2) If E F is a system on T and U ≤ T then NE (U) NF (U), CE (U) CF (U),andInnE (U) InnF (U). Proof. Part (1) follows from 1.2 in [A1]. Assume the hypothesis of (2), let U ≤ T ,letQ = NT (U), and let (A, B)be(NE (U),NF (U)), (CE (U),CF (U)), or (InnE (U),InnF (U)). Then A and B are saturated by (1), while A is B-invariant by 3.6.1 in [A1]. Observe CT (Q) ≤ CT (U) ≤ TU ≤ Q,soQ = TQ. By 6.10.1 in [A1], ˆ N(D(U), E(U)) is satisfied. Thus each φ ∈ AutE(U)(Q) extends to φ ∈ AutD(U)(VQ) ˆ such that [φ, CS(Q)] ≤ Z(Q). Let A = Q ∩A.ThenA = Q, CT (U), TU for A equal to NE (U), CE (U), InnE (U), respectively. In particular A is φ-invariant. Let ψ ∈ AutA(A). In the third case where A = TU , ψ certainly acts on TU .In the second φ extends to ψ˜ centralizing U as ψ ∈ CE (U). In the first case, A = Q and ψ acts on U as ψ ∈ NF (U), so again ψ acts on TU . Thus in each case an extension ψ˜ of ψ acts on TU and U,soψ˜ ∈E(U). In particular by an earlier remark, in ˆ ˆ the first case, ψ extends to ψ ∈ AutD(U)(VQ) such that [ψ, CS(Q)] ≤ Z(Q). Thus ψˆ ∈ AutB(VQ), so (N1) is satisfied by (A, B) in this case, and hence A B. ˜ So assume we are in case two or three, where ψ ∈ AutE (U)(TU ). Let H = H(U) ˜ be the normal subgroup of G(U) supplied by Theorem 2 in [A1]. Then ψ = ch|TU for some h ∈ H.Now[h, CS(TU )] ≤ CH (TU )=Z(TU )=Z(A), so [h, CD(A)] ≤ Z(A) ∩B where D = S ,sinceCD(A)=CS(TU ). Thus ch|ACD (A) is an extension of ψ to ACD(A), so again (A, B) satisfies (N1), and hence A B. (8.24). ∈Ff p Let Q and define NF0 (Q)=NS0 (Q)O (NQF0 (Q)).ThenNF0 (Q) NF (Q). fc Proof. Set D = N (Q)andD = NF (Q). Define P = {P ∈D : Q ≤ P }, S D0 0 ∈Ff and for P 0 define V0(P )=PNS0 (P ), V (P )=V0(P )CS(V0(P )), let M(P )be amodelforNNF (V (P ))(P ), and K(P ) the normal subgroup of M(P ) which is a model for NF0 (V0(P )). Let P1 ∈Pand observe that [Q, P1] ≤ Q0 ≤ P ,soQ acts on P1.SetW1 = W (P1)=P1CD(P1) and let G(P1)beamodelforND(W1)=N(Q, P1). For

~~8. T F0 59~~

U a saturated fusion system on U and R ≤ U,writeAU (R) for the set of α ∈ f homU (NU (R),U) with Rα ∈U .Letα1 ∈ AF (P1)andsetP = P1α1 and W = → W (P ). Then 2.2 in [A4]saysα1 : N(U, P1) N(P, Qα1)=NNF (W )(Qα1)is an isomorphism which induces an isomorphismα ˇ1 : G(P1) → X(P )=X1,where X(P )=NM(P )(Qα1) ∩ N(W ). ≤ ≤ Claim Y1 = Y (P )=NK(P )(Qα1) X1.For[CDα1 (P ),NK(P )(Qα1)] ∩ ∩ CK(P )(P ) N(Qα1), which has Sylow p-subgroup CD0α1 (P )=Z(P ), so CK(P )(P ) ≤ ≤ N(Qα1)=Z(P )and[CDα1 (P ),NK(P )(Qα1)] Z(P ) CDα1 (P ), establishing the −1 claim. This allows us to deﬁne H(P1)=Y1αˇ1 .ThusH(P1) G(P1)andone can check that H(P1) is independent of α1. f Now let P1 ≤ P2 ≤ D0 with P2 ∈ ND(W1) and let β ∈ AD(P2). As P1 ≤ P2, W1 acts on P2,soW1β = W (P1β). Similarly CD(P2β) ≤ CD(P1β), −1 so W (P2β)actsonP1β and W (P2β)β = P2CD(P2)=W2 ≤ ND(P1). Set → N(Q, P1,P2)=NN(Q,P1)(W2); by 2.2 in [A4], β : N(Q, P1,P2) N(Q, P2β,P1β) ˇ → is an isomorphism inducing an isomorphism β : NG(P1)(P2) NG(P2β)(P1β). Let α2 ∈ AF (P2β), ζ = βα2, X2 = X(P2ζ), and Y2 = Y (P2ζ); we’ve seen that H(P2β) G(P2β). Next applying the isomorphism α1 : N(Q, P1) → N(P, Qα1), we get an iso- → morphism α1 : N(Q, P1,P2) N(P, Qα1,P2α1) with NG(P1)(P2)ˇα1 = NX1 (P2α1), and similarly NG(P2β)(P1β)ˇα2 = NX2 (P1ζ). Finally −1 ∩ ∩ → ∩ ∩ γ = α1 ζ : NS (W ) N(W2α1) N(Qα1) NS(W1ζ) N(W2ζ) N(Qζ) → F F induces an isomorphismγ ˇ : NX1 (P2α1) NX2 (P1ζ). As 0 is -invariant,

~~NY1 (P2α1)ˇγ = NY2 (P1ζ). Thus ˇ −1 ˇ −1 −1 NH(P1)(P2)β =NY1 (P2α1)ˇα1 β =NY1 (P2α1)ˇγαˇ2 =NY2 (P1ζ)ˇα2 = NH(P2β)(P1β).~~

Pick a set O of representatives in P for the orbits of D on overgroups of Q0 in D0,andforR ∈O,setA(R)=AutH(R)(R). As H(R) G(R), A(R) AutD(R). Extend A to a constricted D-invariant map A on D0 using construction (3) in section 5 of [A1]. Set E = E(A)asinsection5of[A1]. We next prove A is a normal map on D0 by verifying the two hypotheses of 7.7.2 in [A1], and appealing to that lemma. Hypothesis (i) holds by construction ∈P ˆ of H(R)forR . Hypothesis (ii) holds as NH(P1)(P2)β = NH(P2β)(P1β). Let B = QF0. Applying 8.21.3 with T = S and X = QS0, we conclude that p p B is a saturated fusion system on X. Similarly O (B)=O (F0) by 8.21.4. Set J = NB(Q). Applying 8.23.1 to B, we conclude that J is a saturated fusion system p J J J p J on NX (Q)=QNS0 (Q). By 7.7.3, O ( ) , and by 8.21, = O ( )NX (Q)= p p O (J )QD0,soJ = QL,whereL = D0O (J ). By 8.21, L is saturated and J - invariant. As A is a normal map, E = E(A) is a normal subsystem of D on D0 by Theorem 2in[A1]. By construction and the deﬁnition of B in 8.4, E≤NB(Q)=J . Then as E is D-invariant, E is also J -invariant. By 5.5.2 in [A1], AutE (D0)= L A(D0)=AutH(D0)(D0), and by parts (1) and (2) of 8.6 applied to , AutL(D0)= E L AutH(D0)(D0), so = by 5.9 in [A1]. This completes the proof of the lemma.

~~CHAPTER 9~~

~~Components~~

~~In this section F is a saturated fusion system over the ﬁnite p-group S. (9.1). Assume S/Z(F) is abelian. Then (1) S = Op(F). (2) F is not quasisimple.~~

frc Proof. Let U ∈F , G = GF,S(U), Q = NS (U), and Z = Z(F). Then Z ≤ Z(G) ≤ U by 7.11.12, and as S/Z is abelian, [Q, U] ≤ Z. Hence as U = F ∗(G), it follows that Q = U.ThusU = NS(U), so U = S. Therefore S = Op(F)by 7.12.2. Hence (1) holds. Suppose F is quasisimple. By 7.14, F = Op(F), so FS(S) is a proper normal subsystem of F by 7.9. Thus FS (S) ≤ Z(F) by 7.16, so AutF (U) is a 2-group for each U ≤ S, and hence F = Op(F) by 7.12.3, contrary to an earlier remark.

(9.2). Assume Ei F on Ti for i =1, 2,with[T1,T2]=1and T1 ∩T2 ≤ Z(Ei) for i =1, 2.ThenE1E2 is a central product of E1 and E2,soE2 ≤ CF (E1) . Proof. The first remark follows from Theorem 3, and then the second follows from the first, the definition of the central product in section 2, the definition of CF (E1) in 6.8, 6.7.1, Theorem 4, and Alperin’s Fusion Theorem (cf. A.10 in [BLO]).

A subsystem C of F is a component of F if C is a quasisimple subnormal subsystem of F.WriteComp(F) for the set of components of F. (9.3). Let C∈Comp(F) be a system on T and E F asystemonR. Then either C∈Comp(E) or T ∩ R ≤ Z(C). Proof. By 7.2 there is I = C∧E subnormal in C and E on T ∩ R.AsI is subnormal in C and C is quasisimple, either I≤Z(C)orI = C by 7.16. In the latter case C = I is a quasisimple subnormal subgroup of E,soC∈Comp(E). In the former, T ∩ R ≤ Z(C).

~~(9.4). If F is quasisimple then Comp(F)={F}. Proof. If D∈Comp(F) then by 9.3 applied to F, D in the roles of C, E, F∈Comp(D). In particular F≤D≤F, so the lemma holds.~~

p p (9.5). (1) If F = O (F) then O (CAut(F)(S/Z(F))) = 1. (2) Assume F˜ is a saturated fusion system on S˜ with F≤F˜,andX ≤ F ˜ F˜ F≤ CS˜(S/Z( )) with X strongly closed in S with respect to .Then NF˜(X),and p p if in addition F = O (F)=O (F) then F≤CF˜(X).

~~62 MICHAEL ASCHBACHER~~

Proof. Let Z = Z(F). In (1) let α ∈ CAut(F)(S/Z)beap -element. In (2), ∈ ≤ ∈Ffrc set T = CS˜(S/Z) and let α = ct for some t T . By 7.11, Z U for each U , and hence as [α, S] ≤ Z, α acts on U.LetG = GF,S(U)andQ = NS (U). Assume the hypothesis of (1). By 2.3 and 2.4 in [A1], there exists β ∈ Ext(α), an automorphism of G extending α|Q ∈ Aut(Q). We will show that β centralizes Op(G). Thus α centralizes S ∩ Op(G), so as F = Op(F), α centralizes S by 7.7.2. Hence α =1. It remains to show β centralizes Op(G). Form the semidirect product H = GB of G by B = β .Now[Q, α] ≤ Z ≤ Z(G) by 7.11, so B ≤ R = CH (Q/Z) ≤ p O2(G)B.AsB is a p -group, BZ = O (R)Z H, so by a Frattini argument, p p H = ZNH (B). Also [NG(B),B] ≤ G ∩ B =1,soO (G)=O (NG(B))Z = p O (NG(B)) ≤ CG(B). Next assume the hypothesis of (2). Conjugating in F˜, we may assume U ∈ F˜f f and Z ∈ NF˜(U) .Forg ∈ G, ∈ ∩ ≤ ≤ [α, cg] CAutF˜ (U)(U/Z) CAutF˜ (U)(Z) O2(NAutF˜ (U)(Z)) AutS˜(U), ∈ ∈ ≤ so t Ncg and hence cg extends to φ AutF˜(UT). Further X T and X is strongly closed in S˜ with respect to F˜,soφ acts on X.ThusAutG(U) ≤ NF˜(X). Finally assume F = Op(F)=Op (F). Now [S, X] ≤ Z ∩ X,soasF = Op(F), p F≤CF˜(X/(Z ∩ X)). Then as F = O (F) centralizes Z, also F≤CF˜(X), completing the proof of (2).

~~(9.6). Let C∈Comp(F) be a system on T and E F asystemonR. Then either C∈Comp(E) or [T,R]=1.InparticularifC ∈ Comp(F) −{C} is a system on T ,then[T,T]=1and T ∩ T ≤ Z(C) ∩ Z(C).~~

Proof. Note that 9.4 and the ﬁrst statement in the lemma imply the second statement, so it remains to prove the ﬁrst statement. Let C = Fn ··· F0 = F be a subnormal series for C. Assume the lemma fails, and choose a counterexample C, E, F with F minimal. By 9.3 and 9.4, n>0. By minimality of F,foreach ∗ ∗ ∗ C ∈ Comp(F1) −{C} on T ,[T,T ] = 1. As the lemma fails, C ∈/ Comp(E), so E = F,andT ∩ R ≤ Z = Z(C)by9.3.LetE + be a proper normal subsystem of F on T + containing E.IfC∈Comp(E +) then the lemma holds by minimality of F. On the other hand if T centralizes T +, it also centralizes R, so replacing E by E +, we may assume E F. Assume r ∈ R − CS(T ). Then cr ∈ Aut(F1), so C = Ccr ∈ Comp(F1)by 7.3. Suppose C = C;then[T,Tr] = 1 by the previous paragraph. Now [T,r] ≤ r TT ∩ R = R0,andT ∩ R0 ≤ T ∩ R = Z.Thusfort ∈ T ,[T,[t, r]] ≤ Z. −1 r r r But [t, r]=t t with t ∈ T ≤ CS(T ), so for all t ∈ T ,[T,t] ≤ Z.Thisis contrary to 9.1.2, so we conclude that R acts on C, and hence also on T .Thus [T,R] ≤ T ∩ R ≤ Z,soR centralizes T by 9.5.2.

~~(9.7). If C∈Comp(F) is a system on T ,then: (1) C is the unique component of F on T . (2) If Z ≤ CS(T ) and α ∈ Aut(F) acts on ZT and Z,thenCα = C.~~

~~Proof. If C is a second component on T ,thenT is abelian by 9.6 and 9.4. But this contradicts 9.1.2. Hence (1) holds.~~

~~9. COMPONENTS 63~~

Assume the hypothesis of (2). As α ∈ Aut(F), α permutes Comp(F), so Cα is a component on Tα.Asα acts on TZ and Z, TZ = TαZ.IfC = Cα then [T,Tα]=1 by 9.6, so TZ = TαZ centralizes T , and again 9.1.2 supplies a contradiction.

