FUSION SYSTEMS WITH SOME SPORADIC J-COMPONENTS

JUSTIN LYND AND JULIANNE RAINBOLT

Abstract. Aschbacher’s program for the classification of simple fusion systems of “odd” type at the prime 2 has two main stages: the classification of 2-fusion systems of subintrinsic component type and the classification of 2-fusion systems of J-component type. We make a contribution to the latter stage by classifying 2-fusion systems with a J-component isomorphic to the 2-fusion systems of several sporadic groups under the assumption that the centralizer of such a component is cyclic.

1. Introduction

The Dichotomy Theorem for saturated fusion systems [AKO11, II 14.3] partitions the class of saturated 2-fusion systems into the fusion systems of characteristic 2-type and the fusion systems of component type. This is a much cleaner statement than the corresponding statement for finite (simple) groups, and it has a much shorter proof. In the last few years, M. Aschbacher has set out on a program to give a classification of a large subclass of the 2-fusion systems of component type. A lengthy memoir setting down the outline and first steps of such a program is forthcoming [Asc16], but see [Asc15] for a short survey of some of its contents. The immediate goal is to give a simpler proof of roughly half of the classification of the finite simple groups by carrying out most of the work in the category of saturated 2-fusion systems.

Let F be a saturated fusion system over a finite 2-group S, of which the standard example is the fusion system F_{S}(G), where G is a finite group and S is a Sylow 2-subgroup of G. A component is a subnormal, quasisimple subsystem. The system is said to be of component type if some involution centralizer in F has a component. The 2-fusion systems of odd type consist of those of subintrinsic component type or of J-component type. This is a proper subclass of the 2-fusion systems of component type. In focusing attention on this restricted class, one is expected to avoid several difficulties in the treatment of standard form problems like the ones considered in this paper. By carrying out the work in fusion systems, it is expected that certain core difficulties within the classification of simple groups of component type can be avoided, including the necessity of proving Thompson’s B-conjecture.

For the arguments in this paper, we do need to know much about the class of fusion systems of subintrinsic component type. But we do need to know that it contains those fusion systems having a quasisimple subsystem K that is a component in the centralizer of some involution in Z(K). Such a component is called intrinsic. More importantly here, F is said to be of J-component type if it is not of subintrinsic component type, and there is an involution t ∈ S such that the 2-rank of C_{S}(t) is equal to the 2-rank of S, and C_{F}(t) has a component. Such a component in an involution centralizer is called a J-component. These are rough definitions; more detail is given in Section 2.

Date: May 15, 2016.

The research of the first author is partially supported by NSA Young Investigator Grant H98230-14-1-0312 and was supported by an AMS-Simons grant which allowed for travel related to this work.

In this paper, we classify saturated 2-fusion systems having a J-component isomorphic to the 2-fusion system of M_{23}, J_{3}, McL, or Ly under the assumption that the centralizer of the component is a cyclic 2-group. A similar problem for the fusion system of L_{2}(q), q ≡ ±1 (mod 8) was treated in [Lyn15] under stronger hypotheses.

Theorem 1.1. Let F be a saturated fusion system over the finite 2-group S. Suppose that x ∈ S is a fully centralized involution such that F^{∗}(C_{F}(x)) ≅ Q × K, where K is the 2-fusion system of M_{23}, J_{3}, McL, or Ly, and where Q is a cyclic 2-group. Assume further that m(C_{S}(x)) = m(S). Then K is a component of F.

Here F^{∗}(C_{F}(x)) is the generalized Fitting subsystem of the centralizer C_{F}(x) [Asc11], and m(S) := m_{2}(S) is the 2-rank of S – that is, the largest rank of an elementary abelian 2-subgroup of S. We mention that any fusion system having an involution centralizer with a component isomorphic with McL or Ly is necessarily of subintrinsic component type. This means that, when restricted to those components, Theorem 1.1 gives a result weaker than is needed to fit into the subintrinsic type portion of Aschbacher’s program. However, our arguments apply equally well in each of the four cases and, throughout the paper, do not depend on any assumption about the class of known simple 2-fusion systems. So we have included McL and Ly here for completeness, and in case others seeking applications do not desire to have such an assumption.

There is no almost simple group with an involution centralizer having any of these simple groups as a component, but the wreath product G = (K_{1} × K_{2})⟨x⟩ with K_{1}^{x} = K_{2} always has C_{G}(x) = ⟨x⟩ × K with K a component that is diagonally embedded in K_{1} × K_{2}. The strategy for the proof of Theorem 1.1 is to locate a suitable elementary abelian subgroup F in a Sylow 2-subgroup of K, and then to show that the normalizer in S of E := ⟨x⟩F has at least twice the rank as that of F. Thus, the aim is to force a resemblance with the wreath product case, where N_{G}(⟨x⟩F) = (N_{K_{1}}(F_{1}) × N_{K_{2}}(F_{2}))⟨x⟩ is a wreathed subgroup of G (and here F_{i} is the projection of F onto the given factor). Lemma 4.3 is important for getting ahold on the extension of E determined by N_{F}(E) in order to carry out this argument.

We also give, under slightly stronger hypotheses, a classification of 2-fusion systems with J-components of type J_{2} that is still sufficient for applications.

Theorem 1.2. Let F be a saturated fusion system of J-component type over the finite 2group S. Suppose that x ∈ S is a fully centralized involution such that F^{∗}(C_{F}(x)) ≅ Q × K, where K is the 2-fusion system of J_{2} and where Q is a cyclic 2-group. Assume further that m(C_{S}(x)) = m(S). Then K is a component of F.

The automorphism group G = Aut(Suz) of Suzuki’s sporadic simple group has an involution t such that C_{G}(t) ≅ ⟨t⟩ × Aut(J_{2}), and indeed this is the only almost simple group having an involution centralizer with a component isomorphic to J_{2}. Since m_{2}(C_{G}(t)) = 5 while m_{2}(G) = 6, the fusion system of J_{2} is not a J-component in this situation.

In his treatment of this standard form problem in the group case [Fin76b], L. Finkelstein made frequent use of 3-local information in the group G, and in particular information about the centralizer in G of a 3-central element in the J_{2} component. The prospect of switching primes is not available in the fusion system setting. We are able to side-step 3-local arguments in the case where the fusion system of Aut(J_{2}) is involved in C_{F}(x), i.e. the case in which Aut(Suz) is a “shadow”. However, our track in the complementary case, where only the fusion system of J_{2} is involved, requires the “global” information that the centralizer of a 2-central involution (different from x) does not have a component containing that involution in its center. This is ensured by the assumption in Theorem 1.2 that F is not of subintrinsic component type.

We make use of transfer arguments in the proof of Theorem 1.2, again away from the shadow of Aut(Suz), in the form of a lemma whose proof in the group case is attributed to R. Lyons. Lyons’ Lemma says that if an involution lies in the commutator subgroup of a group G, then it also lies in the commutator subgroup of the centralizer of a 2-central involution, under an assumption that is often satisfied in practice. A version of Lyons’ Lemma for fusion systems, applicable at any prime, is proved in Section 3, and we believe this to be of independent interest. The proof is based on the Thompson-Lyons transfer lemma for fusion systems [Lyn14].

Finally we mention that the strategy outlined for the proof of Theorem 1.1 does not apply for Theorem 1.2, since it requires that m(F) ≥ 3 and that the action of Aut_{K}(F) be transitive on the nonzero elements of F (and there is no such F in the J_{2} situation). Instead, attention focuses in Section 8 on a subgroup E ≅ D_{8} ∗ Q_{8} of the Sylow subgroup of K, where Out_{K}(E) ≅ A_{5} acts irreducibly on E/Z(E). The aim is to locate a dihedral subgroup D of S of order 8 commuting with E which contains x and has Z(D) = Z(E). Then with U = D ∗ E = D_{8} ∗ D_{8} ∗ Q_{8}, the aim is to determine the structure of Out_{F}(U) and then the extension of U by Out_{F}(U). Ultimately, the extension is determined enough to show that m(S) ≥ 6 (while m(C_{S}(x)) = 5), giving a contradiction. It may be instructive to compare this to the case of Aut(Suz), where the centralizer of E in an appropriate Sylow 2-subgroup is a dihedral group D_{1} of order 16. One maximal dihedral subgroup of D_{1} is contained in the simple group Suz, while the other (D) is not, and we focus on the latter.

Acknowledgements. We would like to thank the Department of Mathematics and Computer Science at Saint Louis University and the Departments of Mathematics at Rutgers University and Ohio State University for their hospitality and support during mutual visits of the authors. We would also like to thank R. Solomon and R. Lyons for helpful discussions.

