A CHARACTERIZATION of the 2-FUSION SYSTEM of L4(Q)

A CHARACTERIZATION OF THE 2-FUSION SYSTEM OF L4(q)

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of

Philosophy in the Graduate School of the Ohio State University

Justin Lynd, B.S.

Graduate Program in Mathematics

The Ohio State University

2012

Dissertation Committee:

Dr. Ronald Solomon, Advisor

Dr. Matthew Kahle

Dr. Jean-Francois Lafont

Dr. Richard Lyons c Copyright by

Justin Lynd

2012 ABSTRACT

We study saturated fusion systems F on a ﬁnite 2-group S with involution centralizer having a unique component on a dihedral group and containing the Baumann

2 subgroup of S. Assuming F = O (F), O2(F) = 1, and the centralizer of the component is a cyclic 2-group, it is shown that F is uniquely determined as the 2-fusion system of L4(q) for some q ≡ 3 (mod 4). This should be viewed as a contribution to a program recently outlined by Aschbacher for the classification of simple fusion systems at the prime 2. The analogous problem in the classification of finite simple groups of component type (the L2(q), A7 standard component problem) was one of the last to be completed, and was ultimately only resolved in an inductive context with heavy machinery. Thanks primarily to the hypothesis concerning the Baumann subgroup and the absence of cores, our arguments by contrast require only 2-fusion analysis and transfer. We prove a generalization of the Thompson transfer lemma in the context of fusion systems, which is applied often.

ii To David Reid Dillon

iii ACKNOWLEDGMENTS

To:

Ronald Solomon, for knowing precisely when to push and when to let be, providing continual support, being a careful mentor and constant advocate, sharing his over- whelming breadth of knowledge and insight, and dedicating large amounts time and eﬀort throughout this project (and others prior); I always knew I made the right choice.

Richard Lyons, for carefully reading this thesis and oﬀering many probing questions and helpful suggestions; here’s to the next three years.

Matthew Kahle and Jean Lafont, for serving on my dissertation committee.

Mathematical mentors, colleagues, and friends: Andy Chermak, Dan File, Mike

Geline, George Glauberman, David Green, Ellen Henke, Ian Leary, Bob Oliver, Silvia

Onofrei, Sejong Park, Julianne Rainbolt, Radu Stancu, Matt Welz, erica Whitaker,

James Wilson, among others for mathematical discussions, guidance and support, and allowing me the pleasure of your company.

The OSU Mathematics Department for providing a comfortable, yet stimulating en- vironment in which to work.

The OSU Graduate School for a Presidential Fellowship which allowed for the completion of this dissertation.

iv Kristen, for patience, unending support, and many sleepless nights and days.

Mackenzie and Liam, for continually teaching me about myself.

Mom, for love, wisdom, and guidance.

David Dillon, the primary reason I stand where I stand.

Moy, for friendship, conversations, and good times (minus the retinal tear).

Thank you.

v VITA

1981 ...... Born in Huntington, West Virginia

2004 ...... B.S. Mathematics, Marshall University

2004-2010, 2012 ...... Graduate Research/Teaching Associate, Math- ematics, The Ohio State University

2011 ...... Presidential Fellow, The Ohio State Uni- versity

PUBLICATIONS

Adam Allan, Michael Dunne, Harold Ellingsen, and John Jack. Classiﬁcation of the group of units in the Gaussian integers modulo n, The Pi Mu Epsilon Journal, Issue 12:9, (2008), 513-519.

Justin Lynd and Sejong Park. Analogues of Goldschmidt’s thesis for fusion systems, J. Algebra 324 (2010), 3487–3493.

Justin Lynd. 2-subnormal quadratic oﬀenders and Oliver’s p-group conjecture, Pro- ceedings of the Conference on Algebraic Topology, Group Theory, and Representation Theory, Proc. Edin. Math. Soc., accepted.

vi FIELDS OF STUDY

Major Field: Mathematics

Specialization: Finite Group Theory

vii TABLE OF CONTENTS

Abstract ...... ii

Dedication ...... ii

Acknowledgments ...... iv

Vita ...... vi

CHAPTER PAGE

1 Introduction ...... 1

1.1 The L2(q), A7 problem in ﬁnite groups ...... 5 1.2 The proof of the main theorem ...... 7 1.3 Outline ...... 9 1.4 Notation ...... 11

2 Fusion and linking systems ...... 12

2.1 Automorphism groups of p-groups ...... 12 2.2 Ln(q) and Un(q) for n 6 4...... 16 2.3 Fusion systems ...... 18 2.4 Subsystems and normality ...... 28 2.5 The hyperfocal and residual subsystems ...... 31 2.6 Centralizers ...... 36 2.7 Simple fusion systems ...... 38 2.8 Linking systems ...... 43

3 Thompson Transfer Lemma for fusion systems ...... 48

4 The involution centralizer ...... 56

4.1 Reduced and tame fusion systems ...... 56 4.2 Structure of the involution centralizer ...... 59

5 Proof of Theorem A ...... 63

5.1 The 2-central case ...... 63 viii 5.2 The 2-rank 3 case ...... 70 5.3 The 2-rank 4 case: |Q| =2 ...... 74 5.4 The 2-rank 4 case: |Q| > 2 ...... 82 5.5 Proof of Theorem A ...... 96

Bibliography ...... 97

ix CHAPTER 1

INTRODUCTION

Let p be a prime number. If G is a ﬁnite group with Sylow p-subgroup S, two subsets

X and Y of S are fused in G if they are conjugate in G, i.e. if there exists g ∈ G with gXg−1 = Y . A subgroup H of G controls fusion in S if whenever two subsets of S are fused in G, then they are fused in H. The study of p-fusion in finite group theory began with the work of Frobenius and Burnside on control of fusion in the last decade of the 19th century. In the latter half of the 20th, the analysis of 2-fusion was a central technique in the classification of the finite simple groups. Now, motivated by categorical models of p-fusion originated by Lluis Puig (see [Pui06]), fusion theory has evolved to become both the foundation for the investigation of p-completed classifying spaces of finite groups and related spaces, as well as a unifying tool which motivates long-standing local-to-global conjectures in modular representation theory. For good introductions to this subject and its applications, see [AKO11] and [Cra11b].

A fusion system on a ﬁnite p-group S is a category F whose objects are the subgroups of S, and whose morphisms satisfy

HomS(P,Q) ⊆ HomF (P,Q) ⊆ Inj(P,Q) for P , Q 6 S in such a way that every morphism of F is the composition of an isomorphism followed by an inclusion. Here, for any group X which contains S, the set HomX (P,Q) is the set of group homomorphisms from P to Q which are

1 induced by conjugation by an element of X, and Inj(P,Q) is the set of injective group homomorphisms from P to Q. If G is a ﬁnite group and S ∈ Sylp(G), then there is a fusion system F = FS(G) with HomF (P,Q) = HomG(P,Q) and we say that

F is realized by G. Sometimes we write Fp(G) for FS(G) when the Sylow p-subgroup S is unimportant. A fusion system as described above is too general an object to say many meaningful things about. For this reason one imposes axioms which force S to act like a Sylow p-subgroup of F, and we speak of a saturated fusion system. These axioms are easily checked via Sylow’s Theorem for the fusion system of a ﬁnite group.

See Chapter 2 for more details.

The emerging “structure theory” of saturated fusion systems, continuing to un- fold as it has over the last decade, parallels that of ﬁnite groups. Axioms and important foundational ideas have originated and have been subsequently simpliﬁed in work of Puig [Pui06], Broto, Levi, and Oliver [BLO04], Stancu [Sta06], and Roberts and Shpectorov [RS09]. Many standard objects and key concepts in group theory have been written in the language of fusion systems. For example, there are notions of the hyperfocal subsystem, Op(F), and the residual subsystem, Op0 (F), of a fusion system [Pui06, BCG+07], constrained fusion systems [BCG+05], normal subsystems [Lin06, Asc08, Cra11b], and the layer E(F) and generalized Fitting subsys-

∗ tem F (F) = Op(F)E(F) of a fusion system [Asc08, Asc11a]. Once saturation is determined for appropriate subsystem analogues of important subgroups as above, classical theorems of group theory often have analogues in fusion systems which ad- mit easier proofs and which require less machinery than their counterparts in group theory. Some examples of this phenomenon lie in the work of Kessar and Linckel- man on Glauberman’s ZJ-theorem [KL08], D´ıaz,Glesser, Mazza, Park, and Stancu on Glauberman-Thompson Theorems on control of fusion [DGMP09] and Yoshida’s transfer theorem [DGPS11], Aschbacher’s E-balance theorem [Asc11a], Park and the

2 author’s work on Goldschmidt’s thesis [LP10], and Onofrei-Stancu on Stellmacher’s

W -functors [OS09], among others to appear or in progress.

One particularly salient result from ﬁnite group theory whose analogue does not hold the context of saturated fusion systems is the following theorem, proved by

Solomon [Sol74]: There is no ﬁnite simple group with an involution centralizer isomorphic to Spin7(q) for any odd q. Indeed, motivated by predictions of Benson [Ben94] that Solomon’s non-existent sporadic groups should nevertheless have 2-completed classifying spaces, Levi and Oliver construct in [LO02] a simple saturated fusion system FSol(q) on the Sylow 2-subgroup Sq of Spin7(q) for each odd q which has ∼ the property that CF (z) = FSq (Spin7(q)) for an involution z ∈ Z(Sq). Moreover, these fusion systems are not realizable as the fusion system of any ﬁnite group by

Solomon’s Theorem. Such fusion systems are called exotic. While exotic fusion systems at odd primes seem to be prevalent and have been constructed by several au- thors [BLO04,BLO06,RV04,Rui07,CP10], the Solomon systems are the only known simple exotic systems at the prime 2.

On the heels of the initial success of the above eﬀorts and to provide a framework for the search for new exotic 2-fusion systems, Michael Aschbacher has proposed

[AKO11, II.14-15] a program for the classification of simple fusion systems at the prime 2 which is analogous to Gorenstein’s program for the classification of finite simple groups. A crucial difference is the use of the Baumann subgroup to provide the separation of simple fusion systems into even and odd cases. Aschbacher suggests likely advantages of partitioning the saturated 2-fusion systems into those of Baumann characteristic 2-type and those of Baumann component type.

Deﬁnition 1.0.1. Let F be a fusion system on a 2-group S. Then F is of Baumann characteristic 2-type if the normalizer fusion system NF (X) is constrained for every fully F-normalized subgroup X of S with the property that NS(X) contains the

3 Baumann subgroup of S. F is of Baumann component type if there exists a fully

F-centralized involution x ∈ S of order 2 such that CS(x) contains the Baumann subgroup of S and CF (x) is not constrained, i.e. CF (x) has a component.

For the precise deﬁnitions of the relevant terms here and below, see Chapter 2, in particular Section 2.7. Here and throughout this thesis, we take the elementary abelian version of the Thompson subgroup: J(S) = hA(S)i where A(S) is the set of elementary abelian subgroups of S of greatest order. Then the Baumann subgroup of S is deﬁned as Baum(S) = CS(Ω1(Z(J(S)))) (see [AS04, Section B.2]), which sometimes contains J(S) properly.

The goal of this thesis is to make a contribution to the classiﬁcation of simple

2-fusion systems of Baumann component type. We consider fusion systems with a involution centralizer having a component on a dihedral 2-group. Our main theorem is the following characterization of the 2-fusion system of L4(q).

Theorem A. Let F be a saturated fusion system on the 2-group S with O2(F) = F and O2(F) = 1. Let x ∈ S be a fully F-centralized involution and set C = CF (x),

T = CS(x), and K = E(C). Assume

(1) K is a fusion system on a dihedral group of order 2k,

(2) Q := CT (K) is cyclic, and

(3) Baum(S) 6 T .

∼ ∼ ∼ Then S = D2k o C2, and F = FS(G) where G = L4(q) for some q ≡ 3 (mod 4) with

ν2(q + 1) = k − 1.

Here the requirement that x be fully F-centralized implies that the centralizer subsystem CF (x) is saturated, and ν2 is the 2-adic valuation.

4 Hypothesis (1) means that K is uniquely determined as the sole perfect saturated fusion system on a nonabelian dihedral group of the given order. Thus, K is isomor-

2 phic to the 2-fusion system of L2(q1) for some q1 ≡ ±1 (mod 8) with ν2(q1 −1) = k+1. Hypothesis (2) implies that K is the unique component of C, and hints that K should be regarded as a “standard component” of F in the sense of [Asc75]. To date, there is no suitable notion of a standard component K of a fusion system available, which presumably would say in particular that CF (K) is a “tightly embedded” subsystem of F. Aschbacher has deﬁned an appropriate notion of tightly embedded subsystem, in work in progress, so the main obstruction seems to be what one means by the centralizer CF (K), and more generally by the normalizer NF (K). In [Asc11a], As- chbacher deﬁnes the centralizer of a normal subsystem, and we rely on this to talk about CT (K) as in (2), K being a normal subsystem of C. At any rate, hypotheses (1) and (2) are equivalent to specifying the structure of the generalized Fitting subsystem

∗ of C as F (C) = FQ(Q) × K, and K should be a standard component of F under any appropriate future deﬁnition. See Section 2.7.

To motivate the third hypothesis of Theorem A involving the Baumann subgroup, it is helpful to have some history of the analogous standard component problem in the classiﬁcation of ﬁnite simple groups, which we outline next.

1.1 The L2(q), A7 problem in ﬁnite groups

The analogous standard component problem in the classification of finite simple groups of component type, the L2(q), A7 problem, was one of the last to be completed and one of the most difficult of its kind. Let G be a finite core-free group with an involution centralizer C = CG(x) having a perfect normal subgroup K with nonabelian ∼ dihedral Sylow 2-subgroups (and hence K/O(K) = L2(q) for some q ≡ ±1 (mod 8)

5 ∗ or A7), such that Q ∈ Syl2(CC (K/O(K))) is cyclic. Let P ∈ Syl2(K). If F (G) is ∼ simple then K/O(K) = A7 and

∗ ∼ F (G) = A9 or He,

∼ or K/O(K) = L2(q) and

∗ ∼ F (G) = J2,J3,HS,

A8, P Sp4(4),L5(2),U5(2),L3(4), (1.1.1)

1 1 1 1 2 2 2 2 − 4 L2(q ),L3(q),U3(q), P Sp4(q ),L4(q ),U4(q ), or Ω8 (q )

1 1 (In all cases aside from L4(q 2 ), U4(q 2 ), and HS, the involution x induces an outer automorphism of F ∗(G), and hence most situations do not occur in a simple group.)

Initial analysis of the above in the case |P | = 8 is already quite involved. This case was handled by Fritz [Fri77] and independently by Harris and Solomon [HS77,

Har77] with both accounts relying on the Gorenstein-Harada classiﬁcation of groups of sectional 2-rank at most 4 [GH74] as well as a classiﬁcation in Ph.D. theses of

Beisiegel and Stingl [Bei77, Sti76] of the simple groups with Sylow 2-subgroups of order at most 210. The methods rely on 2-group and fusion analysis to build up the approximate structure of the Sylow 2-subgroup of G in order to get in a position to apply transfer arguments and bound the Sylow 2-subgroup of F ∗(G). Complicating the picture, L2(q) falls victim to many exceptional isomorphisms for small q:

∼ L2(7) = L3(2)

∼ ∼ 0 ∼ L2(9) = A6 = Sp4(2) = Ω5(2).

In particular, the many faces of L2(9) force the recognition of the characteristic 2 almost simple groups Aut(Sp4(4)), Aut(L5(2)), and Aut(U5(2)), which have large

∗ Sylow 2-subgroups relative to F (CG(x)). In addition, the Higman-Sims group has an involution with a centralizer isomorphic to C2 ×Aut(L2(9)) and Sylow 2-subgroup 6 of order 210. Note, however, that these are examples of ﬁnite groups of Baumann characteristic 2-type.

For the remaining case |P | > 16, Harris [Har81] requires an inductive approach which relies on the solution of a wide variety of additional standard component problems, in particular for components involving most of the simple groups appearing in

(1.1.1) and their “pumpups”. Furthermore, Harris’ analysis is complicated by the presence of cores (odd order normal subgroups) in 2-local subgroups. Indeed, it had emerged in the mid 1970’s via work of Thompson and others that the analysis of a minimal counterexample G to the B-conjecture (and ultimately to the determination of all simple groups X such that CX (t) has a nontrivial odd order normal subgroup for some involution t of X) led inevitably to an involution centralizer of G with an

L2(q) component. Thus the L2(q)-problem and the B-conjecture were inextricably intertwined. For this reason, both Harris and Richard Foote, who handled the L2(q) problem [Foo78] in the case where the centralizer of the component is of 2-rank at least 2, must assume that G is a minimal counterexample to the list of target groups

((1.1.1) in the cyclic centralizer case) as well as a minimal counterexample to the

B-conjecture for the proofs.

1.2 The proof of the main theorem

The heart of the proof of Theorem A relies on 2-group analysis, fusion, and transfer, and requires no inductive hypothesis (on the fusion system F itself, or on automorphism groups in it). In this regard, it is similar in spirit to the initial treatment by

Fritz and Harris-Solomon when the Sylow 2-subgroup of the component is dihedral of order 8. At the beginning, we need some results on the vanishing of higher limits of certain functors related to L2(q) and its extensions (Section 2.8) due to Oliver [Oli06],

7 and recent work of Andersen, Oliver, and Ventura [AOV12] on reduced and tame fusion systems (Section 4.1). At the end, after having determined S as D2k o C2, we quote a result of Oliver [Olia], part of his classiﬁcation of reduced fusion systems over

2-groups of sectional 2-rank at most 4, to identify F as the fusion system of L4(q) for appropriate q.

But in the middle, the hypothesis that the involution centralizer should contain the Baumann subgroup (Hypothesis (3)) allows us to avoid building the Sylow 2- subgroup of groups in characteristic 2 as well as the Higman-Sims group (and Aut(J2),

Aut(J3), etc.), as the fusion systems of these groups are of Baumann characteristic 2-type. Indeed, this is one of the motivations for the use of the Baumann subgroup: one expects that fusion systems of Baumann component type will more directly correspond to fusion systems of groups of Lie type in odd characteristic, and fusion systems of Baumann characteristic 2-type to those in characteristic 2. Theorem A provides a key conﬁrmation of this expectation.

In addition, the analysis in the context of fusion systems is greatly simpliﬁed by the absence of cores. For a ﬁnite group G and prime p, the subgroup Op0 (G) centralizes any p-subgroup that it normalizes. But more generally, for any p-local subgroup N of G, Op0 (N) centralizes any p-subgroup of G that N normalizes. So cores of local subgroups induce no nontrivial p-fusion, and hence do not appear in the fusion system context.

Complementing the 2-group and fusion analysis, we make frequent use of the transfer map in fusion systems. The transfer map is deﬁned by way of an S-S biset associated to a saturated fusion system on S, which plays the role of the S-S biset

G in the case of a ﬁnite group, and depends on a set of morphisms which may be regarded as a set of S-S double coset representatives of the fusion system. Our main

8 application of transfer is via the following version of the Thompson Transfer Lemma

(or Thompson’s Fusion Lemma), which we prove in fusion systems and use often.

Theorem B. Suppose that F is a saturated fusion system on the p-group S, and that

T is a proper normal subgroup of S with S/T abelian. Let u ∈ S − T and let I be the set of fully F-centralized F-conjugates of u in S − T . Assume

(1) u is of least order in S − T ,

(2) the set of cosets IT = {vT | v ∈ I} is linearly independent in Ω1(S/T ), and

(3) F = Op(F).

Then u has a fully F-centralized F-conjugate in T .

The proof of this theorem, found in Section 3, may be applied in the group case to obtain several extensions of Thompson’s original lemma due to Harada, Goldschmidt, and Lyons.

1.3 Outline

We now outline the structure of this thesis and indicate where the various portions of the proof of Theorem A can be found in this document. In Chapter 2 we give the deﬁnition of a saturated fusion system and state known results which will be used later or which are necessary for context. The deﬁnition of the transfer homomorphism in saturated fusion systems is found in Chapter 3 together with a proof of Theorem B.

In order to get oﬀ the ground (Chapter 4) in describing the involution centralizer from knowledge of its generalized Fitting subsystem, we apply in Section 4.1 a recent theorem of Andersen, Oliver, and Ventura. This theorem says that, under certain conditions, extensions of realizable fusion systems are again realizable. Our

9 application of this theorem requires some knowledge of the vanishing of higher limits of certain functors on orbit categories of fusion systems, shown by Oliver in his resolution of the Martino-Priddy conjecture, and of which a discussion is found in

Section 2.8.

In Section 4.2, we begin the analysis of a fusion system F satisfying the hypotheses of Theorem A, recording some results about Aut(L2(q)) and their consequences and showing that S must be of 2-rank either 3 or 4.

Chapter 5 contains the heart of the proof of Theorem A. Sections 5.1 and 5.2 commence the analysis, where we obtain contradictions in the 2-central case (in which x ∈ Z(S) in the notation of Theorem A) and in the case in which S has 2-rank

3, respectively. These two sections are based upon the Gorenstein-Lyons-Solomon

[GLS05] treatment in the group case, mainly for the reason that our hypothesis involving the Baumann subgroup is already easily seen to be satisﬁed in these cases.

In Section 5.3, we treat the case in which Q = CT (K) has order 2. This is the place where the Baumann hypothesis is most crucial; we apply it repeatedly and obtain a contradiction in this case. Finally, in Section 5.4, we handle the case in which S has

2-rank 4 and the order of Q is at least 4. After some preliminary work, we show here that there are at most three possibilities for S. These correspond to the Sylow 2- subgroups of L4(q), P Sp4(q1), and P GL4(q) where q and q1 are as in Theorem A and the discussion after it. Then we compute the centralizer of the central involution, ruling out the P GL4(q)-case. A brief analysis of the resulting fusion information rules out the P Sp4(q1)-case. Finally, we appeal to a result of Oliver classifying fusion systems over D2k o C2 to identify F and complete the proof.