Deﬁne E(F) to be the normal subsystem of F generated by Comp(F). (9.8). (1) E(F) is a characteristic subsystem of F. {C ∈ } F ⊆ (2) Let i : i I be the set of components of . Then for each J I, C C F J = j∈J j is a normal subsystem of E( ), which is a central product of the C ∈ C ≤ C ⊆ − components j , j J,and J CE(F)( K ) for each K I J. E F E C F E { ∈ C ∈ (3) If ,thenE( )= j∈J(E) j ,whereJ( )= j I : j Comp(E)}. Proof. Assume F is a minimal counterexample and let D be a proper normal subsystem of F. Then by minimality of F, E(D)= Cj j∈J(D) satisﬁes the conclusions of the Theorem. In particular E(D)charD,soE(D) F by 7.4. Pick D so that J = J(D) is of maximal order m.IfJ = I then (1) and (2) hold. Further if F = E F, then we’ve seen that E(E) is the product of the components Ck with k ∈ J(E)andE(E) F.Thus(3)holdstoo.Therefore ∈ − F C J = I,sowecanpickk0 I J. The lemma follows from 9.4 if = k0 ,sowe C ∈ E may take k0 Comp( ). Let E¯ = E(E), and E, D, E¯ be systems on E,D,E¯ ≤ S, respectively, K = J(E) and U = K − J.AsE F, AutF (E) ≤ Aut(E) by 3.3 in [A1], so α ∈ AutF (E) permutes Comp(E). For k ∈ J ∩ K, Ek = S ∩Ck ≤ D,soasD is strongly closed in S with respect to F, Ekα ≤ D.IfCkα/∈ Comp(D), then [Ekα, D]=1 by 9.6, contrary to Ekα ≤ D and 9.1.2. Thus Ckα ∈ Comp(D), so AutF (E) {C ∈ ∩ } {C ∈ } ¯ permutes j : j J K , and hence also Δ = u : u U . Similarly AutE (E) Uˆ C permutes Δ. Let = u∈U u be the direct product of the members of Δ, and ˆ U = Δ ≤E¯. From 2.3, Uˆ is generated by the maps φu of 2.3, with φu a Cu-map, u ∈ U, so its image U is generated by the images of these maps under the natural morphism θ of 2.11. Therefore as AutF (E)andAutE (E¯) permute Δ, it follows that these groups are subgroups of Aut(U). As (2) holds in E, U E¯,andthenas AutE (E¯) ≤ Aut(U), it follows from 7.4 that U E.ThenasAutF (E) ≤ Aut(U), another application of 7.4 says that U F.FurtherifE(D)andU are systems on TJ and TU , respectively, then [TJ ,TU ] = 1 by 9.6. Thus F = E(D)U F by Theorem 3. Thus as k0 ∈ U, F = F by maximality of m. In particular F = E(F), so (1) holds. Next by Theorem 1, V = E(D)∧U is a normal subsystem of F on V = TU ∩TJ . frc As [TU ,TJ ]=1,V is abelian. Thus V = {V },soOp(V)=V by 7.12.2. Thus V ≤ Op(Y)forY∈{E(D), U}. As (2) holds in Y,eachC∈Comp(Y)isnormal Y ∩C ∈Yfc in .LetT = S and W T .ThenV is strongly closed in CTY (W ), so V GY,T (W ), and then setting H = HY,C(W ), [V,H] ≤ V ∩ H ≤ Z(H)as p V ∩ T ≤ Z(C) by 9.3. Thus H = O (H)NT (W ) centralizes V ,soC = AutC(W ): W ∈Cfc centralizes V .ThusV ≤ Z(Y). Therefore F is the central product of E(D)andU by 9.2. In particular Ci F for each i ∈ J ∪ U,soU = {k0} by maximality of m.Ask0 was arbitrary in I −J, it follows that I = J ∪U, and hence

~~64 MICHAEL ASCHBACHER~~

~~(2) holds. Finally if E F, then (3) holds for E trivially if E = F,andifE is proper, we saw that (3) holds earlier.~~

~~(9.9). (1) E(F) centralizes Op(F),andE(F)Op(F) is a central product of E(F) and Op(F). ∗ (2) F (F)=E(F)Op(F) is a characteristic subsystem of F.~~

Proof. Let C∈Comp(F)onT and R = Op(F). By 9.6, either C∈Comp(R) or T centralizes R, and by 7.13.1 the latter holds. Thus R centralizes E,where E(F) is a system on E ≤ S.NextV = R ∩ E E(F), so arguing as in the proof of 9.8, V ≤ Z(E(F)). Then (1) follows from 9.2. By (1) and Theorem 3, ∗ F (F) F.ThenasE(F)andOp(F) are characteristic, (2) holds.

~~We call F ∗(F)thegeneralized Fitting subsystem of F.~~

~~(9.10). If E = Op(E) F then E≤Op(F). Proof. This follows by induction on the length of a subnormal series for E.~~

∗ ∗ (9.11). CF (F (F)) = Z(F (F)). ∗ Proof. Let R = Op(F), E = CF (F (F)) be a system on T ,andE(F)a system on Q. Then by 9.9.2 and Theorem 4, E F. Next E(E) ≤ E(F) by 9.8, so as T centralizes Q, T ∩ Q is abelian. Thus ∗ E(E) = 1 by 9.1. Similarly Z0 = Op(E) ≤ R,soasE centralizes F (F), Z0 ≤ Z = Z(F ∗(F)), and of course the opposite inclusion is trivial, so F ∗(E)=Z.Further as E centralizes F ∗(F), Z = Z(E). Thus replacing F by E, it remains to show that if F ∗(F)=Z(F)=Z,thenF = Z. Assume otherwise and choose D to be minimal subject to D F and D Z. In particular each proper normal subsystem of D is contained in Z.IfOp(D) = D p ∗ then O (D) ≤ Z,soD = Op(D) by 7.12.3, contrary to 9.10 and F (F)=Z.Thus D = Op(D), D is subnormal in F, and each proper normal subsystem of D is in Z; hence, using 7.16, D∈Comp(F), contradicting F ∗(F)=Z.

~~Observe that Theorem 6 follows from 9.8, 9.9, and 9.11.~~

(9.12). Let E = CF (E(F)) and Q = CS(E(F)).Then ∗ (1) F (E)=Op(F)=R. ∗ (2) CQ(R)=Z(F (F)) = Z(R). (3) E is constrained, so E = FQ(G),whereG ∈G(E), Q ∈ Sylp(G),and R = F ∗(G). Proof. If C is a component of E then C is also a component of F,andweobtain a contradiction from 9.1 as in the proof of the previous lemma. Thus F ∗(E)= ∗ Op(E). Hence F (E) ≤ R by 9.10. On the other hand R ≤E,soR ≤ Op(E), and hence (1) holds. ∗ ∗ Next CQ(R) ≤ CQ(F (E)) ≤ Z(F (E)) = Z(R)=Z by 9.11 and (1). Thus (2) holds. By (2), E is constrained, so E = FQ(G)forG ∈G(E), and Q ∈ Sylp(G)by ∗ 2.5 in [A1]. Then as R = Op(E), also R = Op(G)=F (G). (9.13). E F F − E D C Let , J = Comp( ) Comp( ),and = C∈J .Then D F,andED is the central product of E and D.

~~9. COMPONENTS 65~~

Proof. By 9.8, D E(F). Let E = S ∩E and D = S ∩D.AsE F, AutF (S ∩ E(F)) acts on J,soD F by an argument in the proof of 9.8. By 7.2.2, V = E∧D D. By deﬁnition of J, V contains no component of ∗ ∗ D,soF (V)=Op(V) ≤ Op(D). By 9.9.1, Op(D) ≤ Z(D), so F (V)=Z(V), and hence V = Z(V). Thus V≤Z(D). Next [D, E] ≤ D ∩ E = V ≤ Z(D), so E centralizes D by 9.5.2. By 9.2, it remains to show that V ≤ Z(E). If not as V E and V centralizes # E, there is a p element φ ∈ AutE (V ) .AsV ≤ Z(E), φ lifts to α ∈ AutE (E). Then as D centralizes E and E F, α lifts to β ∈ AutF (ED)with[β,D] ≤ Z(E). As φ is a p-element, we may choose β to be a p-element. As D is strongly closed, [β,D] ≤ E ∩ D = V , so by 9.5.1, β centralizes D, and hence also V , for our ﬁnal contradiction. (9.14). Let D = E(F) and E be a normal constrained subsystem of F.Then ED is the central product of E and D. ∗ Proof. As E is constrained, F (E)=Op(E), so E contains no components of F. Thus the lemma is a special case of 9.13.

~~CHAPTER 10~~

~~Balance~~

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

~~f (10.1). Let U ∈F and suppose C is a component of NF (U).ThenC≤ CF (Op(F)).~~

Proof. Let R = Op(F), D = NF (U), and Q = Op(D). By 1.2.1 in [A1], D is saturated. As C∈Comp(D), C centralizes Q by 9.9. Now NR(U) ≤ Q,soC fc ˜ centralizes NR(Q). Let W ∈C .Theneachφ ∈ AutC(W ) extends to φ =1onQ, andthenasR F, φ˜ extends to φˆ on WQR.Moreoverifφ is a p-element, then ˆ ˆ we can choose φ to be a p -element. Next as CR(Q) ≤ CR(U) ≤ Q ≤ CQ(φ), we conclude from the Thompson A×B-Lemma (cf. 24.2 in [FGT]) that the p-element φˆ centralizes R. Let P = S ∩C. By 7.8.1, fc P = [W, φ]:W ∈C ,φ∈ AutC(W ),φ a p -element . Then as [W, φ]=[W, φˆ]andφˆ centralizes R, it follows that P centralizes R,so p AutC(W )=O (AutC(W ))AutP (W ) ≤ CF (R). Thus fc C = AutC(W ):W ∈C ≤CF (R), completing the proof.

~~(10.2). Let U ∈Ff .Then (1) CF (U),andInnF (U) are normal subsystems of NF (U). (2) E(NF (U)) = E(CF (U)) = E(InnF (U)).~~

~~Proof. It is straightforward to check that A∈{CF (U),InnF (U)} is invariant under B = NF (U), and that (A, B) satisﬁes (N1). By 8.23.1, A and B are saturated, so (1) holds.~~

~~Next E(B) ≤Aby 9.9.1, so (1) and Theorem 9.8.3 imply (2).~~

~~f (10.3). Let U ∈F and suppose U centralizes E(F).ThenE(F)=E(NF (U)).~~

Proof. Let D = NF (U), E = E(F), and E = S ∩E.AsU centralizes E, E≤D.AsE F, E is saturated, and by 3.6.1 in [A1], E is D-invariant. Each ˆ ˆ φ ∈ AutE (E) extends to φ ∈ AutF (VE)with[φ, CS(E)] ≤ Z(E). Further if φ is a p-element, we may pick φˆ to be a p-element, and

~~[φ,ˆ CS(E)] ≤ CS(E) ∩ Z(E)=Z(E) ≤ CE(φ)=CE(φˆ), so φˆ centralizes CS(E). Then as U ≤ CS(E), φˆ is a D-map. Therefore E D,so E≤E(D).~~

~~68 MICHAEL ASCHBACHER~~

~~Conversely suppose C∈Comp(D). Then C centralizes Op(F) by 10.1, and if ∗ C E then C centralizes E,soC≤CF (F (F)) = Z(F), contradicting C perfect. Hence E = E(D).~~

f Theorem 10.4. Let U ∈F .ThenE(NF (U)) ≤ E(F). The proof involves a series of reductions. f Let P = P(F)bethesetofpairs(U, C)whereU ∈F and C∈Comp(NF (U)), but C E(F). Assume the Theorem is false and pick F such that P(F) = ∅,andwithF minimal subject to this constraint. Let (U, C) ∈P. (10.5). U does not centralize E(F). Proof. This is a consequence of 10.3.

(10.6). Assume V NS (U) centralizes C, V is not normal in F,andei- ther V ≤ Op(NF (U)) or V = S ∩E for some E NF (U). Then for β ∈ f ∗ homF (NS (V ),S) with Vβ ∈F , Cβ ≤ E(NF (Vβ)),sothereexistsacomponent CV of NF (Vβ) with (Vβ,CV ) ∈P.

f Proof. Let N = NF (U). As V NS (U), V ∈N .IfV ≤ Op(N )thenC is a component of NN (V ) by 10.3. If V = S ∩E for some E N then C E by 9.1 and the assumption that V centralizes C. By 9.13, the product M of the components of N not in E is normal in N ; then the proof of 10.3 shows M NN (V ), so again C is a component of NN (V ). Let D = NF (Vβ) and pick α ∈ homD(Uβ,NS(Vβ)) f ∗ ∗ with U = Uβα ∈D .AsC is a component of NN (V ), C = Cβ α is a component of ND(U ). By hypothesis, D = F. Thus by minimality of F, C ≤ E(D). Then as E(D) D, Cβ∗ ≤ E(D). Thus as C E(F), also E(D) E(F), so the lemma follows.

(10.7). Z(F)=1. Proof. Assume Z = Z(F) =1,set F + = F/Z, and let θ : F→F+ be the natural map discussed in section 8 of [A1]. By 10.5, U Op(F), so V = UZ Op(F). Then appealing to 10.6 and replacing (U, C)by(Vβ,CV )asinthatlemma, we may assume Z ≤ U. By 8.5 in [A1], F + is saturated, and by 8.4.2 in [A1], U + = Uθ ∈F+f .As + + Z ≤ U, NF (U)θ = NF + (U ) by 8.4.1 in [A1]. Then by 8.9.2 in [A1], C = Cθ + + is subnormal in NF + (U ), and as C is quasisimple, so is C using 8.10 in [A1]. Therefore by minimality of F, C+ ≤ E(F +). Let D+ be a component of F + and D its preimage in F. By 7.15, D = Op(D)Z. Further using 8.10 in [A1], Op(D) is quasisimple and subnormal in F,soOp(D) ∈ Comp(F). Thus E(F)θ = E(F +)andC = Op(C) ≤ E(F), contrary to the choice of C.

~~(10.8). Op(F)=1.~~

Proof. Assume R = Op(F) = 1, and pick V of order p in Z(S) ∩ R.Then V ∈Ff centralizes C and U,soifV is not normal in F, then by 10.6, there exists CV ∈ Comp(NF (V )) with (V,CV ) ∈P. But this is contrary to 10.5.

~~10. BALANCE 69~~

Thus V F.LetE = CF (V ). By 10.7, E is proper in F.AsV ≤ Z(S), S = E∩S. By 10.2, E F. By 8.23.2, NE (U) NF (U), so by 9.8, C∈ Comp(NE (U)). Then by minimality of F, C≤E(E), contrary to 10.2.2.

~~In the remainder of the proof let E = E(F)andsetE = S ∩E.LetT = S ∩C, EU = NE (U), Q = NS (U), QT = NQ(T ), FU = NF (U), and deﬁne EU = NE (U) as in 8.24.~~

~~(10.9). EU FU . Proof. This follows from 8.24.~~

~~(10.10). (1) C centralizes EU . f (2) There exists β ∈ homF (NS(EU ),S) such that EU β ∈F,andthereis (EU β,CE) ∈P.~~

~~Proof. As C E, (1) follows from 9.13 applied to FU , EU in the roles of F and E. f Let β ∈ homF (NS (EU ),S) with EU β ∈F . Now (2) follows from 10.6.~~

Appealing to 10.10.2 and replacing (U, C)by(EU , CE), we may assume U ≤ E f and U = NE (U), so U = E.LetZ be of order p in Z(S) ∩ E.ThenZ ∈F .As Z centralizes C and E acts on Z, 10.6 says that C≤E(FZ ), where FZ = CF (Z), and (Z, CZ ) ∈Pfor some CZ ∈ Comp(FZ ). Replacing (E,C)by(Z, CZ ), we may C∈ F E E F assume Comp( Z ). Set Z = CE (Z). By 8.23.3, Z Z . F {C ∈ } { ∈ C E } D C Let Comp( Z )= i : i I , J = j I : j Z ,and = j∈J j .By 9.13, D FZ and EZ D is the central product of EZ and D. By the choice of C, C C ∈ = j0 for some j0 J. ∈Ffc F ≤ Let X E and X = NF (X). Then Z X,soCF (X)=CFZ (X). Then D≤ D D CFZ (X)=CF (X)). Claim CF (X). First is CF (X)-invariant by 3.6.1 in [A1], and D is saturated. Set D = D∩S. We must show each φ ∈ ˜ ∈ ˜ ≤ AutD(D) lifts to some φ AutCF (X)(DCS(DX)) with [φ, CS(DX)] Z(D), and as Inn(D) ∈ Sylp(AutD(D), we may take φ to be a p -element. But φ lifts to ˆ ∈ ˆ ≤ ˆ φ AutFZ (DCS(DZ)) with [φ, CS(DZ)] Z(D), and we may take φ to be a p-element. Then [E,φˆ] ≤ E ∩ Z(D) ≤ Z(D)asE is strongly closed in S,and ˆ ˆ ˆ φ|Z(D) = φ|Z(D) =1,soφ centralizes E as φ is a p -element. Thus we may take ˜ ˆ C∈ F φ = φ|DCS (XD), to complete the proof of the claim. By the claim, Comp( X ). E F ∗ E E ∈Ffc Next X = NE (X) X by 8.23. Further F ( X )=Op( X )asX E . C≤ F ≤ E ∈Ffc Thus E( X ) CFX ( X ) by 9.14. As this holds for each X E , it follows from the deﬁnition of CF (E(F)), that T ≤ CF (E(F)). This is impossible by 9.12.1 and 10.8. This completes the proof of Theorem 10.4. Observe that Theorem 7 is just a restatement of Theorem 10.4, so we have established Theorem 7. (10.11). ∈Ff {O ∈ Assume X = x is of order p in S and X .Let i : i I) be the orbits of X on Comp(F), F = B,andS = S ∩F.Then i B∈Oi i i (1) NE(F)(X) NF (X). F F (2) i XE( ) and NFi (X) NE(F)(X). (3) Suppose C∈Comp(NF (X)). Then either:

~~70 MICHAEL ASCHBACHER~~

(a) C is a full diagonal subsystem of F for some orbit O of length p.In ∼ i i particular C/Z(C) = B/Z(B) for B∈Oi. C∈ O (b) Comp(NXFi (X)) for some orbit i of length 1. Proof. As X induces a group of automorphisms of F, X permutes Comp(F). As |X| = p the orbits of X on Comp(F) are of length 1 or p. By 9.8, Fi E(F). Let E = S ∩ E(F)andEi = S ∩Fi. Parts (1) and (2) follow from 8.24. By 8.21, XE(F) is a saturated fusion system, and by 10.4 and 9.8, components of NF (X) are components of NXE(F)(X). Thus to prove the remainder of the lemma, we may replace F by XE(F), and assume F = XE(F). By 9.8, E(F) is the central product of its components. Further x induces an automorphism of E(F)oforder1orp permutating the components, so by 2.11, x lifts to an automorphismx ˆ of the direct product Eˆ(F) of the components of F. Let Z = Z(Eˆ(F)). Next as X permutes Oi, X acts on Ei and induces a group of automorphism of Fi.ThusasFi E(F)andF = XE(F), we conclude from 7.4 that Fi F, completing the proof of (2). Finally assume the hypothesis of (3), and let T = S ∩C.ThenT ≤ E.LetTˆ be the preimage of T in Eˆ(F)andTî the projection of Tˆ on Eî. By 9.1, T/Z(C) is not abelian, so for some i, Tî/(Z ∩ Tî) is nonabelian. Thus asx ˆ centralizes Tˆ, ˆ [T,CEˆ (ˆx)] Z,soT does not centralize CEi (x). We conclude from 9.13 applied i F E C≤F O to NF (X)andNFi (X) in the roles of and ,that i. In particular if i is of order 1 then (3b) holds, while if Oi is of order p then (3a) holds by 2.12.