2. Background on fusion systems

We assume some familiarity with notions regarding saturated fusion systems, as can be found in [AKO11] or [Cra11], although some items are recalled below. Most notation is standard. Whenever G is a group, we write G^{#} for the set of nonidentity elements of G. For g ∈ G, denote by c_{g} the conjugation homomorphism c_{g}: x ↦ x^{g} and its restrictions. Morphisms in fusion systems are written on the right and in the exponent. That is, we write x^{φ} (or P^{φ}) for the image of an element x (or subgroup P) of S under a morphism φ in a fusion system, by analogy with the more standard exponential notation for conjugation in a group.

2.1. Terminology and basic properties. Throughout this section, fix a saturated fusion system over the p-group S. For a subgroup P ≤ S, we write Aut_{F}(P) for Hom_{F}(P, P), and Out_{F}(P) for Aut_{F}(P)/ Inn(P). Whenever two subgroups or elements of S are isomorphic in F, we say that they are F-conjugate. Write P^{F} for the set of F-conjugates of P. If E is a subsystem of F on the subgroup T ≤ S and α: T → S is a morphism in F, the conjugate of E by α is the subsystem E^{α} over T^{α} with morphisms φ^{α} for φ a morphism in E.

We first recall some of the terminology for subgroups and common subsystems in a fusion system.

Definition 2.1. Fix a saturated fusion system over the p-group S, and let P ≤ S. Then • P is fully F-centralized if |C_{S}(P)| ≥ |C_{S}(Q)| for all Q ∈ P^{F},

• P is fully F-normalized if |N_{S}(P)| ≥ |N_{S}(Q)| for all Q ∈ P^{F},

• P is F-centric if C_{S}(Q) ≤ Q for all Q ∈ P^{F},

• P is F-radical if O_{p}(Out_{F}(P)) = 1,

• P is weakly F-closed if P^{F} = {P },

• P is strongly F-closed if each F-conjugate of a subgroup of P is contained in P,

• the centralizer of P in F is the fusion system C_{F}(P) over C_{S}(P) with morphisms those φ ∈ Hom_{F}(Q, R) such that there is an extension φ̃ ∈ Hom_{F}(P Q, P R) that restricts to the identity on P,

• the normalizer of P in F is the fusion system N_{F}(P) over N_{S}(P) with morphisms those φ ∈ Hom_{F}(Q, R) such that there is an extension φ̃: P Q → P R in F such that P φ̃ = P.

We write F^{f}, F^{c}, and F^{r} for the collections of fully F-normalized, F-centric, and F-radical subgroups of S, respectively.

Whenever P ≤ S, we write 𝔄(P) for the set of α ∈ Hom_{F}(N_{S}(P), S) such that P^{α} is fully F-normalized.

Lemma 2.2. For each P ≤ S, 𝔄(P) is not empty. Moreover, for each Q ∈ P^{F} ∩ F^{f}, there is α ∈ 𝔄(P) with P^{α} = Q.

Proof. This is [BLO03, A.2(b)], applied with K = Aut(P). □

By a result of Puig, the centralizer C_{F}(P) is saturated if and only if P is fully F-centralized, and the normalizer N_{F}(P) is saturated if and only if P is fully F-normalized. We write O_{p}(F) for the (unique) largest subgroup P of S satisfying N_{F}(P) = F, and Z(F) for the (unique) largest subgroup P of S satisfying C_{F}(P) = F.

2.2. The Model Theorem. A subgroup P ≤ S is F-centric if and only if C_{S}(P) ≤ P and P is fully F-centralized. If P is F-centric and fully F-normalized, then the normalizer fusion system M := N_{F}(P) is constrained – that is, O_{p}(M) is M-centric. By the Model Theorem [AKO11, Proposition 5.10], there is then a unique finite group M with Sylow p-subgroup N_{S}(P) such that O_{p}(M) = O_{p}(M) and F_{N_{S}(P)}(M) ≅ M.

2.3. The focal and hyperfocal subgroups. Whenever x ∈ S and φ ∈ Hom_{F}(⟨x⟩, S), write [x, φ] = x^{−1}x^{φ} ∈ S. The focal subgroup of F is

𝔣𝔬𝔠(F) = ⟨[x, φ] | x ∈ S, φ ∈ Hom_{F}(⟨x⟩, S)⟩ ≤ S.

By the Focal Subgroup Theorem, 𝔣𝔬𝔠(F) = S ∩ [G, G] whenever G is a finite group and F = F_{S}(G).

The hyperfocal subgroup is

𝔥𝔶𝔭(F) = ⟨[x, φ] | P ≤ S, x ∈ P, φ ∈ O^{p}(Aut_{F}(P))⟩ ≤ S.

By the Hyperfocal Subgroup Theorem of Puig, 𝔥𝔶𝔭(F) = S ∩ O^{p}(G) whenever G is a finite group and F = F_{S}(G).

There is a unique normal subsystem of F over each of these subgroups, which in the case of 𝔥𝔶𝔭(F) is called the hyperfocal subsystem and is denoted by O^{p}(F). In addition, the normal subsystems of F containing O^{p}(F) are in one-to-one correspondence with the normal subgroups of S containing 𝔥𝔶𝔭(F).

2.4. Fusion systems of component type. We next discuss briefly Aschbacher’s partition of the class of odd fusion systems, as can be found in the memoir [Asc16]. See also [Asc13] and [Asc15] for expository accounts. Take p = 2 for this subsection.

A fusion system K is quasisimple if K = O^{2}(K) and K/Z(K) is simple. A component is a quasisimple subnormal subsystem. The layer E(F) is the commuting product of the components of F, and the generalized Fitting subsystem is a commuting product F^{∗}(F) = O_{p}(F)E(F) [Asc11].

Currently, the class 𝒦 of “known” quasisimple 2-fusion systems consists of the 2-fusion systems of quasisimple groups (aside from a few whose fusion systems are solvable) and the Benson-Solomon fusion systems. Write 𝒦_{int} for the class of intrinsic members of 𝒦 – that is, those K ∈ 𝒦 such that Z(K) ≠ 1. For example, the 2-fusion systems of 2A_{n} are in 𝒦_{int} for n ≥ 6. Roughly, a member K ∈ 𝒦 is subintrinsic if some involution in K has centralizer with a component containing that involution in its center. Write 𝒦_{sub} ⊆ 𝒦 for the class of subintrinsic members. Each intrinsic member is subintrinsic, but, for example, the 2fusion systems of McL and Ly are also in 𝒦_{sub} as they have involutions with centralizers the 2-fusion systems of 2A_{8} and 2A_{11}, respectively.

For a saturated 2-fusion system F, by ℭ(F) it is meant the set of conjugates in F of components of centralizers of fully F-centralized involutions of S.

Definition 2.3. F is of subintrinsic component type if some member of ℭ(F) is in 𝒦_{sub}.

By a J-component, we mean some conjugate of a component of a centralizer of a fully F-centralized involution x ∈ S such that m(C_{S}(x)) = m(S). Write ℭ_{J}(F) for the collection of J-components of F.

Definition 2.4. F is of J-component type if F is not of subintrinsic component type and ℭ_{J}(F) is not empty.

These definitions are needed only for the proof of Lemma 8.7.

2.5. Tame fusion systems. The main hypothesis of Theorem 1.1 and Theorem 1.2 is that the generalized Fitting subsystem of the involution centralizer C is the fusion system of a finite group Q × K, where Q is (the fusion system of) a cyclic 2-group, and K is simple. In this situation, C_{F}(x) is itself the fusion system of a finite group C with F^{∗}(C) = Q×K, with K ≅ M_{23}, McL, J_{2}, J_{3}, or Ly, since each of these simple groups tamely realizes its 2-fusion system [AOV12, Oli16c]. Roughly, a finite group tamely realizes its fusion system if every automorphism of its fusion system is induced by an automorphism of the group. Moreover, a fusion system is said to be tame if there is some finite group that tamely realizes it. We refer to [AOV12] for more details.

The importance of tameness in the context of standard form problems was pointed out in [Lyn15, §§1.5]. The discussion there is centered around the notion of strong tameness, which was needed for proofs of the results of [AOV12], but the contents of [Oli13, GL16] imply that a fusion system is tame if and only if it is strongly tame. Recently, Oliver has established the following useful corollary of the results in [AOV12], which we state for our setup here.