10 1.4 Notation

Our notation is mostly standard. We follow [Gor80,GLS05] for group theoretic terms and [AKO11] for fusion system concepts. Some items that will not be elaborated upon are as follows. Here n is a positive integer, p is a prime, G is a ﬁnite group, and

P is a p-group.

• Cn is the cyclic group of order n, D2n (n > 2), Q2n (n > 3), SD2n (n > 4) are the dihedral, quaternion, and semidihedral groups, respectively, of order 2n

• G# is the set of nonidentity elements of G

•Ip(G) is the set of elements of G of order p

pn • Ωn(P ) = hx ∈ P | x = 1i

n pn • f (P ) = hx | x ∈ P i

n • Epn (P ) is the set of elementary abelian subgroups of P of order p

• Nonstandardly, for P with Z(P ) cyclic, P o∗ C2 is the quotient of P o C2 by its center (a “commuting wreath product”).

11 CHAPTER 2

FUSION AND LINKING SYSTEMS

After some lemmas on automorphism groups of p-groups and the structure of Ln(q) and Un(q) for n 6 4, we present in this section the deﬁnition of a saturated fusion system and many of the deﬁnitions and theorems which have appeared in the literature over the last decade. In addition, we prove a couple of additional lemmas which, to our knowledge, have not appeared. Our main references for group theoretic material are [Gor80], [Suz82], [Asc00] and [GLS05]. For the background on fusion systems, we follow [AKO11] and [Cra11b].

2.1 Automorphism groups of p-groups

In this section, we list some results about automorphism groups of p-groups needed later. The ﬁrst one is well-known and well-worn in the area.

Theorem 2.1.1. Let A be a p0-group of automorphisms of the p-group S which stabilizes a normal series 1 = S0 6 S1 6 ··· 6 Sn = S and acts trivially on each factor

Si+1/Si. Then A = 1.

Proof. See for example [Gor80, Theorem 3.2].

A ﬁnite group is indecomposable if it is not the direct product of two proper subgroups.

12 Proposition 2.1.2. An automorphism of a direct product of indecomposable ﬁnite groups permutes the commutator subgroups of the factors.

Proof. This is a consequence of the Krull-Schmidt theorem for ﬁnite groups, found in [Suz82, Theorem 2.4.8]. A proof of this precise statement is given in [Olib, Propo- sition 3.1].

Next we describe the outer automorphism group of a nonabelian dihedral 2-group

D, and prove a couple of additional statements which basically express the fact that a noncentral involution of D cannot be a commutator or a square in a 2-group containing D as a normal subgroup. As a starting point, we take for granted that the automorphism group of a cyclic 2-group is an abelian 2-group [Asc00, 23.3].

Lemma 2.1.3. Let D be a 2-group isomorphic to D2k+1 for some k > 2. Fix the presentation hb, c | b2 = c2k = 1, b−1cb = c−1i for D and let C = hci be the cyclic maximal subgroup of D. Let S be any 2-group containing D as a normal subgroup.

Then

∼ ∼ ∼ (a) Out(D) = A × B where A = C2 and B = C2k−2 is the kernel of the action of Out(D) on the D-classes of fours subgroups of D.

(b) [S,S] 6 CS(C), and

1 (c) if S0 is the preimage of Ω1(S/CS(D)D) in S, then f (S0) 6 CS(D)C.

Proof. D has three conjugacy classes of involutions bD = bC = C2b,(cb)D = (cb)C =

C2cb, and zD = {z}, where z = c2k−1 generates the center of D. Also, D has three maximal subgroups, two of which, C2hbi and C2hcbi, are dihedral, and the last one

C = hci is cyclic.

It is straightforward to prove by induction that 52k−2 ≡ 1 (mod 2k), and that

2k−3 k−1 k 5 k−2 5 ≡ 2 + 1 (mod 2 ) (when k > 3). Thus the map c 7→ c has order 2 as 13 an automorphism of C, and no power of this automorphism inverts C. Consider the automorphisms of D:    −1  b 7→ c b b 7→ b η = and ϕ =   5 c 7→ c c 7→ c

. Then [η, ϕ] is the automorphism of D sending b 7→ c−4b and c 7→ c, and hence is the inner automorphism conjugation by c−2. Also, η2 is the map b 7→ c−2b, and c 7→ c, which is also inner. Thus, writing [α] for the class in Out(D) of an automorphism of D and A = h[η]i, B = h[ϕ]i, we have by these observations and the congruences above that ∼ h[η], [ϕ]i = A × B = C2 × C2k−2 in Out(D). Furthermore, the stabilizer in A × B of any D-class of fours groups in D is B.

It remains to show that A × B is all of Out(D). Let α be any automorphism of

D. We will show that [α] ∈ A × B. Composing with conjugation by an appropriate

4 element of hbi, we obtain α1 centralizing C/C . Composing with some power of

ϕ, we get an automorphism α2 centralizing C. Composing with conjugation by an appropriate element of C, we obtain an automorphism α3 which is either the identity or, in addition to centralizing C, takes b to c−1b, i.e. is equal to η. Tracing back these choices, we have shown that [α] ∈ B ∪ B[η], and the proof of (a) is complete.

Now ﬁx a 2-group S containing D as a normal subgroup. Then C E S as C is a characteristic subgroup of D. Hence (b) follows from the exact sequence 1 →

CS(C) → S → AutS(C) → 1 and the fact that Aut(C) is abelian.

Let S0 be the preimage of Ω1(S/CS(D)D) in S as in (c). Thus, S0 consists of the elements of S which square into CS(D)D. Let s ∈ S0. If s ∈ CS(D)D we have that s

1 2 squares into CS(D)f (D) = CS(D)C , so we may assume that s induces a nontrivial

14 (involutory) outer automorphism of D. If s embeds into the coset B[η] as in (a), then

2 s centralizes no noncentral involution of D, and hence s ∈ CS(D)C as claimed. So we may further assume that s induces an involutory outer automorphism in B. Then

s −1 2 c = cz or c z, and so s ∈ CCS (D)D(C) = CS(D)C as claimed.

The next lemma shows that some related 2-groups also have no automorphism of odd order.

Lemma 2.1.4. Suppose k > 3 and let D be a 2-group isomorphic to Q2k+1 , SD2k+1 ,

C2 × D2k , D2k × D2k , or D2k o C2. Then Aut(D) is a 2-group.

∼ Proof. If D = Q2k+1 or SD2k+1 with k > 3, then the cyclic maximal subgroup is characteristic in D. Thus, Aut(D) is a 2-group by [Asc00, 23.3] and Theorem 2.1.1.

Now ﬁx 2-groups D1 and D2 each isomorphic to D2k with k > 3, and let Zi be the cyclic maximal subgroup of Di for i = 1, 2. Let α be an automorphism of D of odd order. ∼ First suppose that D = C2 ×D1. Then α ﬁxes [D,D]∩Z(D) = Z(D1) and so acts trivially on the fours group Z(D). The quotient D/Z(D) is dihedral of order 2k−1 and so α is the identity on D by Theorem 2.1.1 and Lemma 2.1.3 in case k > 3. In any case, it is easy to see that Z(D)Z1 is the unique subgroup of D with its isomorphism

2 type (C2 × C2k−1 ), so α must ﬁx a point of the fours group D/Z(D)Z1 and thus act trivially on it. It follows that α is the identity by Theorem 2.1.1.

2 2 Suppose that D = D1 × D2. Then α permutes Z1 = [D1,D1] and Z2 = [D2,D2]

2 2 by Proposition 2.1.2. Hence, α acts trivially on Z1 Z2 because Aut(Zi) is a 2-group

2 2 for i = 1, 2. It follows that α also acts trivially on Z1Z2/Z1 Z2 , since elements in

2 2 2 2 distinct nonidentity cosets of Z1 Z2 power to distinct elements of Z1 Z2 . Therefore α centralizes Z := Z1Z2 by Theorem 2.1.1. But α also centralizes D/Z since the three

15 nonidentity elements of D/Z have diﬀerent actions on Z. Thus α is the identity by another application of Theorem 2.1.1. ∼ a Now suppose that D = D2k o C2. Let D0 = D1 × D2 with D2 = D1 for some a ∈ D − D0. Since ND(D1) = D0, each element t of D − D0 interchanges D1 and D2. t ∼ Thus for every involution t ∈ D − D0, the centralizer CD0 (t) = {xx | x ∈ D1} = D1, and so CD(t) has 2-rank 3. Since D0 is generated by its elementary abelian subgroups of order 16, J(D) = D0. Thus, α stabilizes D0, and must induce the identity on it by the preceding case. Because D/D0 has order 2, one last application of Theorem 2.1.1 shows that α is the identity and completes the proof of the lemma.

2.2 Ln(q) and Un(q) for n 6 4

In this section, let q be an odd prime power. We collect here some properties of

− − Ln(q) and Un(q) for 2 6 n 6 4. We sometimes write Ln (q) for Un(q) and GLn (q) for

GUn(q).

The ﬁrst lemma records some information about Aut(L2(q)).

Lemma 2.2.1. Let K be a ﬁnite group isomorphic to L2(q) and P ∈ Syl2(K). Let

H = Aut(K), and T ∈ Syl2(H). Then

∼ (a) H = KhhiF with h ∈ I2(T ), F cyclic, and Out(K) = hhi × F ,

(b) P is dihedral of order 2k where ν(q2 − 1) = k + 1,

∼ (c) Khhi = P GL2(q) and P hhi is dihedral,

(d) F is isomorphic to the Galois group of Fq inducing ﬁeld automorphisms on K,

k−2 (e) if FT := T ∩ F , then P hhi ∩ FT = 1, and |FT | 6 2 ,

(f) all involutions of P are K-conjugate as are all involutions of P h,

16 (g) if f is an involution in FT := T ∩ F , then P hfi = P × hfi, q is a square and ∼ 1/2 k−1 E(CK (f)) = L2(q ), and P ∩ E(CK (f)) is dihedral of order 2 .

2 (h) if f is an involution in FT := T ∩ F and hzi = Z(P ), then [f, h] = (fh) = z, P hfhi is semidihedral, and all involutions of T lie in P ∪ P h ∪ P f.

Proof. See Chapter 10, Lemma 1.2 of [GLS05].

Lemma 2.2.2. Let K be a ﬁnite group isomorphic to L3(q) for q ≡ − (mod 4).

k Then a Sylow 2-subgroup of K is semidihedral of order 2 , where ν2(q + ) = k − 2,

and K has one class of involutions. If z ∈ I2(K), then CK (z) is a quotient GL2(q) by a group of odd order.

Proof. For the description of the Sylow subgroups, see Theorem 4.10.5(c) of [GLS98].

The description of the centralizer of the central involution is found in Chapter 10,

Lemma 4.1 of [GLS05].

Lemma 2.2.3. Let K be a ﬁnite group isomorphic to L4(q) with q ≡ − (mod 4). Then

(a) a Sylow 2-subgroup of K is isomorphic to D2k o C2 where ν2(q + ) = k − 1.

(b) K has two classes of involutions. If z is a central involution then C := CK (z) =

Mhfi where M is a normal subgroup of C isomorphic to SL2(q) ∗ SL2(q), and

f is an involution interchanging the two SL2(q) factors. If x is a noncentral

involution of K, then C := CK (x) = Mhfi such that M = Q × K is a normal k−1 ∼ 2 subgroup of C with Q is cyclic of order 2 , with K = L2(q ), and f is an involution which inverts Q and acts as an involutory ﬁeld automorphism on K.

ν2(q1 + 1) = ν2(q2 − 1) (i.e. if they have isomorphic Sylow 2-subgroups), then ∼ F2(L4(q1)) = F2(U4(q2)). 17 Proof. For (a), see Theorem 4.10.5(f) of [GLS98]. Concerning (b), see Chapter 10,

Lemma 4.11 of [GLS05] for the description of the centralizer of a central involution, and see Tables 4.3.1 and 4.5.1 of [GLS98] for a noncentral involution. Part (c) is a special case of a theorem of Broto, Møller, and Oliver; see [BMO12, Proposi- tion 3.3(a)].

2.3 Fusion systems

In this section, we deﬁne the notion of a saturated fusion system, which forms the main object of study. For a group G with subgroups H and K, let NG(H,K) be

g the transporter set: {g ∈ G | H = K}. This admits an action of NG(H) on the right, and of NG(K) on the left via multiplication in G. Denote by HomG(H,K) the set of group homomorphisms from H to K induced by conjugation by elements of the group G. Thus, there is a bijection NG(H,K)/CG(H) → HomG(H,K) which sends a group element to the homomorphism it induces. Also write AutG(H) for

HomG(H,H). When ϕ : H → K is any isomorphism, we write the induced map from Aut(H) → Aut(K) as α 7→ ϕα.

For any category C and A, B ∈ Ob(C), we write MorC(A, B) for the set of morphisms from A to B, and Iso(A, B) for the set of isomorphisms in C. Often however, when it is to be emphasized that the morphisms are to be thought of as group homomorphisms, we write HomC(A, B) for MorC(A, B). Now ﬁx a prime p. For a ﬁnite group G with Sylow p-subgroup S, the fusion system of G at p is the category FS(G) with objects the subgroups of S, and with morphisms the group homomorphisms between subgroups of S induced by conjugation by elements of the group G. That is,

HomFS (G)(P,Q) = HomG(P,Q).

18 Thus we throw away the individual group elements and retain only the conjugation homomorphisms between subgroups of S induced by those elements.

Now we describe what we mean by an abstract fusion system and by the saturation axioms which make fusion systems come closer to what is seen in the fusion system of a ﬁnite group.

Deﬁnition 2.3.1. Let S be a ﬁnite p-group. A fusion system on S is a category F in which the objects are the subgroups of S, and whose morphism sets HomF (P,Q) for P , Q 6 S consist of injective group homomorphisms from P to Q satisfying the following conditions.

1. HomS(P,Q) ⊆ HomF (P,Q), and

2. every morphism in F factors (in F) as an isomorphism followed by an inclusion

of subgroups.

When P and Q are subgroups of S which are F-isomorphic, we also say that P and Q are F-conjugate and write P F for the set of F-conjugates of P in F. Note that

AutP (P ) = Inn(P ) E AutF (P ) 6 Aut(P ). Write OutF (P ) = AutF (P )/ AutP (P ) for the F-outer automorphism group of P .

As it stands with the previous deﬁnition, the notion of a fusion system is rather general. For example, for any p-group S, there is the universal fusion system U(S) on S. This has morphism sets consisting of the set Inj(P,Q) of all injective group homomorphisms between P and Q. In particular, AutU(S)(P ) = HomU(S)(P,P ) =

Aut(P ) for every P 6 S. Thus, U(S) has “too many p-automorphisms” in the following sense. Unless S has order at most p, Out(S) is divisible by p [Suz82,

Theorem 8.14] and so AutS(S) is not a Sylow p-subgroup of AutF (S), as is the case when F is the fusion system of a ﬁnite group G with Sylow subgroup S.

19 0 Here is another sort of example. Let S = Cp × Cp, let A be a cyclic p -group of automorphisms of S, and set G = S o A. Deﬁne a fusion system F on S as

HomF (P,Q) = HomFS (G)(P,Q) if P is of order p, and HomF (S,S) = {idS}. It is easy to see that F is in fact a fusion system, and that AutS(S) = 1 is a Sylow p-subgroup of AutF (S) = 1, but F is missing some automorphisms of S (by design) that should be present if F is to model the fusion system of a ﬁnite group (cf. Lemma 2.3.15 below). Loosely, we may think of F as having “too few p0-automorphisms”.

Before presenting the saturation axioms which remedy these deﬁciencies in fusion systems, we need a couple of deﬁnitions.

Deﬁnition 2.3.2. Let F be a fusion system on S. A subgroup P of S is said to be

F • fully F-centralized if |CS(P )| > |CS(Q)| for every Q ∈ P ,

F • fully F-normalized if |NS(P )| > |NS(Q)| for every Q ∈ P ,

• fully F-automized if AutS(P ) is a Sylow p-subgroup of AutF (P ).

We will often say an element x ∈ S is fully F-centralized if hxi is fully F- centralized, especially when x is an involution.

Deﬁnition 2.3.3. For a fusion system F on the p-group S, and an isomorphism

ϕ ∈ IsoF (P,Q) between P and Q, set

ϕ Nϕ = {s ∈ NS(P ) | cs ∈ AutS(Q)}.

The subgroup Q is said to be receptive if for every ϕ ∈ IsoF (P,Q), there exists

ϕ˜ ∈ HomF (Nϕ,S) withϕ ˜|P = ϕ.

When ϕ is an arbitrary morphism in F (not necessarily an isomorphism), Nϕ is defined with respect to the induced isomorphism P → ϕ(P ). The set Nϕ in the preceding definition is the largest subgroup of NS(P ) to which ϕ could possibly 20 extend, for in the case that ϕ does extend to someϕ ˜ defined on the element s ∈ NS(P )

ϕ we have that cs = cϕ˜(s) ∈ AutS(ϕ(P )).

Deﬁnition 2.3.4. Let F be a fusion system on the p-group S. Then F is said to be saturated if the following two axioms hold.

(I) (Sylow Axiom) Every fully F-normalized subgroup of S is fully F-centralized

and fully F-automized.

(II) (Extension Axiom) Every fully F-centralized subgroup of S is receptive.

Note also that it is always the case that CS(P )P 6 Nϕ for any morphism ϕ, so every map P → Q with Q receptive extends automatically to CS(P )P in a saturated fusion system. In fact, in a great number of the instances where we apply the extension axiom explicitly in this work, it will be in this special case.

As designed, the fusion system of a ﬁnite group is saturated. The proof of this is a straightfoward application of Sylow’s theorem and can be found in many places, e.g. [AKO11, Theorem 2.3].

Theorem 2.3.5. Let G be a ﬁnite group and S ∈ Sylp(G) for some prime p. Then the fusion system FS(G) is saturated.

Deﬁnition 2.3.6. A fusion system F on the p-group S is realizable if there exists a

ﬁnite group G such that S ∈ Sylp(G) and F is isomorphic to FS(G). In this case, G is a model for F. Otherwise, it is said that F is exotic.

See Deﬁnition 2.3.16 for the precise deﬁnition of isomorphism.

Deﬁnition 2.3.7. Let F be a fusion system on S. A subgroup P of S is F-centric if

F CS(Q) = Z(Q) for each Q ∈ P . The subgroup P is F-radical if Op(OutF (P )) = 1.

21 Following Aschbacher, we will sometimes write F c, F r, and F f for the set of

F-centric, F-radical, and fully F-normalized subgroups of S, respectively. Concate- nation in the superscript will denote the intersection of the relevant sets. For example,

F cr is the set of subgroups which are both F-centric and F-radical.

The importance of these notions lies in the fact that a saturated fusion system

F is determined by the F-automorphism groups of the subgroups which lie in F fcr.

This is Alperin’s fusion theorem [Alp67]. Alperin’s fusion theorem in this setting is originally due to Puig; we present a proof here from [BLO04, Theorem A.10] in the setting of saturated fusion systems. First, the following lemma gives a couple immediate consequences of saturation.

Lemma 2.3.8. Let F be a saturated fusion system on the p-group S.

(a) If Q is a fully F-normalized subgroup of S and P ∈ QF , then there exists

η ∈ HomF (NS(P ),NS(Q)) such that η(P ) = Q.

(b) A subgroup P of S is F-centric if and only if CS(P ) 6 P and P is fully F- centralized.

Proof. Let ϕ ∈ IsoF (P,Q) in part (a). We have AutS(Q) ∈ Sylp(AutF (Q)) by the

αϕ Sylow axiom, and so there exists α ∈ AutF (Q) such that AutS(P ) 6 AutS(Q) by

Sylow’s theorem. Then with η0 = αϕ, we have NS(P ) = Nη0 by construction and so

η0 has an extension η ∈ HomF (NS(P ),NS(Q)) by the Extension axiom. Furthermore, η(P ) = Q and so (a) holds.

For part (b), suppose that P is F-centric. Then for Q ∈ P F with Q fully

F-centralized, |CS(P )| = |Z(P )| = |Z(Q)| = |CS(Q)|, and so P is also fully F- centralized. Now suppose that CS(P ) 6 P and P is fully F-centralized. For any

ϕ ∈ IsoF (Q, P ), the Extension axiom gives an extension of ϕ mapping CS(Q) into

22 F CS(P ) = Z(P ). Hence, CS(Q) = Z(Q) holds for any Q ∈ P , completing the proof of (b).

Theorem 2.3.9 (Alperin’s Fusion Theorem). Suppose F is a saturated fusion system

0 on the p-group S, and let ϕ ∈ IsoF (P,P ). Then there exists an integer n > 1, 0 subgroups Pi 6 S for 0 6 i 6 n with P0 = P and Pn = P , subgroups Qi 6 S for

1 6 i 6 n, and automorphisms αi ∈ AutF (Qi) satisfying the following conditions:

(a) Pi−1, Pi 6 Qi and α(Pi−1) = Pi for each i,

(b) each Qi is fully F-normalized, F-centric, and F-radical, and

Proof. We will say that an isomorphism in F is decomposable if it satisﬁes the conclusion of the theorem in place of ϕ. Note that inverses, restrictions, and compositions of decomposable morphisms are again decomposable. The proof is by induction on the index |S : P |. By the Sylow axiom, S itself is F-radical, and also is clearly fully

F-normalized and F-centric; hence, there is nothing to show in the case P = S. So assume that P < S, and that all isomorphisms in F on subgroups of strictly larger order than |P | are decomposable.

0 00 00 −1 Let ψ ∈ IsoF (P ,P ) with P fully F-normalized. Then ϕ = ψ (ψϕ) is decomposable if ψ and ψϕ are decomposable, and both of the latter have target P 00, so we may assume that P 0 itself is fully F-normalized.

0 0 Since P is fully F-normalized, there exists a morphism η : NS(P ) → NS(P )

0 with η(P ) = P by Lemma 2.3.8. The theorem holds for η : NS(P ) → η(NS(P )) by

−1 −1 0 induction, and as ϕ = (ϕη |P 0 )η|P 0 with ϕη |P 0 ∈ AutF (P ) we are reduced to the

0 case in which P = P and ϕ ∈ AutF (P ). Since P is fully F-normalized, it is receptive by the Sylow and extension axioms.