~~CHAPTER 11~~

~~The fundamental group of F c~~

In this section F is a fusion system over the ﬁnite p-group S.WriteF c for the set of all F-centric subgroups, and for the full subcategory of F on the set of centrics. Set Ω=Ω(F)= homF (P, Q), P,Q∈F c c and write F = F (F)forthefreegrouponΩ.Letπ = π1(F )=F/K,where K = K(F) is the normal subgroup of F generated by: c (F1) The inclusion maps ιP,Q : P → Q, for all P, Q ∈F with P ≤ Q,and c (F2) For all P, Q, R ∈F , α ∈ homF (P, Q), and β ∈ homF (Q, R), the elements (αβ) · β−1 · α−1. For f ∈ F ,setf˜ = Kf ∈ π, and write η : F → π for the natural map η : f → f˜.Letθ = θF :Ω→ π be the function θ : α → α˜.Thusθ = η ◦ ιΩ,F ,where ιΩ,F :Ω→ F is inclusion. (11.1). Let P, Q, R ∈Fc.Then (1) If P ≤ Q then θ(ιP,Q)=1,whereιP,Q : P → Q is inclusion. (2) θ(1P )=1,where1P is the identity map on P . −1 −1 (3) For α ∈ homF (P, Q), θ(α )=θ(α) . (4) For all α ∈ homF (P, Q) and β ∈ homF (Q, R), θ(αβ)=θ(α)θ(β).

Proof. Part (1) follows as K contains all elements of type (F1). Then as 1P = −1 −1 −1 ιP,P , (1) implies (2). Take R = P and β = α in (F2) to get (αα ) · α · α ∈ K. −1 −1 Then as αα =1P ,itfollowsfrom(2)that1=θ(α)θ(α ), so (3) holds. Finally by (F2) and (3), 1 = θ(αβ)θ(β−1)θ(α−1)=θ(αβ)θ(β)−1θ(α)−1, so (4) follows. Given a group G, deﬁne hom(F,G) to be the set of all functions ρ :Ω→ G satisfying (M1) and (M2): c (M1) For all P, Q ∈F with P ≤ Q, ρ(ιP,Q) = 1, and c (M2) For all P, Q, R ∈F, α ∈ homF (P, Q), and β ∈ homF (Q, R), ρ(αβ)= ρ(α)ρ(β). For example θ ∈ hom(F,π) by 11.1. (11.2). Suppose G is a group and ρ ∈ hom(F,G).Then c (1) There exists a unique group homomorphism ψ(ρ):π1(F ) → G such that ψ(ρ) ◦ θF = ρ. (2) The map ψ = ψF,G : ρ → ψ(ρ) is a bijection of hom(F,G) with c hom(π1(F ),G).

Proof. Let ι = ιΩ,F .AsF is free on Ω, there exits a group homomorphism ϕ : F → G with ϕ ◦ ι = ρ.By(M1)and(M2),K ≤ ker(ϕ), so ϕ induces φ : π → G with φ ◦ η = ϕ.Thenasθ = η ◦ ι,wehave φ ◦ θ = φ ◦ η ◦ ι = ϕ ◦ ι = ρ.

~~72 MICHAEL ASCHBACHER~~

As π = θ(Ω) , φ is the unique homomorphism with φ ◦ θ = ρ,soψ :hom(F,G) → hom(π, G) is an injection. If μ ∈ hom(π, G)thenasθ ∈ hom(F,π), μ = μ ◦ θ ∈ hom(F,G) with ψ(μ)=μ,soψ is a surjection, completing the proof of (2). (11.3). Suppose G is a group and ρ ∈ hom(F,G). Assume P, Q, R ∈Fc with R ≤ P ,andα ∈ homF (P, Q).Then (1) α|R ∈ homF (R, Q) and ρ(α|R)=ρ(α). (2) Let β : P → Pα be the isomorphism induced by α.ThenPα ∈Fc, β ∈ homF (P, Pα),andρ(β)=ρ(α).

~~Proof. First ι = ιR,P ∈ homF (R, P ), so α|R = ια ∈ homF (R, Q)and~~

ρ(α|R)=ρ(ια)=ρ(ι)ρ(α)=ρ(α), by (M1) and (M2). That is (1) holds. c c Similarly as P ∈F , also Pα ∈F and β ∈ homF (P, Pα), while i = ιPα,Q ∈ homF (Pα,Q). Then α = βi,so ρ(α)=ρ(βi)=ρ(β)ρ(i)=ρ(β), by (M1) and (M2), so (2) holds. In the remainder of the section, assume that F is saturated. Definition 11.4. ∈ F ∈ ∈Fc U Suppose G is a group and ρ hom( ,G). For s S and U ,writecs for the U → s s conjugation map cs : x x in homF (U, U ). Define { ∈ S } S(ρ)= s S : ρ(cs )=1 , set homρ(U, S)={α ∈ homF (U, S):ρ(α)=1},andsetAρ(U)=Aut(U) ∩ homρ(U, S). (11.5). Assume G is a group and ρ ∈ hom(F,G). Then for U ∈Fc, x ∈ S, and α ∈ homF (U, x ,S): U ∗ Uα Uα U ρ(α) (1) (cx )α = cxα ,soρ(cxα )=ρ(cx ) . U S (2) ρ(cx )=ρ(cx ). (3) If x ∈ S(ρ) then xF ⊆ S(ρ). Proof. For y ∈ Uα, U ∗ −1 x −1 xα xα Uα y(cx α )=((yα ) )α =((yα )α) = y = ycxα , so the first statement in (1) holds. Then the second statement follows from the first by applying ρ and using (M2). U S As cx =(cx )|U , (2) follows from 11.3.1. ∈ ∈ U S Suppose x S(ρ)isinU and α AutF (U). Then by (2), ρ(cx )=ρ(cx )=1, U U ρ(α) ρ(α) so by (1), ρ(cxα)=ρ(cx ) =1 = 1. Then by another application of (2), S U ∈ ρ(cxα)=ρ(cxα)=1,soxα S(ρ). Then Alperin’s Fusion Theorem (cf. A.10 in [BLO]) implies (3). (11.6). Assume G is a group and ρ ∈ hom(F,G).ThenS(ρ)=S.

Proof. Let x ∈ S, X = x , and pick α ∈ homF (X, S) such that U = VXα is of maximal order. Then for β ∈ homF (U, S), CS(Uβ) ≤ CS(Xαβ), so VUβ ≤ VXαβ, and hence |VUβ|≤|VXαβ|≤|VXα| = |U|.

~~11. THE FUNDAMENTAL GROUP OF F c 73~~

c Thus CS(Uβ) ≤ Uβ,soU ∈F . U ∈ S U Next cxα =1asxα Z(U), so by 11.5.2, ρ(cxα)=ρ(cxα) = 1. Therefore xα ∈ S(ρ), so x ∈ S(ρ) by 11.5.3. Notation 11.7. We recall some definitions and notation from [A1]and[BCGLO2]. For U ∈Ffc, p define B(U)=O (AutF (U)). By parts (1) and (3) of 5.2 in [A1], B defines a F B p F B constricted -invariant map on S.Set = O0 ( )=E(B). By 5.5 in [A1], is an F-invariant subsystem on S. Define 0 c AutF (S)=α ∈ AutF (S):α|P ∈ homB(P, S)forsomeP ∈F . 0 As B is F-invariant, AutF (S) AutF (S), so we can define 0 Γ=Γp (F)=AutF (S)/AutF (S). 0 c Let ζ : AutF (S) → Γ be the natural map ζ : α → AutF (S)α.ForP ∈F,and α ∈ homF (P, S), define

G(α)={(ϕ, φ):ϕ ∈ AutF (S),φ∈ homB(Pϕ,S), and α = ϕφ}. (11.8). Assume G is a group and ρ ∈ hom(F,G).Then fc (1) For P ∈F , Aρ(P ) AutF (P ),soAρ extends to a constricted F- invariant map Aρ on S. (2) E = E(ρ)=E(Aρ) is a normal subsystem of F on S. c (3) For R ∈F , homE (R, S)=homρ(R, S). (4) AutE (P )=Aρ(P ). B p F ≤E (5) = O0 ( ) . → (6) ψ(ρ)(π)=ρ(AutF (S)) and ρ0 = ρ|AutF (S) : AutF (S) ψ(ρ)(π) is a 0 surjective group homomorphism with AutF (S) ≤ ker(ρ0). fc Proof. Let P ∈F , D = AutF (P ), and B = Aρ(P ). Then B is the kernel of the group homomorphism ρ : D → π,soB D. Then (1) follows from 5.2.3 in [A1]. By (1) and 5.5.1 in [A1], E is an invariant subsystem of F on S. We claim A = Aρ is a normal map, as defined in Definition 7.4 in [A1]. To establish the claim, we must verify conditions (SA1)-(SA3) of 7.4 in [A1]forP . As the strongly closed subgroup T of 7.4 in [A1]isS, condition (SA1) is trivially satisfied. c c Let Q = NS (P ). As P ∈F and P ≤ Q, Q ∈F so CS(Q) ≤ Q. Hence for φ ∈ AutA(Q)(P ), φ is an extension of φ to QCS(Q)=Q,and[CS(Q),φ]= [Z(Q),φ] ≤ Z(Q), so condition (SA3) holds. c c Suppose P ≤ Q ≤ S.AsP ∈F, also Q ∈F.Letβ ∈ NA(Q)(P ). Then β ∈ Ω with ρ(β) = 1, so by 11.3, ρ(β|P ) = 1 and hence β|P ∈ NA(P )(AutQ(P )). On the other hand if γ ∈ NA(P )(AutQ(P )) then Q ≤ Nγ ,soγ extends toγ ˆ ∈ AutF (Q). Further by 11.3, ρ(ˆγ)=ρ(γ)=1,soˆγ ∈ NA(Q)(P ). That is NA(Q)(P )= NA(P )(AutQ(P )), so (SA2) holds, completing the proof of the claim. Observe (2) follows from the claim and Theorem 3 in [A1]. c Let R ∈F and α ∈ homE (R, S). As E = E(A), α = α1 ···αn with αi ∈ A(Ui) fc for some Ui ∈F .Thenρ(αi)=1,soρ(α)=ρ(α1 ···αn)=ρ(α1) ···ρ(αn)=1, so α ∈ homρ(R, S). Thus (3) holds, and (3) implies (4). By 5.3 in [A1], for each U ∈Ffc, B(U) ≤ A(U). Then by 5.5.6 in [A1], B = B(U):U ∈Ffc ≤A(U):U ∈Ffc = E,

74 MICHAEL ASCHBACHER so (5) holds. c Let R ∈F and α ∈ homF (R, S). By 3.3 in [A1], there is ϕ ∈ AutF (S) and φ ∈ homE (Rϕ, S) such that α = ϕφ.Thenρ(α)=ρ(ϕ)ρ(φ)=ρ(ϕ)by(3). → Hence ρ(Ω) = ρ(AutF (S)), while by (M2), ρ1 = ρ|AutF (S) : AutF (S) G is a group homomorphism. By 11.2.1, ρ(Ω) = ψ(ρ)(π), so ρ0 : AutF (S) → ψ(ρ)(π)is a surjective group homomorphism. Finally for γ ∈ AutF (S) such that δ = γ|P ∈ homB(P, S), also δ ∈ homE (P, S) by (5). Then by 11.3, ρ(γ)=ρ(δ)=1,so 0 AutF (S) ≤ ker(ρ), completing the proof of (6). The construction in parts (2) and (3) of the following lemma comes from the proof of 5.2 in [BCGLO2]. c (11.9). For each P ∈F and α ∈ homF (P, S): (1) G(α) = ∅. (2) For each (ϕi,φi) ∈ G(α), i =1, 2, ζ(ϕ1)=ζ(ϕ2). (3) Define ρ :Ω→ Γ=Γp (F) by ρ(α)=ζ(ϕ),for(ϕ, φ) ∈ G(α).Thenρ is a well defined member of hom(F, Γ). (4) ρ(Ω) = Γ. 0 (5) AutF (S)=AutE(ρ)(S). (6) E(ρ)=Op (F). c (7) ψF,Γ(ρ):π1(F ) → Γ is an isomorphism. Proof. Part (1) follows from 3.3 in [A1] and the observation in 11.7 that B is F-invariant. Assume the setup of (2), and regard φi as an isomorphism from Pϕi to Pα. −1 ∈ −1 ∈ Then ϕ1φ1 = α = ϕ2φ2,soμ = ϕ2 ϕ1 AutF (S), and ν = φ2φ1 homB(Pϕ2,S) ∈ 0 with ν = μ|Pϕ2 .Thusμ AutF (S)=ker(ζ), so ζ(μ) = 1. Therefore ζ(ϕ1)= ζ(ϕ2), establishing (2). By (2), ρ :Ω→ Γ is well defined. Suppose β ∈ homF (Pα,S). Let (ϕ, φ) ∈ G(α)and(Ψ,ψ) ∈ G(β). Then αβ = ϕφΨψ =(ϕΨ)(φΨ∗Ψ), with ξ = φΨ∗Ψ ∈ homB(PϕΨ,S), so (ϕΨ,ξ) ∈ G(αβ). Thus ρ(αβ)=ζ(ϕΨ) = ζ(ϕ)ζ(Ψ) = ρ(α)ρ(β), so ρ satisfies (M2). Next (1S, 1P ) ∈ G(ιP,S), so ρ(ιP,S)=ζ(1S) = 1. Therefore ρ satisfies (M1), completing the proof of (3). For ϕ ∈ AutF (S), (ϕ, 1) ∈ G(ϕ), so ρ(ϕ)=ζ(ϕ), and hence (4) holds. Moreover 0 this shows that Aρ(S)=ker(ζ)=AutF (S), so (5) follows from 11.8.4. By 11.8.2, E = E(ρ) is a normal subsystem on S, so the smallest such subsystem D = Op (F) is contained in E. On the other hand for each U ∈Ffc, B(U) ≤ 0 AutD(U), so as D is saturated, AutF (S) ≤ AutD(S). Then AutD(S)=AutE (S) by (5), and therefore by 5.9 in [A1], D = E, establishing (6). Finally by 11.1, θ ∈ hom(F,π), while by 11.2.1, ψ(θ)=1π is the identity → map on π.Thusπ = ψ(θ)(π), and by 11.8.6, θ0 = θ|AutF (S) : AutF (S) π 0 is a surjective group homomorphism with AutF (S) ≤ ker(θ0). Therefore |π|≤ 0 |AutF (S)/AutF (S)| = |Γ|. On the other hand ψ(ρ):π → Γ is surjective homomorphism by (4) and 11.2, so (7) follows. We are now in a position to prove Theorem 8. Part (1) of Theorem 8 follows from 11.9.7. We next prove part (2) of Theorem 8. By parts (5) and (6) of 11.9, AutOp (F)(S) 0 = AutF (S). Let Λ be the set of normal subsystems of F on S,andforE∈Λ, set

~~11. THE FUNDAMENTAL GROUP OF F c 75~~

0 p 0 μ(E)=AutE (S)/AutF (S). As O (F)isthesmallestmemberofΛ,AutF (S)= ≤ E F AutOp (F)(S) AutE (S). Further as , AutE (S) AutF (S). Therefore μ(E) is contained in the set Σ of normal subgroups of Γ. By 5.9 in [A1], μ is injective. Let Δ ∈ Σ, G =Γ/Δ, and β :Γ→ G the natural map. Form the map ρ ∈ hom(F, Γ) of 11.9.3, and set γ = β ◦ ρ,sothatγ ∈ hom(F,G). Hence by 11.8.2, E = E(γ) ∈ Λ, with AutE (S)=Aγ (S), where Aγ (S) is the preimage of Δ in AutF (S) under ζ.Thusμ(E)=Δ,soμ :Λ→ Σ is a bijection, establishing part (2) of Theorem 8. Part (3) of Theorem 8 follows from parts (1) and (2). Finally F is simple iﬀ condition (a) of part (4) of Theorem 8 holds and F = Op (F). Then (3) completes the proof of (4).