Theorem 2.5 ([Oli16b, Corollary 2.5]). Let C be a saturated fusion system over a 2-group. Assume that F^{∗}(C) = O_{2}(C)K, where K is simple and tamely realized by a finite simple group K. Then C is tamely realized by a finite group C such that F^{∗}(C) = O_{2}(C)K.

The effect of Theorem 2.5 for our purposes is that we may work in the group C, where Q is a normal subgroup. In particular, in the setup of the main theorems, C/Q is isomorphic to a subgroup of Aut(K) containing Inn(K), where K is one of the simple groups appearing in Theorems 1.1, 1.2.

3. Lyons’ Lemma for fusion systems

In this section, we establish a transfer result that we believe will be useful in the study of simple fusion systems of component type. The proof in the group case is attributed to Lyons and carries over to the fusion system setting [GLS02, Ch.2, Lemma 6.1]. The proof here uses the following special case of the Thompson-Lyons transfer lemma for fusion systems [Lyn14].

Lemma 3.1. Let F be a saturated fusion system over the finite p-group S. Let S_{1} be a proper normal subgroup of S such that S/S_{1} is cyclic, and let x be an element of least order in S − S_{1}. Assume that each F-conjugate of x in S − S_{1} lies in the coset S_{1}x. If x ∈ 𝔣𝔬𝔠(F), then x has a fully F-centralized conjugate in S_{1}.

Lemma 3.2 (Lyons’ Lemma). Let F be a saturated fusion system over the finite p-group S. Fix a fully F-centralized element x ∈ S of order p, let T = C_{S}(x), and let t ∈ Z(S). Assume that

t^{F} ∩ Z(T) = t^{C_{F} (x)} ∩ Z(T). (3.3)

If x ∈ 𝔣𝔬𝔠(F), then x ∈ 𝔣𝔬𝔠(C_{F}(t)).

Proof. Set S_{0} = 𝔣𝔬𝔠(C_{F}(t)) for short and assume by way of contradiction that x ∉ S_{0}. Since t ∈ Z(S), we have S_{0} ≥ [S, S]. Choose S_{1} ≤ S containing S_{0} such that x ∉ S_{1} and S/S_{1} is cyclic. By Lemma 3.1, there is a morphism φ ∈ Hom_{F}(T, S) such that x^{φ} is fully F-centralized, x^{φ} ∈ S_{1}, and T^{φ} ≤ C_{S}(x^{φ}). In fact

T^{φ} = C_{S}(x^{φ}) (3.4)

since x is itself fully centralized. Since t ∈ Z(S) by assumption, we see from (3.4) that t ∈ Z(T^{φ}), and so t^{φ^{−1}} ∈ Z(T). By assumption (3.3), there is a morphism ψ ∈ Hom_{C_{F} (x)}(⟨x, t^{φ^{−1}}⟩, T) such that t^{φ^{−1}ψ} = t. Consequently,

ψ^{−1}φ ∈ C_{F}(t) and x^{φ} = x^{ψ^{−1}φ},

so that [x, φ] = [x, ψ^{−1}φ] ∈ S_{0} ≤ S_{1}. We conclude that x ≡ x^{φ} (mod S_{1}), which is the desired contradiction as x ∉ S_{1} but x^{φ} ∈ S_{1}. □

4. Structure of the components

In this section, we recall some properties of the simple systems appearing in Theorem 1.1 that are required for the remainder.

Lemma 4.1. Let E ≅ D_{8} ∗ Q_{8} and A = Aut(E). Then (a) A ≅ E_{16}S_{5} is a split extension of Inn(E) by O_{4}^{−}(2) ≅ S_{5};

(b) Any two subgroups of A isomorphic to A_{5} are conjugate in A.

(c) If B is a subgroup of A isomorphic to A_{5}, then the isomorphism type of EB is uniquely determined, and each subgroup of EB isomorphic to A_{5} is conjugate to B.

Proof. See [GH74, II, Lemma 2.7]. □

Lemma 4.2. Let G be A_{7} or GL_{2}(4), and V a faithful F_{2}[G]-module of dimension 4. Then (a) G acts transitively on the nonzero vectors of V,

(b) C_{GL(V)}(G) ≤ G,

(c) H^{1}(G, V) = 0, and

(d) if G acts on a homocyclic 2-group Y with Ω_{1}(Y) = V, then Y = V.

Proof. In each case, V is uniquely determined up to isomorphism. Point (a) is well-known. Point (b) holds for G = A_{7} by absolute irreducibility. Similarly for G = GL_{2}(4), one has that C_{GL(V)}(G) = End_{F_{2}[G]}(V)^{×} = F_{4}^{×}, and so (b) follows in this case as Z(G) ≅ C_{3}. Point (c) for G = A_{7} holds, for example, by [AG72]. Point (c) for G = GL_{2}(4) holds because, by coprime action, Z(G) (and so G) has a fixed point on any 5-dimensional module containing V as a submodule.

We now turn to (d), which follows from a special case of a result of G. Higman [Hig68, Theorem 8.2]. This says that if SL_{2}(4) acts faithfully on a homocyclic 2-group Y in which an element of order 3 acts regularly on Y, then Y is elementary abelian. In case G = GL_{2}(4), V is the natural module for G and certainly SL_{2}(4) ≤ G has (in an appropriate basis) a diagonal element of order 3 acting regularly. In case G = A_{7}, we have G ≤ A_{8} ≅ GL_{4}(2), and the action of G on V is the restriction of the natural action of GL_{4}(2) on V. Restriction of V to GL_{2}(4) ≅ C_{3} × SL_{2}(4) ≅ C_{3} × A_{5} shows that SL_{2}(4) is embedded in G as an A_{5} moving 5 points in the natural permutation action. So, as before, SL_{2}(4) ≤ G has an element of order 3 acting regularly on V. Hence, (d) holds in this case as well by Higman’s Theorem. □

For a vector space E over the field with two elements, the next lemma examines under rather strong hypotheses the structure of extensions of E by certain subgroups of the stabilizer in GL(E) of a hyperplane.

Lemma 4.3. Let E ≅ E_{2^{n+1}} (n ≥ 3), A = Aut(E), V ≤ E with V ≅ E_{2^{n}}, P = N_{A}(V), and U = O_{2}(P). Let L be a complement to U in P acting decomposably on E, x ∈ E − V the fixed point for the action of L, and G ≤ L. Let H be an extension of E by UG with the given action and let X be the preimage in H of U under the quotient map H → UG. Assume that

(a) G acts transitively on V^{#};

(b) C_{GL(V)}(G) ≤ G; and

(c) H^{1}(G, V) = 0. Then there is a subgroup Y of X that is elementary abelian or homocyclic of order 2^{2n} and a complement to ⟨x⟩ in X.

Proof. Let X̄ = X/V. Since the commutator map

(4.4) [x, −] determines a G-equivariant linear isomorphism X/E → V,

G is transitive on the nonzero vectors of X/E by (a), and ⟨x̄⟩ ≤ Z(X̄). Hence if X̄ is not elementary abelian, then it is extraspecial with center ⟨x̄⟩, and G preserves the squaring map X̄ → Z(X̄). This is not the case, because G is transitive on the nonzero vectors of X/E and n ≥ 3. Therefore,

(4.5)  X̄ is elementary abelian.

Assumption (c) now yields that there is a G-invariant complement Ȳ to ⟨x̄⟩. Let Y be the preimage of Ȳ in X. We claim that Y is abelian; assume on the contrary. Then [Y, Y] and ℧^{1}(Y) are contained in V since Ȳ is elementary abelian, and by assumption, neither of these are trivial. Similarly, V is contained in Z(Y), which is not Y. Therefore,

V = [Y, Y] = Φ(Y) = ℧^{1}(Y) = Z(Y), (4.6)

by (a) and (4.4).