So ϕ extends to an elementϕ ˜ ∈ AutF (CS(P )P ) which is decomposable if CS(P )P > 23 P , in which case ϕ =ϕ ˜|P is decomposable as well. Thus, we may assume CS(P ) 6 P , and P is then F-centric by Lemma 2.3.8(b).

If P is F-radical, then we are ﬁnished. Assume that P is not F-radical. Now

P is fully F-automized by the Sylow axiom, and so Op(AutF (P )) 6 AutS(P ) ∈

Sylp(AutF (P )). Let N be the preimage in NS(P ) of Op(AutF (P )) under the surjection

NS(P ) → AutS(P ). Then as Op(AutF (P )) is invariant under ϕ, we have N 6 Nϕ.

Since AutP (P ) < Op(AutF (P )), we have P < N. It follows that ϕ extends to N > P because P is receptive. Thus, ϕ is decomposable by the induction hypothesis, completing the proof.

A subsystem of a fusion system F on S is a fusion system E on a subgroup of

S all of whose morphisms are morphisms in F. Given two fusion systems on the p-group S, we may regard them as subsystems of the universal fusion system U(S) on S, mentioned at the beginning of this section. Then it is easy to see from the definitions that for two fusion systems on S, their intersection, defined in the obvious way, is again a fusion system on S. Given any collection of monomorphisms X of F, we may therefore speak of the fusion system on S generated by X, which is defined as the intersection of all fusion systems on S which contain compositions of restrictions of morphisms in X ∪ Inn(S). We denote this fusion system by hXiS. Thus another way to state Alperin’s fusion theorem is

frc F = hAutF (P ) | P ∈ F iS.

We now turn to the deﬁnition of weak and strong closure, and of the normalizer and centralizer subsystems of a p-group.

Deﬁnition 2.3.10. Fix a fusion system F on S, and let T be a subgroup of S. Then

• T is weakly F-closed if ϕ(T ) = T for every overgroup P of T in S and ϕ ∈

HomF (P,S), and 24 • T is strongly F-closed if ϕ(P ∩ T ) 6 T for every P 6 S and ϕ ∈ HomF (P,S).

The most important weakly closed subgroups for our purposes are the Thompson subgroup J(S) and the Baumann subgroup Baum(S) = CS(Ω1(Z(J(S)))) of a p- group S. Each of these are weakly closed in any fusion system over S.

Deﬁnition 2.3.11. Let F be a fusion system on the p-group S. Fix a subgroup T of S.

• The normalizer NF (T ) is the fusion system on NS(T ) with morphism sets

HomNF (T )(P,Q) consisting of those ϕ ∈ HomF (P,Q) having an extensionϕ ˜ ∈

HomF (TP,TQ) withϕ ˜(T ) = T .

• The centralizer CF (T ) is the fusion system on CS(T ) with morphism sets

HomCF (T )(P,Q) consisting of those ϕ ∈ HomF (P,Q) having an extensionϕ ˜ ∈

HomF (TP,TQ) withϕ ˜|T = idT .

A subgroup T is normal in F if F = NF (T ), and central if F = CF (T ).

It is straighforward to show from the deﬁnition that if T1 and T2 are both normal or central, then so is their product. Thus, there is a unique largest normal p-subgroup of F, denoted Op(F), and a unique largest central subgroup, written Z(F) and called the center of F. Also, Op(F) E S and Z(F) 6 Z(S) and both are strongly F-closed. The following fundamental result will usually be applied without comment. The

ﬁrst part is due to Puig [Pui06, Proposition 2.15].

Proposition 2.3.12. Suppose that F is a saturated fusion system on the p-group S.

Then NF (T ) is saturated whenever T is fully F-normalized, and CF (T ) is saturated whenever T is fully F-centralized. If in addition F = FS(G) for some ﬁnite group ∼ ∼ G, then NF (T ) = FNS (T )(NG(T )) and CF (T ) = FCS (T )(CG(T )).

25 Proof. See Propositions I.5.4 and I.5.5 of [AKO11], applied there with K = Aut(T ) and K = 1 for normalizers and centralizers, respectively.

Deﬁnition 2.3.13. A fusion system F is constrained if Op(F) is F-centric.

Of fundamental importance is the Model Theorem for saturated fusion systems

[BCG+05, Proposition C] which follows. We do not use this theorem explicitly in this work, but it is often used implicitly when citing many of Aschbacher’s results.

Theorem 2.3.14 (Model Theorem). Let F be a saturated fusion system on the p- group S. If F is constrained, then F is realizable by a ﬁnite group G such that

Op(F) = Op(G) and Op0 (G) = 1.

We shall need a version of Burnside’s fusion theorem, which, when applied in the case where T = S is abelian, yields that S is normal in any saturated fusion system over it. More often, however, we will apply it in the case T = J(S).

Lemma 2.3.15 (Burnside’s Fusion Theorem). Let F be a saturated fusion system on the p-group S, and suppose that T is a weakly F-closed subgroup of S. Then any morphism in F between subgroups of Z(T ) lies in NF (T ).

Proof. Suppose P and Q are subgroups of Z(T ), and let ϕ ∈ HomF (P,Q). Let

0 0 ψ ∈ IsoF (Q, Q ) with Q fully F-centralized. By the extension axiom, ψϕ and ψ have extensions to CS(P ) and CS(Q) respectively, and these subgroups both contain T . Restricting these extensions to T and using the fact that T is weakly F-closed, we

−1 get automorphisms α, β ∈ AutF (T ) such that (β α)|P = ϕ, which is what was to be shown.

We now deﬁne morphisms of fusion systems.

Deﬁnition 2.3.16. Let F1 on S1 and F2 on S2 be fusion systems. A morphism

F1 → F2 is a pair (α, α˙ ) such that α : S1 → S2 is a group homomorphism and

α˙ : F1 → F2 is a functor satisfying 26 1.α ˙ (P ) = α(P ) for each P 6 S1, and

2. α|Q ◦ϕ =α ˙ (ϕ)◦α|P (as homomorphisms P → α(Q)) for each ϕ ∈ HomF1 (P,Q).

The morphism (α, α˙ ) is injective (resp. an isomorphism) if α is injective (resp. an isomorphism) andα ˙ is injective (resp. bijective) on morphism sets. We say

(α, α˙ ) is surjective if α is surjective, and for each P , Q 6 S2, the mapα ˙ P0,Q0 :

HomF1 (P0,Q0) → HomF2 (P,Q) is surjective where P0 and Q0 are the inverse images in S1 of P and Q.

Note that the functorα ˙ is determined on objects by α from (1), and is also determined on morphisms by α from (2). In other words, one may think of a morphism of fusion systems as simply a group homomorphism α of the Sylow subgroups which preserves fusion. This helps explain why the resultant functor is denoted so similarly, and usually we will abuse notation saying α itself is a morphism of fusion systems.

For a fusion system F, write Aut(S, F) for those α ∈ Aut(S) which preserve fusion,

α i.e for which HomF (P,Q) = HomF (α(P ), α(Q)) for subgroups P and Q of S. Naturally, the image of a surjective morphism (θ, θ˙) out of a saturated fusion system F is saturated, and is isomorphic to quotient of F by the kernel of θ, which is a strongly F-closed subgroup [AKO11, II.5]. We now describe these quotient systems.

For a strongly closed subgroup T of S and a morphism ϕ ∈ HomF (P,Q), we have

ϕ(P ∩ T ) 6 Q ∩ T . Denote by HomF (P,Q)/T the image of the canonical map ˙ θP,Q : HomF (P,Q) → Hom(P T/T, QT/T ), induced by this inclusion.

Deﬁnition 2.3.17. Let F be a saturated fusion system on S, and T a strongly

F-closed subgroup of S. The quotient system F + = F/T is the fusion system on

S+ = S/T with morphisms

+ + HomF + (P ,Q ) := HomF (P,Q)/T. where P and Q are the inverse images of P + and Q+ under the quotient map S → S+. 27 By construction, there is a surjective morphism (θ, θ˙) from F to F +, induced by the projection θ : S → S+.

We will also need the notions of the direct and central products of fusion systems.

Deﬁnition 2.3.18. Let F1 and F2 be saturated fusion systems on the p-groups S1 and S2, respectively.

1. The direct product, denoted F1 × F2, is the fusion system on S1 × S2 with

morphism sets as follows. Let P 6 S1 × S2, and denote by Pi the projection of

P onto Si (followed by inclusion into the product). Then

HomF1×F2 (P,Q) := {(ϕ1, ϕ2)|P | ϕi ∈ HomFi (Pi,Si)), (ϕ1, ϕ2)(P ) 6 Q}.

2. A fusion system F is the central product of F1 and F2 if it is of the form

(F1 × F2)/Z for some 1 6= Z 6 Z(F1) × Z(F2) which intersects each Z(Fi) in

the identity. We write F1 ×Z F2 for F in this case, or simply F1 ∗ F2 when Z is uniquely determined.

Of course, one may deﬁne a direct product (or central product) with more than two factors analogously. A direct product (and hence, a central product) of two saturated fusion systems is saturated. See, e.g. [BLO04, Lemma 1.5].

2.4 Subsystems and normality

We have deﬁned what it means for a subgroup of S to be normal in a fusion system

F over S. In this section, we consider notions of normality for fusion subsystems.

There are at least three notions of normality in the literature, which have come to be called, in increasing strength, F-invariance [Pui06, 6.4], weak normality [Lin06,

Deﬁnition 3.1], and normality [Asc08, Section 6].

28 Deﬁnition 2.4.1. Fix a saturated fusion system F on the p-group S and a fusion subsystem E of F on the strongly F-closed subgroup T .

1. E is F-invariant if, whenever P 6 Q 6 T , ϕ ∈ HomE (P,Q), and ψ ∈ ψ HomF (Q, S), then ϕ ∈ HomE (ψ(P ), ψ(Q)).

2. E is weakly normal in F if E is F-invariant and saturated, and

3. E is normal in F, written E E F, if E is weakly normal and every element

α ∈ AutE (T ) has an extensionα ˜ ∈ AutF (CS(T )T ) such that [CS(T )T, α] 6 T .

Roughly, F-invariance is the condition that morphisms in E should be invariant under conjugation by morphisms in F, whenever this makes sense. In practice,

F-invariant subsystems are rarely saturated. As an example, let E be the full subcategory of F on a proper fully F-automized, F-centric subgroup T of S. Then being a full subcategory, E is F-invariant, but | AutT (T )| < | AutS(T )| so AutT (T ) is not a

Sylow p-subgroup of AutE (T ) = AutF (T ). Thus, E fails to satisfy the Sylow axiom for saturation.

Example 2.4.2. Here is an example (and a standard one) showing that a weakly normal subsystem need not be normal. Consider a group isomorphic to A4 × A4, and let G be a subgroup of index 3 containing neither A4 factor. Then G has Sylow 2- subgroup S = V1 ×V2, a product of two fours groups and of index 3 in G. An element g ∈ G − S of order 3 acts ﬁxed point freely on each Vi. Set T = V1, H = T hgi,

F = FS(G), and E = FT (H). Then it is straightfoward to see that T is strongly F-closed, and E is a weakly normal subsystem of F. However, every extension of an odd order automorphism in AutE (T ) to S acts nontrivially on V2, and thus the extra extension condition for normality is not satisﬁed.

The following result of Aschbacher, which exhibits the behavior of strongly closed subgroups and normal subsystems under quotient maps, will be useful. 29 Lemma 2.4.3 ( [Asc08, Lemmas 8.9, 8.10]). Assume that θ : F → F + is a surjective

+ morphism of saturated fusion systems, and S0 is the kernel of θ : S → S . Then the following statements hold.

(a) The map T 7→ θ(T ) is a bijection between the strongly F-closed overgroups of

+ + S0 in S and the set of strongly F -closed subgroups of S .

(b) Let E be a weakly normal subsystem of F on an overgroup T of S0 in S. Then E + = θ(E) is a weakly normal subsystem of F + on T + and if θ : E → E + is

+ + surjective and E E F, then E E F .

systems of F on overgroups of T and the normal subsystems of F +.

Although part (c) gives a partial analogue of the correspondence theorem for normal subgroups, notice that, more generally, part (b) does not claim a one-to-one correspondence between the normal subsystems of F + and the normal subsystems of

F based on overgroups of T . This is because such a correspondence does not exist in the setting of saturated fusion systems. Here is an example illustrating the failure at the prime 2. Let V be elementary abelian of order 16, and H a group isomorphic to

A5 acting on V as the heart of the standard permutation module. Let G = V oH and + let S ∈ Syl2(G), a group of order 64 with S = S/V a fours group. Set F = FS(G), + + + ∼ and F = F/V . Then S is a normal 2-subgroup of F = FS+ (A5) by Burnside’s

+ + fusion theorem (Lemma 2.3.15). Hence, FS+ (S ) is a normal subsystem of F as can be easily seen. But the preimage E = FS(S) in F is not F-invariant. In the deﬁnition of F-invariance, one may take P = Q = V , ϕ = cs ∈ AutS(S) where s ∈ S − V is of

ψ order 2, and ψ ∈ AutF (V ) of order 5 to see that ϕ ∈ AutF (V ) is not conjugation by

ψ any element of S, i.e. ϕ∈ / AutE (V ).

30 Despite the lack of a correspondence theorem for normal subsystems, the following theorem of Craven gives analogues of two of the isomorphism theorems.

Theorem 2.4.4 ( [Cra10, Theorem E]). Let F be a saturated fusion system on S, and suppose T1 6 T2 are strongly F-closed subgroups of S. Let E be a saturated subsystem of F on the subgroup T of S. Writing T1E/T1 for the image of E in F/T1, we have that

∼ ∼ T1E/T1 = E/(T ∩ T1) and F/T1 = (F/T1)/(T2/T1).

2.5 The hyperfocal and residual subsystems

For a ﬁnite group G, denote by Op(G) the smallest normal subgroup N of G for which G/N is a p-group; equivalently Op(G) is the subgroup of G generated by the elements of p0-order in G. Similarly, Op0 (G) is the smallest normal subgroup

N of G for which G/N is a p0-group; equivalently, Op0 (G) is the subgroup of G generated by the elements of p-power order. Appropriate subsystem analogues of these subgroups in the context of saturated fusion systems were originally considered by Puig [Pui06, Sections 6 and 7] and studied by Broto, Castellana, Grodal, Levi, and Oliver in [BCG+07]. Somewhat diﬀerent approaches are given by Aschbacher in [Asc11a, Sections 7 and 11]. In this section, we give a treatment of the main results. Other accounts appear in [AKO11] and [Cra11b].

If G has normal subgroup N with quotient G = G/N a p-group, then letting ˆ ˆ N = N[G, G], the quotient G/N is an abelian p-group. If S ∈ Sylp(G), then the standard isomorphism theorems show that any abelian p-quotient of G is a quotient of S/(S ∩ [G, G]). The subgroup S ∩ [G, G] is the focal subgroup of S in G. Although the focal subgroup a priori relies on global information, the focal subgroup theorem

31 [Gor80, Theorem 7.3.4] says that S ∩ [G, G] depends only on S and G-fusion inside it:

S ∩ [G, G] = h[s, cg] | s ∈ S, g ∈ G with cg(s) ∈ Si.

Here, for any group H with K 6 H and an injective group homomorphism ϕ : K → H, we write [k, ϕ] = k−1ϕ(k) for the commutator of k with ϕ. Analogously, the hyperfocal subgroup theorem of Puig [Pui00, 1.1] (see also [BCG+07, Lemma 2.2]) gives a fusion theoretic description of the hyperfocal subgroup S ∩ Op(G) of S in G:

p p S ∩ O (G) = h[s, cg] | s ∈ P 6 S, g ∈ O (NG(P ))i.

This motivates the following deﬁnitions.

Deﬁnition 2.5.1. Let F be a saturated fusion system on the p-group S.

(a) The F-focal subgroup is the subgroup of S deﬁned by

foc(F) = h[s, ϕ] | ϕ ∈ HomF (hsi,S)i

= h[s, ϕ] | s ∈ P 6 S and ϕ ∈ AutF (P )i.

(b) The F-hyperfocal subgroup is the subgroup of S deﬁned by

p hyp(F) = h[s, ϕ] | s ∈ P 6 S and ϕ ∈ O (AutF (P ))i.

The equivalence between the two descriptions of foc(F) in the deﬁnition can be seen easily via Alperin’s fusion theorem (Theorem 2.3.9).

The following lemma shows that quotients of a fusion system by these subgroups are as one would expect. Part (c) comes down to the elementary fact that if P is a p-group with P/[P,P ] cyclic, then [P,P ] = 1.

Lemma 2.5.2. Let F be a saturated fusion system on the p-group S. Then

(a) hyp(F) 6 foc(F) and both subgroups are strongly F-closed, 32 (b) for any strongly F-closed subgroup T of S, the quotient F/T is the fusion

system of the (resp. abelian) p-group S/T if and only if T > hyp(F) (resp.

T > foc(F)), and

Proof. Strong closure in part (a) is a straighfoward calculation and the inclusion follows from the deﬁnitions. We present a concise proof for (b) due to Craven. By

Alperin’s fusion theorem (Theorem 2.3.9), F/T is the fusion system of the p-group

S/T if and only if there are no p0-automorphisms in F/T of subgroups of S/T . Under the surjective morphism F → F/T , this happens if and only if, for each subgroup

0 p P of S, each p -automorphism α of P , we have [P, α] 6 T . Since O (AutF (P )) is 0 generated by the p elements in AutF (P ), we conclude that F/T is the fusion system

p of S/T if and only if T > hyp(F) = h[P,O (AutF (P ))] | P 6 Si. A similar argument establishes that in addition, S/T is abelian p-group if and only if T > foc(F). Now suppose S/foc(F) is cyclic as in (c). Set S+ = S/hyp(F) and F + =

F/hyp(F). Then part (b) and Craven’s (second) isomorphism theorem (Theorem 2.4.4) imply that foc(F)+ = foc(F +), and the latter is just the commutator subgroup

[S+,S+] because F + is the fusion system of the p-group S+. Therefore, the commutator quotient S+/[S+,S+] = S+/foc(F)+ ∼= S/foc(F) is cyclic. It follows that [S+,S+] = 1, i.e. foc(F) = hyp(F) as claimed.

Deﬁnition 2.5.3. Let F be a saturated fusion system on the p-group S. A fusion subsystem F0 on the subgroup S0 6 S has

p • p-power index in F if S0 > hyp(F), and AutF0 (P ) > O (AutF (P )) for every

P 6 S0, and

p0 • p-prime index in F if S0 = S and AutF0 (P ) > O (AutF (P )) for every P 6 S0.

33 The next theorem describes the subsystems of a saturated fusion system of p-power index, and the ﬁrst part is due originally to Puig [Pui09, Theorem 13.6]. Reﬁnements are found in [BCG+07, Theorem 4.3] and [Asc11a, Section 7].

Theorem 2.5.4 ( [AKO11, Theorem 7.4]). Let F be a saturated fusion system on the p-group S. For any overgroup T of hyp(F), there is a unique saturated fusion subsystem FT on T of p-power index in F, and it is given by

p FT = hAutT (T ),O (AutF (P )) | P 6 T iT .

Moreover, FT is normal in F if and only if T is normal in S and FU ⊆ FT whenever

U 6 T . In particular, there is unique minimal normal subsystem of F of p-power index, based on hyp(F), which is denoted by Op(F) and called the hyperfocal subsystem.

When F = Op(F), it is sometimes said that F is perfect by analogy with p-perfect groups. We will require the following theorem of Aschbacher, which allows one to consider the product of a p-group with a normal subsystem.

Theorem 2.5.5 ( [Asc11a, 8.21]). Let F be a saturated fusion system on the p-group

S and let S0 and T be strongly F-closed subgroups of S with S0 6 T . Suppose F0 is a normal subsystem of F on S0. Then there exists a saturated fusion subsystem T F0 of F with the following properties.

(a) F0 E T F0,

∼ + + (b) T F0/S0 = FT + (T ) where T = T/S0, and

S0 and the set of saturated subsystems of T F0 containing F0.

34 p Note the particular case of the preceding theorem: if F0 = O (F0), then as

p hyp(F0) 6 hyp(T F0) 6 S0 by (b), we have F0 = O (T F0). We now give the description of the subsystems of index prime to p due to Broto,

Castellana, Grodal, Levi, and Oliver [BCG+07, Theorem 5.4]. See also [Pui06, The- orem 6.11].

Theorem 2.5.6 ( [AKO11, Theorem I.7.7]). Let F be a saturated fusion system on the p-group S. Then there exists a subgroup Γ 6 AutF (S) of index prime to p with the property that there is a one-to-one correspondence between saturated fusion subsystems of index prime to p and overgroups of Γ in AutF (S). In particular, there is a unique minimal such subsystem, corresponding to Γ itself, called the residual subsystem and denoted Op0 (F).

The following corollary to Theorem 2.5.6 will suﬃce for the purposes of this thesis.

Corollary 2.5.7. Let S be a ﬁnite p-group with automorphism group a p-group. Then

F = Op0 (F) for every saturated fusion system F on S.

Proof. By the Sylow axiom, each saturated fusion system F over S has Γ = AutF (S) =

AutS(S) in Theorem 2.5.6.

Here are a few elementary facts regarding the relationship between the hyperfocal and residual subsystems, surjective morphisms, and direct products.

Lemma 2.5.8. Suppose that F is a saturated fusion system on the p-group S.

(a) If θ : F → F + is a surjective morphism of fusion systems, then θ(Op(F)) =

Op(F +) and θ(Op0 (F)) = Op0 (F +).

p p p (b) If F = F1 × F2 with each Fi saturated, then O (F) = O (F1) × O (F2) and

p0 p0 p0 O (F) = O (F1) × O (F2).

35 Later in Section 5.4, we will make use of the following special case of a much more general theorem of Oliver.

Theorem 2.5.9. [Olib, Theorem C] Let F be a perfect saturated fusion system on a direct product D1 × D2 of two nonabelian dihedral 2-groups of the same order. Then

F = F1 × F2 where Fi is a perfect fusion system on Di.