~~(11.10). Op (Op (F)) = Op (F).~~

~~Proof. Set E = Op (F). Then Op (E)charE F,soOp (E) F by 7.4. Thus Op (E) contains the smallest normal subsystem E of F on S,so Op (E)=E.~~

~~The next lemma and its proof are due to B. Oliver.~~

~~(11.11). Assume T ≤ S such that T = CS(T ) and T is weakly closed in S with respect to F. Assume Y ≤ T ∩ Z(S) with Y F ⊆ T .Set~~

AutF (T ) Δ= CAutF (T )(Y ) , Γ=AutF (T )/Δ, and ζ : α → Δα the natural map from AutF (T ) onto Γ.Then ∈ F (1) There exists ρ hom( ,AutF (T )) with ρ|AutF (T ) = ζ. (2) E(ρ) is a normal subsystem of F on S. (3) Let

p Σ={α ∈ AutF (S):α|T ∈ Δ} and Ξ={α ∈ AutF (S):α|T ∈ O (AutF (T ))}. 0 Then Ξ ≤ AutF (S) ≤ Σ=AutE(ρ)(S). c Proof. Let P, Q ∈F and α ∈ homF (P, Q). Then Y ≤ Z(S) ≤ P .Set F f −1 P0 = Y ∩ P , and let γ ∈ homF (P0,S) with U = P0γ ∈F .Thenβ = α γ ∈ −1 F ∈Ff homF (P0α, U)andα|P0 = γβ .As is saturated and U ,thereexists ˆ extensionsγ ˆ ∈ homF (Nγ ,S)andβ ∈ homF (Nβ,S)ofγ and β, respectively. Since ≤ F ≤ ∈ T is abelian and P0 Y T , it follows that for each t T , ct|P0 =1,so ˆ T ≤ Nγ . Similarly P0α ≤ T ≤ Nβ.ThusT γˆ = T β = T ,asT is weakly closed. −1 −1 ˆ | ˆ ∈ F | Therefore δ =ˆγ T β|T Aut (T ) extends γβ = α P0 . Therefore P { ∈ } ∅ (α)= φ AutF (T ):φ|P0 = α|P0 = .

Let A = AutF (T )andB = CA(Y ), so that B ≤ Δ. If φ, ψ ∈P(α)thenasY ≤ P0, φψ−1 ∈ B ≤ Δ, so that ζ(φ)=ζ(ψ). Thus we can deﬁne ρ(α)=ζ(φ)forφ ∈P(α), to obtain a map ρ :Ω→ Γ. c F Let R ∈F , μ ∈ homF (Q, R), and Q0 = Y ∩ Q .Then F F P0α = Y ∩ Pα ≤Y ∩ Q = Q0. Let φ ∈P(α)andψ ∈P(μ). Then · · (φψ)|P0 = φ|P0 ψ|Q0 = α|P0 μ|Q0 =(αμ)|P0 ,

76 MICHAEL ASCHBACHER so φψ ∈P(αμ). Therefore ρ(αμ)=ζ(φψ)=ζ(φ)ζ(ψ)=ρ(α)ρ(μ), so ρ satisfies (M2). Further 1T ∈P(ιP,S), so ρ(ιP,S)=ζ(1T ) = 1, and hence (M1) is also satisfied. Therefore (1) is established. Further (1) and 11.8.2 imply (2). Let D = AutF (S). Observe Σ = ker(ρ0), where ρ0 = ρ|D : D → Γ. By 0 (1) and 11.8.4, ker(ρ0)=AutE(ρ)(S), while by 11.8.6, AutF (S) ≤ ker(ρ0). Further 0 0 Ξ ≤ AutF (S) by definition of AutF (S) in 11.7. This completes the proof of (3).

~~CHAPTER 12~~

~~Factorizing morphisms~~

~~In this section F is a saturated fusion system over the ﬁnite p-group S,andS0 is a subgroup of S strongly closed in S with respect to F. For P, Q ≤ S,writeP0 for P ∩ S0 and deﬁne~~

Φ(P, Q)={φ ∈ homF (P, Q):[P, φ] ≤ S0}. Recall for x ∈ P ,[x, φ]=x−1 · xφ ∈ S,and[P, φ]=[x, φ]:x ∈ P ≤S. For α ∈ homF (P, S) deﬁne F(α)tobethesetofpairs(ϕ, φ) such that ϕ ∈ homF (PS0,S), φ ∈ Φ(Pϕ,S), and α = ϕφ.WecallF(α)thesetofF/S0- factorizations of α. (12.1). If α ∈ Φ(P, Q) and β ∈ Φ(Q, R) then αβ ∈ Φ(P, R). Proof. For x ∈ P , −1 −1 −1 [x, αβ]=x (x(αβ)) = x · xα · (xα) · (xα)β =[x, α][xα, β] ∈ S0, as [x, α]and[xα, β]areinS0.

~~(12.2). Let φ ∈ Φ(P, Q).ThenPφS0 = PS0.~~

~~Proof. For x ∈ P , xφ = x[x, φ] ∈ xS0.~~

~~(12.3). Let α ∈ homF (P, S), β ∈ homF (Pα,S), (ϕ, φ) ∈ F(α),and(Ψ,ψ) ∈ F(β).Then (1) φΨ∗ ∈ Φ(PϕΨ,S). (2) (ϕΨ, (φΨ∗)ψ) ∈ F(αβ).~~

Proof. As α ∈ homF (P, S)and(ϕ, φ) ∈ F(α), α = ϕφ and φ ∈ Φ(Pϕ,S). Thus PαS0 = PϕφS0 = PϕS0 by 12.2. As β ∈ homF (Pα,S)and(Ψ,ψ) ∈ F(β), Ψ ∈ homF (PαS0,S). Then as PαS0 = PϕS0,alsoΨ∈ homF (PϕS0,S). Next for x ∈ Pϕ,[x, φ] ∈ S0,so ∗ −1 ∗ −1 −1 [xΨ,xΨ(φΨ )] = (xΨ) · (xΨ)(φΨ )=x Ψ · xφΨ=(x · xφ)Ψ ∈ S0Ψ=S0, ∗ and hence [Pϕ,φΨ ] ≤ S0. Therefore (1) holds. As ϕ ∈ homF (PS0,S)andΨ∈ homF (PαS0,S) with PαS0 = PϕS0,wehave ∗ ∗ ϕΨ ∈ homF (PS0,S). By (1), φΨ ∈ Φ(PϕΨ,S). Observe that Ψ(φΨ )=φΨ, so (PϕΨ)(φΨ∗)=PϕφΨ=PαΨ. Then as ψ ∈ Φ(PαΨ,S), we have (φΨ∗)ψ ∈ Φ(PϕΨ,S) by 12.1. Finally (ϕΨ)(φΨ∗ψ)=ϕ(Ψ(φΨ∗))ψ = ϕφΨψ = αβ, so (2) holds.

(12.4). Assume αi ∈ homF (Pi,S) for 1 ≤ i ≤ n,withPiαi = Pi+1,andset α = α1 ···αn. Assume (ϕi,φi) ∈ F(αi) for 1 ≤ i ≤ n.Thenα ∈ homF (P1,S) and there exists (ϕ, φ) ∈ F(α) with ϕ = ϕ1 ···ϕn.

~~78 MICHAEL ASCHBACHER~~

~~Proof. This follows from 12.3 by induction on n.~~

Theorem 12.5. For each P ≤ S and α ∈ homF (P, S), F(α) = ∅. The proof of Theorem 12.5 involves a series of reductions. Assume the Theorem is false, and pick a counterexample α with m = |S : P | minimal.

~~(12.6). P0 = S0.~~

~~Proof. If P0 = S0 then (α, 1) ∈ F(α), contrary to the choice of α as a counterexample. (12.7). m>1. Proof. This follows from 12.6.~~

~~(12.8). If P~~

~~Proof. Assume β ∈ homF (Q, S) extends α. By minimality of m,thereexists ∈ ∈ (ϕ, φ) F(β). Then (ϕ|PS0 ,φ|Pϕ) F(α), contrary to the choice of α. (12.9). We may choose P ∈Ffrc.~~

~~Proof. By Alperin’s Fusion Theorem (cf. A.10 in [BLO]), there exists Pi, frc Ui, βi,1≤ i ≤ n, such that Ui ∈F , βi ∈ AutF (Ui), Pi ≤ Ui, Pj+1 = Pj βj for ··· j~~

~~(12.10). Q0 = P0. Proof. ∩ By 12.6, P0~~

(12.12). Let P ≤ S, α ∈ homF (P, S),and(ϕ, φ) ∈ F(α).Then + + + (1) ϕ ∈ homN (P, S) and ϕ ∈ homF + (P ,S ). (2) For x ∈ P , (xα)+ = x+ϕ+. Proof. As (ϕ, φ) ∈ F(α), ϕ ∈N, so by deﬁnition of the +-notation, x+ϕ+ = (xϕ)+. That is (1) holds. Further (xα)+ =(xϕφ)+ =(xϕ · [xϕ, φ])+ =(xϕ)+[xϕ, φ]+ =(xϕ)+, + + as φ ∈ Φ(Pϕ,S), so [xϕ, φ] ∈ S0,thekernelofθ : S → S ,whereθ : x → x = xS0. Thus (2) holds.

~~12. FACTORIZING MORPHISMS 79~~

Definition 12.13. + + + For P ≤ S and α ∈ homF (P, S), define αΘ ∈ homF + (P ,S )byαΘ=ϕ for (ϕ, φ) ∈ F(α). Observe that Θ is well defined: Namely by 12.12.1, ϕ+ ∈ + + + + homF + (P ,S ). Further if (Ψ,ψ) ∈ F(α)andx ∈ P , then by 12.12.2, x ϕ = (xα)+ = x+Ψ+, so the definition of αΘ is independent of the choice of (ϕ, φ)in F(α). Next define Θ : S → S+ to be the natural map Θ : s → s+. F F Write ΘF,S0 for this map from to /S0. (12.14). F→F (1) Θ=ΘF,S0 : /S0 is a surjective morphism of fusion systems. (2) θ is the restriction of Θ to N = NF (S0). + Proof. Let P ≤ S.Forγ ∈ homN (P, S), (γ,1) ∈ F(γ), so γΘ=γ = γθ. Then as θ = Θ as a map of groups on S,(2)holds. Let α ∈ homF (P, S)andβ ∈ homF (Pα,S). By 12.3.2, (αβ)Θ = αΘ · βΘ. Let (ϕ, φ) ∈ F(α)andx ∈ P . Then by 12.12.2, (xα)Θ=(xα)+ = x+ϕ+ =(xΘ)(αΘ), so Θ is a morphism of fusion systems. By definition, Θ : S → S+ is surjective. By (2), Θ extends θ,soasθ is surjective, so is Θ.

(12.15). Suppose E is a saturated subsystem of F on T ≤ S,andsetT0 = ∩ T S0, Θ=ΘF,S0 ,andΔ=ΘE,T0 .Then (1) T0 is strongly closed in T with respect to E. + (2) Define ρ : EΘ →E/T0 by (tS0)ρ = tT0 and α ρ : tT0 → tαT0 for t ∈ T , P ≤ T ,andα ∈ homE (P, T).Thenρ is an isomorphism of fusion systems, and Θ|E ρ =Δ. (3) If S0 ≤ T then Θ|E =Δ,soinparticularΘ|E : E→E/S0 is a surjective morphism. Proof. Part (1) is straightforward. From basic group theory, ρ : T + = TS0/S0 → T/T0 is a well defined isomorphism of groups, such that Θ|T ρ =Δ + as a map of groups, and α ρ : PT0/T0 → T/T0 is a well defined group homomorphism. Next for t ∈ T , + (tT0)(αΔ) = (tα)T0 =(tT0)(α ρ)=((tT0)(αΘ))ρ =(tT0)(α(Θρ)), so αΔ=α(Θρ). As this holds for all α,asamapoffusionsystems,Δ=Θ|E ρ.In + particular as Δ is surjective, so is ρ : EΘ →E/T0.Thenasρ : T → T/T0 is an isomorphism, ρ is an isomorphism and (2) holds. Assume the hypothesis of (3). Then ρ : T/S0 → T/S0 is the identity map, so Θ|E = Δ by (2). Then as Δ is surjective by 12.14.1, also Θ|E is surjective, establishing (3). (12.16). Assume α : F→F˜ is a surjective morphism of fusion systems, with S˜ = Sα and S0 is contained in the kernel S1 of the map α : S → S˜.Then (1) For P, Q ≤ S and φ ∈ Φ(P, Q), φα =1. (2) For P ≤ S, β ∈ homF (P, S),and(ϕ, φ) ∈ F(β), βα = ϕα. (3) Define π : F + → F˜ by x+π = xα for x ∈ S,andϕ+π = ϕα,forϕ an N -map. Then π is a surjective morphism of fusion systems with Θπ = α. + (4) If S0 = S1 then π : F → F˜ is an isomorphism of fusion systems.

~~80 MICHAEL ASCHBACHER~~

Proof. First assume the setup of (1), and let x ∈ P .Then (xα)(φα)=xφα =(x · [x, φ])α = xα · [x, φ]α = xα, as [x, φ] ∈ S0 ≤ S1 and S1α =1.Thus(1)holds. Next assume the setup of (2). Then by (1), βα =(ϕφ)α =(ϕα)(φα)=ϕα, establishing (2). + As S1 is the kernel of α : S → S˜ and S0 ≤ S1, π : S → S˜ is a well defined surjective group homomorphism with Θπ = α as a map of groups. Further if S0 = S1,thenπ is an isomorphism. Let P ≤ S and η, μ ∈ homN (P, S). Then ηα = μα iff for all x ∈ P ,(xη)α = + + (xα)(ηα)=(xα)(μα)=(xμ)α iff xη ∈ xμS1.Thusifη = μ then as S0 ≤ S1, + + + ηα = μα,soπ :homN (P ,S ) → homF˜(Pα,S˜)=homF˜(P π, S˜) is well defined. + + Further if S0 = S1, η = μ iff ηα = μα,soπ is injective. For x ∈ P , (x+π)(η+π)=(xα)(ηα)=(xη)α =(xη)+π =(x+η+)π.

For ξ ∈ homN (Pη,S), (η+ξ+)π =(ηξ)+π =(ηξ)α = ηα · ξα = η+π · ξ+π. Thus π : F + → F˜ is a morphism of fusion systems. Further by (2), βΘπ = ϕ+π = ϕα = βα. Then as α is a surjection, so is π, completing the proof of (3).

+ Finally assume that S0 = S1. Then by remarks above, π : S → S˜ is an + + + isomorphism, as is π :homN (P S ) → homF˜(P π, S˜), so (4) follows from (3). We can now prove Theorem 9. Part (1) of Theorem 9 follows from 12.14.1. Part (2) of Theorem 9 is a consequence of 12.14.1 and 12.16.4.

(12.17). Assume S0 F, S0 ≤ T ≤ R ≤ S,andE is a subsystem of F on R such that T + is strongly closed in R+ with respect to E +.ThenT is strongly closed ∼ in R with respect to E,andE +/T + = E/T .

+ Proof. By 8.3 in [A1], θ : E→E/S0 = E is a surjective morphism of fusion systems, so as T + is strongly closed in R+ with respect to E +, T is strongly closed in R with respect to E by 8.9.1 in [A1]. Let E ! = E +/T + and E − = E/T .Thusforr ∈ R, r! = r+T + and r− = rT. − ! − From the group homomorphism theorems, ρ0 : r → r is an isomorphism of R with R!. − − − − − Let P ≤ R and α ∈ homE− (P ,R ). Take P to be the preimage of P − ∈ ! ∈ ! ! in R.Thenα = β for some β homNE (T )(P, R). Deﬁne β homE! (P ,R)by r!β! =(r+β+)T +.Then ! ∗ − − ! + + + + + ! ! r (αρ0)=r αρ0 =(rβ) ρ0 =(rβ) =(rβ) T =(r β )T = r β . ! ! ◦ + + ∈ E ∈ + Further for γ hom (P ,R), γ = δ for some δ homNE+ (T )(P ,R ), where ! ◦ + + + ◦ + ◦ ! r δ =(r δ)T . Also δ = μ for some μ ∈ homE (P, R), so γ = δ =(μ ) = μ . As T ≤ P and T is strongly closed in R with respect to E, μ is in NE (T ). Thus we have shown: ! ! ! − − ∗ { ∈ } − (*) homE! (P ,R)= μ : μ homNE (T )(P, R) =homE (P ,R )ρ0.