By (4.6), the squaring map Ȳ → V is G-equivariant linear isomorphism; let \sqrt{−} be its inverse. Then the map ξ: V → V given by ξ(v) = [x, \sqrt{v}] is a linear isomorphism commuting with the action of G, and so ξ ∈ ρ(G) by (b), where ρ: G → GL(V) is the structure map. Let g ∈ G map to ξ^{−1} under ρ. Then y ↦ [x, y^{g}] is the squaring map. This means that for each y ∈ Y, we have y^{gx} = y^{2}y^{g}. Hence, for each pair w, y ∈ Y,

w^{2}y^{2}w^{g}y^{g} = w^{gx}y^{gx} = (wy)^{gx} = (wy)^{2}w^{g}y^{g}

which gives w^{2}y^{2} = (wy)^{2} = w^{2}y^{2}[y, w]. Thus Y is abelian after all. It follows that Ω_{1}(Y) = V or Y by (a), and this completes the proof of the lemma. □

We now examine the structure of the simple systems occupying the role of K in Theorems 1.1 and 1.2. Let T_{0} be a 2-group isomorphic to a Sylow 2-subgroup of L_{3}(4). This is generated by involutions t_{1}, t_{2}, a_{1}, a_{2}, b_{1}, and b_{2} such that Z(T_{0})) = ⟨t_{1}, t_{2}⟩ and with additional defining relations:

[a_{1}, a_{2}] = [b_{1}, b_{2}] = 1, [a_{1}, b_{1}] = [a_{2}, b_{2}] = t_{1}, [a_{2}, b_{1}] = t_{2}, [a_{1}, b_{2}] = t_{1}t_{2}.

A Sylow subgroup of M_{23} or McL is isomorphic to a Sylow 2-subgroup of an extension of L_{3}(4) by a field automorphism; this is a semidirect product T_{0}⟨f⟩ with

f^{2} = 1, [a_{1}, f] = [a_{2}, f] = a_{1}a_{2}, [b_{1}, f] = [b_{2}, f] = b_{1}b_{1}, [t_{2}, u] = t_{1}.

A Sylow subgroup of J_{2} or J_{3} is isomorphic to a Sylow 2-subgroup of L_{3}(4) by a unitary (i.e., graph-field) automorphism; this is a semidirect product T_{0}⟨u⟩ with

u^{2} = 1, [a_{1}, u] = a_{1}b_{1}, [a_{2}, u] = a_{2}b_{2}, [t_{2}, u] = t_{1}.

A Sylow subgroup of Ly is isomorphic to a Sylow 2-subgroup of Aut(L_{3}(4)); this is a semidirect product T_{0}⟨f, u⟩ with [f, u] = 1 and the relations above.

Denote by T_{1} a 2-group isomorphic to one of T_{0}⟨f⟩, T_{0}⟨u⟩, or T_{0}⟨f, u⟩.

Lemma 4.7. Let K be M_{23}, McL, J_{2} J_{3}, or Ly, with Sylow 2-subgroup T_{1} as above. Then (a) Z(T_{1}) = ⟨t_{1}⟩ is of order 2;

(b) A(T_{1}) = {F_{1}, F_{2}} where F_{1} = ⟨t_{1}, t_{2}, a_{1}, a_{2}⟩ and F_{2} = ⟨t_{1}, t_{2}, b_{1}, b_{2}⟩, and after suitable choice of notation, one of the following holds:

(i) K = M_{23}, McL, or Ly, and Aut_{K}(F_{1}) ≅ A_{7}, (ii) K = J_{3} and Aut_{K}(F_{1}) ≅ GL_{2}(4), or (iii) K = J_{2} and Aut_{K}(F_{1}) ≅ C_{3} × A_{4} is the stabilizer in GL_{2}(4) of the one-

dimensional F_{4}-subspace ⟨t_{1}, t_{2}⟩.

(c) one of the following holds:

(i) there is F ∈ A(T_{1}) such that the pair (Aut_{K}(F), F) satisfies assumptions (a)-(c) of Lemma 4.3 in the role of (G, V), or

(ii) K = J_{2}; and

(d) T_{0} = J(T_{1}) with Z(T_{0}) = ⟨t_{1}, t_{2}⟩, and all involutions in ⟨t_{1}, t_{2}⟩ are K-conjugate.

Proof. Point (a) holds by inspection of the relations above. For part (b), F_{1} and F_{2} are elementary abelian of rank 4. An involution in Aut(L_{3}(4)) − L_{3}(4) has centralizer in L_{3}(4) one of U_{3}(2) ≅ (C_{3} × C_{3})Q_{8}, L_{3}(2), or SO_{3}(2) ≅ A_{4}, which have 2-ranks 1, 2, and 2, respectively. It follows that J(T_{1}) ≤ T_{0}, and from the relations above, each involution in T_{0} is contained in one of F_{1} or F_{2}, so the first statement in (b) holds. The second part of (b) is well-known; see, for example, [Fin76a, Fin76b, Wil84]. However, we point out that in K = Ly, one has N_{K}(F) = F L with L the triple cover of A_{7} and F Z(L) = F × Z(L). Now point (c) follows from (b) and Lemma 4.2. The first part of (d) follows from (b), while the second part follows from (c)(i) and (b)(iii). □

Lemma 4.8. Let K be one of the sporadic groups M_{23}, McL, J_{2}, J_{3}, or Ly, and let T_{1} be a Sylow 2-subgroup of K. Then

(a) Out(K) = 1 if K ≅ M_{23} or Ly, and Out(K) ≅ C_{2} otherwise; and

(b) for each involution α ∈ Aut(K) − Inn(K), (i) C_{K}(α) ≅ M_{11} if K ≅ McL, (ii) C_{K}(α) ≅ P GL_{2}(7) if K ≅ J_{2}, and (iii) C_{K}(α) ≅ L_{2}(7) if K ≅ J_{3}; and

(c) each automorphism of K centralizing a member of A(T_{1}) is inner.

Proof. Points (a) and (b) follow by inspection of Table 5.3 of [GLS98]. By Lemma 4.7(b), the 2-rank of K is 4, while each of the centralizers in (b)(i-iii) is of 2-rank 2, so (c) holds. □

Lemma 4.9. Let K = J_{2} with Sylow 2-subgroup T_{1} as above, and let

E = ⟨t_{2}, u, a_{1}b_{1}t_{2}, a_{2}b_{2}t_{2}⟩ ≅ D_{8} ∗ Q_{8}.

Then (a) N_{K}(E) = C_{K}(t_{1}) ≅ EA_{5}, and N_{Aut(K)}(E) = C_{Aut(K)}(t_{1}) ≅ ES_{5};

(b) each involution of E is K-conjugate to t_{1}, and Ω_{1}(E) = E; and

(c) E is the unique subgroup of T_{1} isomorphic to D_{8} ∗ Q_{8}.

Proof. We refer to Table 5.3g of [GLS98] for part (a). K has two classes of involutions. An involution of T_{1} not K-conjugate to t_{1} has centralizer C_{2}×C_{2}×A_{5} in K, and so has centralizer an elementary abelian group in T_{1}. As each involution in E centralizes a quaternion group, the first part of (b) holds, and the second part is clear. By (a) and Lemma 4.1(c), all involutions of T_{1} − E are N_{K}(E)-conjugate, so (c) follows from (b). □

5. Preliminary lemmas

We now begin in this section the proofs of the main theorems, and so we fix the notation and hypotheses that will hold throughout the remainder of the paper.

Let F be a saturated fusion system over the 2-group S, and let x ∈ S be a fully Fcentralized involution with m(C_{S}(x)) = m(S) and with F^{∗}(C_{F}(x)) = Q × K, where Q is cyclic. Set C = C_{F}(x) and T = C_{S}(x), so that C is saturated over T. Let T_{1} be a Sylow subgroup of K, and set

R = Q × T_{1} ≤ T.

Assume that K is the fusion system of one of the sporadic groups K = M_{23}, J_{2}, J_{3}, McL, or Ly. Since K tamely realizes K in each case, the quotient

(5.1) T/R induces a 2-group of outer automorphisms of K

by Theorem 2.5. Arguing by contradiction, we assume

K is not a component of F.

We fix the presentation in Section 4 for T_{1} in whichever case is applicable, and we note that

Ω_{1}(Q) = ⟨x⟩

by assumption.

Lemma 5.2. Notation may be chosen so that T and J(T) are fully F-normalized.

Proof. We repeatedly use Lemma 2.2. Let α ∈ 𝔄(T). Then as |C_{S}(x^{α})| ≥ |T^{α}| = |T | = |C_{S}(x)|, we have x^{α} is still fully F-centralized. Thus we may assume that T is fully Fnormalized after replacing x, T, and J(T) by their conjugates under α. Now let β ∈ 𝔄(J(T)). Then as N_{S}(T) = N_{N_{S}(J(T))}(T), it follows that

|N_{S}(T^{β})| = |N_{N_{S}(J(T)^{β})}(T^{β})| ≥ |N_{N_{S}(J(T))^{β}}(T^{β})| = |N_{N_{S}(J(T))}(T)^{β}| = |N_{S}(T)^{β}| = |N_{S}(T)|

and so equality holds because T is fully F-normalized. Hence T^{β} is still fully F-normalized. As before, x^{β} is still fully F-centralized. □

Lemma 5.3. ⟨x⟩ is not weakly F-closed in T.