The actual hypotheses of Oliver’s theorem require also that F = O20 (F), but this holds automatically in the case stated above by Lemma 2.1.4 and Corollary 2.5.7.

Also, the Fi must be perfect by Lemma 2.5.8(b).

2.6 Centralizers

In Section 2.3, we saw the definition of the normalizer and centralizer of a p-subgroup in a fusion system. To date, however, an appropriate notion of the normalizer or centralizer of an arbitrary saturated fusion subsystem has been elusive. Aschbacher has shown via his theory of normal maps (which we do not define here) that in the special case that a subsystem E is normal in F, one can define the centralizer CF (E) in F of E, which enjoys many of the properties one would like. The key result underlying the definition is the following theorem.

Theorem 2.6.1 ( [Asc11a, (6.7)]). Let F be a saturated fusion system on S and let

E be a normal subsystem on T . Let X denote the set of subgroups X 6 CS(T ) for which CF (X) contains E. Then X has a unique maximal element, denoted by CS(E) and called the centralizer in S of E. Moreover, CS(E) is strongly F-closed.

We will need a result examining in a special case how centralizers behave under quotienting by a strongly closed subgroup. But ﬁrst, the following lemma shows that for the purposes of computing the centralizer in S of a normal subsystem E of F, we may restrict to the subsystem SE of Theorem 2.5.5. 36 Lemma 2.6.2. Suppose F is a saturated fusion system on S and E is a normal subsystem of F on T . Let Q 6 CS(T ). Then CF (Q) > E if and only if CSE (Q) > E.

Proof. Let ϕ ∈ HomE (U, V ) for subgroups U and V of T . If ϕ lies in CSE (Q), then it clearly lies in CF (Q). Suppose ϕ ∈ CF (Q). Then ϕ extends to a morphism

ϕ˜ ∈ HomF (QU, QV ) withϕ ˜|Q = idQ, and it suﬃces to show thatϕ ˜ ∈ SE. By Alperin’s fusion theorem (Theorem 2.3.9) applied in E, it is enough to show this when

fc U = V and U ∈ E . But thenϕ ˜ ∈ AutN (CS (U)U)(U) with N (CS(U)U) in the sense of [Asc11a, Notation 8.4], and SE is generated by such automorphism groups.

Proposition 2.6.3. Let F be a saturated fusion system on the p-group S. Suppose that E is a normal subsystem of F on the strongly F-closed subgroup T of S. Set

Q = CS(E), and assume that Q ∩ T = 1. Then CS/Q(SE/Q) = 1.

Proof. Without loss of generality we will assume that F = SE by Lemma 2.6.2. By

Theorem 2.6.1, Q is strongly F-closed so factoring by Q makes sense. Let θ : F →

F/Q be the surjective morphism of fusion systems and denote passage to the quotient by bars. Since Q and T are normal in S, we have [Q, T ] 6 Q ∩ T = 1. Also, for every

U, V 6 T ,

' θU,V : HomF (U, V ) −→ HomF (U, V ). (2.6.4) is a bijection since Q ∩ T = 1.

Suppose the proposition is false and let Q1 = CS(E) 6= 1, where Q1 > Q is the preimage of Q1 under θ. By Theorem 2.6.1, Q1 is strongly F-closed. By

Lemma 2.4.3(a), Q1 is strongly F-closed. Since T E S, we have [Q1,T ] 6 T . But

[Q1,T ] 6 Q as well because [Q1, T ] = 1. Therefore, [Q1,T ] 6 Q ∩ T = 1.

We will show that E 6 CF (Q1), supplying a contradiction. Let U be a fully

E-normalized, E-centric subgroup of T and let ϕ ∈ AutE (U). Since U is fully E 37 0 0 0 normalized, we have that ϕ = ϕ ct for some ϕ ∈ AutE (U) of p -order and some t ∈ T .

0 If ϕ ∈ CF (Q1), then so is ϕ since Q1 centralizes T , and so we may assume that ϕ has order prime to p.

Let ϕ = θ(ϕ). Then by deﬁnition of Q1, ϕ extends to ϕ1 ∈ AutF (Q1U) such that

ϕ | = id . Let ϕ1 be a morphism in AutF (Q1U) such that θ(ϕ1) = ϕ . Then by 1 Q1 Q1 1 0 (2.6.4), ϕ1 restricts to ϕ on U. Letϕ ˜ be the p -part of ϕ1. As ϕ has order prime to p,

ϕ˜ still restricts to ϕ on U. Furthermore, ϕ1|Q1 stabilizes the series 1 6 Q 6 Q1 and centralizes Q1/Q, so the same is true forϕ ˜. Recall that we have assumed as we may that F = SE. Quotienting now by T and ∼ applying (2.6.4) with the roles of Q and T interchanged, we have that AutF (Q) = ∼ AutSE/T (QT/T ) = AutS/T (Q) is a p-group, since SE/T is the fusion system of the p-group S/T by Theorem 2.5.5(b). It follows thatϕ ˜|Q = idQ. By Theorem 2.1.1,

fc ϕ˜ = idQ1 . We have thus produced for arbitrary U ∈ E and ϕ ∈ AutE (U), an extensionϕ ˜ ∈ AutF (Q1U) of ϕ which restricts to the identity on Q1. Therefore,

E 6 CF (Q1) by Alperin’s fusion theorem, contradicting the maximality of Q.

2.7 Simple fusion systems

We now turn to a discussion of simple fusion systems and Aschbacher’s program for the classiﬁcation of simple 2-fusion systems mentioned in the introduction. Most of the material here can be found in the papers [Asc08, Asc11a], but see also [Cra11b] and [AKO11, Chapter II].

A saturated fusion system is simple if it has no nontrivial proper normal subsystems. Recall the two competing notions for normality of saturated subsystems

(weakly normal and normal). A result of Craven shows that a fusion system has a nontrivial proper normal subsystem if and only if it has a nontrivial proper weakly normal subsystem:

38 Theorem 2.7.1 ( [Cra11a, Theorem A]). Suppose that F is a saturated fusion system and E is a weakly normal subsystem of F. Then Op0 (E) is a normal subsystem (based on the same subgroup as E).

Hence the two competing notions of simplicity are equivalent. Although we do not use Craven’s theorem explicitly, it is at least of clear psychological importance for any investigation simple fusion systems.

A saturated subsystem E of F is subnormal if there is a sequence E = E0 E E1 E

··· E En = F.

Deﬁnition 2.7.2. A saturated fusion system E is quasisimple if E = Op(E) and

E/Z(E) is simple. E is a component of F if E is a subnormal, quasisimple subsystem of F.

An example similar to Example 2.4.2 shows that the intersection of two normal subsystems Ei on Ti need not be normal. However, in [Asc11a, Theorem 1], As- chbacher locates a normal subsystem E := E1 ∧ E2 E F on T := T1 ∩ T2 which is contained in E1 ∩ E2. Then E is the largest subsystem of E1 ∩ E2 on T which is normal in E1 and E2. This allows one to speak of the normal subsystem generated by a collection of subsystems.

Deﬁnition 2.7.3. Let Comp(F) denote the set of components of F. The layer of F, denoted E(F), is the normal subsystem of F generated by Comp(F). The generalized

∗ Fitting subsystem is F (F) = Op(F)E(F).

Theorem 2.7.4 ( [Asc11a, Theorem 6]). The layer E(F) is a central product of the

∗ components of F. Furthermore, F (F) is a central product of Op(F) and E(F), and

∗ ∗ CF (F (F)) = Z(F (F)).

Recall that a fusion system N is constrained if it has an N -centric, normal subgroup. 39 Deﬁnition 2.7.5. Let F be a saturated fusion system on the p-group S, and let W be a weakly F-closed subgroup of S. Then F is said to be of W -characteristic p-type if for each fully F-normalized subgroup P 6 S such that NS(P ) > W , the normalizer

NF (P ) is constrained. F is of W -component type if there exists a fully F-normalized subgroup X of order p such that CS(X) > W and CF (X) has a component.

In the above deﬁnition, we note the special cases in which W ∈ {1, Baum(S),S}, and we say that F is of Baumann characteristic p-type or Baumann component type when W = Baum(S). Deﬁne F to be of characteristic 2-type or component type if

W = 1, and of even characteristic or even component type when W = S.

Using his E-balance theorem [Asc11a, Theorem 7], i.e. that E(NF (X)) 6 E(F) for every fully F-normalized subgroup X of S, Aschbacher proves a Dichotomy The- orem for saturated fusion systems:

Theorem 2.7.6. Let F be a saturated fusion system on the p-group S. Then for

W ∈ {1, Baum(S),S}, F is either of W -characteristic p-type or of W -component type.

Proof. A proof appears in [AKO11, Theorem II.14.3] for W = 1. The other cases follow by similar arguments.

For the remainder of this section, we specialize to p = 2 and outline Aschbacher’s program for the classification of simple 2-fusion systems of W -component type for some W ∈ {1, Baum(S),S}. Aschbacher proposes in [AKO11] the possibility of classifying simple 2-fusion systems by transferring concepts and strategies from the classification of finite simple groups to the domain of 2-fusion systems, with the results of this section forming the infrastructure for such a task.

Fix a simple fusion system F at the prime 2. The first step in Aschbacher’s program involves considering the situation where an involution centralizer in F has a 40 subnormal perfect subsystem K on a quaternion group (cf. [Asc77a, Asc77b]); hence ∼ K = F2(SL2(q)) for some odd q (see Lemma 3.0.7 below). Aschbacher’s preliminary analysis of this situation leads him to extract the notion of a “quaternion fusion packet” ([AKO11, Definition II.14.9], cf. [Asc77a, Hypothesis Ω]), which generically arises in 2-fusion systems of groups of Lie type in odd characteristic. A classification of such quaternion fusion packets should provide a characterization of such simple

2-fusion systems aside from F2(L2(q)). The second step is to show that if an involution centralizer in F has a quaternion fusion packet, then aside from perhaps a small number of exceptions, F has one as well or F is exotic and isomorphic to FSol(q) for some odd q. (Recall that FSol(q) is the family of simple exotic fusion systems at the prime 2 constructed by Levi and Oliver in [LO02]; these have involution centralizers isomorphic to F2(Spin7(q)), which have quaternion fusion packets.) Thus, the ﬁrst two steps would suﬃce to characterize the simple 2-fusion systems with an involution centralizer having a component which is the 2-fusion system of a group of Lie type in odd characteristic aside from that of

L2(q). The third step is to prove an analogue of Aschbacher’s Component Theorem

[Asc75] for W -components for some W ∈ {1, Baum(S),S}. We will say a W - component is a component in an involution centralizer which contains W , as in Def- inition 2.7.5. Aschbacher’s Component Theorem says that if K is a component of some involution centralizer in a ﬁnite group G (satifying the B-conjecture) and maximal under a certain partial ordering on the set of such components, then either K has quaternion Sylow 2-subgroups (a case already covered above), or K commutes with none of its distinct conjugates and CG(K) is tightly embedded in G. In the latter case, K is called a standard component of G. A subgroup of G is tightly embedded if it has even order but its intersection with each distinct conjugate has odd

41 order. Aschbacher [Asc11b] has a deﬁnition and preliminary classiﬁcation of tightly embedded subsystems in saturated fusion systems, and analogously to the group case, such subsystems are (generically) either subnormal or based on 2-groups which are of

2-rank 1 or 2, or which are elementary abelian. Thus, an analogue of the Component

Theorem for fusion systems would put strong restrictions on subsystems centralizing a “standard” W -component.

The ﬁnal step is to solve the standard W -component problems for W -components involving F2(L2(q)), FSol(q), or simple 2-fusion systems of ﬁnite groups not of Lie type in odd characteristic, i.e alternating groups, sporadic groups, and characteristic

2 Lie-type groups. Of course, if any other simple exotic systems arise, these must be considered as standard W -components as well. In this thesis, we consider a portion of the standard Baum(S)-component problem for F2(L2(q)). Namely the case when the centralizer of the component is cyclic. The case in which an F2(L2(q))-component is centralized by a 2-group of 2-rank greater than 1 is currently being considered by

Matthew Welz [Wel].

In the classification of finite simple groups, the analysis of such standard component problems with W = 1 was complicated by the identification of certain nearly simple groups which morally should be thought of as characteristic 2-type groups, but which are actually of component type. For example, let H be a simple group of ∼ Lie type in characteristic 2, hti = C2, and let G be either Hhti, where t is an invo-

∗ lutory ﬁeld automorphism, or the wreath product of H by hti. Set H1 = F (CG(t)),

S ∈ Syl2(G), and S1 ∈ Syl2(H1). Then generally H1 consists of a single component which is a simple group of Lie type of characteristic 2; so G is usually of component type. Moreover, |S| is approximately the square of |S1|. Thus, determining the structure of S from knowledge of S1 in order to apply transfer arguments is quite diﬃcult as S1 grows. To avoid diﬃculties in cases like these, which arise most notably in the

42 analysis of groups with standard components which are of Lie type in characteristic

2 (cf. [Sei79a,Sei79b]), it could be more advantageous to consider only fusion systems having W -components with W = Baum(S) or W = S.

Indeed, this is the point of view which is taken in this thesis, where we consider fusion systems with Baum(S)-components of type F2(L2(q)). From the deﬁnition of

W -component type, this covers also the case of an S-component of type F2(L2(q)) (see Section 5.1). As the remarks in the introduction suggest, this relieves many of the complications which arose in the analogous problem in the group case.

2.8 Linking systems

Let F be a saturated fusion system on S. Denote by F c both the set of F-centric subgroups of F, and the full subcategory with objects the F-centric subgroups.

Deﬁnition 2.8.1. A centric linking system associated to F is a category L with objects the F-centric subgroups of S, together with distinguished monomorphisms

c c δP : P → AutL(P ) for each P ∈ F , and a functor π : L → F which is the identity on objects and surjective on morphisms, subject to the following conditions.

c ∼ (A) for each P,Q ∈ F , the center Z(P ) = δP (Z(P )) acts freely on MorL(P,Q) by

composition, and π induces an bijection MorL(P,Q)/Z(P ) → HomF (P,Q).

c (B) for each P ∈ F and x ∈ P , the composite π(δP (x)) is equal to cx in AutF (P ).

as morphisms in MorL(P,Q).

For a ﬁnite group G the FS(G)-centric subgroups are the p-centric subgroups of

G, i.e. those subgroups of S for which Z(P ) ∈ Sylp(CG(P )). For a p-centric subgroup

43 P of G we have CG(P ) = Z(P ) × Op0 (CG(P )), and so we obtain a centric linking

c system L = LS(G) associated to FS(G) by setting

Mor c (P,Q) = N (P,Q)/O 0 (C (P )). LS (G) G p G

c From the pioneering work of Broto, Levi, and Oliver [BLO03], the category LS(G) contains precisely the algebraic data required to recover, up to homotopy equivalence, the p-completion of the classifying space BG as the p-completion of the nerve of

c LS(G). Let O(F c) be the orbit category of F-centrics. This has objects F c and morphisms given by

MorO(F c)(P,Q) = Inn(Q)\ HomF (P,Q).

c The center functor is the (contravariant) functor ZF from O(F ) to abelian groups,

c which sends P ∈ F to Z(P ) and a morphism [ϕ] ∈ MorO(F c)(P,Q) to the map

−1 ∗ c ϕ |Z(Q) from Z(Q) → Z(P ) for ϕ ∈ HomF (P,Q). Denote by H (O(F ); ZF ) the co-

c homology of the category O(F ) with coeﬃcients in the functor ZF . This has a standard bar resolution and is naturally isomorphic with the derived functors lim∗(Z ) ←− F O(F c) of the inverse limit lim (Z ). See [AKO11, Section II.5.1] for more details. ←− F O(F c) Theorem 2.8.2 ( [BLO04, Proposition 3.1]). Let F be a saturated fusion system

3 c on S. Then there is a class η ∈ H (O(F ); ZF ) such that F has an associated

2 c centric linking system if and only if η = 0. The group H (O(F ); ZF ) acts freely and transitively on the set of isomorphism classes of centric linking systems for

F when this set is nonempty. Hence, F has a unique associated linking system if

2 c 3 c H (O(F ); ZF ) = 0 = H (O(F ); ZF ).

Oliver has shown in [Oli04,Oli06] that for a ﬁnite group G and Sylow p-subgroup

c S, the linking system LS(G) considered above is the unique centric linking system associated to F = FS(G). In fact, his proof shows the stronger statement 44 i c H (O(F ); ZF ) = 0 for all i > 2. In full generality, Oliver’s proof relies on the classification of finite simple groups. Since we shall need in Sections 4.1 and 4.2 some knowledge of the vanishing of these groups in a special case, we describe briefly his reduction to (quasi)simple groups, along with the precise statement we will require.

Fix F = FS(G) for short, as above. The idea of Oliver’s proof is to ﬁlter G by normal subgroups 1 6 H1 E H2 E ··· E Hn = G with Hr+1/Hr a minimal normal subgroup of G/Hr, and deﬁne subfunctors by

Hr ZF (P ) = Z(P ) ∩ Hr when P is F-centric. Then the short exact sequences of quotient functors

Hs Hr Ht Hr Ht Hs 0 → ZF /ZF → ZF /ZF → ZF /ZF → 0 when r 6 s 6 t induce long exact sequences in cohomology. In this way the vanishing

∗ c ∗ c Hr+1 Hr of H (O(F ); ZF ) for ∗ > 2 is reduced to the vanishing of H (O(F ); ZF /ZF ) for each r.

Definition 2.8.3. For a prime p, let Lb(p) be the set of all nonabelian finite simple groups K with the following property: whenever G is a finite group with nor- ∼ m mal subgroups H1 E H2 E G such that H2/H1 = K for some m > 1, then

i c H2 H1 H (O(F ); ZF /ZF ) = 0 for all i > 2 where F = Fp(G). Let G(p) be the set of ﬁnite groups whose nonabelian composition factors lie in Lb(p).

In Chapter 2 of [Oli06], Oliver shows that, for any nonabelian ﬁnite simple group

K, the property of lying in Lb(p) depends only on central extensions of K and their automorphism groups (see Proposition 2.7 there), thereby reducing the computation to quasisimple groups. We will need the special case that G ∈ Lb(2) for G = L2(q) with q ≡ ±1 (mod 8) later. In general, for a path analogous to the one taken in this thesis toward the resolution of other standard component problems (in which 45 the component is realizable), one will need Oliver’s computations in [Oli06] for those components.

Proposition 2.8.4. Let K be a ﬁnite simple group isomorphic to L2(q) for q ≡ ±1

(mod 8). Then K ∈ Lb(2).

Proof. It is shown in [Oli06, Proposition 7.5] that classical groups in odd characteristic lie in Lb(2). Alternatively, Propositions 4.2 and 4.6(b) of [Oli06] show that any ﬁnite simple group of 2-rank at most 3 lies in Lb(2) from general considerations.

We end this section with a discussion of automorphisms of fusion and linking systems and the relation between the two. Let F be a saturated fusion system on the p-group S, and let L be an associated centric linking system. Recall that an automorphism of F can be thought of as an automorphism α of S which is fusion preserving in that

−1 α HomF (P,Q)α = HomF (α(P ), α(Q)) (2.8.5) for each pair of subgroups P , Q of S. Denote by Aut(S, F) 6 Aut(S) the group of all fusion preserving automorphisms of S. From (2.8.5), AutF (S) is a normal subgroup of Aut(S, F). Write Out(S, F) for the quotient Aut(S, F)/ AutF (S).

Denote by Auttyp(L) the group of isotypical equivalences of L. An equivalence

α is isotypical if it commutes with the distingished monomorphisms: α(δP (P )) =

c δα(P )(α(P )) ∈ AutL(α(P )) for each P ∈ F . Let Outtyp(L) denote the equivalences classes in Auttyp(L) under natural isomorphism. Broto, Levi, and Oliver have shown that Outtyp(L) is the most natural group of outer automorphisms of L, in the sense

∧ ∧ that Outtyp(L) is isomorphic with Out(|L|p ) and with Out(BGp ) in the case that L is the linking system of a ﬁnite group G [BLO03, Theorems A and B].

For motivation in Section 4.1, we record a result of Broto, Levi, and Oliver which

46 describes the diﬀerence between these outer automorphism groups of fusion and linking systems. When the fusion system is that of a ﬁnite group, a complete proof can be found in [BLO03, Theorem E].

Theorem 2.8.6 ( [AKO11, Proposition 5.12]). Let F be a saturated fusion system on S and L an associated centric linking system. Then there is group homomorphism

Outtyp(L) → Out(S, F) ﬁtting into an exact sequence

1 c 2 c 1 −→ H (O(F ); ZF ) −→ Outtyp(L) −→ Out(S, F) −→ H (O(F ); ZF )

Thus, each outer automorphism of a fusion system lifts to an outer automorphism

2 c of the linking system if and only if H (O(F ); ZF ) = 0.

47 CHAPTER 3

THOMPSON TRANSFER LEMMA FOR FUSION

SYSTEMS

The classical Thompson transfer lemma is a tool in ﬁnite group theory which allows one to gain information about 2-fusion in a ﬁnite group G with G = O2(G) via the transfer map from a Sylow 2-subgroup S of G to X. It says that if T is a maximal subgroup of S and u is an involution in S − T , the element u has G-conjugate in T .

Thompson’s lemma has been generalized in a number of ways since its appearance in his N-group paper [Tho68, Lemma 5.38]. Harada showed [Har68, Lemma 16] that the same conclusion holds provided one takes u to be of least order in S − T .

Unpublished notes of Goldschmidt extended this to show that one may ﬁnd an G- conjugate of u in T which is extremal under the same conditions. An element t ∈

S is said to be extremal in S with respect to G if CS(t) is a Sylow subgroup of

CG(t). In other words, hti is fully FS(G)-centralized. Much later, Thompson’s result and its extensions were generalized to all primes via an argument of Lyons [GLS96,

Proposition 15.15]. We prove here a common generalization of Lyons’ extension and a similar transfer result [GLS05, Chapter 2, Lemma 3.1] (which relaxes the requirement that S/T be cyclic) in the context of fusion systems.