~~12. FACTORIZING MORPHISMS 81~~

~~∗ E − →E! Deﬁne αρ = αρ0.Thenby(*),ρ : is a surjective morphism of fusion systems, and then as ρ0 is an isomorphism of groups, ρ is an isomorphism of fusion systems.~~

~~CHAPTER 13~~

~~Composition series~~

In this section F is a saturated fusion system over the finite p-group S. Definition 13.1. We recursively define the set S = S(F)ofsupranormal series of F.Themembers of S are sequences λ =(λi :0≤ i ≤ n), such that for each i, λi is a subsystem of p S on Ti ≤ S,1=λ0 ≤ λ1 ≤···≤λn = O (F), and: (SS) If the length n = l(λ)ofλ is greater than 1, then there exists 0

(13.2). Let λ =(λi :0≤ i ≤ n) ∈Sbe of length n>1. Then for each 0 < p i ≤ n, Ti−1 is strongly closed in Ti with respect to λi,andλi/Ti−1 = O (λi/Ti−1) is saturated. Proof. Assume otherwise and pick a counterexample with n minimal. Re- p p placing F by O (F), we may assume F = O (F). As n>1, there is λj ∈ N(λ). p Then λj = O (λj ) F,soλj is saturated and Tj is strongly closed in S with respect to F. In particular using 8.5 in [A1], the lemma holds if n =2,son>2 by the choice of λ as a counterexample. Deﬁne λ and λ as in 13.1. Then l(λ) l(λ), and by 13.1, λ and λ are supranormal series. Therefore by minimality of n, the lemma holds for these series. In particular if 0

~~For λ =(λi :0≤ i ≤ n) ∈Sand 0~~

(13.4). Let λ =(λi :0≤ i ≤ n) ∈S, and assume for some 1 ≤ j ≤ n that p 1 = E/Tj−1 = O (E/Tj−1) Fj (λ).LetT = E∩S and μ =(μi :0≤ i ≤ n +1), where λi = μi for i ≤ j, μj = E,andμi = λi−1 for i>j.Thenμ ∈S and λ ≺j μ.

~~84 MICHAEL ASCHBACHER~~

Proof. Assume otherwise and pick a counterexample with n minimal. Again we may assume F = Op (F). If n =1,then1= E F and μ =(1, E, F) ∈Swith λ ≺1 μ, contrary to the choice of λ.Thusn>1. ∈ ≤ Let λm N(λ). If j m then by minimality of n,theseriesμ ,formedfrom λ via the construction above, is in S with λ ≺j μ .Thusμ = μ λ ,theseries S ≺ formed as in 13.1 at λj = μj+1 is in , and then by construction, λ j μ. Similarly if m

(13.5). λ =(λi :0≤ i ≤ n) ∈S is a composition series for F iﬀ all factors of λ are simple. Proof. If λ is maximal in S, then all factors of λ are simple by 13.4. Suppose λ ≺j μ.Thenμj+1/Tj λj+1/Tj ,andTj+1 >T >Tj ,whereμj+1 is a system on T ,soFj+1(λ)isnotsimple. Theorem 13.6 (Jordon-H¨older Theorem for fusion systems). If λ and μ are composition series for F,thenl(λ)=l(μ) and F (λ)=F (μ).

Proof. As usual we may assume F = Op (F). Assume the theorem is false and pick a counterexample λ, μ, F with n = l(λ) minimal. If n = 1 then by 13.5, F = F/1issimple,soS = {λ} and the theorem holds. Thus n>1

~~13. COMPOSITION SERIES 85~~

Deﬁnition 13.7. By 13.6, we may deﬁne the family CF(F)ofcomposition factors of F to be the set F (λ) of factors of any composition series λ of F. (13.8). For each normal subsystem E of F, CF(F)=CF(E) CF(F/E).

~~p Proof. We may choose a composition series λ for F with O (E) ∈ N(λ). Then CF(F)=F (λ) F (λ) with CF(E)=F (λ)andCF(F/E)=F (λ).~~

~~CHAPTER 14~~

~~Constrained systems~~

In this section F is a saturated fusion system over the ﬁnite p-group S. (14.1). Let Q ≤ S. Then the following are equivalent: (1) Q F. (2) Q is strongly closed in S with respect to F and Q is contained in each member of F frc. (3) There exists a series 1=Q0 ≤···≤Qn = Q such that for each 0 ≤ i1. As (1) and (3) are equivalent, T = Qi−1 F.Thusby(4),Qi is strongly closed in NS (T )=S with respect to NF (T )=F, contrary to the choice of i.

~~(14.2). The following are equivalent: (1) F is constrained. ∗ (2) F (F)=Op(F). (3) E(F)=1.~~

Proof. Set R = Op(F). By Theorem 6, E(F) centralizes R.ThusifF is constrained, then E = S ∩ E(F) ≤ CS(R)=Z(R). Hence E(F) = 1 by 9.1.2, so (1) implies (3). By deﬁnition of F ∗(F), (2) and (3) are equivalent. Suppose (3) holds. Then F = CF (E(F)), so (1) holds by 9.12.3.

(14.3). Suppose F is constrained and set R = Op(F).Then ∗ (1) F (F)=Op(F). (2) Z(R) is an abelian subgroup of S,stronglyclosedinS with respect to F. f ∗ (3) For all U ∈F , F (NF (U)) = Op(NF (U)),soNF (U) is constrained.

~~88 MICHAEL ASCHBACHER~~

~~Proof. Part (1) follows from 14.2. As Z(R)charR F, (2) holds. Assume f U ∈F . Then by Theorem 7, E(NF (U)) ≤ E(F) = 1, so (3) holds by 14.2.~~

(14.4). Assume F = FS (G) for some finite group G with S ∈ Sylp(G). Assume G˜ is a finite group and α : G → G˜ is a group homomorphism. Let Sα ≤ S˜ ∈ ˜ F˜ F ˜ ≤ } Sylp(G) and = S˜(G). Define α˙ =(α, αP,Q : P, Q S by cgαP,Q = cgα for g ∈ NG(P, Q).Then (1) α˙ : F→F˜ is a morphism of fusion systems. (2) If α : G → G˜ is surjective, then so is α˙ : F→F˜. (3) If G˜ = G/Op (G) and α : G → G˜ is the natural map, then α˙ : F→F˜ is an isomorphism. (4) Suppose L G and set G¯ = G/L.ThenE = FS∩L(L) F,and F E ∼ F ¯ / = S¯(G). Proof. ≤ As NG(P, Q)α NG˜ (Pα,Qα), (1) follows. Assume α : G → G˜ is surjective, let K =ker(α), and T = S ∩ K.Let ≤ ∈ → ˜ P, Q S, P0 = TP,andQ0 = QT .Leth NG˜ (Pα,Qα). As α : G G is surjective, there is g ∈ G with gα = h.Then(P g)α =(Pα)h ≤ Qα,soP g ≤ KQ. g ≤ g g ≤ ∈ ∈ Then P0 (PK) = P K QK,soasQ0 Sylp(QK), there is k K with gk ≤ g ≤ P0 Q0.As(gk)α = gα = h, replacing g by gk, we may assume P0 Q0.Thus → αP0,Q0 :homF (P0,Q0) homF˜(Pα,Qα) is a surjection, establishing (2). Assume the hypothesis of (3), and adopt the notation of the previous paragraph. Then T =1,soP = P0 and Q = Q0.Thenby(2),αP,Q :homF (P, Q) → homF˜(Pα,Qα) is a surjection. Finally α : P → Pα is an isomorphism, so for x, y ∈ NG(P, Q), cxα = cyα on Pαiff cx = cy on P ,soαP,Q is a bijection, completing the proof of (3). Let L G, G¯ = G/L,andβ : G → G¯ the natural map. First, from 6.3 in [A1], E = FS∩L(L) F. Next, by (1) and (2), β induces a surjective ˙ F→F ¯ → ¯ ∩ morphism β : S¯(G). The kernel of β : S S is S L, so by 12.16.4, F E F ∩ ∼ F ¯ / = /(S L) = S¯(G). Thus (4) is established. (14.5). Assume Z ≤ Z(F),setF + = F/Z,andletE bethepreimageinF of E(F +).Then (1) If B is a component of F then B+ is a component of F +. (2) Suppose D+ is a component of F +,letD bethepreimageinF of D+,and set C = Op(D).ThenC is a component of F and C+ = D+. (3) E = E(F)Z. (4) F is constrained iff F + is constrained. Proof. As B is subnormal in F, also B+ is subnormal in F + by 8.10.2 in [A1]. + + + Let Z0 be the preimage in B of Z(B )andobserveZ(B) ≤ Z(B) ,soZ(B) ≤ Z0. B B F B Thus by 7.17, either Z( )=Z0 or = Z0 (Z0)anthelatterisimpossibleas is quasisimple. ∼ By 12.16, B+ = B/(B∩Z), so as B∩Z ≤ Z(B), we conclude from 12.17 that + + ∼ p + B /Z(B ) = B/Z0, and hence is simple. Let A be the preimage in B of O (B ). By 8.10.2 in [A1], A B, so by 7.17 either A = B or A≤Z0. In the former case + p + p + + + + B = O (B ), so (1) holds. In the latter, O (B ) ≤ Z(B ), so B = Op(B )by 7.12.3, contradicting B+/Z(B+) simple. So (1) is established. Assume the setup of (2) and let R be the preimage in S of Z(D+). By 8.10.2 in [A1], D is subnormal in F,soC is subnormal in F.

~~14. CONSTRAINED SYSTEMS 89~~

Let C = C∩S, U ∈Dfc,andG = G(U). Then [G, R] ≤ Z,soOp(G) centralizes R, and hence [Op(G),U] centralizes R.ThusC centralizes R by 7.17.2. Now for ∈Dfc ∈Dfc p p P C , U = PCS∩D(P ) ,soH (U)=O (G)C centralizes R, and hence from Deﬁnition 7.6, C = Op(D)=E(Ap) centralizes R ∩ C,soR ∩ C ≤ Z(C). By 7.7.4, C = Op(C). As C D, also C+ D+ by 8.10.2 in [A1]. Therefore + + + + by 7.16, either D = C or C ≤ R . In the latter case, C≤FR∩C (R ∩ C), and then as R ∩ C ≤ Z(C), we conclude from 7.12.3 that C =1.ButthenD = Op(D) by another application of 7.12.1, contradicting D+ quasisimple. Therefore C+ = D+. Then an argument in the proof of (1) shows that C/Z(C)= ∼ C/(R ∩ C) = D+/Z(D+), completing the proof of (2). Observe (3) follows from (1) and (2). By 14.2, F is constrained iﬀ E(F)=1, so (3) implies (4).

(14.6). Assume F is constrained. Then (1) Each subnormal subsystem of F is constrained. f (2) Assume G is a ﬁnite group with S ∈ Syl2(G) and F = FS(G).LetU ∈F , ∗ ∗ L a component of NG(U),andL = L/Op,Z (L).ThenF(S∩L)∗ (L ) is constrained. Proof. Let E be subnormal in F. Then by 9.8.3, E(E) ≤ E(F)=1,so(1) follows from 14.2. Assume the setup of (2). By 14.3.3, NF (U) is constrained, so as NF (U)= F F NS (U)(NG(U)), replacing , G by NF (U), NG(U), we may assume U G. Similarly passing to G/Op (G) and appealing to 14.4.3, we may assume Op (G)=1. ∗ ∼ Now E = FS∩L(L) is constrained by 14.4.4 and (1), and then F(S∩L)∗ (L ) = E/Z(L) is constrained by 14.5.4, completing the proof.

(14.7). Assume G is a ﬁnite group, S ∈ Sylp(G),andF = FS(G). Assume in addition any one of the following hold: (a) S is abelian, or (b) NG(S) is strongly p-embedded in G,or (c) S is of class 2 and Z(S) is strongly closed in S with respect to G. Then F = FS (NG(S)) and S = Op(F).

Proof. Observe that F = FS (NG(S)) iff S = Op(F)iffNG(S) controls fusion in S. The first two statements are equivalent by 7.12.2, while the first and the third are equivalent by definition of control of fusion. By a result of Burnside (cf. 7.7 in [SG]), if S is abelian then NG(S) controls fusion in S. Thus the lemma holds under condition (a). If (b) holds then S ∩Sg =1 for g ∈ G − NG(S), so NG(S) controls fusion in S, and the lemma holds in this case. Finally suppose that (c) holds. As Z(S) is strongly closed in S with respect to G,andasF = FS (G), Z(S) is strongly closed in S with respect to F.Thenas S is of class 2, the series 1

(14.8). Assume p =2, G is a nonabelian ﬁnite simple group, S ∈ Syl2(G), and F = FS (G). Then the following are equivalent: (1) F is constrained. (2) S = O2(F). (3) G is a Goldschmidt group. In particular either S is abelian or G is a Bender group.

~~90 MICHAEL ASCHBACHER~~

Proof. If G is Bender then the Borel group NG(S)ofG is strongly embedded in G. Thus (3) implies (2) by 14.7. Trivially, (2) implies (1). Finally suppose (1) holds. Then there is a nontrivial abelian subgroup of S, strongly closed in S with respect to G by 14.3.2. Thus (3) holds by a theorem of Goldschmidt in [Go1].

~~CHAPTER 15~~

~~Solvable fusion systems~~

~~In this section F is a saturated fusion system over the ﬁnite p-group S. Deﬁnition 15.1.~~

Recall that F is of order p if F = FG(G) for some group of order p. Define F to be solvable if all composition factors of F are of order p. n F 0 F Define the series Op ( ) of subgroups of S recursively by Op( ) = 1, and for n F F n−1 F n>0, Op ( )isthepreimageinS of Op( /Op ( )). (15.2). (1) For each normal subsystem E of F, F is solvable iff E and F/E are solvable. (2) If F is solvable then F is constrained. Proof. Part (1) is immediate from 13.8. Suppose F is solvable, but F is not constrained. Then E(F) = 1 by 14.2. By (1), E(F) is solvable, so we may take F = E(F). Then each component C of F is normal in F,soC is solvable by (1). Hence by another application of (1), we may take F = C to be quasisimple. Replacing F by F/Z(F) and appealing to (1), we may assume F is simple. Thus F is of order p, contrary to 9.1. This completes the proofof(2).

~~(15.3). The following are equivalent: (1) F is solvable. n F (2) Op ( )=S for some nonnegative integer n. (3) There exists a series 1=S0 ≤ ···Sm = S of subgroups of S such that for each 0 ≤ i~~

Proof. Assume (1) holds. Then F is constrained by 15.2.2, so Q = Op(F) = F | | n F 1. By 15.2.1, /Q is solvable, so by induction on S , Op ( /Q)=S/Q for some n+1 F n.ThusOp ( )=S, so (1) implies (2). Assume (2) holds, and let 1 = S0 ≤ ···Sk = Op(F) be the ascending central series for Op(F). By induction on |S|, there exists a series 1 = Sk+1/Sk ≤ ··· ≤ S /S = S/S such that S /S is strongly closed in S/S with respect to F/S and m k ∼ k i k k k Si+1/Si = Si+1/Sk/Si/Sk is abelian for each k

~~92 MICHAEL ASCHBACHER~~

It remains to show (1) and (4) are equivalent. From 2.5 in [A1], F is constrained ∗ iff there exists G ∈G(F); in particular, S ∈ Sylp(G), F (G)=Op(G), and F = FS(G). Further if F is solvable then F is constrained by 15.2.2. Thus we may assume F = FS(G) for some finite group G with S ∈ Sylp(G). ¯ E F F F E ∼ F ¯ Let L G and G = G/L. By 14.4.4, = S∩L(L) and / = S¯(G). By 15.2.1, F is solvable iff E and F/E are solvable, so F is solvable iff FS∩L(L)and F ¯ S¯(G) are solvable. Then continuing this process, (1) and (4) are equivalent. (15.4). Assume F is solvable. Then each saturated subsystem of F is solvable.