Proof. Assume on the contrary, in which case T = S, otherwise N_{S}(T) contains T properly and moves x. It follows that x ∈ Z(S), which is contained in the center of every F-centric subgroup. Hence x ∈ Z(F) by Alperin’s fusion theorem [BLO03, Theorem A.10], and we conclude that K is a component of C = F, contrary to hypothesis. □

Lemma 5.4. If T = S, then Ω_{1}Z(S) = ⟨x, t_{1}⟩. If T < S, then Ω_{1}(Z(S)) = ⟨t_{1}⟩.

Proof. Note first that Ω_{1}(Z(S)) ≤ T. By Lemma 4.8(c) and (5.1), Ω_{1}(Z(S)) ≤ R, and so Ω_{1}Z(S) ≤ Ω_{1}(Z(R)) = ⟨x, t_{1}⟩ by Lemma 4.7(a). Thus, the lemma holds in case T = S. In case T < S, let a ∈ N_{S}(T) − T with a^{2} ∈ T. Then a normalizes Ω_{1}Z(T) = ⟨x, t_{1}⟩ and Ω_{1}Z(T) ∩ [J(T), J(T)] = ⟨x, t_{1}⟩ ∩ ⟨t_{1}, t_{2}⟩ = ⟨t_{1}⟩, but a does not centralize x. Thus, Ω_{1}Z(S) = ⟨t_{1}⟩ as claimed. □

Lemma 5.5. The following hold.

(a) J(T) = J(R) = ⟨x⟩ × J(T_{1}); and

(b) C_{T} (T_{1}) = Q⟨t_{1}⟩.

Proof. From Lemma 4.7(a), C_{R}(T_{1}) = Q⟨t_{1}⟩. Hence C_{T} (T_{1}) = C_{R}(T_{1}) again by Lemma 4.8(c) and (5.1).

Suppose (b) does not hold. Choose A ∈ A(T) with A ≰ ⟨x⟩ × J(T_{1}). Then A ≰ R by the structure of R, and A acts on K by Theorem 2.5. In particular, by Lemma 4.8, Out(K) = 2 and m_{2}(C_{R}(A)) ≤ 3. Hence m(A) ≤ 4 while m(R) = 5, which contradicts A ∈ A(T) and establishes (b). □

Lemma 5.6. The following hold.

(a) x is not F-conjugate to t_{1}; and

(b) x is conjugate to xt_{1} if and only if T < S.

Proof. For part (a), let φ ∈ F with x^{φ} ∈ Z(T). Assume first that T = S. By the extension axiom, φ extends to φ̃ ∈ Aut_{F}(T), which restricts to an automorphism of J(T). Lemma 5.5(b) shows that x ∉ [J(T), J(T)], while t_{1} ∈ [J(T), J(T)] from Lemma 4.7(d), a contradiction.

Now assume that T < S. Then x ∉ Z(S), whereas Z(S) = ⟨t_{1}⟩ by Lemma 5.4. Hence, x is not fully F-centralized, a contradiction that establishes (a).

If T < S, then x is N_{S}(T)-conjugate to xt_{1} by (a), while if T = S, then (a) and Burnside’s fusion theorem imply that x is weakly F-closed in Ω_{1}(Z(S)) = ⟨x, t_{1}⟩. □

6. The 2-central case

In this section it is shown that T < S; that is, x is not 2-central. We continue the notation set at the beginning of Section 5.

Lemma 6.1. Assume T = R. Then T < S. In particular, T < S in case K = M_{23}, J_{3}, or Ly.

Proof. Assume that T = S. Then Ω_{1}(Z(T)) = ⟨x, t_{1}⟩ by Lemma 5.4. By assumption, T_{1} ⊴ R = T = S and T/T_{1} is cyclic with x ∈ T − T_{1}. Now Lemma 3.1 yields that x is conjugate into Z(T_{1}) contradicting Lemma 5.6. □

Lemma 6.2. Assume K is the fusion system of McL. Then T < S.

Proof. Assume T = S. Then R < T by Lemma 6.1. By Burnside’s fusion theorem and Lemma 5.6(b), ⟨x⟩ is weakly closed in Ω_{1}(Z(T)). As K has one class of involutions, there are three C-classes of involutions x^{C} = {x}, (xt_{1})^{C}, and t_{1}^{C} in R. So

(6.3) ⟨x⟩ is weakly F-closed in R.

However, x is not weakly closed in T by Lemma 5.3. Fix f ∈ x^{F} ∩ (T − R). The extension K⟨f⟩ := KT_{1}⟨f⟩ of K is defined by [Asc11, §8], and K⟨f⟩/Q is the 2-fusion system of Aut(McL) by Theorem 2.5. Conjugating in K⟨f⟩ := KT_{1}⟨f⟩ if necessary, we may assume f is fully K⟨f⟩-centralized. By Lemma 4.8(b)(i), all involutions of T_{1}⟨f⟩ − T_{1} are K⟨f⟩- conjugate and C_{K⟨f⟩}(f) ≅ ⟨f⟩ × F_{2}(M_{11}). In particular, C_{T_{1}}(f) is semidihedral of order 16 with center ⟨t_{1}⟩. Fix a four subgroup V ≤ C_{T_{1}}(f). Then f is conjugate to ft_{1} and hence conjugate to each element of fV by the structure of F_{2}(M_{11}).

Let α ∈ F with f^{α} = x. Then α extends to ⟨f⟩V ≤ C_{T} (f), and hence x is F-conjugate to each element in xV^{α}. Since |T: R| = 2, this is a contradiction to (6.3) as V^{α} ∩ R ≠ 1 and x is not in V. □

Lemma 6.4. Assume K is the fusion system of J_{2}. Then T < S.

Proof. Suppose on the contrary that T = S. Recall that we have adopted the presentation of T_{1} from Section 4. From 5.6(b),

(6.5) x is weakly closed in Ω_{1}(Z(S)) = ⟨x, t_{1}⟩.

We claim that

(6.6)  ⟨x⟩ is weakly F-closed in R.

Assume to the contrary that a_{0} ∈ x^{F} ∩ R. As Ω_{1}(R) = ⟨x⟩T_{1}, we may assume a_{0} = x_{0}a where ⟨a⟩ ∈ K^{f} and x_{0} = 1 or x. By (6.5), a is not 2-central in K, so C_{T_{1}}(a) is elementary abelian of order 16. The two members of A(T) are conjugate under u ∈ T_{1}. Hence, by the extension axiom there is some φ ∈ Aut_{F}(E_{1}) with x^{φ} = a_{0}, where E_{1} := ⟨x⟩F_{1} ∈ A(T). There are five Aut_{C}(E_{1})-classes of nonidentity elements of E_{1} with representatives x, xt_{1}, t_{1}, xa_{1} and a_{1} and respective sizes 1, 3, 3, 12, and 12 by Lemma 4.7(b)(iii). We may assume a = a_{1} without loss, and then by (6.5), x has 13 or 25 conjugates under Aut_{F}(E_{1}). Neither of these numbers divides |GL_{5}(2)|, a contradiction that establishes (6.6).

As before, x ∉ Z(F), since otherwise K is a component of C = F. By Alperin’s fusion theorem, x has an F-conjugate f ∈ T − R. Conjugating in K⟨f⟩, we may assume that f is fully K⟨f⟩-centralized. Then C_{K⟨f⟩}(f) = ⟨f⟩ × F_{2}(P GL_{2}(7)) and so there is a four subgroup V ≤ C_{T_{1}}(f) such that f is F-conjugate to each member of fV. We now argue as in the proof of Lemma 6.2 to get the same contradiction as we did there. □

7. Proof of Theorem 1.1

Continue the notation and hypotheses set at the beginning of Section 5. In this section, we finish the proof of Theorem 1.1, and so we assume that

K is the fusion system of M_{23}, J_{3}, McL, or Ly.

Fix F ∈ A(T_{1}) satisfying assumptions (a)-(c) of Lemma 4.3, as guaranteed by Lemma 4.7(c)(i), and set

E = ⟨x⟩F ∈ A(T).

By Lemmas 6.1 and 6.2,

T < S.

Lemma 7.1. | Aut_{F}(T) : Aut_{C}(T)| = 2. Hence x is N_{S}(T)-conjugate to xt_{1}.

Proof. Since T < S, also T < N_{S}(T). Represent Aut_{F}(T) and N_{S}(T) on Ω_{1}Z(T) = ⟨x, t_{1}⟩ and apply Lemma 5.6. □

Lemma 7.2. The following hold.