The proof we present is modeled on Lyons’ argument mentioned above, and relies on the transfer in saturated fusion systems. Transfer in the fusion system setting is deﬁned by way of a characteristic biset associated to a saturated fusion system,

48 which is an S-S biset Ω satisfying certain properties ﬁrst outlined by Linckelmann and

Webb [LW]. We motivate these properties now; more details can be found in [AKO11,

Section I.8].

For groups H and K, an H-K biset is a set X = H XK together with an action of H on the left and K on the right, such that (hx)k = h(xk) for each x ∈ X, h ∈ H, and k ∈ K. An H-K biset X may be regarded as a left (H × K)-set via

(h, k) · x = hxk−1. A transitive biset is a biset with a single orbit under the action of

H × K.

Now let G be a ﬁnite group and S a Sylow p-subgroup of G. Then G = SGS is an S-S biset with left and right action given by multiplication in G. The transitive subbisets of G are the S-S double cosets of G. For g ∈ G, the stabilizer in S × S of g is the graph subgroup

cg g ∆S∩Sg = { (cg(s), s) | s ∈ S ∩ S } of S × S. Thus, the double coset

∼ cg SgS = (S × S)/∆S∩Sg as a left (S × S)-set. In general, for P 6 S and an injective group homomorphism ϕ ϕ : P → S, we write ∆P for the graph subgroup { (ϕ(t), t) | t ∈ P }. Let ψ be a group homomorphism from S onto an abelian group A, and let T be

G the kernel of ψ. The transfer map trS (ψ): G → A is the group homomorphism given by

G Y −1 trS (ψ)(u) = ψ(hu(h · u) ) for u ∈ G, h∈[S\G] where [S\G] is a set of representatives for the right cosets of S in G and h·u ∈ [S\G] is

G the chosen representative for the coset Shu. Restricting trS (ψ) to S and decomposing

49 this product by the right action of S on the set of cosets S\G gives rise to the Mackey decomposition (cf. [GLS96, Lemma 15.13]) of the transfer map:

G Y S trS (ψ)(u) = trS∩Sg (ψ ◦ cg)(u) for u ∈ S. g∈[S\G/S]

S where [S\G/S] is a set of representatives for the S-S double cosets in G and trS∩Sg (ψ◦ cg) is the transfer of the composite

c ψ|g S ∩ Sg −→g gS ∩ S −−−−→S∩S (gS ∩ S)T/T.

G g g Thus, trS (ψ)|S is determined by the collection of morphisms { cg : S ∩ S → S ∩ S | g ∈ [S\G/S] | } in FS(G).

Let P 6 S. For an S-S biset X, denote by SXP the set X considered as an S-P biset upon restriction on the right to P . More generally, for an injective group homomorphism ϕ : P → S, denote by SXϕ the S-P biset with action s · x · t = sxϕ(t) for x ∈ X, s ∈ S, and t ∈ P . In the case, of X = SGS and x ∈ G with ϕ = cx−1 , the change of right coset representatives given by g 7→ gx gives an isomorphism of S-P bisets G ∼ G . This property determines from fusion data the information S P = S cx−1 that

G G −1 trS (ψ)(u) = trS (ψ)(x ux) for [u, x] ∈ S,

G G i.e. that S ∩ [G, G] 6 ker(trS (ψ)|S). So trS (ψ) induces a group homomorphism

S/foc(FS(G)) → S/T .

Deﬁnition 3.0.1. Let F be a saturated fusion system on the p-group S. An S-S biset Ω is said to have the Linckelmann-Webb properties and to be a characteristic biset for F if

ϕ (a) each transitive subbiset of Ω is isomorphic to (S × S)/∆P as a left (S × S)-set

for some ϕ ∈ HomF (P,S),

50 (b) for each P 6 S and each ϕ ∈ HomF (P,S), the S-P bisets SΩP and SΩϕ are isomorphic, and

Theorem 3.0.2 ( [BLO03, Proposition 5.5]). Let F be a saturated fusion system on the p-group S. There exists an S-S biset Ω associated to F satisfying the Linckelmann-

Webb properties.

In the case of a fusion system of a ﬁnite group G, the above discussion indicates one may take Ω = G.

For a saturated fusion system F over the p-group S, let Ω = ΩF be a characteristic biset for F. From Definition 3.0.1(a), fix a decomposition of Ω into transitive S-S bisets: a Ω = (S × S)/∆ϕi . Si i∈I where Si 6 S and ϕi ∈ HomF (Si,S) for each i ∈ I. For a normal subgroup T of S with S/T abelian and a group homomorphism ψ : S → S/T , define the transfer map relative to Ω by Y tr (ψ)(u) = trS (ψ ◦ ϕ )(u). Ω Si i i∈I for u ∈ S, where trS is the ordinary transfer. From Definition 3.0.1(b), this deter- Si mines a homomorphism trΩ(ψ) from S → S/T with

ker(trΩ(ψ)) > foc(F). (3.0.3)

We are now in a position to prove Theorem B.

Theorem 3.0.4. Let F be a saturated fusion system on the p-group S. Assume T is a proper normal subgroup of S with S/T abelian. Fix u ∈ S − T and let I be the set of fully F-centralized F-conjugates of u in S − T . Assume

51 (1) u is of least order in S − T ,

(2) the set of cosets IT = { vT | v ∈ I} is linearly independent in Ω1(S/T ), and

(3) F = Op(F).

Then u has a fully F-centralized F-conjugate in T .

Proof. Let ψ : S → S/T be the quotient map. Suppose that u has no fully F- centralized F-conjugate in T . We shall show that u lies outside the kernel of the transfer map trΩ(ψ). Once this is done, it follows from (3.0.3) that the focal subgroup foc(F) is properly contained in S. By Lemma 2.5.2 then, hyp(F) < S contradicting

(3).

Without loss of generality we may assume that u itself is fully F-centralized. Let

P = CS(u). Applying the Mackey formula for, and the deﬁnition of ordinary transfer, we have

Y tr (ψ)(u) = trS (ψ ◦ ϕ )(u) Ω Si i i∈I

Y Y P = tr t (ψ ◦ ϕi ◦ ct)(u) Si ∩P i∈I t∈[Si\S/P ]

Y Y Y −1 = (ψ ◦ ϕi ◦ ct)(ru(r · u) ) i∈I t t∈[Si\S/P ] r∈[P/Si ∩P ]

t t where r · u is the representative in [P/Si ∩ P ] corresponding to the coset (Si ∩ P )ru. As r commutes with u, we have ru(r · u)−1 = rur−1 = u. It follows that

Y Y −1 |P :St∩P | trΩ(ψ)(u) = ϕi(tut ) i mod T.

i∈I t∈[Si\S/P ] t Suppose i and t are such that the index |P : Si ∩ P | is divisible by p. Then the

−1 |P :St∩P | corresponding factor ϕi(tut ) i has order less than that of u, so lies in T by 52 t assumption and contributes nothing to the transfer. On the other hand, |P : Si ∩P | =

t t −1 1 if and only if P 6 Si . In this case, ϕi is deﬁned on P = CS(tut ), and so the −1 −1 corresponding factor ϕi(tut ) is fully F-centralized. By assumption, ϕi(tut ) ∈/ T .

Write IT = {ujT }j∈J with uj a fully F-centralized F-conjugate of u for each j ∈ J. Let

t T = {(i, t) | i ∈ I, t ∈ [Si\S/P ],P 6 Si }.

Then by the above remarks,

Y kj trΩ(ψ)(u) = uj T j∈J X with kj = |T |. j∈J We now ﬁnish by showing that the cardinality of the set T is prime to p. Notice

t that P 6 Si if and only if P ﬁxes the right coset Sit in its action from the right.

Furthermore, Ω decomposes as a disjoint union of orbits of the form Si\S as a left S-set. Therefore |T | = |(S\Ω)P |, the number of ﬁxed points of P in its right action on this set of orbits. Since |(S\Ω)| is prime to p by Deﬁnition 3.0.1(c), it follows X that |T | is also prime to p. Since, kj = |T |, there exists j0 ∈ J with p - kj0 . j∈J By linear independence of {ujT }j∈J , we have trΩ(ψ)(u) 6= T and so u∈ / ker(trΩ(ψ)), completing the proof.

Suppose that F = FS(G) for some ﬁnite group G with Sylow p-subgroup S.

G Then taking Ω = G, the transfer map relative to Ω is the ordinary transfer trS (ψ) restricted to S, and so we can specialize Theorem 3.0.4 to obtain generalizations of

Lyons’ extensions [GLS96, Proposition 15.15] and [GLS05, Lemma 2.3.1]. We point out in the following corollaries the analogues of these versions for subsequent ease of reference.

53 Corollary 3.0.5 (cf. [GLS96], Proposition 15.15). Let F be a saturated fusion system on a p-group S with F = Op(F). Suppose T is a proper normal subgroup of S with

S/T cyclic, and let u be an element of least order in S − T . Assume that every fully F-centralized F-conjugate of u lies in the coset T or in uT . Then u has a fully

F-centralized F-conjugate in T .

Note that if p = 2 in Corollary 3.0.5, then uT is the unique involution in the quotient S/T , and so the condition that each F-conjugate of u lies in T ∪ uT is automatically satsiﬁed.

Corollary 3.0.6 (cf. [GLS05], Lemma 2.3.1). Let F be a saturated fusion system on a ﬁnite 2-group S with F = O2(F). Suppose T is a proper normal subgroup of S with S/T abelian. Let I be the set of fully F-centralized involutions in S − T , and suppose that the set IT = {vT | v ∈ I} is linearly independent in Ω1(S/T ). Then each involution u ∈ S − T has a fully F-centralized F-conjugate in T .

To end this section, we illustrate how Theorem 3.0.4 can be applied together with Alperin’s fusion theorem to describe the perfect fusion systems on nonabelian

2-groups of maximal class.

Lemma 3.0.7. Let F be a saturated fusion system on the 2-group S with F =

O2(F). If S is nonabelian of maximal class, then F is uniquely determined by S up to isomorphism and one of the following holds.

∼ 2 (a) S = D2k with k > 3, and for any odd prime power q with ν2(q − 1) = k + 1, ∼ ∼ we have F = FS(G) with G = L2(q),

∼ (b) S = SD2k with k > 4, and for any odd prime power q ≡ 3 (mod 4) with ∼ ∼ ν2(q + 1) = k − 2, we have F = FS(G) with G = L3(q).

54 ∼ 2 (c) S = Q2k with k > 3, and for any odd prime power q with ν2(q − 1) = k, we ∼ ∼ have F = FS(G) with G = SL2(q).

Proof. By Alperin’s fusion theorem (Theorem 2.3.9), it suﬃces to describe the F- automorphism groups of the F-centric, F-radical subgroups. Let S be a nonabelian

2-group of maximal class. Then S is dihedral, semidihedral or quaternion by [Gor80,

Theorem 4.5]. Let C = hci be the cyclic maximal subgroup of S, and let Z(S) = hzi 6 C. In all cases, if s ∈ S − C, then s inverts hc2i. Hence, if P is a subgroup of S, then

P is cyclic, dihedral, semidihedral, or quaternion. By [Asc00, 23.3], Lemma 2.1.3, ∼ and Lemma 2.1.4, P has automorphism group a 2-group unless P = C2 × C2 or Q8. ∼ Hence, if P is a proper F-radical subgroup of S, then P = C2 × C2 or Q8 with

AutF (P ) isomorphic to S3 or S4, respectively. cr ∼ Let P ∈ F and suppose that P = C2 × C2. Let u ∈ P − Z(S). Then u lies outside the cyclic maximal subgroup T of S. By Corollary 3.0.5, u is F-conjugate into

T , and therefore F-conjugate to z. By the above description of the members of F cr,

P is the unique proper F-centric and F-radical subgroup containing u. Therefore, u ∼ is in fact AutF (P )-conjugate to z by a morphism of order 3, and AutF (P ) = S3. cr ∼ Let P ∈ F with P = Q8. Then S is semidihedral or quaternion, and P = hu, z1i with z1 ∈ P ∩ C. If S is semidihedral, let T = Ω1(S), the dihedral maximal subgroup of S. If S is quaternion, let T = C. In either case, u is of least order outside T ,

−1 and the only elements of T of order 4 are z1 and z1 . Hence by Corollary 3.0.5, u ∼ is F-conjugate to z1. As in the previous paragraph, it follows that AutF (P ) = S4 ∼ unless S = P , in which case AutF (P ) = A4.

cr We have determined the automorphism groups AutF (P ) for P ∈ F . Therefore, F is therefore uniquely determined, and as is described in (a)-(c).

55 CHAPTER 4

THE INVOLUTION CENTRALIZER

4.1 Reduced and tame fusion systems

In this section we address the problem of determining the structure of extensions of fusion systems, which is resolved via recent work of Andersen, Oliver, and Ven- tura [AOV12]. Under the conditions of Theorem A, and in other standard component scenarios to be handled, this problem manifests itself in the determination of an involution centralizer from the description of its generalized Fitting subsystem. Suppose given a saturated fusion system on a 2-group and assume C is an involution centralizer

∗ on the subgroup T with F (C) = O2(C)E where E is quasisimple. In this situation, one is confronted with the possibility that C is exotic even when E = E/Z(E) is the fusion system of a simple group. Assume, for instance, that E is realizable, but that there is an automorphism of 2-power order of the fusion system E which is not the restriction to the Sylow 2-subgroup of E of any automorphism of a ﬁnite group realizing E. Then one might have, say, that C = C/O2(C) is an exotic, almost simple fusion system for which O2(C) = E.

In order to describe the resolution of this problem in [AOV12], namely that

C/O2(C) and C are in fact realizable under certain conditions, we need a couple of deﬁnitions.

Deﬁnition 4.1.1. Let F be a saturated fusion system on the p-group S. We say

56 p p0 that F is reduced if Op(F) = 1, and F = O (F) = O (F). The reduction red(F) of F is deﬁned as follows. Set F0 = CF (Op(F))/Z(Op(F)), and recursively deﬁne

p p0 Fi = O (Fi−1) if i is odd, and Fi = O (Fi−1) if i is even. Deﬁne red(F) = Fm if

Fm = Fm+1 = Fm+2.

By [AOV12, Lemma 2.3], we have that Op(Fi) = 1 for any term Fi in the reduction sequence. In particular red(F) is a reduced fusion system, and if F is already reduced, then red(F) = F.

In the hypothetical situation concluding the above discussion, E is the reduction of

C and of C. Now let F be a saturated fusion system whose reduction F0 := red(F) is realizable by a ﬁnite group G0, and assume in addition that Op(F) = 1. Very roughly, Andersen, Oliver, and Ventura show F is realizable by ﬁrst constructing a linking system for F, and then applying ideas related to a general proposition of Oliver [Oli10,

Theorem 9] to construct a ﬁnite group G realizing F. As mentioned in the background material (Theorem 2.8.2), the obstruction to the existence and uniqueness of a linking

2 c system for a saturated fusion system F lies in H (O(F ); ZF ). The obstruction to being able to lift an outer automorphism of F0 to an outer automorphism of the

2 c linking system for F0 also lies in H (O(F0 ); ZF0 ) (Theorem 2.8.6). So at a minimum, it is needed that this group vanishes for F0 = Fp(G0) and its extensions. Secondly, as a necessary condition to F being realizable [AOV12, Theorem B], one should have that for which every (isotypical) outer automorphism of the linking system for F0 is induced by an element of Out(G0). That is, F0 is a tame fusion system.

Definition 4.1.2 ( [AOV12, Definition 2.5]). A saturated fusion system F0 on a ∼ p-group S0 is said to be tame if there exists a finite group G0 with F0 = FS(G0) and

c such that the natural map Out(G0) → Outtyp(LS(G0)) is split surjective. The fusion system F0 is strongly tame if G0 can be chosen to lie in G(p). In these cases, we say that F0 is (strongly) tamely realizable by G0. 57 See Deﬁnition 2.8.3 for the deﬁnition of G(p). So in particular, a tame fusion system is realizable. An easy example of a pair (F0,G0) for which F0 = FS(G0) but ∼ F0 is not tamely realizable by G0 is obtained with G0 = A7.

As indicated above, [AOV12, Theorem A] says that if F0 = red(F) is strongly tamely realizable by G0, then F is also tame, and in particular realizable by a ﬁnite group G. Later, we will need the following proposition, which is extracted from the proof of [AOV12, Theorem A], and speciﬁes in a special case that one may take G with Inn(G0) 6 G 6 Aut(G0). We will apply this proposition shortly in the case ∼ F = T K/Q and F0 = QK/Q = K (in the setting of this thesis’ Theorem A).

Proposition 4.1.3. Let F be a saturated fusion system on S. Suppose that F0 =

p O (F), CS(F0) = 1, and that F0 is strongly tamely realized by G0 with Op0 (G0) = 1.

Then F is strongly tamely realized by a ﬁnite group G such that Inn(G0) 6 G 6

Aut(G0).

Proof. A tame fusion system can always be realized by a ﬁnite group with no nontrivial normal p0 subgroups by [AOV12, Lemma 2.19], and if the fusion system is strongly tame, then the group can be chosen to lie in G(p) as well.

Since F0 ∈ G(p), F0 has a unique centric linking system L0. In addition, there is a unique centric linking system L associated to F. From [AOV12, Proposition 2.12(a)]

p and the assumption that F0 = O (F), L0 is normal in L and L/L0 is a p-group in the sense of [AOV12, Deﬁnition 1.27].

Next observe that Z(F) = Z(F0) = 1, since CS(F0) = 1. As Op0 (G0) = 1, any central subgroup of G0 must be a p-subgroup of S central in F0, so it follows that Z(G0) = Z(F0) = 1. Now [AOV12, Proposition 1.31(a)] and [AOV12, Proposi- tion 2.16] apply together to give that F is tamely realized by a ﬁnite group G such ∼ that G0 E G and L/L0 = G/G0.

As G/G0 is a p-group, we have that G ∈ G(p) by [AOV12, Proposition 2.11(c)], 58 and so F is strongly tamely realized by G. Now CS(G0) = 1 follows from CS(F0) = 1. ∼ Thus G is isomorphic to a subgroup of Aut(G0) containing Inn(G0) = G0.

4.2 Structure of the involution centralizer

In this section we lay the groundwork for the study of F as in Theorem A by studying some consequences of Proposition 4.1.3, and ﬁxing notation. The following hypothesis simply extracts those of Theorem A and will be used as a reference point.

Hypothesis 4.2.1. Suppose that F is a saturated fusion system on the 2-group S

2 with F = O (F) and O2(F) = 1. Assume x is a fully F-centralized involution in

S with Baum(S) 6 CS(x). Write T = CS(x) and C = CF (x), and assume that k K = E(C) is a fusion system on a dihedral group P of order 2 . Assume Q = CT (K) is cyclic. Set R = QP = Q × P .

Unless otherwise speciﬁed, we assume for the remainder of this thesis that F is a fusion system satisyﬁng Hypothesis 4.2.1, adopting the notation there.

Proposition 4.2.2. Let P be a nonabelian dihedral group of order 2k. There is a unique reduced saturated fusion system K on P and it is strongly tamely realized by

2 K = L2(q1) for any odd prime power q1 with ν2(q1 − 1) = k + 1.

Proof. We saw in Lemma 3.0.7(a) that there is a unique fusion system K on P with

K = O2(K). This has O20 (K) = K by Lemma 2.1.3 and Corollary 2.5.7. Also,

O2(K) = 1 since K one class of involutions (which generates P ). Therefore K is reduced. The fact that K is tamely realized by any K isomorphic to L2(q1) is the content of [AOV12, Proposition 4.3]. Then K is strongly tamely realized by such a

K by Proposition 2.8.4.

59 2 For the remainder, we fix an odd prime power q1 ≡ ±1 (mod 8) and with ν2(q1 − 1) = k + 1. Also we fix a finite group K realizing K as in Proposition 4.2.2. Thus, K has Sylow 2-subgroups isomorphic to D2k with k > 3.

Since K = E(C) is a normal subsystem of C, for each T1 6 T with P 6 T1 we may 2 2 form the product T1K as in Theorem 2.5.5. Then O (T1K) = O (K) = K.

Proposition 4.2.3. For each T1 6 S with R 6 T1, the quotient T1K/Q is isomorphic to the 2-fusion system of a subgroup of Aut(K) containing K.

Proof. We verify the hypotheses of Proposition 4.1.3 with F = T1K/Q and F0 = QK/Q ∼= K. Denote quotients by Q with bars. By Proposition 4.2.2, K is strongly

2 tamely realized by K and O20 (K) = 1. By Lemma 2.5.8, we have that K = O (T1K). Since Q∩P = 1, C (T K) = 1 by Proposition 2.6.3. Therefore T K/Q is the 2-fusion T 1 1 1 system of a subgroup of Aut(K) containing Inn(K) ∼= K.

Thus by Proposition 4.2.3, T := T/Q embeds into a Sylow 2-subgroup of Aut(K), which is described in Lemma 2.2.1. In particular, we may write T = P F h, where ∼ P = P and F and h are as follows. F is a cyclic group with F ∩P = 1, [Ω1(F ), P ] = 1, and F is the kernel of the action of T/P on the P -classes of fours subgroups of P .

The element h squares to the identity and P hhi is dihedral of order 2k+1 if h 6= 1.

Let F be the preimage of F in T , and let F1 be the preimage of Ω1(F ) in T . Then

[F1,P ] = 1. The following is used most heavily in the analysis below.

Lemma 4.2.4. Let f ∈ T be an involution. If f ∈ CT (P ) but f∈ / R, then

(a) P hfiK is the fusion system of Khfi where f is an involutory ﬁeld automorphism

of K, and

60 i (b) if V is a fours subgroup of P , there exists a unique i ∈ {0, 1} such that CC(fz )

0 contains AutK(V ). In this case, if V is another fours group of P not P -

1−i 0 conjugate to V , then CC(fz ) contains AutK(V ).

Proof. By Theorem 2.5.5(c) the set of saturated subsystems of T K is in one-to-one correspondence with the set of subgroups of T containing P via the bijection X 7→

XK. Factoring by Q induces an isomorphism of P hfiK with the fusion system of an extension of K (by Proposition 4.2.3) containing an involution outside K centralizing a Sylow 2-subgroup of K. This is unique and the required extension by Lemma 2.2.1, proving (a).