Proof. By 15.3 we can choose a series 1 = S0 ≤ ··· ≤ Sm = S as in 15.3.3. Let E be a saturated subsystem of F on T ≤ S and for 1 ≤ i ≤ m,setTi = S ∩ T . Then the series {Ti :0≤ i ≤ m} satisfies the hypotheses in 15.3.3 with respect to E, so the lemma follows from 15.3. Definition 15.5. We extend the definition of “Goldschmidt groups” to odd primes, by defining the notion of a p-Goldschmidt group. The Goldschmidt groups are the 2-Goldschmidt groups. Define a nonabelian finite simple group G with p ∈ π(G)tobeap-Goldschmidt group if for S ∈ Sylp(G), FS(G)=FS (NG(S)). The following two results follow from work of Flores and Foote in [FF]and Foote in [F]; we prove the two results together. Theorem 15.6.. Let G be a nonabelian finite simple group with p ∈ π(G),and S ∈ Sylp(G).ThenG is p-Goldschmidt iff one of the following holds: (a) S is abelian. (b) L is of Lie type in characteristic p of Lie rank 1. ∼ (c) p =5and L = Mc. ∼ (d) p =11and L = J . ∼ 4 (e) p =3and L = J . ∼ 2 (f) p =5and G = HS, Co ,orCo . ∼ 2 3 (g) p =3and G = G2(q) for some prime power q prime to 3 such that q is not congruent to ±1 modulo 9. ∼ (h) p =3and G = J3. Remark 15.7. In cases (b)-(d) of Theorem 15.6, we say G is p-Bender. In those cases, N (S)is ∼ G strongly p-embedded in G. Observe that in cases (c)-(g), S = p1+2. Theorem 15.8 (Flores-Foote). Assume G is a nonabelian finite simple group and S ∈ Sylp(G).SupposeT is a proper nontrivial subgroup of S strongly closed in S with respect to G. Then either (1) G is p-Goldschmidt, or ∼ (2) p =3, G = G2(q) with q ≡±1mod9,andT = Z(S) is of order 3. When p is odd, Theorem 15.8 is due to Flores and Foote, and can be retrieved from Theorem 2.2 in [FF]. When p = 2 the result is due to Foote in [F], and is stated as Theorem 2.1 in [FF]. More precisely, these two results show that, under the hypothesis of Theorem 15.8, either G appears in the list of Theorem 15.6, or G appears in part (2) of Theorem 15.8. Then Theorem 15.6 follows from the following slightly stronger result:

~~15. SOLVABLE FUSION SYSTEMS 93~~

Theorem 15.9. Let G be a nonabelian finite simple group with p ∈ π(G), S ∈ Sylp(G),andF = FS(G). Then the following are equivalent: (1) F is constrained. (2) F is solvable. (3) S = Op(F). (4) G is a p-Goldschmidt group. (5) G satisfies one of conditions (a)-(h) of Theorem 15.6. Proof. Conditions (3) and (4) are trivially equivalent. We next show that (5) implies (3), so assume G appears on the list of Theorem 15.6. As in the proof of 14.8, if G satisfies condition (a) or (b) in 15.6, then (3) holds by 14.7. If one of ∼ conditions (c)-(f) hold then G is sporadic and from the Tables in [GLS3], S = p1+2 with Z(S) strongly closed in S with respect to G. Hence condition (c) of 14.7 is ∼ satisfied, so (3) holds in this case by 14.7. If G = G (q) satisfies condition (g) of 2 ∼ 15.6, then from [FF]and[F], Z(S) is strongly closed in S and S = 31+2, so again ∼ (3) holds by 14.7. This leaves case (h) in 15.6, where p =3andG = J .Inthiscase ∼ 3 from the Tables in [GLS3], Z = Z(S) = E9 is strongly closed in S with respect to G,andS = O3(NG(S)). Therefore S = O3(F) by the equivalence of parts (1) and (4) of 14.1. This completes the proof that (5) implies (3). Trivially, (3) implies (2), while (2) implies (1) by 15.2.2. Thus we may assume that F is constrained, and it remains to show that (5) holds. Assume otherwise. As F is constrained, 14.3.2 says there is a nontrivial abelian subgroup T of S strongly closed in S with respect to F. Therefore by Theorem 15.8, and as (5) does not ∼ hold, conclusion (2) of Theorem 15.8 holds. That is p =3andG = G2(q) with q ≡±1 mod 9. But then by 16.11.5, F is quasisimple, contrary to 14.3.1. This completes the proof of the Theorem. Observe that Theorem 11 follows from the equivalence of parts (1) and (4) of 15.3, and the equivalence of parts (2) and (4) of 15.9.

~~CHAPTER 16~~

~~Fusion systems in simple groups~~

In this section p is a prime, G is a finite group, S ∈ Sylp(G), and F = FS(G). Thus F is a saturated fusion system over the finite p-group S. Define frc Ξ(G, S)=NG(R):R ∈F . (16.1). Let H =Ξ(G, S).Then (1) FS(H)=FS (G). (2) Op(F)=Op(H). Proof. Part (1) follows from Alperin’s Fusion Theorem, A.10 in [BLO]. Let frc Q = Op(F). By 14.1, Q ≤ R for each R ∈F and Q is strongly closed in S with respect to G,soQ NG(R). Thus Q ≤ Op(H). On the other hand by (1), Op(H) ≤ Q,so(2)holds. (16.2). Assume G is simple and (a) G is not p-Goldschmidt, and if p =3, G is not G2(q) with q ≡±1mod9. frc (b) Aut (S)=Aut p (S):R ∈F . G O (NG(R)) Then FS (G) is simple. Proof. First, by condition (a) of the lemma and 14.9, S has no nontrivial proper strongly closed subgroups. Hence condition (a) of part (4) of Theorem 8 holds. Second, Aut (S)=AutF (S)andAut p (S) consists of the conju- G O (NG(R)) p gation maps α = cg : S → S such that g ∈ O (NG(R)) ∩ NG(S). For each such α, p α| ∈ Aut p (R)=O (AutF (R)) = B(R) ≤ AutB(R), R O (NG(R)) 0 so α ∈ AutF (S). Therefore condition (b) of the lemma implies condition (b) of part (4) of Theorem 1. Hence the lemma follows from part (4) of Theorem 1. (16.3). Assume G is simple of Lie type and characteristic p,orp =2and 2 G = F 4(2) is the Tits group. Let l be the Lie rank of G.Then (1) F frc is the set Q of unipotent radicals of proper parabolics containing S. (2) If l>1 then F is simple. Proof. Let R ∈Ffrc.AsG is of Lie type and characteristic p,ortheTits group, the Borel-Tits Theorem says there exists Q ∈Qsuch that R ≤ Q and frc NG(R) ≤ NG(Q). As R ∈F , NS (R) ∈ Sylp(NG(R)), CS(R) ≤ R,and Inn(R)=Op(AutG(R)). As NG(R) ≤ NQ(R), AutQ(R) ≤ Op(AutG(R)), so NQ(R) ≤ RCS(R)=R, and hence Q = R. This establishes (1). It remains to prove (2), so we may assume l>1. Thus condition (a) of 16.2 is satisfied, so by 16.2, it suffices to verify condition (b) of 16.2. We will work in the universal Chevalley group of type G, so replacing G by that group, we assume that G is universal rather than simple. Then G = G¯σ for some simple simply connected

96 MICHAEL ASCHBACHER algebraic group G¯ and some Steinberg endomorphism σ of G¯.ThenG¯ has a root system Σ with simple system π.Weappealto[St]. In particular G hasrootsystem Σ=Σˆ / ∼ with simple systemπ ˆ = π/ ∼, for a suitable equivalence relation ∼ on Σ, G is generated by root groups Uαˆ,ˆα ∈ Σ,ˆ and there are simple reﬂections rαˆ, αˆ ∈ πˆ, and a Cartan subgroup H = H¯σ for G,whereH¯ is the Cartan subgroup of G¯ determined by Σ. We may choose S = Uαˆ, αˆ∈Σˆ + and B = SH a Borel subgroup of G. ∈ ∈ # From part (b) of Theorem 33 in [St](anditsproof),forˆα πˆ and x U−αˆ, p rαˆ = nαˆ(x)H for some nαˆ(x) ∈ NG(H) with nαˆ(x) ∈Uαˆ,U−αˆ = O (Lαˆ), where Lαˆ is a Levi factor of Pαˆ = B,rαˆ , the minimal parabolic of G determined by B andα ˆ. Further from step (1) in the proof of Lemma 64 in [St],

~~p (*) Hαˆ = H ∩Uαˆ,U−αˆ = H ∩ O (Pαˆ). Now~~

(**) Hαˆ :ˆα ∈ πˆ ≤H, | | | | and from (*) and the order of H listed in [GLS3], H = αˆ Hαˆ , so it follows from (**) that H = αˆ Hαˆ.ThusasH is a Hall p -subgroup of B = NG(S), condition (b) of 16.2 holds, completing the proof of the lemma. Given a subgroup X of the symmetric group on a set Ω, and Δ ⊆ Ω, write M(X) for the set of points of Ω moved by X, Fix(X) for the ﬁxed point set of X, and XΔ for the subgroup of X ﬁxing each point of Δ. ∼ (16.4). Assume G = An is an alternating group on Ω={1,...,n} with n ≥ 5 and S =1 .LetM be the symmetric group on Ω and S ≤ SM ∈ Sylp(M).Then (1) p ≥ n. (2) Let n = ap + b with a, b ∈ N and 0 ≤ bp,andEpc = V ≤ SΩ−θ regular on θ such that V ∩ E = Zp and NM (V )= W × J where W = N (V ), J = M , W is the split extension of V by GL (p), ∼ MΩ−θ θ c and J = Sn−pc . (8) Assume S is nonabelian. Let Δ=M(S).If(p, n) ∈{(2, 8), (2, 9)} let ∼ ∼ R1 = O2(NG(E)) and R2 = U1U2,whereE4 = Ui ≤ S with Ui ∩ E = Z2 and Ui is regular on M(Ui)=Δ− M(U3−i). Otherwise set R1 = E and R2 = V (S ∩ J). frc Then Ri ∈F for i =1, 2,andΞ(G, S)=NG(R1),NG(R2) = NG(Δ).

~~16. FUSION SYSTEMS IN SIMPLE GROUPS 97~~

~~(9) Assume the hypothesis and notation of (8) with b =0.Then~~

Aut (S)=Aut p (S),Aut p (S) . G O (NG(R1)) O (NG(R2)) Proof. As S =1, p ∈ π(G), so as |G| = n!/2, n ≥ p.Thatis(1)holds. Let X be the set of subgroups of SM generated by p-cycles. Then EM = X is an abelian subgroup with Λ = {M(X),Fix(E):X ∈X}the partition described in (2). Then (3) is straightforward, and implies (4)-(6). 2 ∼ Assume S is nonabelian. By (6), n ≥ p ,soθ exists and V = Epc can be embedded in MΩ−θ so that V is regular on θ.BySylowwemaychooseV ≤ S. Then V acts on Γ, so θ is a union of blocks Ω ,...,Ω c−1 of Λ, and as V is regular 1 p ∼ on θ, V is selfcentralizing in M − ,soC ··· (V )=V ∩ E Z .Then(7) Ω θ X1 Xpc−1 = p follows. Assume the hypothesis and notation of (8), and set H = NG(R1),NG(R2) . Then NG(R1)=NG(E) is transitive on Δ, Λ = {Ωi :1≤ i ≤ a} is the unique nontrivial NG(E)-invariant partition of Δ, and NG(E) induces Sb on Ω0.Next frc if (p, n) ∈{/ (2, 8), (2, 9)},thenfrom(7),R2 ∈F ,andNG(R2)actsonΔ,so H ≤ NG(Δ). The same remark holds in the exceptional case. Further NG(R2) NG(Λ ), so H is primitive on Δ, and then as E contains p-cycles, (or involutions 2 of cycle type 2 if p = 2), it follows that HΩ0 is the alternating group on Δ, and if b>1thenH induces the symmetric group on Δ. Therefore H = NG(Δ). ∈Ffrc ⊆ ≤ Let Q .ThenM(Q) M(S)=ΔandQ GΩ0 .IfM(Q) =Δ, then Ωi ⊆ Fix(Q)forsome1≤ i ≤ a.ButthenXi ≤ CS(Q) ≤ Q, contradicting Ωi ⊆ Fix(Q). Therefore M(Q)=Δ,soNG(Q) acts on Δ, and hence Ξ(G, S) ≤ NG(Δ) = H,soH =Ξ(G, S), completing the proof of (8). We next observe that, when p = 2, for each nonnegative integer n, a Sylow 2-subgroup S of M is selfnormalizing in M. We prove this by induction on n; M ∼ the case n = 1 is trivial. Now EM = K NM (SM ), and from (3), NM (E)/K = Sa, so by induction on n, S/K is selfnormalizing in NM (E)/K, and hence SM is selfnormalizing in M. Assume the hypothesis of (9). Suppose first that p = 2. Again E = K ∩ G ∼ NG(S) with NG(E)/E = Sa, so by the previous paragraph, S/E is selfnormalizing in NM (E)/E, and hence S is selfnormalizing in G. Therefore (9) is trivial when p = 2, so we may assume p is odd. p Let Q = NM (S), and for i =1, 2, let Hi = NM (Ri)andLi = O (Hi). Let ∩ ¯ D be a Hall p -subgroup of H1 K and H = NH1 (D). By a Frattini argument, ¯ ¯ ≤ ≤ H1 = EH, and we may choose notation so that H = DB.For1 i a,set ∩ → Di = D Yi,sothat D = i Di. Regard Di as Zp−1, define χ : D Zp−1 by → ¯ χ :(d1,...,da) i di, and let D0 =ker(χ). Then χ is a H-equivariant surjection, p with D0 =[D, O (B)]. Therefore: p ∼ (!) L1 = ED0O (B)andH1 = L1Dt , with H1/L1 = Zp−1 × Z2,where t ∈ B − G induces a transposition on X . Set I = MΩ−θ and for U ∈{I,J} and P ∈{Q, S},setPU = P ∩ U.Then Q = QI × QJ , S = SI × SJ ,andSU ∈ Sylp(U). Choose notation so that θ = {Ωi : 1 ≤ i ≤ pc−1}. c−1 Set E1 = E ∩ I and X1 = {Xi :1≤ i ≤ p }, and proceeding recursively, for i>1 define Xi to be the set of subgroups of S/Ei−1 moving exactly p points of Xi−1,andEi/Ei−1 = Xi .Then1=E0 < ···

~~98 MICHAEL ASCHBACHER~~

Next I2 = H2 ∩ I = VB2,whereB2 acts faithfully as GLa(p)onV .Now S2 = S ∩ I2 stabilizes the ﬂag V = {1 1, AutT2 (V1)=Aut(V1) = Zp−1. c−1 For d ∈ Zp−1,letd∗ =(d1,...,da)wheredi = d for i ≤ p and di =1for c−1 pc−1 i>p .LetD∗ = {d∗ : d ∈ Zp−1}.Thenχ(d∗)=d = d,soD∗ ∩ D0 =1,and hence D∗ is a complement to D0 in D. By (!), H1 = L1D∗t with D0 = D ∩ L1. Then as D∗ ≤ QI and we may take t ∈ QI , Q =(Q ∩ L1)D∗t =(Q ∩ L1)QI . Then as L1 ≤ G: (!!) Q ∩ G =(Q ∩ L1)(QI ∩ G). As Q = Q × Q acts on E = E ∩ I, it also acts on C (S)=V .Nextwe I J 1 E1 ∼ 1 may choose B so that NI (V1)=E1D∗BI ,wheret ∈ BI = B ∩ I = Spc−1 .We’ve ∩ seen that D∗ is a complement to D0 in D,andD0 = L1 D,soNL1∩I (V1)=E1B1, p ∼ where B1 = O (BI ) = Apc−1 .AsE1BI centralizes V1 and D∗ acts faithfully as Aut(V1)onV1, E1BI = CI (V1), and then NL1∩I (V1)=CG∩I (V1). Then as ≤ ≤ D∗ QI NI (V1), QI = D∗CQI (V1)and ∩ (+) QI L1 = CQI ∩G(V1). We saw that T2 ≤ QI ∩ G acts faithfully as Aut(V1)onV1,soby(+),QI ∩ G = ≤ ∩ ∩ ∩ ∩ ∩ T2CQI ∩G(V1) (L2 QI )(L1 QI ). Then by (!!), Q G =(Q L1)(QI G)= (Q ∩ L1)(Q ∩ L2), completing the proof of (9). ∼ (16.5). Assume G = An is an alternating group on Ω={1,...,n} with 6 ≤ 2 n ≥ p .Writen = ap + b with 0 ≤ bp .Then (1) S is nonabelian iﬀ either: (a) p ∈ Π and (G, p) =( ON,3),or (b) p =2and G is not J1. (2) If G is M11, M22, M23,orJ1,thenΠ=∅.