(a) | Aut_{F}(J(T)): Aut_{C}(J(T))| = 4 and

(b) | Aut_{F}(E): Aut_{C}(E)| = 16.

Proof. Since Aut_{K}(F) is transitive on F^{#} and F ∈ A(T_{1}), Aut_{K}(J(T_{1})) is transitive on Ω_{1}ZJ(T_{1}). So x^{Aut_{F} (J(T))} = Ω_{1}ZJ(T_{1})x is of size 4 by Lemma 5.6(b). Thus (a) holds.

From Lemma 5.6(a), |A(T)| = 2. By (a) and Lemma 5.2, |N_{S}(J(T)) : T | = 4. Representing N_{S}(J(T)) on A(T), we see that N_{S}(J(T))/T has a nontrivial kernel on A(T). Thus, N_{T} (E) < N_{S}(E). Another appeal to Lemmas 7.1 and 5.6(b) yields x^{Aut_{F} (E)} = F x has size 16, which establishes (b). □

Lemma 7.3. The following hold:

(a) Q = ⟨x⟩.

(b) E = C_{S}(E).

Proof. Suppose on the contrary that Q > ⟨x⟩ and choose w ∈ Q with w^{2} = x. Fix a ∈ N_{S}(T) − T such that a^{2} ∈ T. Then x^{a} = xt_{1} and also (w^{a})^{2} = xt_{1}. Further ⟨w^{a}⟩ is normal in T since ⟨w⟩ is. Thus, [⟨w^{a}⟩, T_{1}] ≤ ⟨w^{a}⟩ ∩ T_{1} = 1, whereas C_{T} (T_{1}) = Q⟨t_{1}⟩ by Lemma 5.5(a). It follows that xt_{1} = ℧^{1}(⟨w^{a}⟩) ≤ ℧^{1}(Q⟨t_{1}⟩) = x, a contradiction that establishes (a).

Next, C_{S}(E) = C_{T} (E) as x ∈ E. By Lemma 4.8(c), C_{T} (E) = C_{R}(E) = QF by Lemma 4.8(c). Hence C_{S}(E) = QF and this is equal to E by part (a). □

Since we will be working in N_{F}(E) for the remainder of this section, we may assume, after replacing E by an F-conjugate if necessary, that E is fully F-normalized. Hence E ∈ F^{fc} by Lemma 7.3(b). Let H be a model for N_{F}(E) (cf. §§2.2).

Lemma 7.4. H satisfies the hypotheses of Lemma 4.3.

Proof. Set G = Aut_{C}(E) and observe that Aut_{F}(E) = H/E by Lemma 7.3(b). Now x is Aut_{F}(E)-conjugate to xt_{1} but not F-conjugate to t_{1}, so as G acts transitively on F^{#} and centralizes x, it follows that xF and F^{#} are the orbits of Aut_{F}(E) on E^{#}. Hence Aut_{F}(E) ≤ N_{Aut(E)}(F), the unique nontrivial split extension of an elementary abelian 2group U of order 16 by C_{GL(E)}(x).

We claim that U ≤ Aut_{F}(E). Suppose that this is not the case. Since G acts transitively on F^{#}, it acts transitively on U^{#}. So as U is normalized by Aut_{F}(E), we see that Aut_{F}(E)∩ U = 1. In particular, Aut_{F}(E) embeds into C_{GL(E)}(x) ≅ GL(F). Now G ≅ A_{7} or GL_{2}(4) in the cases under consideration, and by Lemma 7.2(b), Aut_{F}(E) is therefore a subgroup of GL_{4}(2) containing G with index 16. However, A_{7} has index 8 in GL_{4}(2), and GL_{2}(4) is contained with index 2 in a unique maximal subgroup of GL_{4}(2), a contradiction. Therefore, U ≤ Aut_{F}(E) as claimed.

It has thus been shown that Aut_{F}(E) is an extension of U = O_{2}(N_{Aut(E)}(F)) by G. Assumptions (1)-(3) of Lemma 4.3 hold by Lemma 4.7(c) and choice of F. □

Proof of Theorem 1.1. By Lemma 7.4 and Lemma 4.3, there is an Aut_{C}(E)-complement Y to ⟨x⟩ in O_{2}(H) that is homocyclic of order 2^{8} with Ω_{1}(Y) = F, or elementary abelian of order 2^{8}. Now Aut_{C}(E) ≅ A_{7} or GL_{2}(4) with faithful action on F, so the former case is impossible by Lemma 4.2. Hence m_{2}(T) = 5 < 8 ≤ m_{2}(S), contrary to hypothesis. □

8. Proof of Theorem 1.2

In this section, we finish the proof of Theorem 1.2, so we assume

K is the fusion system of J_{2}, and

(8.1) F is not of subintrinsic component type.

Recall the notation and hypotheses fixed at the beginning of Section 5.

By Lemma 5.6,

x^{F} ∩ Z(T) = {x, xt_{1}}. (8.2)

Let E = ⟨t_{2}, u, a_{1}b_{1}t_{2}, a_{2}b_{2}t_{2}⟩ denote the unique subgroup of T_{1} isomorphic to D_{8} ∗ Q_{8}, as described in Lemma 4.9.

Lemma 8.3. E is weakly F-closed in T, as is ⟨x⟩E.

Proof. This is essentially [Fin76b, Lemma 9.4], but the hypotheses are a little different. Let P ≤ T be an F-conjugate of E. Now x is not F-conjugate to t_{1}, while each involution of E is conjugate to t_{1} by Lemma 4.9(b). It follows that x ∉ P and consequently that Q ∩ P = 1 since Q is cyclic. Set T̄ = T/Q, so that P̄ = P and so that T̄ is isomorphic to a Sylow 2-subgroup of J_{2} or of Aut(J_{2}). The group P GL_{2}(7) has dihedral Sylow 2-subgroups, so by Lemma 4.8(b)(ii), no element of T̄ − T̄_{1} centralizes a quaternion subgroup of T̄_{1}. Since P̄ is generated by its involutions (each of which centralizes a quaternion group) we see that P̄ ≤ T̄_{1}. By Lemma 4.9(a), we conclude that P̄ = Ē and that P = Ω_{1}(P) ≤ Ω_{1}(QE) = ⟨x⟩E. By (8.2) and one more application of Lemma 4.9(b), we have P = E. Thus the first claim of the lemma holds. The second claim then follows as any morphism sending ⟨x⟩E back in T will send E to E, and will send x into Ω_{1}(C_{T} (E)) = ⟨x, t_{1}⟩. □

Lemma 8.4. The following hold.

(a) Q = ⟨x⟩.

(b) C_{S}(E) is dihedral or semidihedral.

Proof. By Lemma 8.3, N_{S}(T) ≤ N_{S}(E), and so C_{C_{S}(E)}(Ω_{1}Z(T)) = Q⟨t_{1}⟩ is invariant under N_{S}(T). If Q > ⟨x⟩, then ⟨x⟩ = Ω_{1}(℧^{1}(Q⟨t_{1}⟩)) is invariant under N_{S}(T), a contradiction to T < N_{S}(T). Thus (a) holds. Next, C_{C_{S}(E)}(x) = C_{T} (E) ≤ C_{R}(E) = Q⟨t_{1}⟩, with the inequality by Lemma 4.8(b)(ii) and the last equality by Lemma 4.9(c). We conclude by part (a) that ⟨x, t_{1}⟩ is a self-centralizing four subgroup of C_{S}(E), and (b) follows directly from this by a theorem of Suzuki (see [GLS96, Lemmas 10.24,10.25]). □

Lemma 8.5. Notation may be chosen so that ⟨x⟩, T, E, and ⟨x⟩E are each fully Fnormalized.

Proof. Notation has been chosen so that x and T are fully normalized (Lemma 5.2). By Lemma 8.3, ⟨x⟩E is normal in N_{S}(T), and so

N_{S}(T) ≤ N_{S}(⟨x⟩E) ≤ N_{S}(⟨x, t_{1}⟩) = N_{S}(T).

Equality holds here and thus

C_{S}(x) = T ≤ N_{S}(⟨x⟩E) = N_{S}(T) ≤ N_{S}(E) (8.6)

Choose β ∈ 𝔄(⟨x⟩E) and γ ∈ 𝔄(E^{β}). Arguments similar to those in the proof of Lemma 5.2 then show with (8.6) that all the βγ-conjugates of the named subgroups are fully F-normalized. □

We now focus attention on D_{1} := C_{S}(E). Recall from Lemma 8.4(b) that D_{1} is dihedral or semidihedral, and our first objective is to show that D_{1} is nonabelian. Thus we must rule out the case where D_{1} is a four group, namely when D_{1} = C_{T} (E) = ⟨x, t_{1}⟩.