Now by Lemma 2.2.1(g), CP hfiK(f) contains AutK(U) for each U in some unique P -class of fours groups in P . Furthermore, there is an (abstract) P hfiK-fusion preserving isomorphism ch of P hfi which swaps the P -classes of fours groups of P and in-

0 terchanges f and fz by Lemma 2.2.1(c,h). So either AutK(V ) or Aut(V ) is contained in CP hfiK(f), and the other K-automorphism group is contained in CP hfiK(fz).

Although we have singled out Khfi in Lemma 4.2.4 a similar argument in part

(a) applies to give that RhhiK/Q is the 2-fusion system of P GL2(q1) if h ∈ T is any element with Rhhi/Q dihedral of order 2k+1, i.e. if h ∈ T maps onto h as above.

Similarly, in this case, RhfhiK/Q is the fusion system of the twofold extension of K with semidihedral Sylow 2-subgroups (cf. Lemma 2.2.1(c,h)).

Let Rd denote the largest subgroup of T which contains R and with the property that Rd/Q is dihedral. Thus, Rd contains R with index 1 or 2. In view of the Lemma 4.2.4, we also make the following deﬁnition in the situation of Hypothesis 4.2.1.

Deﬁnition 4.2.5. We say that an involution f ∈ T is an f-element on K, and that

61 it induces a nontrivial f-automorphism on K, if f ∈ CT (P ) but f∈ / QP . We also write Khfi for P hfiK.

Note that we are abusing terminology in this deﬁnition, for an involutory f- element induces the trivial automorphism on the fusion system K since it induces a trivial automorphism on P by conjugation. However, conjugation by f induces a nontrivial automorphism of the centric linking system associated K, which is essentially the content of Lemma 4.2.4 (cf. [AOV12, Proposition 4.2] and the example after it).

Lemma 4.2.6. Assume Hypothesis 4.2.1. If T contains an involution f inducing a nontrivial f-automorphism on K, then T (and hence S) has 2-rank 4 and J(S) =

J(Rhfi. Otherwise T (and hence S) has 2-rank 3.

Proof. Let π : T → T = T/Q be the projection. By Proposition 4.2.3, we have that

T K/Q is isomorphic to the 2-fusion system of a ﬁnite group H with K 6 H 6 Aut(K) and O2(H) = K. In particular, by Lemma 2.2.1(a), we can write T = P hhiF where ∼ P = P is nonabelian dihedral, h ∈ I2(T ) induces a diagonal automorphism on K (or h = 1), and F is a cyclic group of ﬁeld automorphisms on K (or F = 1). Let F be the preimage in T of F , and let h be an element of T mapping to h under π (with h = 1 if h = 1). Then Rd = Rhhi, F ∩ Rd 6 Q and T = RF hhi. Also, let F1 be the preimage of Ω1(F ) in T .

Since P is normal in T ,[F1,P ] 6 Q ∩ P = 1, and so F1Z(P ) = CT (P ) by

Lemma 2.2.1. Thus, there is an f-element of T if and only if F 6= 1 and F1 splits over

Q. If h1 ∈ I2(T − RF ), then CT (h1) is of 2-rank 3 by Lemma 2.2.1(c,h). Suppose f is an f-element of T . Then as f ∈ CT (P ) the 2-rank of RF is 4, and hence J(T ) = J(RF ) in this case, and T is of 2-rank 4. If T contains no f-element, then

J(RF ) = J(R) is of 2-rank 3, and so T is of 2-rank 3 as well. Since Baum(S) 6 T , we have J(S) = J(T ), and the 2-rank of S is the 2-rank of T .

62 CHAPTER 5

PROOF OF THEOREM A

5.1 The 2-central case

We begin now the heart of the analysis of a fusion system F satsifying Hypothe- sis 4.2.1. The objective of the current section is to consider the case in which x lies in the center of S, i.e. in which S = T . Eventually, in Proposition 5.1.18, we will reach the conclusion that there is no such F in this case. This section and the next are modeled on the treatment in [GLS05], in particular Proposition 3.4 of Chapter 2 and Section 12 of Chapter 3 there.

Adopt the notation of Section 4.2 and in particular of Hypothesis 4.2.1. Thus

C = CF (x), K is a component of C on the dihedral group P , T = CS(x) and Q =

CT (K). Recall the definition of f-element from Definition 4.2.5 and the definitions of

Rd and F . Also set Z(P ) = hzi, T0 = Ω1(T ), and Z = Ω1(Z(T0)). Directly from the deﬁnition of the centralizer and the fact that K has a single class of involutions, we have

all involutions of P #x are C-conjugate (5.1.1)

We begin with two lemmas which apply throughout this section, after which we state the main technical result of the present case.

Lemma 5.1.2. z is weakly F-closed in Z.

63 Proof. As T0 = Ω1(T ) is weakly F-closed in T , AutF (T0) controls fusion in Z =

Ω1Z(T0) by Lemma 2.3.15. Suppose z is not weakly F-closed in Z, and let ϕ ∈

AutF (T0) such that z 6= ϕ(z) ∈ Z. Since z ∈ Z(T0), we may take ϕ to be of odd

i order. Set Pi = ϕ (P ) for i ∈ Z>0. As P E T and Pi = Ω1(Pi) for all i, we have

Pi E T0 for all i. Then

Z(Pi) ∩ Z(Pj) = 1 for each i 6= j ∈ {0, 1, 2} (5.1.3) because ϕ has odd order and ϕ(z) 6= z. Furthermore, Z(Pi) 6 Z for all i.

Let i 6= j ∈ {0, 1, 2}, and suppose that Pi ∩ Pj 6= 1. As Pi ∩ Pj E Pj, Z(Pj) 6

Pi ∩ Pj. Hence Z(Pj) 6 Pi ∩ Z = Z(Pi), contrary to (5.1.3). As [Pi,Pj] 6 Pi ∩ Pj, we ∼ have P0P1P2 = P0 × P1 × P2. But T has 2-rank at most 4 by Lemma 4.2.6, so this is a contradiction.

Lemma 5.1.4. If ϕ ∈ HomF (P,T ), then ϕ(z) = z.

Proof. First recall that P E T . Let C be the cyclic maximal subgroup of P . Then

C E T . Now [T,T ] 6 R from Proposition 4.2.3, and in fact [T,T ] 6 CR(C) = QC by Lemma 2.1.3(b). So T/QC is abelian. Thus

Ω1([T,T ]) 6 Ω1(QC) = hx, zi 6 Z. (5.1.5)

Let ϕ ∈ HomF (P,T ). Then [u1, u2] = z for a pair of involutions u1 and u2 of P , and so ϕ(z) = [ϕ(u1), ϕ(u2)] 6 Ω1([T,T ]) 6 Z by (5.1.5). Now Lemma 5.1.2 shows that ϕ(z) = z.

Proposition 5.1.6. Suppose F satisﬁes Hypothesis 4.2.1 with S = T . Then no involution of T is an f-element.

Assume the hypotheses and notation of the proposition, but that the statement is false. To that end, let f be an involutory f-element in T . We proceed in a series of lemmas. 64 Lemma 5.1.7. zF ∩ Rhfi ⊆ P .

Proof. We ﬁrst show that

F z ∩ CT (P ) = hzi. (5.1.8)

F Let y ∈ z ∩ CT (P ) and choose ϕ ∈ F with ϕ(y) = z. Since z ∈ Z(T ), ϕ extends to a morphism on P 6 CT (y). Therefore y = z by Lemma 5.1.4, and (5.1.8) holds. F Now we suppose the lemma fails and let y ∈ z ∩ (Rhfi − P ∪ CT (P )) be arbitary. We claim that

y is C-conjugate to an element of CT (P ). (5.1.9)

Together with (5.1.8) and the fact that P is strongly C-closed, this will yield a contradiction. Recalling that Rhfi = Qhfi × P , write y = uf0v with u ∈ Q, f0 ∈ hfi,

2 and v ∈ P . Since y is an involution outside P ∪ CT (P ), we have uf0 6= 1 = (uf0) ,

2 v∈ / Z(P ), and v = 1. Let V be the fours group of P containing v. Then AutK(V ) 6 i CKhfi(f0z ) for some i ∈ {0, 1} by Lemma 4.2.4, and so there exists a morphism

i i i ϕ ∈ AutKhfi(V hfi) with ϕ(f0v) = ϕ(f0z )ϕ(z v) = f0z · z ∈ hf, zi. As Q E C, this ϕ extends to a morphismϕ ˜ ∈ C ﬁxing Q, and henceϕ ˜(y) =ϕ ˜(uf0v) ∈ Qhf, zi = CT (P ) conﬁrming (5.1.9).

Lemma 5.1.10. P is weakly F-closed.

Proof. Suppose not. Choose by Alperin’s fusion theorem a fully F-normalized subgroup D 6 T containing P and an automorphism ϕ ∈ AutF (D) with ϕ(P ) 6= P . i Since P is normal in T , we can choose such a ϕ of odd order. Set Pi = ϕ (P ) for each i. Thus the subgroups P0 = P , P1 and P2 are distinct by choice of ϕ, whereas

Z(Pi) = Z(P0) = hzi for all i by Lemma 5.1.4.

Now we examine the images of the Pi in T = T/Q. Since z 6= 1, we have that

65 ∼ F Pi = D2k for all i. Furthermore, Pi = Ω1(Pi) 6 hz i as all involutions of P are

F-conjugate. Thus there exists h ∈ Rd − R squaring into Q, and

Pi 6 P hhi for all i by Lemma 5.1.7 and the fact (Lemma 2.2.1(h)) that there are no involutions in P hf.

Suppose that P = Pi for some i. Then PQ > Pi, and so I2(Pi) ⊆ R. By F Lemma 5.1.7 then, Pi 6 hz ∩ Pii 6 P . So P = Pi. This shows that P0 6= P1 and P0 6= P2. But P hhi is dihedral and the Pi are among the two dihedral maximal subgroups of P hhi so

P1 = P2. (5.1.11)

∼ Set S0 = P0P1 and S1 = P1P2, so that ϕ(S0) = S1. Then S0 = P hhi = D2k+1 but

∼ S1 = D2k (5.1.12) from (5.1.11).

As P0 6= P1 are dihedral maximal subgroups of P hhi, we have [P0, P1] is the cyclic maximal subgroup of P0. So [P0,P1] is the cyclic maximal subgroup of P0. But

[P0,P1] 6 P1 because the Pi normalize each other, and hence [P0,P1] is the cyclic maximal subgroup of P1 as well. It follows that P0 ∩ P1 has index 2 in P0 and P1, k+1 ∼ ∼ and |S0| = 2|P0| = 2 . Hence, S0 = S0 = D2k+1 and S1 = ϕ(S0) is also isomorphic ∼ to D2k+1 with center ϕ(hzi) = hzi. As z 6= 1, S1 = D2k+1 . This contradicts (5.1.12) and completes the proof.

Lemma 5.1.13. Let u be an involution in CT (P ). If u is fully F-centralized, then so is uz.

66 Proof. Let ϕ ∈ HomF (huzi,T ) with ϕ(uz) fully F-centralized. Then ϕ extends to a morphismϕ ˜ on CT (uz) = CT (u) > P , andϕ ˜(z) = z by Lemma 5.1.4. Since u is fully F-centralized, we have

|CT (uz)| = |CT (u)| > |CT (ϕ ˜(u))| = |CT (ϕ ˜(u)z)| = |CT (ϕ ˜(uz))| and so uz is fully F-centralized as well.

Lemma 5.1.14. There exists f0 ∈ fhzi and ϕ ∈ NF (P ) such that ϕ(f0) = x.

Proof. We have R E T with T/R abelian. By Lemma 2.2.1(h), all involutions in 2 T − R lie in Rh ∪ Rf. As F = O (F), there exists ϕ ∈ HomF (hfi,R) with ϕ(f) fully F-centralized by Corollary 3.0.6. Then ϕ extends (by the extension axiom) to a morphismϕ ˜ on CT (f) > P normalizing P by Lemma 5.1.10. Thus,ϕ ˜ ∈ NF (P ) and soϕ ˜(f) ∈ Ω1(CR(P )) − hzi = xhzi as [P, f] = 1. Since one ofϕ ˜(f) orϕ ˜(fz) =ϕ ˜(f)z equals x, we are ﬁnished.

Now the next two lemmas give a contradiction in the proof of Proposition 5.1.6.

Lemma 5.1.15. xz ∈ xF .

Proof. Suppose not. Since x is not F-conjugate to z by Lemma 5.1.7, and all involutions of P #x are F-conjugate by Lemma 5.1.1, we have that

hxi is weakly F-closed in hxi × P. (5.1.16)

Replacing f by fz if necessary, there exists a subgroup D 6 T with P 6 D and a morphism ϕ ∈ HomF (D,T ) with ϕ(x) = f and ϕ(P ) = P by Lemma 5.1.14. By Lemma 4.2.4(b), hfi is not weakly closed in Khfi and so there exists ψ ∈

−1 HomF (hfi, hfi × P ) with ψ(f) ∈ P f − {f}. Then ϕ ψϕ(x) ∈ P x − {x}, which contradicts (5.1.16) and completes the proof.

67 Lemma 5.1.17. xz∈ / xF .

Proof. Let ϕ ∈ F with ϕ(x) = xz. Then as x and z lie in Z(T ) we may assume

ϕ ∈ AutF (T ) is of odd order by the extension and Sylow axioms. Then ϕ(z) = z by Lemma 5.1.10, and so ϕ induces an automorphism of hx, zi of order 2, a contradiction.

This completes the proof of Proposition 5.1.6. We now can prove the main result of this section.

Proposition 5.1.18. There is no saturated fusion system F satisfying Hypothe- sis 4.2.1 with S = T .

Proof. Suppose there is. From Proposition 5.1.6, there is no involutory f-element in

T , and hence there are no involutions in Rf. By Lemma 2.2.1(h), it follows that

T0 6 Rd (5.1.19) and hence Z = hx, zi 6 Z(T ) in the present case. Furthermore,

T is of 2-rank 3 (5.1.20) by Lemma 4.2.6.

In view of Lemma 5.1.2, we know z is not F-conjugate to x or xz. Since fusion in Z(T ) is controlled in AutF (T ),

x, xz, and z are pairwise not F-conjugate. (5.1.21)

Our assumption that O2(F) = 1 yields that x has an F-conjugate outside Z. Apply Alperin’s fusion theorem to obtain a fully F-normalized, F-centric subgroup D of T , and an automorphism α ∈ AutF (D) with α(x) ∈/ Z. Set h = α(x).

68 Note that h∈ / R, as otherwise h would lie in Ω1(R) = hxi × P . Because all involutions of P ∪ P #x are F-conjugate to xz or z, (5.1.21) would yield h = x, contrary to the choice of h. Therefore, by (5.1.19):

h ∈ Rd − R and P hhi is dihedral. (5.1.22)

Since D is F-centric, it contains Z = hx, zi 6 Ω1(Z(T )), and hence Ω1(Z(D)) = Zhhi as T is of 2-rank 3. Set A = Zhhi. Then h is NP (A)-conjugate to hz by (5.1.22). If h is AutF (A)-conjugate to hx or hxz then it is AutF (A)-conjugate to both, so x has exactly ﬁve conjugates under AutF (A) by (5.1.21), which is not the case. So

hAutF (A) = {x, h, hz}. (5.1.23)

Since Q is cyclic and normal in T and h is not NQ(A)-conjugate to hx, it follows that [Q, h] = 1 and hence

Ω1(T ) = Ω1(Rd) = hxi × P hhi. (5.1.24)

We claim that

P hhi is normal in T. (5.1.25)

If this does not hold, then from (5.1.24) and the fact that P is normal in T , there exists t ∈ T with ht ∈ P hx ∪ P hxz. But all involutions in P h are P -conjugate (5.1.22), and hence multiplying t by a suitable element of P , we have that h is NT (A)-conjugate to hx or to hxz, contradicting (5.1.23).

We now complete the proof via transfer arguments. Note that F is cyclic or quaternion by Proposition 5.1.6. If F is cyclic, then as it covers T/P hhi, we can apply Corollary 3.0.5 to get that x is F-conjugate to z, an immediate contradiction to (5.1.21). So F is quaternion and consequently, |F : Q| = 2. Let w ∈ F − Q of

2 order 4, so that w = x. In the present situation, Rd = Q × P hhi, w is of least 69 order in T − Rd by (5.1.24), and T/Rd is cyclic of order 2. So Corollary 3.0.5 yields

2 a morphism ϕ ∈ F with ϕ(w) ∈ Rd, and hence ϕ(w) ∈ Z by the structure of Rd. As w2 = x, ϕ(x) = x by (5.1.21). Thus ϕ ∈ C and so ϕ extends to a morphismϕ ˜ on a subgroup of T containing Q because Q E C. This forcesϕ ˜(F ) 6 hQ, ϕ˜(w)i to be abelian, a ﬁnal contradiction.

5.2 The 2-rank 3 case

Continuing the notation from Section 5.1, we prove here the following reduction.

Theorem 5.2.1. Let F be a fusion system on S satifying Hypothesis 4.2.1. Then T is of 2-rank 4.

Throughout this section, assume to the contrary that T is of 2-rank 3. By Hy- pothesis 4.2.1, S is also of 2-rank 3. From Proposition 5.1.18, we may assume that

x∈ / Z(S). (5.2.2)

Recall F1 is the preimage of Ω1(F ), so that F1 contains Q with index 1 or 2. Then

CT (P ) = F1 × hzi. (5.2.3)

By Lemma 4.2.6, there exists no involution in T which is an f-element. Set

J = J(S) = J(T ) for short. We have the inclusions P 6 J 6 Ω1(T ) 6 Rd. This shows that Z = Ω1(Z(Ω1(T ))) 6 Ω1(CT (P )) = Ω1(F1 × hzi) = hx, zi. So Z = hx, zi and Z coincides with Ω1(Z(J)). Therefore, by the Baumann hypothesis and (5.2.2), we have that T = Baum(S) and

T is of index 2 in S. (5.2.4)

Fix a ∈ S − T .

70 Lemma 5.2.5. z∈ / xF ∪ (xz)F .

Proof. Since x is fully F-centralized and not central in S, we need only show that z ∈ Z(S). Suppose that za 6= z and hence P a 6= P . Since P and P a are normal in

a a a a T , we have [P ,P ] 6 P ∩ P is normal in both P and P . Furthermore, [P ,P ] 6= 1 since otherwise P aP = P a × P is of 2-rank 4. Therefore [P a,P ] contains both Z(P a) and Z(P ). But [Z(P a),P ] = 1, forcing Z(P a) = Z(P ) contrary to assumption.

It follows in particular that xa = xz.

Lemma 5.2.6. Q = hxi and F is cyclic.

Proof. Suppose that Q > hxi and let u ∈ Q with u2 = x. Then huai is a normal

a 2 a a subgroup of T . Since (u ) = xz, we have [hu i,T ] 6 hxzi. But [hu i,P ] 6 P a as P is normal, and it follows that u ∈ CT (P ) = F1 × hzi by (5.2.3). Therefore,

1 a 1 xz ∈ f (hu i) 6 f (F1 × hzi) 6 Q, which is absurd. So Q = hxi and consequently F is cyclic (since F/Q is cyclic).

Lemma 5.2.7. Let V be a fours subgroup of P and set E = hxi × V . Then Ea is not T -conjugate to E.

Proof. Suppose that Ea is T -conjugate to E. Modifying a if necessary, we may assume that a normalizes E. Now the subgroup N = hca, AutC(E)i of AutF (E) lies

a in L3(2) and does not act transitively on I2(E) by Lemma 5.2.5. As x = xz, N does not stabilize a point of E. So N must ﬁx a line, which is then V . It follows ∼ that N = AutF (E) = S4. Now | AutT (E)| = 2, and we can obtain a contradiction to

(5.2.4) by showing that | AutS(E)| = 8, i.e. that E is fully automized in F. Suppose that E is not fully F-automized. Either J = hxi × P or there exists an involution h ∈ Rd − R and J = hxi × P hhi. In either case, there are exactly two

F S-classes of elementary abelian subgroups of order 8. Moreover, if E1 ∈ E is fully

71 S F-automized, then E1 6= E , and so hE,E1i = J. By Alperin’s fusion theorem, there

fc is a subgroup D ∈ F and an automorphism α ∈ AutF (D) of odd order such that

E1 := α(E) is fully F-automized. But then J = hE,E1i 6 D, and consequently α restricts to a nontrivial (odd order) automorphism of J. On the other hand, Aut(J) is a 2-group by Lemma 2.1.4, a contradiction.

Lemma 5.2.8. Rd > R.

Proof. Suppose on the contrary that Rd = R. If |T : PF | = 2, then J = hxi × P , and

T acts transitively on E23 (T ) contrary to Lemma 5.2.7. So T = PF . If |F | > 2, then

1 Z(T ) = Ω2(F ) × hzi and so f (Z(T )) = hxi is normal in S, at odds with (5.2.2). So T = R = J = hxi × P .

We now obtain a contradiction by a transfer argument. Note that as all involutions of P x are F-conjugate to x, we have P = hzF ∩ T i is normal in S by Lemma 5.2.5.

F Moreover, the quotient S/P is abelian. If b ∈ x is fully F-centralized, then CS(b) has 2-rank 3, whence b ∈ T ∩xF ⊆ P x. Corollary 3.0.6 now says that x is F-conjugate into P , contradicting Lemma 5.2.5 and completing the proof.

As a consequence of the previous lemma, T is transitive on E23 (R). Fix a fours

a group V of P . By Lemma 5.2.7 and the preceding remark, V ∈ Rd − R. Fix an

a involution h ∈ V − P . Then P1 := P hhi is dihedral of order 2|P |, and therefore is generated by F-conjugates of z. We have at this point that J = hxi × P1 = Rd. As every involution in P hx is P -conjugate to hx, and hence S-conjugate into P x, it follows that

F F P1 = hz ∩ P1i = hz ∩ T i. (5.2.9)

P1 is normal in S. (5.2.10) 72 Recall now that CT (P ) = Ω2(F ) × hzi from (5.2.3) and Lemma 5.2.6. Moreover, if

Ω2(F ) > hxi, then [Ω2(F ), h] = hzi by (5.2.10) and Lemma 2.2.1(h). Consequently

CS(P1) = Z and so

S/P1Z is abelian (5.2.11)

by Lemma 2.1.3 because P1 is nonabelian dihedral.