~~16. FUSION SYSTEMS IN SIMPLE GROUPS 99~~

(3) If G is M12, M24, J2, J3, Suz, F22,orF23,thenΠ={3}. (4) If G is a Conway group, Mc, Ru, Ly, F5, F3,orF2,thenΠ={3, 5}. (5) If G is HS then Π={5}. (6) If G is He, O N,orF24 then Π={3, 7}. (7) If G is J4 then Π={3, 11}. (8) If G is F1 then Π={3, 5, 7, 13}. Proof. We appeal to the Tables in [GLS3] for the local structure of G.In particular from those Tables, a Sylow 2-subgroup of G is abelian iﬀ G is J1.Further groups of order p and p2 are abelian, so if p isanoddprimenotinΠ,thenSylowp- subgroups of G are abelian. Finally by inspection of the Tables in [GLS3], if p ∈ Π ∼ then either S is nonabelian and (2)-(8) hold, or (G, p)=(O N,3) and S = E81. (16.7). Let M be the set of maximal overgroups of S in G and M ∈M. frc frc (1) If Q ≤ S with NS (Q) ∈ Syl2(NG(Q)) and Q ∈FS (M) ,thenQ ∈F . frc frc (2) If R ∈FS (M) with R S,thenR ∈F . ∗ frc (3) If F (M)=Op(M) then Op(M) ∈F . f Proof. Assume the setup of (1). As NS(Q) ∈ Sylp(NG(Q)), Q ∈F. frc Further NS (Q) ≤ M and as Q ∈FS(M) , CS(Q) ≤ CM (Q)=Z(Q), so c frc fc Q ∈F . Finally as Q ∈FS (M) , Inn(R)=O2(AutM (R)), while as Q ∈F , O2(AutG(R)) ≤ AutS(R), so O2(AutG(R)) ≤ O2(AutM (R)). This completes the proof of (1). Observe (2) is a special case of (1), and (3) is a special case of (2).

(16.8). Assume G is a sporadic simple group, but not J1,andp =2.Then FS(G) is simple. Proof. As in 16.7, let M be the set of maximal overgroups of S and G.We claim that either: (i) S is selfnormalizing in G,or 2 (ii) G is J2 or J3 and NG(S) ≤ M ∈M, with M = NG(R), M = O (M), and frc R = O2(M) ∈F . In particular condition (b) of 16.2 is satisfied. First, from [GLS3], either Z(S)= ∼ z = Z2,orG is F23 and each involution z ∈ Z(S) is weakly closed in Z(S). Second, if G is not J2 or J3 then S/O2(CG(z)) is selfnormalizing in CG(z)/O2(CG(z)), either from the proof of this fact for Sn and Am, m>5, during the proof of 16.4, the fact that it holds for groups of Lie type over F2, or the fact that it holds for smaller sporadics by induction on |G|. Thus (i) follows unless G is J2 or J3. On the other hand if G is J2 or J3,thenfrom[A2], M = CG(z) ∈Msatisfies (ii), establishing the claim. We must prove F is simple, so by 16.2, it suffices to verify conditions (a) and (b) of 16.2. Condition (b) follows from the claim. As p =2andG is not J2, S is nonabelian by 16.6.1. Therefore G is not 2-Goldscmidt, so condition (a) holds too. This completes the proof of the lemma. ∼ (16.9). (1) If (G, p)=(Ru, 3) then X =Ξ(G, S) = 2F (2). ∼ 4 (2) If (G, p)=(M , 3) then X = Aut(M ). 24 12 ∼ ∼ (3) If (G, p)=(Ru, 5) then S ∈ Syl5(L) with L = Aut(L3(5)) and FS (G) = F (L). S ∼ ∼ (4) If (G, p)=(J , 3) then S ∈ Syl (L) with L = 2F (2) and F (G) = F (L). 4 3 4∼ S S ∼ (5) If (G, p)=(Co1, 5) then S ∈ Syl5(L) with L = PO5(5) and FS(G) = FS(L).

100 MICHAEL ASCHBACHER ∼ Proof. Suppose ﬁrst that (G, p)=(Ru, 3). Then from [GLS3], S = 31+2 with ∼ frc NG(S)=SD,whereD = SD16 is faithful on S,andF = {S, Ui :1≤ i ≤ 4}, where |S : U | =3andN (U ) is the split extension of U by GL (3) acting faithfully i G i i ∼ 2 on U .Moreoverfrom[A2], there is X ∈Mwith X = 2F (2), and from the i ∼ 4 ∼ subgroup structure of X,Ξ(X, S)=X, NG(S) = NX (S), and NX (Ui) = NX (Ui) for each i. It follows that Ξ(G, S)=X, establishing (1). ∼ 1+2 Next suppose that (G, p)is(M24, 3). Then from [GLS3], S = 3 with frc NG(S) the extension of S by D8,andF = {S, U1,U2},where|S : Ui| =3and NG(Ui) is the split extension of Ui by GL2(3). From [A2], there is X ∈Mwith ∼ frc X = Aut(M12), and X contains NG(R) for each such R in F ,so(2)holds. ∼ 1+2 Suppose (G, p)is(Ru, 5) or (J4, 3). Then from [GLS3], S = p , and, up ∼ frc to conjugation in NG(S), there are two members S and R = Ep2 of F ,so F = NF (R),NF (S) .SetG1 = NG(R), G2 = NG(S)andG1,2 = G1 ∩ G2. Suppose ﬁrst that G is Ru.ThenG2 is the split extension of S by Z4 wr Z2 and G1 is the split extension of R by GL2(5). In particular Aut(G2)=Inn(G2), so the amalgam A =(G1,2 → Gi : i =1, 2) is determined up to isomorphism. Moreover F F NF (R)= NS (R)(G1)andNF (S)= S(G2), so (cf. 3.5 and 3.7 in [ACh]) F N F F F A = F (R),NF (S) = NS (R)(G1), S(G2) = S(F ( )), where F (A) is the free amalgamated product of the amalgam A and S is a “Sy- low 5-subgroup” of F (A). As L = Aut(L (5)) has the same amalgam A of fully 3 ∼ ∼ normalized radical centrics, it follows that F = FS(F (A)) = FS (L), establishing (3). So take G to be J4. The situation is a bit more complicated in this case, but the argument is essentially the same. For J ⊆{1, 2},setQJ = O2(GJ ) ¯ and GJ = GJ /QJ .ThenQ2 is of order 2 and Q2 = CG2 (R)=CG1 (S), so the inclusion maps G → G induce injections α : G¯ → G¯ . Form the amalgam 1,2 i i 1,2 i ∼ (α : G¯ → G¯ : i =1, 2) and F (A). Using 14.4.3, F ∩ (G ) F (G¯ ) , i 1,2 i S Gi i = S∩Gi i F ∼ F A ¯ ¯ ¯ so = S¯(F ( )). This time G2 is the split extension of S by SD16 and G1 is the split extension of R¯ by GL2(3), so again A is determined up to isomorphism. 2 Finally L = F 4(2) has the same amalgam of fully normalized radical centrics, so (4) holds. Finally suppose (G, p)=(Co , 5). Then, using [GLS3], F frc = {S, R ,R }, ∼ 1 ∼ 1 2 where Z = Z(S) = Z ; R = O (N (Z)) = 51+2,andN (Z) is the split extension 5 1 ∼ 5 G G of R1 by GL2(5); and E53 = R2 = J(S)andNG(R2) is the split extension of R2 by Z4 × S5.LetGi = NG(Ri), G1,2 = G1 ∩ G2, and form the amalgam A = (G1,2 → Gi : i =1, 2). Observe that G1,2 = NG(S), so F = FS(Gi):i =1, 2 ,and therefore, as usual, F = FS (F (A)). Again Aut(G2)=Inn(G2), so A is determined up to isomorphism. In particular, A is the amalgam of parabolics in PO5(5), so (5) follows.

(16.10). Assume G is a sporadic simple group, but not p-Goldschmidt. Then one of the following holds: (1) F (G) is simple. S ∼ (2) (G, p)=(Ru, 3) or (M24, 3),andFS(Y ) FS (X) = FS (G),whereX = ∼ 2 ∼ ∗ Ξ(G, S) = F 4(2) or Aut(M12) for G = Ru or M24, respectively, and Y = F (X) is of index 2 in X. ∼ ∼ (3) (G, p)=(Ru, 5) and S ∈ Syl5(L) with L = Aut(L3(5)) and FS (G) = FS(L).

16. FUSION SYSTEMS IN SIMPLE GROUPS 101 ∼ ∼ (4) (G, p)=(J , 3) and S ∈ Syl (L) with L = 2F (2) and F (G) = F (L). 4 3 ∼ 4 S ∼S (5) (G, p)=(Co1, 5) and S ∈ Syl5(L) with L = PO5(5) and FS(G) = FS(L). Proof. Assume (G, p) is a counterexample. By 16.8, p is odd. As G is not p-Goldschmidt, S is nonabelian, so p ∈ Π by 16.6.1. If (G, p) appears in one of (2)-(5) then the corresponding conclusion holds by 16.9. Therefore we may assume (G, p) is not one of these pairs. In the remaining cases we appeal to 16.2, where it suﬃces to verify conditions (a) and (b) of that lemma. By hypothesis, G is not p-Goldschmidt, so condition (a) holds. Thus it remains to verify condition (b) of 16.2. Let frc D = N p (S):R ∈F . O (NG(R))

To verify condition (b) of 16.2, it suﬃces to show that D = NG(S). We appeal to the Tables in [A2] for the list M of maximal overgroups of S and G,andthe structure of those overgroups. Suppose ﬁrst that: (i) NG(S) ≤ M ∈Mwith M simple and frc N (S)=N p (S):R ∈F (M) and R S . M O (NM (R)) S

In that event by 16.7.2, NG(S)=NM (S) ≤ D, so condition (b) of 16.2 holds. We also observe that (i) holds when (G, p)is(F , 3), (Ly, 5), and (F , 3), with ∼ 22 2 M = Ω7(3), G2(5), and F23, respectively. In the ﬁrst two cases, (i) holds by the proof of 16.3. In the third it follows from the discussion of F23 below. Thus we may assume (G.p)isnoneofthesethreepairs. Suppose next that: ∗ p (ii) NG(S) ≤ M ∈Mwith Op(M)=F (M)andM = O (M). frc p In this case Op(M) ∈F by 16.7.3, so as M = O (M)andNG(S) ≤ M, NG(S)=NM (S)=D.Moreoverif(G, p)is(Suz,3), (Co1, 3), (F2, 5), or (F24, 3) then we can check in [A2] that (ii) holds, so we may assume that (G, p)isnoneof these pairs. Suppose that: ∗ (iii) Z(S)=Z is of order p, H = NG(Z) ∈M, F (H)=Op(H), |H : ∗ p | − ∈M O (H) = p 1, and M with F (M)=Op(M)andAutOp (M)(Z)=Aut(Z). If (iii) holds we claim that NG(S)=NOp (H)(S)NOp (M)(S) with Op(H)and frc Op(M)inF , so once again D = NG(S). First S ≤ CM (Z), so by (iii) and a ≤ Frattini argument, NOp (M)(S) induces Aut(Z)onZ.ThenasNG(S) NG(Z)= p p H,asO (H) ≤ CG(Z), and as |H : O (H)| = p − 1=|Aut(Z)|, it follows that p ≤ p O (H)=CG(Z)andNG(S) NOp (M)(S)O (H), so that the claim holds. Also we can check in [A2] that (iii) holds when p =3andG is F23, Mc, F5 or F3,and when G is F and p = 13. Thus we may assume (G, p) is none of these pairs. 1 ∼ We next assume that S = p1+2.LetU be the set of subgroups U of index p ∗ in S such that Z = Z(S)isnotnormalinNG(U), T = NG(S), and T = T/S. p frc Then LU = O (NG(U)) is the split extension of U by SL2(p), so U⊆F ,and DU = TU : U ∈U ≤D,whereTU = T ∩ LU .ThustoshowT = D in this case, it suﬃces to show that T ≤ DU . We establish this when p =3andG is M12 or He, or p =7andG is He, O N,orp =13andG is F24.LettU be an involution in ∈U−{ } ∗ ∗ TU and observe that U =[S, tU ], so if V U then tU = tV and tU does not invert S/Z(S).

~~102 MICHAEL ASCHBACHER~~

∗ ∼ First consider the two cases where p =3.From[GLS3], T = E4 and |U| =2. U { } ∗ ∗ Now setting = U1,U2 and taking ti = tUi , t1 = t2 by the previous paragraph, so indeed T = S, t1,t2 = D. ∗ ∼ Next consider the three cases where p =7.Herefrom[GLS3], T = Z3 × D2m ∼ and |U| = m,wherem =3, 4, 6forG = He, O N, F24, respectively. Then as ∗ ∼ ∗ ∗ ∗ ∈ ∗ TU = Z6,astU = tV for U = V ,andastU / Z(T )astU does not invert S/Z(S), it follows that T = D. ∗ ∼ ∗ ∗ ∼ Finally suppose (G, p)=(F1, 13). Here T = Z2/(Z12 SL2(3)), TU = Z12, and |U| = 6, so, arguing as usual, T = D. We have reduced to the following cases: (G, p)=(Co3, 3), (Co2, 3), or (Ly, 3). Suppose first that (G, p)=(Co3, 3) and let Z = Z(S). Then from [A2], H = NG(Z) ∼ 1+4 ∼ is the split extension of Q = 3 by Z /(Z ∗ SL (9)), R = J(S) = E 5 with 2 4 2 ∼ 3 M = NG(R) the split extension of R by Z2 × M11,andNG(S)/S = SD16.Let 3 3 X be the subgroup of order 8 in O (NH (S)); we show X O (M), so that ≤ NG(S)= X, NO3 (M)(S) D, and hence condition (b) and the lemma hold in this case. Namely X centralizes Z, but subgroups Y of O3 (M) normalizing S do not centralize Z. For example there are exactly two 5-dimensional irreducible F3M11-modules, these are duals of each other, and from the discussion in the proof of 48.5 of [SG] of such a module V , Y is nontrivial on points Z of V fixed by Y . Suppose next that (G, p)=(Co2, 3). Let Z = Z(S). Then from [A2], H = ∼ 1+4 3 NG(Z) = S5/Q8D8/3 ∈M,soTZ = O (H) ∩ NG(S) is of index 2 in NG(S). ∗ 3 Further J(S)=F (NG(J(S)), and Z is inverted in O (NG(J(S)), so D = NG(S) in this case as in (iii). This leaves the case (G, p)=(Ly, 3). From [A2], there is M ∈Msuch that ∗ ∼ ∼ frc R = J(S)=F (M) = E35 and M/R = Z2 × M11. From 16.7.3, R ∈F ,and from the 3-local structure of M11 in [GLS3], NG(S)=ST where T = t ×T0 and 3 ∼ ∼ ∼ T0 = T ∩ O (M) = SD16.NextZ = Z(S) = E9 and from [A2], H = NG(Z) = ∗ ∼ ∼ Z /(Z ∗ SL (5))/Q,whereQ = F (H) = 32+4, T = T ∩ O3 (H) = Z ,and 2 8∼ 2 ∼H 4 T/TH = D8. In particular TH T0, or else T/TH = E8.Thusas|T : T0| =2, T = T0TH ,soNG(S)=ST = D, completing the verification of condition (b) of 16.2 and the proof of the lemma. ∼ (16.11). Assume p =3and G = G2(q).LetZ = Z(S) and H = NG(Z).Then 3 ∼ ≡ (1) H is O (H)=L = SL3(q), q mod 3, extended by a graph automorphism t. (2) Z is strongly closed in S with respect to F. (3) F = FS(H) and FS(L) F. (4) If q is not congruent to modulo 9, then F = FS(NH (S)), so that S = Op(F). (5) If q ≡ mod 9 then FS(L) is quasisimple with center Z. Proof. As discussed in [A3], G possesses a 7-dimensional indecomposable module V over F = Fq, and there exists a hyperplane U of V such that HU = 3 ∼ ≡ NG(U)istheextensionofLU = O (HU ) = SL3(q), q mod 3, by a graph automorphism t,andHU = NG(ZU ), where ZU = Z(LU ). Then |G|3 = |LU |3,so we may take S ∈ Syl3(LU ). Therefore Z = ZU , and (1) is established. Suppose first that =1.Thenfrom[A3], U = U1 ⊕ U2 where Ui is a 3- ∼ ∗ dimensional irreducible FL-module with U2 = U1 the dual of U1 as an FL-module. In particular a generator z for Z acts via scalar multiplication on U1 with eigenvalue

~~16. FUSION SYSTEMS IN SIMPLE GROUPS 103~~

2 ω of order 3, and on U2 via ω . Further 1 is an eigenvalue for z of multiplicity 1. On the other hand if x ∈ S −Z is of order 3, then x is free on Ui, so the multiplicity of ω as an eigenvalue of x is 3. Thus (2) holds in this case. − ˜ ∼ 2 ≤ ˜ Now suppose = 1. Then G embeds in G = G2(q ) with H H = NG˜ (Z) 2 the extension of SL3(q )byt. Therefore by the previous paragraph, Z is strongly closed in H˜ with respect to G˜, and hence Z is strongly closed in H with respect to G, so (2) holds in this case also. By (2), H controls fusion in S, so (3) holds. Under the hypothesis of (4), S/Z is abelian, so (4) follows from (2) and 14.7.c. Finally (5) follows from he simplicity of FS/Z (L/Z)whenq ≡ mod 3, an exercise which is best done in the context of a more complete discussion of the fusion systems of groups of Lie type.