Lemma 8.7. |D_{1}| ≥ 8.

Proof. Assume on the contrary that |D_{1}| = 4, so that D_{1} = C_{T} (E) = ⟨x, t_{1}⟩. Then ⟨x⟩E contains its centralizer in S and is fully F-normalized by Lemma 8.5, so

(8.8) ⟨x⟩E is F-centric.

A model H for N_{F}(⟨x⟩E) has the structure (⟨x⟩ × E)B⟨s⟩, in which C_{H}(x) = (⟨x⟩ × E)B and s ∈ N_{S}(T) − T. It follows that B ≅ A_{5}, since otherwise C_{H}(x) induces a full S_{5} on E, and then some element of the coset C_{T} (E)s centralizes E, contrary to assumption. Hence H/⟨x⟩E ≅ S_{5}.

In particular, it follows from the structure of H that every strongly N_{F}(⟨x⟩E)-closed subgroup of N_{S}(⟨x⟩E) contains t_{1}, and that

x ∉ 𝔣𝔬𝔠(N_{F}(⟨x⟩E)). (8.9)

Set M = C_{F}(t_{1}). Then N_{F}(⟨x⟩E) is a subsystem of M. We claim that M is constrained. Suppose on the contrary that N := E(M) ≠ 1, and let N be the corresponding Sylow group. Since F is of J-component type, t_{1} ∉ Z(N). Hence, as N ⊆ C_{F}(t_{1}),

t_{1} ∉ N. (8.10)

Since N is strongly M-closed, it follows that N_{S}(⟨x⟩E) ∩ N is strongly N_{F}(⟨x⟩E)-closed. We conclude that N ∩ N_{S}(⟨x⟩E) = 1. However, C_{S}(x) = T ≤ N_{S}(⟨x⟩E), and so

C_{N}(x) ≤ N ∩ N_{S}(⟨x⟩E) = 1.

As x acts on N and has nontrivial fixed points there, this is a contradiction. Therefore,

E(M) = 1

and M is constrained, as desired.

Set Y = O_{2}(M). From the structure of Aut_{F}(⟨x⟩E) = Aut_{M}(⟨x⟩E) in the first paragraph, ⟨x⟩E is M-radical, and ⟨x⟩E is M-centric by (8.8). Hence Y ≤ ⟨x⟩E by [LP10, Proposition 1] (see also [BCG^{+}05, Proposition 1.6]), from which it follows that Y = ⟨x⟩E since Y contains its centralizer in S. Therefore,

M = N_{F}(⟨x⟩E). (8.11)

By Lemma 5.6, hypothesis (3.3) of Lyons’ Lemma holds. By (8.9) and (8.11), x is not in 𝔣𝔬𝔠(M). Hence,

x ∉ 𝔣𝔬𝔠(F), (8.12)

by Lemma 3.2.

By (8.11), H is a model also for C_{F}(t_{1}) and so S is Sylow in H. Further T_{1} is a Sylow subgroup of O^{2}(H), so normal in S, and T_{1} has index 4 in S.

Assume first that S/T_{1} is cyclic. Then Ω_{1}(T/S_{1}) is generated by the coset S_{1}x, and so as 𝔥𝔶𝔭(F) ≥ 𝔥𝔶𝔭(K) = T_{1}, it follows from (8.12) that 𝔥𝔶𝔭(F) = T_{1}. Thus, there is an index 2 subsystem over T = ⟨x⟩T_{1} in which x is 2-central, and we obtain a contradiction by Lemma 6.4.

Thus, S/T_{1} is a four group. According to (8.12), we may choose a normal subsystem F_{1} of F of index 2 over a subgroup S_{1} with T_{1} < S_{1} < S and with x ∉ S_{1} (see §§2.3). Let H_{1} be the corresponding subgroup of H (i.e. a model for N_{F_{1}}(E)). Then H_{1}/E ≅ S_{5} by the structure of H, and so in particular,

(8.13) S_{1}/E is dihedral of order 8.

Observe that T_{0} = J(T_{1}) is characteristic in T_{1} and so normal in S_{1}. We next show that

(8.14) T_{0}/Z(T_{0}) is self-centralizing in S_{1}/Z(T_{0}) via an argument supplied to us by R. Solomon. Suppose on the contrary that S_{1} = T_{1}C_{S_{1}}(T_{0}/Z(T_{0})). Let F = F_{1} as in the notation of Section 4, and temporarily denote quotients by Z(T_{0}) with bars. Then for each a ∈ F,

[ā, S̄_{1}] = [ā, T̄_{1}] ≤ \overline{[T_{1}, T_{1}]} ≤ Ē

with the last inclusion since [T_{1}, T_{1}] ≤ E. Hence T_{1}/E = F E/E is of order 4 and central in S_{1}/E, and this supplies the desired contradiction with (8.13).

By (8.14) and Lemma 6.2 of [Oli16a], there are two possibilities for the structure of S_{1} as an extension of T_{1}. In both cases, there is s ∈ S_{1} − T_{1} such that

c_{s}|_{T_{0}} = c_{f}|_{T_{0}}, (8.15)

where f is in the notation of Section 4. In particular,

t_{2}^{s} = t_{1}t_{2}. (8.16)

Also, as S_{1} ∩ T = T_{1} is of index 2 in S_{1} and s ∈ N_{S_{1}}(T) − T, we have

x^{s} = xt_{1}. (8.17)

We can now obtain the final contradiction. Recall from Lemma 4.7(b)(iii) that Aut_{K}(F_{1}) ≅ C_{3}×A_{4} is the stabilizer of the 2-dimensional F_{4}-subspace Z(T_{0}) of F_{1}. Let Z(Aut_{K}(F_{1})) ≅ C_{3} be generated by β; this subgroup permutes the involutions of Z(T_{0}) transitively, and we may assume t_{2}^{β} = t_{1}. By (8.15), s normalizes F_{1}, and the restriction of c_{s} to F_{1} normalizes ⟨β⟩. Thus, we have on the one hand that x^{β^{cs}} = x^{β^{−1}} = x. On the other hand, (8.17) and (8.16) yield x^{β^{cs}} = (xt_{1})^{βc_{s}} = (xt_{2})^{c_{s}} = xt_{1}t_{1}t_{2} = xt_{2}. These two facts are incompatible. □

By Lemma 8.4(b) and Lemma 8.7, we have that D_{1} is nonabelian dihedral or semidihedral. Set

D := N_{D_{1}}(⟨x, t_{1}⟩),

which is dihedral of order 8 with center ⟨t_{1}⟩, and fix an involution s_{1} ∈ D − ⟨x, t_{1}⟩. Finally, let

U = DE ≅ D_{8} ∗ D_{8} ∗ Q_{8}.

We fix this notation for the remainder of the section.

Lemma 8.18. x^{F} ∩ U = x^{Aut_{F} (U)}.

Proof. Let y ∈ x^{F} ∩ U and φ ∈ F with y^{φ} = x. By the extension axiom, φ extends to a morphism C_{S}(y) → T, which we also call φ. Witt’s Lemma (applied with respect to Out(U) acting on the orthogonal space U/Z(U)) gives C_{U}(y) = ⟨y⟩ × E_{0} with E_{0} ≅ E, and we have E_{0}^{φ} ≤ ⟨x⟩ × E by Lemma 8.3. Hence we may choose E_{0} mapping to E under φ.

Now set D_{0} = C_{U}(E_{0}) and select an involution y_{0} ∈ D_{0} with D_{0} = ⟨y_{0}, y⟩. Then s_{1} acts as c_{y_{0}}^{φ} on ⟨x⟩E, and so viewing φ as an isomorphism ⟨y⟩E_{0} → ⟨x⟩E, it follows that y_{0} ∈ N_{φ}. Lemma 8.5 and the extension axiom imply that φ|_{⟨y⟩E_{0}} extends to a map φ̃ ∈ Hom_{F}(U, S). Finally, yφ̃ ; 0 ∈ N_{C_{S}(E)}(⟨x, t_{1}⟩) = D, by Lemma 8.4(b) and so Dφ̃ ; 0 = D. □

Lemma 8.19. C_{F}(D) ⊇ Aut_{K}(E).