Notice FP1/P1 is a cyclic normal subgroup of S/P1 of index 2. So S/P1 is either abelian or modular, or else |F | = 4 by (5.2.11) and S/P1 is dihedral or quaternion. We rule out each of these cases in turn.

Lemma 5.2.12. S/P1 is not abelian.

F Proof. Suppose S/P1 is abelian. For any b ∈ x which is fully F-centralized, CS(b)

F is of 2-rank 3, and so b ∈ J ∩ x ⊆ P1x by (5.2.9) and Lemma 5.2.5. Now x has an

F-conjugate in P1 by Corollary 3.0.6, and this contradicts Lemma 5.2.5.

The next lemma shows that S/P1 is not quaternion.

Lemma 5.2.13. There exists an involution b in S − T .

2 Proof. Let b1 ∈ S − T . Modifying b1 by an element of F , we may assume b1 ∈ P1Z because S/P1 is not cyclic (by the previous lemma). But then P1Zhb1i/hxi is dihedral or semidihedral because b1 swaps the P1-classes of fours groups of P1 by Lemma 5.2.7.

2 Modifying b1 by an element of P1 then, we may assume b1 ∈ Z. Now CZ (b1) = hzi

2 so b1 ∈ hzi. Set b = b1 if b1 is an involution, and set b = xb1 otherwise. Then b is an involution.

Lemma 5.2.14. S/P1 is not modular.

Proof. Suppose it is. Then S/P1 has a unique fours subgroup, covered by hx, bi.

2 Hence, S0 := Ω1(S) = P1Zhbi and S/S0 is cyclic. Let w ∈ F with w = x. Then 73 w is of least order outside S0 and centralizes FP , whence |S : CS(w)| 6 4. Apply

Corollary 3.0.5 to obtain a morphism ϕ in F with ϕ(w) in S0 and fully F-centralized.

Any element of S0 − P1Z interchanges the two classes of fours subgroups of P1.

2 Hence if b1 ∈ S0−P1Z is of order 4, then b1 ∈ Z and b1 induces an involutory automorphism of P1 interchanging the two classes of fours subgroups of P1. So CP1 (b1) = hzi and |S : CS(b1)| > |P1 : CP1 (b1)| > 8 as |P1| > 16.

As ϕ(w) is fully F-centralized, the preceding paragraph implies ϕ(w) ∈ P1Z = hxi × P1, and consequently ϕ(w) = x0v for some x0 ∈ hxi and v ∈ P1 of order 4. Now ϕ(x) = ϕ(w)2 = z, contrary to Lemma 5.2.5.

Therefore by the previous three lemmas, |F | = 4 and S/P1 is dihedral of order 8. We now obtain the ﬁnal contradiction, completing the proof of Theorem 5.2.1.

Lemma 5.2.15. S/P1 is not dihedral.

2 Proof. Suppose it is. Again let w ∈ F with w = x. Then F = hwi. Since CT (F ) = R is of index 2 and [F, h] = hzi, we have F 6 Z2(T ) in the present situation. Moreover,

T/Z = F/Z × P1Z/Z with the second factor dihedral of order at least 8, and so

Z2(T ) = F × V where V is cyclic of order 4 in P1. Now b inverts P1w by assumption;

b b hence ww ∈ Z2(T ) ∩ P1 = V . As [w, w ] = 1, we have on the one hand that

b 2 b 2 2 b 2 2 2 b ww ∈ CV (b) = hzi, because b = 1, and on the other (ww ) = w (w ) = w (w ) = x(xz) = z. These two facts are incompatible, and the proof is complete.

5.3 The 2-rank 4 case: |Q| = 2

For a fusion system F on S satisfying Hypothesis 4.2.1, T has 2-rank 3 or 4 by

Lemma 4.2.6. By Theorem 5.2.1, there are no such fusion systems with T of 2-rank

3. We begin now the study of F when the rank of T is 4. The content of the current section will be devoted to the proof of the following reduction. 74 Theorem 5.3.1. Assume that F satisﬁes Hypothesis 4.2.1 with T of 2-rank 4. Then

|Q| > 2.

The notation follows that begun in Section 4.2, in particular that of Hypoth- esis 4.2.1. For instance, K is the unique component of the involution centralizer

C = CF (x), and is a fusion system of a ﬁnite group K isomorphic with L2(q1) for some q1 ≡ ±1 (mod 8). The Sylow subgroup of K is denoted by P , a dihedral group

k 2 of order 2 (k = ν2(q1 − 1) − 1 > 3). Consistent with Sections 5.1 and 5.2, we also set Z(P ) = hzi. Denote by C the cyclic maximal subgroup of P . By Hypothesis 4.2.1, the

Thompson subgroup J(T ) = J(S) and so this common subgroup is denoted simply by J.

By Lemma 4.2.6,

there exists an involutory f-element f ∈ CT (P ). (5.3.2)

We ﬁx such an involution f.

Before beginning the proof of Theorem 5.3.1, we collect some facts seen before, and which hold throughout 2-rank 4 case. In particular,

J 6 Rhfi = Qhfi × P (5.3.3) from Lemma 4.2.4, and

CT (P ) = Qhfi × hzi (5.3.4) from Lemma 4.2.3 and the structure of Aut(K) in Lemma 2.2.1. Finally,

T < S (5.3.5) by Proposition 5.1.18.

Assume for the remainder of this section that Q = hxi is of order 2, as we prove

Theorem 5.3.1 by way of contradiction in a series of lemmas. Thus, J = hx, fi × P by (5.3.3) and Z(J) = hx, f, zi is elementary abelian of order 8. 75 Lemma 5.3.6. zF ∩ Z(J) = hzi.

Proof. The Thompson subgroup J = hx, fi×P is weakly F-closed. By Lemma 2.3.15

1 fusion in Z(J) is controlled in AutF (J). But f (J) ∩ Z(J) = hzi is characteristic in J, so the statement follows.

Lemma 5.3.7. Let y ∈ Z(J). Then each involution of yP is C-conjugate to y or to yz. In particular, zF ∩ J = zF ∩ P .

Proof. Since K has one class of involutions and x ∈ CT (K) the lemma holds for y = x, z, and xz. So we may assume that y is an f-element on K, that is, y centralizes P but y∈ / hx, zi. Let t be an involution of P so that yt is also an involution. If t = z then the statement is obvious, so assume t is a noncentral involution of P . Set U = ht, zi, and let ϕ ∈ AutK(U) of order 3 such that ϕ(t) = z. Then ϕ extends toϕ ˜ ∈ C on Uhyi and centralizes either y or yz by Lemma 4.2.4. In the former case,ϕ ˜(yt) = yz, and in the latter,ϕ ˜2(yt) =ϕ ˜2(yztz) = y. This completes the proof of the ﬁrst statement.

The second statement now follows from Lemma 5.3.6.

Lemma 5.3.8. The following hold.

(a) P is normal in S,

(b) there exists a fully F-centralized fours group in P ,

(d) no element of S squares into J − Z(J)C.

Proof. Let s ∈ S. Then zs ∈ Z(J), and so zs = z. But all involutions of P s are

s s s Ks K -conjugate by Lemma 2.2.1(f). Hence P = Ω1(P ) = hz i 6 P by Lemma 5.3.7 and (a) holds.

76 Let U be a fours group of P , and let ψ ∈ HomF (U, S) such that ψ(U) is fully

F-centralized. By the extension axiom, ψ extends to CS(U), which contains an elementary abelian subgroup of maximal rank. Thus, ψ(U) 6 J and the nonidentity elements of ψ(U) consist of F-conjugates of z. It follows that ψ(U) 6 P by Lemma 5.3.7, proving (b).

Part (c) is Lemma 2.1.3(b). For (d), suppose that s ∈ S with s2 ∈ J − Z(J)C.

2 2 Then s must lie in CS(P )C by Lemma 2.1.3(c). Thus, s ∈ CJ (P )C = Z(J)C.

Lemma 5.3.9. J = PCS(P ).

2 Proof. Suppose the lemma is false and choose a ∈ CS(P ) − J with a ∈ J. Let

2 Za = CZ(J)(a), which contains z and is of order 4. Then a ∈ Z(J) ∩ CS(a) = Za since a centralizes P . By (5.3.4) and (5.3.3), J = PCT (P ), and so a does not centralize x.

Fix a fully F-centralized fours group U in P guaranteed by Lemma 5.3.8(b), a

K-automorphism ϕ of U of order 3, and an extensionϕ ˜ ∈ HomF (CS(U),S) (by the extension axiom). Observe that

Z(J)a contains no involution, (5.3.10)

because each element of Z(J)a lies outside J and centralizes the subgroup ZaU, which is of 2-rank 3. Also,ϕ ˜ is deﬁned on a; it follows that

a2 6= z, (5.3.11) since otherwiseϕ ˜(a) is an element of S squaring to a noncentral involution of P , contrary to Lemma 5.3.8(d).

If [x, a] = z, then [ϕ ˜(x), ϕ˜(a)] = ϕ(z) is a noncentral involution of P , contradicting

Lemma 5.3.8(c). Finally, we consider the case in which [x, a] = y 6= z. Here, a2 ∈

2 2 Za = hy, zi. If a = y, then x inverts a, and so (xa) = 1 contrary to (5.3.10). Hence 77 a2 = yz by (5.3.11). In this case, we may replace a by xa to obtain a2 = z, again contradicting (5.3.11), and completing the proof.

Let Ω be the two element set consisting of the P -classes of fours subgroups of P .

Let N be kernel of the action of S on Ω. Then J 6 N. By the previous two lemmas S/J embeds into Out(P ). Thus by Lemma 2.1.3(a), S/J has a cyclic subgroup B with index 1 or 2 and with B = N/J cyclic of order dividing 2k−3. Thus, N is of index 1 or 2 in S.

Lemma 5.3.12. We have J < N. In particular, |P | > 16.

Proof. Suppose to the contrary that J = N. Then N 6 T and T < S by (5.3.5).

Since |S : N| 6 2, it follows that N = J = T and |S : T | = 2. Fix a ∈ S −T . As a acts on Z(J) = Z(T ) and does not centralize x, Z(S) = hy, zi for some y ∈ Z(J) − hx, zi. Since a normalizes P and acts nontrivially on Ω, we have

[P, a] = C. Thus, we have two possibilities for the commutator subgroup of S. Either

[x, a] = z and [S,S] = C, or else [S,S] = Z(S)C.

Assume ﬁrst that [S,S] = C. We have that y ∈ Z(S) − hzi is not F-conjugate to z by Lemma 5.3.6. But then Corollary 3.0.6, applied with C representing the T of that Corollary, forces y ∼F z anyway, a contradiction. Now suppose [S,S] = Z(S)C, so that [S,S] ∩ Z(J) = Z(S) is of order 4. We will show in this case that x∈ / foc(F). We claim

every fully F-centralized conjugate of x lies in Z(J). (5.3.13)

Let s be a fully F-centralized conjugate of x. Then s lies in a elementary abelian subgroup of rank 4 by the extension axiom, so s ∈ J = Z(J)P . If s ∈ Z(J)(P − C), then s has at least four conjugates under P hai because Z(J) ∩ P = hzi and P hai is transitive on the involutions in P − C. So |CS(x)| > |CS(s)|, and (5.3.13) holds.

78 Thus, (5.3.13) implies Z(S)Cx is the unique nonidentity element of S/Z(S)C containing a fully F-centralized F-conjugate of x. This allows us to apply Corol- lary 3.0.6, with Z(S)C = [S,S] in the role of T , to obtain an F-conjugate of x in

Ω1([S,S]) = Z(S), contradicting the assumption that x is fully F-centralized. We conclude that J < N, and the ﬁrst statement of the lemma holds.

For the last statement, suppose |P | = 8. Every element inducing an outer automorphism on P interchanges the two classes of fours subgroups of P . Thus N induces inner automorphisms on P , i.e. N = PCS(P ) = J, contrary to J < N. Therefore,

|P | > 16.

By an earlier remark and Lemma 5.3.12, N/J is nontrivial cyclic. Choose w ∈ N mapping to a generator of N/J. By the deﬁnition of N, we may adjust w by an element of P and assume that w centralizes a fours group U = he, zi of P . Replacing w by ew if necessary, we may assume also that

w centralizes C/C4. (5.3.14)

2 Let f1 ∈ hwi such that f1 ∈/ J but f1 ∈ J. Since J = PCS(P ) by Lemma 5.3.9,

2 we have f1 ∈ CS(C) by Lemma 2.1.3(d) applied with D = P there. Then f1 takes a generator c of C to cz by choice of f1 ∈ hwi and (5.3.14), and hence f1 centralizes

2 C . As |P | > 16 from Lemma 5.3.12, it follows that

2 CP (f1) is the nonabelian dihedral subgroup P1 := hU, C i of P. (5.3.15)

2 So f1 ∈ CS(P1) ∩ J = Z(J), and f1 is of order at most 4. But f1 is not an involution, otherwise hf1,CZ(J)(f1),Ui is an elementary 16 outside J, so f1 is of order 4. In fact, it is shown below that there are no involutions in Jf1. For this, we will need that

f1 does not square to z. (5.3.16)

79 2 2 Assume to the contrary that f1 = z. Then as [C , f1] = 1 and |P | > 16 by 2 Lemma 5.3.12, there exists an element v ∈ CP (f1) with (vf1) = 1. But then CP (vf1) contains the fours subgroup hce, zi of P , and so vf1 ∈ J, yielding the same contradiction as before and thus conﬁrming (5.3.16). We now show

Lemma 5.3.17. There is no involution in Jf1.

2 Proof. By (5.3.16), we may assume that f1 ∈ Z(J) − hzi. If [Z(J), f1] 6 hzi, then ∼ Jhf1i/P = C2 × C4 with P f1 of order 4. In this case, every element of order 2 in

Jhf1i lies in J as claimed. Hence we may assume that [Z(J), f1] = hyi 6= hzi. Then ∼ Z(J)P1hf1i = hy1, f1i×P1 with D1 := hy1, f1i = D8. If E1 is the other fours subgroup of D1, then E1 × P1 has 2-rank 4 and so E1 6 J. But then D1 6 CS(P1) ∩ J = Z(J), a contradiction.

2 In addition, we let h1 ∈ S − N be an element such that h1 ∈ J or set h1 = 1 if such an element does not exist. Note that if h1 = 1, then S/J is cyclic by the structure of Out(P ) (Lemma 2.1.3(a)). In any case, S/Jhh1i is cyclic.

Assume that h1 6= 1. Then

both h1 and h1f1 square into J. (5.3.18)

s Let s ∈ Jh1 ∪ Jh1f1. Then for any e1 ∈ P − C, we have e1 = e1c for some generator c of C because s∈ / N. Therefore,

[P, s] = C and CJ (s) 6 Z(J)C is abelian. (5.3.19)

s2 s Furthermore, as e1 = e1cc , we have

2 if s ∈ CS(P ) then s inverts C. (5.3.20)

With this setup, the next two lemmas contradict each other and complete the proof of Theorem 5.3.1. 80 Lemma 5.3.21. h1 6= 1 and both cosets Jh1 and Jh1f1 contain involutions.

Proof. Suppose either that h1 = 1 or that there are no involutions in Jh1f1. The argument is the same in case h1 6= 1 and Jh1 contains no involutions. (Alterna- tively, swap the roles of h1 and h1f1 in this extra case.) By (5.3.17) and assumption,

Ω1(S) 6 Jhh1i. Also S/Jhh1i is cyclic, and f1 is of least order outside Jhh1i. By

Corollary 3.0.5, there exists a morphism ϕ ∈ F such that ϕ(f1) ∈ Jhh1i is fully

F-centralized, and ϕ(CS(f1)) 6 CS(ϕ(f1)).

Now if h1 6= 1, then ϕ(f1) cannot lie in the coset Jh1. This is because Ω1(CP (f1)) is nonabelian dihedral by (5.3.15), whereas Ω1(CS(s)) is abelian for every s ∈ Jh1 by

1 (5.3.19). So ϕ(f1) ∈ J whether or not h1 = 1. But f1 is of order 4 and Ω1(f (J)) = 2 2 hzi, so ϕ(f1 ) = z. But z is weakly F-closed in Z(J) by Lemma 5.3.6 and so f1 = z, contrary to (5.3.16).

Lemma 5.3.22. Jh1f1 contains no involution.

2 Proof. We may assume h1 6= 1 = h1 by Lemma 5.3.21, and then h1 inverts C by (5.3.20). So

−1 h1f1 sends a generator c of C to c z. (5.3.23)

2 In particular, CC (h1f1) = hzi and so CZ(J)C (h1f1) 6 Z(J)Ω2(C). As (h1f1) ∈ J 2 from (5.3.18), we have (h1f1) ∈ CJ (h1f1) = CZ(J)C (h1f1) 6 Z(J)Ω2(C) with the 2 equality by (5.3.19). But (h1f1) does not lie in CS(P ) = Z(J) by (5.3.20), since h1f1 does not invert C.

Set M = Jhh1f1i and M = M/Z(J). Then M contains the dihedral group P as a maximal subgroup, which is nonabelian as |P | > 16. Furthermore M is of maximal class by (5.3.19) and (5.3.23). As h1f1 is of order 4 squaring into the center of M, we know M is semidihedral. But then M contains no involutions outside its dihedral

81 maximal subgroup P . It follows that M = Jhh1f1i contains no involutions outside J, which is what was to be shown.

5.4 The 2-rank 4 case: |Q| > 2

For this penultimate section, we continue to assume F is a saturated fusion system on the 2-group S satisfying Hypothesis 4.2.1. By the main results of the previous three sections, we are reduced to the following situation in describing F.

1. T = CS(x) is a proper subgroup of S (Proposition 5.1.18),

2. S is of 2-rank 4 (Theorem 5.2.1), and

3. Q = CT (K) is of order at least 4 (Theorem 5.3.1).

Theorem 5.4.1. Let F be a saturated fusion system on the 2-group S. Assume F ∼ satisﬁes Hypothesis 4.2.1 and, in addition, the above three items. Then S = D2k o C2, and F is the fusion system of L4(q) for some q ≡ 3 (mod 4) with ν2(q + 1) = k − 1.

Adopt the notation of Hypothesis 4.2.1 and the set up at the beginning of Sec- tion 5.3. By Lemma 4.2.6,

there exists an involutory f-element f ∈ CT (P ).

We continue to ﬁx such an involution f. As T a proper subgroup of S, we also

2 ﬁx a ∈ NS(T ) − T with a ∈ T.

As usual, we prove Theorem 5.4.1 in a sequence of lemmas. It will emerge quickly

(after Lemma 5.4.8) that J = Rhfi is the product of two dihedral groups Qhfi and

P of the same order, T has index 2 in S, and T/R is of exponent 2. Since T/R embeds into Out(K) (Proposition 4.2.3), this means that either T = J = Rhfi, or 82 T = Rhh, fi for some 1 6= h ∈ T such that h2 ∈ Q and Rhhi/Q is dihedral of order

2|P |. Thus RhhiK/Q is uniquely determined as the fusion system of P GL2(q1) (see the description of Out(K) in Lemma 2.2.1). In anticipation of this we let

h ∈ T − R such that h2 ∈ Q and Rhhi/Q is dihedral, or (5.4.2)

h = 1 if such an element does not exist.

Note in the case h 6= 1,

[Qh, Qf] = Qz (5.4.3) by Lemma 2.2.1(h).

Much of the 2-group and transfer analysis will be dedicated to analyzing whether or not h exists, and if it does, whether Qhhi splits over Q. Together with our target

F = FS(L4(q)) appearing within the case T = J, the following table lists the fusion systems (of ﬁnite groups) which nearly satisfy the conditions of Theorem 5.4.1, and why they are eventually ruled out.

Scenario T S Group Contradiction ∼ h 6= 1 and Qhhi is dihedral Jhhi Q2k+1 o∗ C2 P Sp4(q1) CT (K) = D2k

2 h 6= 1 and Qhhi is cyclic Jhhi SD2k+1 o∗ C2 P GL4(q) O (F) < F ∼ h = 1 J D8 o C2 A10 CT (K) = C2 × C2

h = 1 J D2k o C2 L4(q)

The 2-group and transfer analysis is carried out through Lemma 5.4.23, where it is shown that S is of type P Sp4(q1) or P GL4(q) when h 6= 1. Then we compute the centralizer of a central involution via an argument modeled on that of [GH73,

Lemmas 3.15,3.16], thus ruling out the P GL4(q)-case. Analyzing the resulting fusion information allows us to conclude that that h = 1, and S is then isomorphic to

83 D2k o C2. Lastly, we appeal to a result of Oliver [Olia] to identify F as the fusion system of L4(q). We begin by pinning down the structure of J in the next few lemmas.

Lemma 5.4.4. The following hold.

(a) Qhfi is dihedral or semidihedral,

(b) J = Ω1(Qhfi) × P ,

(d) T = Rhh, fi.

Proof. We claim that Z(Qhfi) = hxi. Suppose this is not the case. Then either f centralizes Q or |Q| > 8 and f acts on Q by sending a generator d of Q to dx. In either case we have that J = hx, fi × P by (5.3.3). Then D = CT (J) is normal in S,

1 and D is equal to Q×hf, zi if f centralizes Q and to f (Q)×hf, zi otherwise. In any case, x is the only involution which is a square in D, so x ∈ Z(S). This contradicts

T < S. Thus Qhfi is of maximal class and f is an involution outside Q, so (a) holds.