∗ (16.12). Assume F is nonabelian and simple. Set G = G/Op (G).Then ∼ ∗ (1) F = FS∗ (G ). (2) L∗ = F ∗(G∗)=Op (G∗) is nonabelian simple. ∗ ∗ ∗ (3) G = L Op (CG∗ (S )). ∼ ∗ (4) F = FS∗ (L ).

Proof. Part (1) follows from 14.4.3, and by (1) we may assume Op (G)=1. Let L be a minimal normal subgroup of G.AsOp (G)=1,1= S ∩ L. By 14.4.4, FS∩L(L) F,soasF is simple, F = FS∩L(L). Therefore S ≤ L and by a Frattini argument, G = LE,whereE is a Hall p -subgroup of NG(S). Applying this argument to a component K of L, S ≤ K,soK = Op (L)charL, and hence K = L by minimality of L. This proves (2) and completes the proof of (4). p As F is simple, F = O (F), so it follows that E =(E ∩ L)CE(S), so G = LOp (CG(S)), establishing (3).

Notation 16.13. Given a ﬁnite group D,writeF(p, D) for the isomorphism type of the fusion system FT (D), where T ∈ Sylp(D). The next two lemmas follow by applying various results in [BMO]. (16.14). Assume p is odd and q is a prime power with (p, q)=1. (1) Suppose D = GUn(q); Spn(q) or SOn+1(q) with n even; or GOn(q) with n even and q not congruent to − modulo p. Then there exists a prime power q with (p, q )=1such that F(p, D)=F(p, GLn(q )). (2) Suppose D is GOn(q) with n even and q not congruent to modulo p.Then there exists a prime power q with (p, q )=1such that F(p, D)=F(p, GLn−2(q )).

Proof. When D is GUn(q), the lemma follows from part (d) of Theorem A in [BMO]. When D is Spn(q), it follows from 5.3 in [BMO], and the discussion following that proposition. In the remaining cases, the lemma follows from 5.2 in [BMO] and the symplectic case.

~~(16.15). Assume p is odd and G is a simple classical group over a ﬁeld Fq with (p, q)=1.ThenF(p, G)=F(p, Lm(q )) for some integer m and prime power q with (p, q)=1.~~

~~Proof. We may choose G =[D, D]/Z([D, D]), where D is GLn(q), GUn(q), F F Spn(q), SO2k+1(q), or GO2k(q). By 16.14, (p, D)= (p, E), where E = GLm(q )~~

104 MICHAEL ASCHBACHER for some prime power q with (p, q) = 1, and some integer m. By 1.4.c in [BMO], F(p, [D, D]) = F(p, [E,E]). Then by parts (a) and (b) of 1.4 in [BMO], F(p, G)=F(p, [D, D]/Z([D, D])) = F(p, [E,E]/Z([E,E])) = F(p, Lm(q )).

~~CHAPTER 17~~

~~An example~~

In this section V is a 20-dimensional vector space over the ﬁeld F of order 2, ∼ c ∼ G = GL(V ) = L20(2), S ∈ Syl5(G), and F = FS (G). We will see that π1(F ) = 5 Z4,sothatO (F)=E is a proper normal subsystem of F of index 4. Indeed we will see that E is simple and exotic. Each of these facts follows from theorems of Ruiz in [R]. However it is relatively easy to verify the observations in this special case, using Oliver’s lemma 11.11 to simplify the calculation of the index of E in F. Set I = {1,...,5} and write V = V1 ⊕ ··· ⊕ V5 as the direct sum of ﬁve 4-dimensional subspaces Vi,andsetV = {Vi : i ∈ I}.SetM = NG(V) and let V ∈ ∈ −{ } K = GV be the kernel of the action of M on .Fori I,setVi = Vj : j I i ∩ ∼ and set Ki = CG(Vi ) NG(Vi), so that Ki acts faithfully as GL(Vi) = L4(2) on V and K = K ×···×K . Further there is a complement B to K in M acting i 1 ∼ 5 faithfully as Sym(V) = S5 on V with NB(Vi)=CB(Vi), so that M is the wreath product of L4(2) by S5. As |G| =56 = |M| ,wemaytakeS ∈ Syl (M), and S = XQ,whereX = ∼ 5 5 5 S ∩ B = Z and Q = S ∩ K.MoreoverQ = S ×···×S ,whereS = S ∩ K is of 5 1 5 ∼ i i order 5, so S is the wreath product of Z5 by Z5,andQ = J(S) = E55 .

(17.1). (1) S = {Si : i ∈ I} is the set of subgroups Y of S of order 5 with dim([V,Y ]) = 4. (2) H = NG(Q)=NG(S)=NM (Q)=BNK (Q). (3) K = N (Q)=H ×···H ,whereH = N (S ) is the extension of F# Q ∼ K 1 5 i Ki i 16 by Aut(F ) = Z . 16 4 ∼ (4) Let Z = Z(S).ThenZ = Z5 is a follow diagonal subgroup of Q = S1 × ···×S5,wemaychooseB to centralize Z,andCM (Z)=CK (Z)B with CK (Z)= F (K ∩ H). (5) ZG ⊆ Q. ∼ (6) Set H¯ = H/F(H∩K).ThenF (H∩K)=CG(Q),soAutG(Q) = H¯ = K¯QB¯ 5 ¯ 5 ¯ ¯ ¯ 4 is the wreath product of Z4 with S5.InparticularO (H)=O (B)[B,KQ] is Z4 extended by A5. H¯ ¯ ¯ ¯ (7) Δ= CH¯ (Z) = B[B,KQ]. Proof. Part (1) is trivial. Then as V = {[V,Y ]:Y ∈S}, we conclude that ∼ (2) holds. Then (3) follows from (2) and the structure of L (2) = A . ∼ 4 8 As S = Z5 wr Z5, Z is of order 5, and (4) follows from (3). As V =[V,Z] while CV (Y ) =0for Y of order 5 in S with Y = Q, (5) holds. Part (6) follows from (2) and (3), and (7) follows from (4) and (6).

~~(17.2). There exists a subgroup P of S such that: ∼ 1+2 ∼ (1) P = 5 with Z = Z(P ) and P ∩ Q = Z2(S) = E25. (2) NG(Z2(S)) ≤ M.~~

~~105~~

106 MICHAEL ASCHBACHER ∼ (3) CG(P )=CF (K∩H)(P ) = Z15. (4) NG(P ) is the split extension of CG(P ) by GL2(5). (5) P ∈Ffrc. 5 ∼ (6) O (NG(P )) ∩ NG(S)=NS(P )CG(P )Y ,whereZ4 = Y centralizes Z. (7) Δ=O5 (H¯ )Y¯ . Proof. Let P be a group isomorphic to 51+2.ThenP has a faithful irreducible representation on V , so without loss, P ≤ S.AsP is nonabelian, P Q,so Z(P )=CQ(P )=Z and |P : P ∩ Q| = |S : Q| =5.AsZ =[P ∩ Q, P ]andX is free on Q, it follows that P ∩ Q = Z2(S), completing the proof of (1). As V is the set of irreducible Z (S)-submodules of V , (2) follows. By (1) and (2), and by 2 ∼ 17.1.4, CG(P )=CM (Z) ∩ CG(P )=BF(H ∩ K) ∩ CG(P )=CF (H∩K)(P ) = Z15, establishing (3). The representation of P on V is determined up to equivalence, so (cf. 1.1 in ∼ [SG]) AutG(P ) = Aut(P ), which implies (4). Then (3) and (4) imply (5). Let T = N (P ). By (3) and (4), O5 (N (P )) ∩ N (T )=TC (P )Y where ∼ S G G G Z4 = Y centralizes Z.NowY acts on Z2(T )=Z2(S), so Y ≤ M by (2). Therefore Y acts on PQ = S, completing the proof of (6). By 17.1.7, Δ = B¯[B,¯ K¯ ], so by 17.1.6, Δ = O5 (H¯ ) is of index 2 in Δ with ∼ Q 0 Δ=ΔY¯ ,whereY = Z is a Sylow 2-subgroup of N (X). By 17.1.4, Y is 0 0 0 4 ∼ B 0 also Sylow in CM (Z), while by (6), Z4 = Y centralizes Z and acts on S,soas ≤ ¯ ¯ CH¯ (Z) Δ it follows that Δ0Y =Δ0Y0 = Δ, establishing (7).

c ∼ 5 (17.3). (1) π1(F ) = Z4,soE = O (F) is of index 4 in F. (2) E is simple. Proof. We first prove (1) via an appeal to Oliver’s lemma 11.11 applied to Q, Z in the roles of the groups “T,Y ” in that lemma. As Q = J(S), Q is weakly closed in S with respect to F, and we have seen that Q = CS(Q). By 17.1.5, ZF ⊆ Q. Therefore the hypotheses of 11.11 are satisfied. By 17.1.6, AutF (Q)=AutG(Q)=H¯ , and then the group denoted by “Δ” in 11.11 is the group Δ of 17.1.6. Define Σ and Ξ as in 11.11.3, and let ψ : AutF (S) → 0 AutG(T )=H¯ be the restriction map. By 11.11.3, Ξ ≤ AutF (S) ≤ Σ, with ¯ ¯ 5 ¯ ¯ Ξψ = NO5 (H¯ )(S)andΣψ = NΔ(S). By 17.2.7, Δ = O (H)Y , and by definition 0 0 −1 of Y in 17.2.6, AutY (S) ≤ AutF (S), so it follows that AutF (S)=NΔ(S¯)ψ =Σ, so, using parts (6) and (7) of 17.1, F c ∼ 0 ∼ ¯ ¯ ∼ π1( ) = AutF (S)/AutF (S) = NH¯ (S)/NΔ(S) = Z4, establishing (1). By 11.10, O5 (E)=E,sotoshowthatE is simple, it suffices, by parts (3) and (4) of Theorem 8, to assume that T is a proper nontrivial subgroup of S strongly closed in S with respect to E, and to exhibit a contradiction. Now 1 = T ∩ Z(S), Δ so as Z = Z(S)isoforder2,Z ≤ T .NowAutE (Q)=ΔandZ = Q,so 5 Ω Q ≤ T . Similarly Ω = O (AutG(P )) ≤ AutE (P ), so P = Z2(S) ≤T , and hence S = PQ = T , a contradiction. This proves (2).

~~It remains to show that E is exotic. Ruiz accomplishes this in [R], in much more general situations, by an appeal to [BM]. We sketch a somewhat diﬀerent proof here.~~

~~17. AN EXAMPLE 107~~

Assume E = FT (D) for some finite group D and T ∈ Sylp(D). Then by 16.12, we may assume D is a nonabelian finite simple group. As S is the wreath product of Z by Z , D is not sporadic or a group of Lie type in characteristic 5. 5 5 ∼ Suppose G = An is an alternating group on a set Ω of order n. Then by 16.5, we may take n =5a for some integer a ≥ p.As|S| =56, it follows from 16.4 that a =5andZ is generated by an element z with Fix(z)=∅.Butfors of order 5 in S − Q, also Fix(s)isempty,sos ∈ZD, contrary to 17.1.5. Therefore D is a group of Lie type over a field Fq with (5,q)=1. From section 4.10 in [GLS3], and in particular from 4.10.2 in [GLS3], the 5-rank of an exceptional group over Fq is not 5. Therefore D is a classical group. Then by 16.14, we may take D to be Ln(q) for some integer n.Hereweadoptthe notation of section 4.8 in [GLS3], except we write U for the n-dimensional vector ∼ space over Fq for Dˆ = GL(U) = GLn(q), and regard D as SL(V )/Z(SL(V )). In # ≤ ˆ particular d is the order of q in Fp and n = n0d + k0 where 0 k0 0, Ki is Fqd extended by Gal(Fqd /Fq)of 4 order d.AsAutF (Q)isZ extended by S5, we conclude that n0 =5andd =4. ∼ 4 Hence Qˆ = Q. ˆ { ∈ # } Next F (K1) acts faithfully on U1 as g(a):a Fqd ,whereweidentifyU1 with Fqd and g(a):b → ab. On the other hand, viewing U1 as a d-dimensional ⊗ qi ≤ ≤ Fq-space, g(a)actsonFqd Fq U1 with eigenvalues a ,0 i d,so d−1 i d det(g(a)) = aq = a(q −1)/(q−1). i=0 ∈ ∈ # It follows that for each b Fq, there exist a Fqd with det(g(a)) = b.There- ˆ ˆ fore as F (K1) centralizes Q, we conclude that D = CDˆ (Q)SL(U), and hence ∼ 4 AutF (Q)=AutD(Q)=AutDˆ (Q). This is a contradiction as O2(AutF (Q)) = Z4, ∼ 5 while O2(AutDˆ (Q)) = Z4.

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[A1] M. Aschbacher, Normal subsystems of fusion systems, Proc. London Math. Soc. 97 (2008), 239–271. MR2434097 (2009e:20044) [A2] M. Aschbacher, Overgroups of Sylow subgroups of sporadic groups,MemoirsAMS343 (1986), 1–235. MR831891 (87e:20037) [A3] M. Aschbacher, Chevalley groups of type G2 as the group of a trilinear form,J.Alg. 109 (1987), 193–259. MR898346 (88g:20089) [A4] M. Aschbacher, Generation of fusion systems of characteristic 2-type, Invent. Math. 180 (2010), 225–299. [ACh] M. Aschbacher and A. Chermak, A group theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver, Ann. Math. (to appear). [ASm] M. Aschbacher and S. Smith, The Classification of the Quasithin Groups. I, II Amer- ican Math. Soc., 2004. MR2097623 (2005m:20038a); MR2097624 (2005m:20038b) [BCGLO1] C. Broto, N. Castellana, J. Grodal, R. Levi, and B. Oliver, Subgroup families control- ling p-local finite groups, Proc. London Math. Soc. 91 (2005), 325–354. MR2167090 (2007e:20111) [BCGLO2] C. Broto, N. Castellana, J. Grodal, R. Levi, and B. Oliver, Extensions of p-local finite groups,Trans.AMS359 (2007), 3791-3858. MR2302515 (2008i:55013) [BLO] C. Broto, R. Levi, and Bob Oliver, The homotopy theory of fusion systems,J.Amer. Math. Soc. 16 (2003), 779–856. MR1992826 (2004k:55016) [BM] C. Broto and J. Møller, Chevalley p-local finite groups,Alg.Geom.Top.7 (2007), 1809–1919. MR2366180 (2009g:55017) [BMO] C. Broto, J. Møller, and B. Oliver, Equivalences between fusion systems of finite groups (to appear). [F] R. Foote, A characterization of finite groups containing a strongly closed 2-subgroup, Comm. Alg. 25 (1997), 593–606. MR1428800 (97k:20042) [FF] R. Flores and R. Foote, Strongly closed subgroups and the cellular structure of classifying spaces (to appear). [FGT] M. Aschbacher, Finite Group Theory, Cambridge University Press, 1986. MR895134 (89b:20001) [Go1] D. Goldschmidt, 2-fusion in finite groups, Ann. Math. 99 (1974), 70–117. MR0335627 (49:407) [Go2] D. Goldschmidt, Strongly closed 2-subgroups of finite groups, Ann. Math 102 (1975), 475-489. MR0393223 (52:14033) [GH] D. Gorenstein and M. Harris, Finite groups with product fusion, Ann. Math. 101 (1975), 45-87. MR0357605 (50:10073) [GLS3] D. Gorenstein, R. Lyons, and R. Solomon, The Classification of the Finite Simple Groups, Number 3, Mathematical Surveys and Monographs, vol. 40, AMS, 1999. MR1303592 (95m:20014) [GW] D. Gorenstein and J. Walter, Balance and generation in finite groups,J.Alg.33 (1975), 224-287. MR0357583 (50:10051) [L] M. Linckelmann, Simple fusion systems and Solomon 2-local groups,J.Alg.296 (2006), 385–401. [P1] L. Puig, Frobenius categories,J.Alg.303 (2006), 309–357. MR2253665 (2007j:20011) [P2] L. Puig, Frobenius categories versus Brauer blocks, Birkhauser, 2009. MR2502803 [R] A. Ruiz, Exotic subsystems of finite index in the fusion systems of general linear groups over finite fields (to appear).

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~~Titles in This Series~~

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