Now (b) follows by (5.3.3). ˜ Suppose some element f1 ∈ T squares into Rf. Write T = T/P and denote ˜ ˜ ˜ ˜ images modulo P similarly. Then D1 := Qhf1i contains D := Qhfi as a normal subgroup. Moreover as Qhfi ∩ P = 1, D ∼= Qhfi. If Qhfi is dihedral in (a), then

Lemma 2.1.3(d), with D1 in the role of S there, gives a contradiction. If Qhfi is semidihedral, then considering the images of D1 and D modulo hx˜i, we obtain the same contradiction to Lemma 2.1.3(d). Therefore, (c) holds. By the structure of

Out(K) and (c), T/R is a fours group covered by hh, fi, yielding (d).

Lemma 5.4.5. xS = {x, z}. Consequently, |S : T | = 2.

84 Proof. By Lemma 5.4.4, S normalizes Z(J) = hx, zi and J = J(Qhfi) × P =

Ω1(Qhfi)×P is the product of two nonabelian dihedral groups. By Proposition 2.1.2, S must permute the commutator subgroups of these factors. Therefore, xS ⊆ {x, z} and as T = CS(x) is a proper subgroup of S, we have equality and |S : T | = 2.

Recall a ∈ S − T has been ﬁxed, squaring into T . By the previous lemma,

T E S = T hai and a swaps x and z. So Z(S) = hxzi, and

x is not F-conjugate to xz (5.4.6) because x∈ / Z(S) is fully F-centralized.

Lemma 5.4.7. [P a,P ] = P a ∩ P = 1.

a a a Proof. Note ﬁrst that both P and P are normal in T , So [P,P ] 6 P ∩ P . Suppose a a a that Z0 := P ∩ P 6= 1. Then Z0 is nontrivial normal in P , and so hxi = Z(P ) 6

Z0 6 P , a contradiction.

Lemma 5.4.8. Qhfi is dihedral of the same order as P .

Proof. Recall from Lemma 5.4.4 that Ω1(Qhfi) is nonabelian dihedral and J =

Ω1(Qhfi) × P . Let C1 and C2 be the cyclic maximal subgroups of Ω1(Qhfi) and

a a P , respectively. Because x = z, we have C1 is a cyclic subgroup of J with z as its unique involution, and so |C1| 6 |C2| by the structure of J. We conclude sim- a ilarly that |C2| 6 |C1| by considering C2 , and hence C1 and C2 are of the same order. Therefore either Qhfi is either dihedral of the same order as P , or Qhfi is semidihedral with |Qhfi| = 2|P |. ∼ a Suppose Qhfi is semidihedral. Then Ω1(Qhfi) = P . Set S1 = (Qhfi) for short.

a a Then S1 centralizes P . But P also centralizes P by Lemma 5.4.7. In fact,

a P × hzi 6 Ω1(CT (P )) = Ω1(Qhfi) × hzi 85 so the above three subgroups are equal, as the outside two are of the same order.

Taking centralizers, we get that

a S1 6 CT (P ) = CT (Ω1(Qhfi)) 6 CT (f) (5.4.9)

Now f centralizes no element in the coset Rh when h 6= 1 by (5.4.3). So CT (f) =

CRhfi(f) = P × hf, xi by Lemma 5.4.4(d,a), and (5.4.9) is a contradiction because P × hf, xi contains no semidihedral subgroup.

In view of the previous lemma, it is now determined that

J = Qhfi × P = P a × P (5.4.10)

a a with a interchanging P and P . Since P 6 CT (P ) = Qhfi × hzi, we may replace f by fz and assume that

f ∈ P a. (5.4.11)

We ﬁx notation for the maximal cyclic subgroup of P , calling it C. Then

a a C 6 Qhzi and Q 6 Chxi (5.4.12) by the above remarks.

The next lemma shows that a may be chosen to be an involution. Part (a) of it will later be shown in Lemma 5.4.25 to rule out the P Sp4(q1)-case mentioned above and determine that in fact h = 1.

Lemma 5.4.13. The following hold.

(a) If C = hci, then ff a is not F-conjugate to f(cf a).

(b) There exists an involution in S − T .

86 Proof. We will show that ff a is not conjugate to f(cf a) from the fact that one of them is F-conjugate to x and the other to xz. Recall from (5.4.11) we have chosen f ∈ P a,

a a a so f ∈ P . Thus, U0 = hf , zi and U1 = hcf , zi are fours groups of P which are not P -conjugate. Since f induces an f-automorphism on K (Deﬁnition 4.2.5), CC(f) contains AutK(Uj) for some j, and CC(fz) contains AutK(U1−j) by Lemma 4.2.4(b).

j a j a Thus, there is an element ϕ ∈ AutC(hfiUj) of order 3 with ϕ(f·c f ) = ϕ(f)ϕ(c f ) = fz. On the other hand, there is a similar element ψ ∈ AutC(hfiU1−j) with ψ(fz ·

1−j a a zc f ) = fz · z = f. Since f is K -conjugate to x in CF (z), and so fz is CF (z)- conjugate to xz, this contradicts (5.4.6).

For (b), suppose that a2 ∈ J. It will be shown ﬁrst that (b) holds in this situation.

2 −1 a a0 a Write a = ts with t ∈ P and s ∈ P . Let a0 = as. Then P = P , and

2 2 a a a 2 a 2 a0 = a s s = ts as [P,P ] = 1. So a0 ∈ P and centralizes a0. Therefore a0 = 1, as claimed.

So it remains to prove that a2 ∈ J. If a does not square into J, then h 6= 1 and a squares into the coset Jh; so S/J is cyclic of order 4 in the present case. Let J denote the set of J-classes of “noncentral diagonal” involutions of J, that is, those involutions in J outside the set I = P hxi ∪ P ahzi. Thus J has cardinality 4, and for any generator c of C, the set {ff a, (caf)f a, f(cf a), caf(cf a)} is a set of representatives for the members of J. Since I is a normal subset of S and J is a normal subgroup,

S acts on J by conjugation. Moreover, any element in Jh swaps the two P -classes of noncentral involutions in P , and so acts nontrivally on J. It follows that hai acts transitively as a four-cycle on J, and hence all involutions in J − I are S-conjugate.

This contradicts part (a) and completes the proof of the lemma.

From now on, we assume a2 = 1. We narrow down the structure of T to two possibilities in the next lemma, depending on whether Qhhi splits over Q or not, as described in the introduction to this section.

87 Lemma 5.4.14. Suppose h 6= 1. Then one of the following holds.

(a) h2 = 1 and Qhhi is dihedral, or

(b) Q = hh2i.

2 Proof. Recall that Q E T and h ∈ Q by the choice of h. Since T = Rhh, fi and J = Rhfi, it follows that ha ∈ Jh. The coset P aha lies outside the dihedral group

J/P a, which is isomorphic to P . And T/P a is a dihedral group containing J/P a as a maximal subgroup. Thus either P aha is an involution in T/P a − J/P a, or P aha squares to a generator of the cyclic maximal subgroup CP a/P a of J/P a.

Suppose that P aha is an involution in T/P a−J/P a. Then P aha inverts CP a/P a ∼=

a a−1 C, and so h inverts C. It follows that h inverts C 6 Q × hzi. Since h normalizes Q, h must invert Q. As P aha is an involution, we have (ha)2 ∈ P a. So h2 ∈ P . But h2 ∈ Q by choice of h. Therefore, h2 ∈ Q ∩ P = 1 giving (a).

Suppose that P aha squares to a generator of CP a/P a. Then ha and hence h has order at least 2|C|. But h2 ∈ Q and |Q| = |C|, so we must have Q = hh2i, giving

(b).

a Set J0 = QC = Q × C = C × C, a homocyclic normal subgroup of S. It will be helpful for what follows to call attention to the action of T on J0, and describe

a what this means for the structure of the quotient S/J0. Recall that C 6 Qhzi from (5.4.12), and so the action of an element of T on Ca is the same as on Q. From

Lemma 5.4.8 and the two possibilities in Lemma 5.4.14, each element in T centralizes

C or inverts it, and the same holds for Ca in place of C. Conjugation by an element in S − T swaps the actions. Moreover, T/J0 is elementary abelian of order 8 when h 6= 1, and a induces an automorphism of T/J0 ﬁxing pointwise a fours group. For

a instance, from the actions of h, f, and f on J0, and since J E S,

a if h inverts Q, then hh, ff i covers CT/J0 (a). (5.4.15)

88 and

a if h centralizes Q, then hfh, ff i covers CT/J0 (a), (5.4.16)

Lastly,

a [S,S] = J0hff i. (5.4.17)

Suppose h is involution as in Lemma 5.4.14(a) from now through the next lemma.

a a From (5.4.15), we have [h, a] ∈ J0 = C C and we may arrange to have [h, a] ∈ C by

a replacing h by an appropriate element in C h 6 Qhzih. Then

2 a [h, a] = (ha) ∈ C ∩ CJ0 (ha) = 1, and h still squares to the identity. Fix this choice for h once and for all. Thus, S is a split extension of J by the fours group hh, ai.

We can now write down a presentation for S. The rest of the following lemma is veriﬁed by direct computation, or by appeal to [GH73, Lemma 3.5].

Lemma 5.4.18. Suppose h is an involution. Then S has presentation

h d, c, f, e, h, a | d2k−1 = c2k−1 = f 2 = e2 = a2 = 1,

[d, c] = [f, e] = 1,

df = d−1, ce = c−1, ca = d, ea = f,

h2 = 1, eh = ec, ha = h i with notation consistent with that ﬁxed. Here, P = hc, ei, P a = hd, fi, J = P aP ,

2k−2 2k−2 J0 = hd, ci, x = d , z = c , Z(S) = hxzi, and T = hd, c, f, e, hi. Furthermore, the following hold.

∼ (a) Jhai = D2k o C2.

89 (b) Q = hdzi.

(d) CS(h) = hhi × B0 where B0 = hx, ai is dihedral of order 8.

a k+1 (e) CS(a) = hai × Ba where Ba = hff h, hi is dihedral of order 2 .

a k+1 (f) CS(ha) = hhai × Bha where Bha = h[h, f]ff h, hi is dihedral of order 2 .

(g) All involutions of Ja are J-conjugate as are all involutions of Jha.

a ∼ a (h) CS(ff ) = hf, x, ai = (C2 × C2) o C2 and all elements of J0ff are S-conjugate.

a a a k+1 (i) D1 := hff h, xff ai and D2 := h[h, f]ff h, hi are quaternion of order 2

f a with [D1,D2] = 1, D1 ∩ D2 = hxzi, D1 = D2, D := D1D2 = J0hff , h, ai = ∼ [S,S]hh, ai, and S = Dhfi = Q2k+1 o∗ C2 is of type P Sp4(q1).

From now through the next lemma, assume Q = hh2i as in Lemma 5.4.14(b).

We adjust h slightly as follows. As a consequence of (5.4.17), we have that ha ∈

a a a 2 a a a J0ff h = CC ff h. Since a = 1, we may write h = c1c1ff h for some c1 ∈ C.

a −1 Replacing h by (c1) h, which lies in Qhzih by (5.4.12), we arrange that

ha = ff ah, (5.4.19) and h still squares to a generator of Q. Fix this choice of h once and for all. Then it follows that

[fh, a] = 1. (5.4.20)

Now ﬁx the generator c = [f a, h] of C and set d = ca. Then

d = ca = [f, ha] = [f, ff ah] = [f, h]. (5.4.21)

90 From (5.4.20) and the fact that (fh)2 ∈ Qz in (5.4.3), we have (fh)2 = xz, and so xz = fhfh = h−2[f, h] = h−2d as f inverts Q. Hence,

h2 = dxz. (5.4.22)

Lastly, from (fh)2 = xz = (xa)2 and (5.4.20), we have

xfha ∈ Jha is an involution.

We can now write down a presentation for S in case Q = hh2i. The rest of the following lemma is veriﬁed by direct computation.

Lemma 5.4.23. Suppose Q = hh2i. Then S has presentation

h d, c, f, e, h, a | d2k−1 = c2k−1 = f 2 = e2 = a2 = 1,

[d, c] = [f, e] = 1,

df = d−1, ce = c−1, ca = d, ea = f

h2 = dd2k−2 c2k−2 , eh = ec, ha = feh i with notation consistent with that ﬁxed. Here P = hc, ei, P a = hd, fi, J = PP a,

2k−2 2k−2 J0 = hd, ci, x = d , z = c , Z(S) = hxzi, and T = hd, c, f, e, hi. Furthermore, the following hold.

∼ (a) Jhai = D2k o C2.

(b) Q = hdxzi = hdzi.

(d) fh inverts J0; all elements of J0fh square to xz and are J-conjugate.

a a k+1 (e) CS(a) = hai × Ba where Ba = hf h, ff i is semidihedral of order 2 with

Z(Ba) = hxzi. 91 2 (f) Set b1 = xfha ∈ Jha. Then b1 = 1 and CS(b1) = hb1i × Bha where Bha =

−1 a k+1 hd f h, xai is semidihedral of order 2 with Z(Bha) = hxzi.

(g) All involutions of Ja are J-conjugate as are all involutions of Jha.

a ∼ a (h) CS(ff ) = hx, f, ai = (C2 × C2) o C2 and all elements of J0ff are S-conjugate.

a a −1 a k+1 (i) D1 = hf h, ff ai and D2 = hd f h, ai are semidihedral of order 2 with

f a [D1,D2] = 1, D1 ∩ D2 = hxzi, D1 = D2, D := D1D2 = J0hff , fh, ai = ∼ [S,S]hfh, ai and S = Dhfi = SD2k+1 o∗ C2 is of type P GL4(q) for q ≡ 3

(mod 4) with ν2(q + 1) = k − 1.

Armed with this data, the centralizer of the central involution is computed next.

Lemma 5.4.24. Suppose h 6= 1. Then CF (xz) is realizable by a ﬁnite group G having Sylow 2-subgroup S and with the property that G contains a normal subgroup isomorphic to SL2(q1) ∗ SL2(q1) of index 2 with f interchanging the two SL2(q1) ∼ factors. In particular, S = Q2k+1 o∗ C2 and h is an involution.

Proof. Assume that h 6= 1. The two possibilities for S in Lemmas 5.4.18 and 5.4.23 will be treated simultaneously. Fix t ∈ J such that tha is an involution as follows.

When in the case of Lemma 5.4.18, we take t = 1. In the other case, we take t = xf as in Lemma 5.4.23(f). In either case th commutes with a and inverts J0. Let b be one of

k+1 a or tha. Then CS(b) = hbi×B where B is dihedral or semidihedral of order 2 with

Z(CS(b))∩[CS(b),CS(b)] = Z(S) by Lemma 5.4.18(e,f) and Lemma 5.4.23(e,f). More- over, all involutions of Jb are J-conjugate by Lemma 5.4.18(g) and Lemma 5.4.23(g).

When h is an involution, all involutions in the coset Jh are J-conjugate by

Lemma 5.4.18(c). From Lemma 5.4.18(d), CS(h) = hhi × B0 where B0 is dihe-

k+1 2 dral of order 8 < 2 , whence |CS(h)| < |CS(b)|. In the case where Q = hh i, there are no involutions in Jh by Lemma 5.4.23(c). This shows that in either case the

92 set of fully F-centralized F-conjugates of b outside J lies in Ja ∪ Jha. By Corol- lary 3.0.6, there exists a morphism ϕ ∈ F such that ϕ(b) ∈ J is fully F-centralized, and ϕ(CS(b)) 6 CS(ϕ(b)). Since CS(b) has nilpotence class k > 3 and [S,S,S] 6 J0 from (5.4.17), we have ϕ(xz) ∈ Ω1(J0) = hx, zi. It follows that ϕ(xz) = xz as xz is not F-conjugate to x or to z. Composing with ca if necessary, we may assume that ϕ(b) = z. Thus, we have shown there exist ϕa, ϕha ∈ CF (xz) such that

ϕa(a) = ϕha(tha) = z.

Set N = CF (xz). We claim that the hyperfocal subgroup hyp(N ) is of index 2 in S. We will show this by ﬁrst demonstrating that the normal closure

S h[a, ϕa], [tha, ϕha]i is the commuting product D := D1 ∗ D2 of two quaternion or two semidihedral subgroups of order 2k+1 as in Lemma 5.4.18(i) or Lemma 5.4.23(i), respectively. Then we shall use a transfer argument inside N to show that in fact foc(N ) = D from which it will follow that hyp(N ) = D as well.

S S Set D0 = h[a, ϕa], [tha, ϕha]i = hxa, xthai . Taking products, D0 contains th inverting J0. All elements of J0th are J-conjugate by Lemma 5.4.18(c) and

a Lemma 5.4.23(d), so D0 contains J0. This shows that [S,S] = J0hff i = J0h[f, a]i 6

D0. But D0 is a proper normal subgroup of S contained in D = [S,S]hth, ai, and so

D0 = D. We conclude that foc(N ) > D is of index 1 or 2 in S.

As f interchanges D1 and D2, fz∈ / D. Suppose that foc(N ) = S. Then by Corollary 3.0.5, there exists a morphism η ∈ N such that η(fz) ∈ D is fully N - centralized and η(CS(fz)) 6 CS(η(fz)). Since CS(fz) = hx, fi × P is of 2-rank 4 we a a have that η(fz) ∈ J ∩ D = J0hff i. Suppose η(fz) ∈ J0ff . By Lemma 5.4.18(h) ∼ 5 and Lemma 5.4.23(h) then, CS(η(fz)) = (C2 × C2) o C2 is of order 2 , forcing |P | = 8

93 contradiction. We conclude that foc(N ) = D is of index 2 in S. As S/foc(N ) is cyclic this shows that hyp(N ) = foc(N ) = D is of index 2 as well by Lemma 2.5.2(c).

Let M = O2(N ), a saturated fusion system on D. Set M+ = M/hxzi, and let

τ : M → M+ denote the surjective morphism of fusion systems. Thus M+ is a

+ + + saturated fusion system on D = D/Z(S), a product D1 × D2 of dihedral groups

k + 2 + + each of order 2 , and with M = O (M ) by Lemma 2.5.8. Since k > 3, Aut(D ) is a 2-group by Lemma 2.1.4, and so it follows that M+ = O20 (M+) as well, by

Corollary 2.5.7. The hypotheses of Theorem 2.5.9 are now satisﬁed and therefore + ∼ + + + M = M1 × M2 , by that theorem, for some pair Mi of saturated fusion systems

+ + 2 + on Di . Note then that the Mi = O (Mi ) are determined as the unique perfect

+ + 2-fusion system on the dihedral group D (Lemmas 2.5.8 and 3.0.7), i.e. as F + (M ) i Di i + ∼ with Mi = L2(q1).

+ Let Mi be the full preimage of Mi under τ; Mi is a saturated fusion system on

2 + Di. Then Mi = O (Mi) since Z(S) = Z(Mi) 6 [Di,Di], and Mi/Z(Mi) = Mi for each i = 1, 2. As there are no perfect fusion systems on a semidihedral group with nontrivial center by Lemma 3.0.7(b), each of the Di is quaternion, and hence

Mi is the 2-fusion system of SL2(q1) by (c) of the same lemma. Furthermore as f interchanges D1 and D2, f interchanges M1 and M2. In particular, we conclude that D is a commuting product of quaternion groups ∼ of the same order, and S = Q2k+1 o∗ C2. Thus S is not isomorphic to SD2k+1 o∗ C2 as the latter has an involution with centralizer isomorphic to C2 × SD2k+1 , whereas the former does not. By Lemma 5.4.18(i) and Lemma 5.4.23(i), h is an involution.

We now extract fusion information from the description of the centralizer of the central involution in Lemma 5.4.24 to show

Lemma 5.4.25. h = 1.

94 Proof. Suppose h 6= 1. Then the structure of S is that of Lemma 5.4.18 and N =

+ CF (xz) is given by Lemma 5.4.24. Let N = N /hxzi as before, and denote passage to the quotient by pluses. Recall S = Di o∗ C2, with f a wreathing element and with each Di quaternion and given as in Lemma 5.4.18(h). We claim

a every element of J0ff is N -conjugate to x, (5.4.26) and once shown, this contradicts Lemma 5.4.13(a).

+ + a To see (5.4.26), note that Z(S ) = hx i, and the image of each element of J0ff =

f f + + J0a a = J0(xa) (xa) in S is an involution which is not contained in either of the Di factors of the base subgroup of S+. Thus, by the structure of N +, each such element has image in S+ which is N + conjugate to x+. Pulling back over the surjective

+ a morphism N → N , it follows that each element of J0ff is N -conjugate into Z(T ) = hx, zi, and hence N -conjugate to x. This ﬁnishes the proof of (5.4.26) and the lemma.

Lemma 5.4.27. F is the fusion system of L4(q) for some q ≡ 3 (mod 4) with ν2(q + 1) = k − 1.

Proof. By Lemma 5.4.25, S = Jhai is isomorphic to D2k o C2. By [Olia, Proposi- tion 4.4], either k = 3 and F is the fusion system of A10, or F is the fusion system of L4(q) for q ≡ 3 (mod 4) with ν2(q + 1) = k − 1. In the case of A10, x is a product of two transpositions with centralizer having a unique component K isomorphic to ∼ the fusion system of A6 = L2(9). But then Q = CT (K) is a fours group, contrary to hypothesis.

Lemma 5.4.27 completes the identiﬁcation of F and the proof of Theorem 5.4.1.

95 5.5 Proof of Theorem A

The results of Sections 5.1-5.4 now give Theorem A, which we restate and prove here for completeness.

Theorem A. Let F be a saturated fusion system on the 2-group S with O2(F) = F and O2(F) = 1. Let x ∈ S be a fully F-centralized involution and set C = CF (x),

T = CS(x), and K = E(C). Assume

(1) K is a fusion system on a dihedral group of order 2k,

(2) Q := CT (K) is cyclic, and

(3) Baum(S) 6 T .

∼ ∼ ∼ Then S = D2k o C2, and F = FS(G) where G = L4(q) for some q ≡ 3 (mod 4) with

ν2(q + 1) = k − 1.

Proof. From Lemma 4.2.6, S is of 2-rank 3 or 4. Proposition 5.1.18 says that x does not lie in the center of S, while Theorems 5.2.1 and 5.3.1 show that S is of 2-rank 4 and Q is of order at least 4. Finally, Theorem 5.4.1 identiﬁes F and completes the proof of the theorem.

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