Proof. We will show that C_{F}(C_{S}(E)) ⊇ Aut_{K}(E). Let α ∈ Aut_{K}(E). By Lemma 8.5, E is fully F-normalized and so α extends to a morphism α̃ ∈ Aut_{F}(C_{S}(E)E) which normalizes C_{S}(E).

Suppose first that α is of odd order. Since C_{S}(E) is nonabelian dihedral or semidihedral, it has automorphism group a 2-group. Hence, α̃|_{C_{S}(E)} has order a power of 2, so some power of α̃ restricts to α on E and to the identity on C_{S}(E). Hence α ∈ C_{F}(D) in this case. However, since Aut_{K}(E) ≅ E_{16}A_{5} is generated by its elements of odd order (and since the restriction Aut_{F}(C_{S}(E)E) → Aut_{F}(E) is a group homomorphism), the general case follows. □

We now begin to pin down the structure of N_{F}(U). Denote quotients by Z(U) with bars. Let U_{2} and U_{4} denote the set of elements of U of order 2 and 4, respectively, and similarly for E_{2} and E_{4}. Thus, Ū_{2} and Ū_{4} are the set of singular and nonsingular vectors in the orthogonal space Ū determined by the squaring map Ū → Z(U), and similarly for Ē_{2} and Ē_{4}. Then |Ē_{2}| = 5 and |Ē_{4}| = 10.

The action of Out_{K}(E) ≅ A_{5} ≅ Ω_{4}^{−}(2) on Ē is the orthogonal action. On Ē_{2}, this is the standard permutation action of A_{5} on 5 letters. Further, the 10 elements of Ē_{4} are the nonsingular vectors in the orthogonal space Ē and are therefore also permuted transitively by Witt’s Lemma.

We want to examine the action of C_{Aut_{F} (U)}(x) on U. Observe that C_{Aut_{F} (U)}(x) normalizes ⟨x⟩E = C_{U}(x). By Lemma 8.3, C_{Aut_{F} (U)}(x) normalizes E and hence D. Now Aut_{C}(E) contains Aut_{K}(E) ≅ ĒA_{5} with index 1 or 2 (according to whether T = R or T > R). Since T = C_{S}(x) normalizes D, by Lemma 8.19, each member of Aut_{C}(E) has an extension to C_{Aut_{F} (U)}(x) which centralizes D or acts on D as conjugation by x. These facts allow us to describe the orbits of C_{Aut_{F} (U)}(x) on elements of Ū as follows.

The C_{Aut_{F} (U)}(x)-orbit decomposition of Ū_{2} is given by

Ū_{2} = Ē_{2} ⊔ {x̄} ⊔ Ē_{2}x̄ ⊔ {s̄_{1}} ⊔ Ē_{2}s̄_{1} ⊔ Ē_{4}x̄s̄_{1}; (8.20)

the sizes of these orbits are 5, 1, 5, 1, 5, and 10, respectively.

The C_{Aut_{F} (U)}(x)-orbit decomposition of Ū_{4} is given by

Ū_{4} = Ē_{4} ⊔ Ē_{4}x̄ ⊔ Ē_{4}s̄_{1} ⊔ {x̄s̄_{1}} ⊔ Ē_{2}x̄s̄_{1}; (8.21)

the respective sizes of these orbits are 10, 10, 10, 1, and 5.

We may now determine the structure of Out_{F}(U).

Lemma 8.22. The following hold.

(a) x^{Out_{F} (U)} = ({1} ∪ E_{2})x ∪ E_{4}xs_{1}, and

(b) Out_{F}(U) ≅ E_{16}A_{5} or E_{16}S_{5} is the full stabilizer in Ω_{6}^{−}(2) or O_{6}^{−}(2) of the singular vector s̄_{1} ∈ Ū.

Proof. Let O be the orbit of x̄ under Out_{F}(U). Since x is conjugate to t_{1}x by Lemma 5.6(b), it is conjugate to t_{2}x ∈ E_{2}x by Lemma 4.7(b)(iii). Thus

Ē_{2}x̄ ⊆ O (8.23)

by Lemma 8.18. Furthermore by Lemma 5.6(a), Ē_{2} is not in O.

The orbit O must have size dividing the order of Out(U) ≅ O_{6}^{−}(2), which is 2^{7} · 3^{4} · 5. With this constraint, a check of the (eight) possible remaining orbit sizes using (8.20) shows that there are only two additional possibilities for O other than the one claimed in (a):

({1} ∪ Ē_{2})x̄ and ({1} ∪ Ē_{2})x̄ ∪ ({1} ∪ Ē_{2})s̄_{1}

with sizes 6 and 12. Assume O is one of these. Then since Out_{F}(U) involves A_{5}, we have Out_{F}(U) ≅ S_{6} or C_{2}×S_{6} by [CCN^{+}85, p.26], the stabilizer in Ω_{6}^{−}(2) or O_{6}^{−}(2) of a nonsingular vector in Ū. By (8.21), that vector must be x̄s̄_{1}.

Assume first that O = ({1} ∪ Ē_{2})x̄. Then as E is generated by E_{2}, we have that ⟨x⟩E = ⟨x^{Aut_{F} (U)}⟩ is invariant under Aut_{F}(U). Hence, E is invariant under Aut_{F}(U) by Lemma 8.3, which is a contradiction, as Aut(E) ≅ S_{5} while Aut_{F}(U) ≅ S_{6} induces at least Out_{K}(E) ≅ A_{5} on E.

Assume now that O = ({1} ∪ Ē_{2})x̄ ∪ ({1} ∪ Ē_{2})s̄_{1}. Since Out_{F}(U) ≅ C_{2} × S_{6} is the full stabilizer in O_{6}^{−}(2) of the vector ⟨x̄s̄_{1}⟩, there is an element in Out_{F}(U) of order 2 (generating the cyclic direct factor) that interchanges x̄ and s̄_{1} and centralizes Ē. Thus, by the argument of the last paragraph, there is an index 2 subgroup of Out_{F}(U) isomorphic to S_{6} which acts on E, giving the same contradiction. This completes the proof of (a).

By (a), Out_{F}(U) is a subgroup of O_{6}^{−}(2) containing A_{5} or S_{5} with index 16. By [CCN^{+}85, p.26], this must be the stabilizer of a singular vector in Ū, which is isomorphic to E_{16}A_{5} or E_{16}S_{5} in the respective cases. The fixed vector is s̄_{1} by (8.20) and (8.23), and this completes the proof of (b). □

We now determine the structure of Aut_{F}(U).

Lemma 8.24. Let H̄ = Aut_{F}(U), Ȳ = O_{2}(H̄). Then O^{2}(H̄) = Ȳ B̄, where B̄ ≅ A_{5} and Ȳ = P̄_{1} ∗ P̄_{2} with P̄_{1} ≅ P̄_{2} ≅ D_{8} ∗ Q_{8} and P̄_{1}^{c_{x}} = P̄_{2}. Furthermore, Z(Ȳ) = ⟨c_{s_{1}}⟩, and [P̄_{i}, B̄] = P̄_{i} for i = 1, 2.

Proof. Lemma 8.22 establishes that H̄ is the stabilizer in Aut(U) or O^{2}(Aut(U)) of the singular vector s̄_{1} of Ū. Hence the general structure is as described by Lemma 4.3 of [Fin76b]. We review Finkelstein’s proof of this result enough to show the additional information on embeddings of c_{x} and c_{s_{1}} in H̄.

Let J_{i} (i = 1, 2) be isomorphic nontrivial extensions of P_{i} ≅ D_{8} ∗ Q_{8} by B_{i} ≅ S_{5}. Let G be generated by J_{1} × J_{2} and an involution x_{0} which interchanges J_{1} and J_{2} by conjugation. Set U_{0} = ⟨x_{0}, Z(O_{2}(G)), C_{O_{2}(G)}(x_{0})⟩, let D_{0} be generated by x_{0} and Z(O_{2}(G)), and let E_{0} = C_{O_{2}(G)}(x_{0}). Finally, set H = N_{G}(U_{0}). Then D_{0} ≅ D_{8}, E_{0} ≅ D_{8} ∗ Q_{8}, and U_{0} is the central product of D_{0} and E_{0}. Further, H = P_{1}P_{2}C_{G}(x_{0}), and denoting the generator of J_{1} by z_{1}, we have that

(8.25) Z(U_{0})z_{1} is a fixed point under H/Z(U_{0}).

Identify U with U_{0}, E with E_{0}, and H̄ with H/Z(U) via the injection H/Z(U_{0}) → Aut(U). By (8.25), we may further identify s_{1} with z_{1}, and x with x_{0}. The lemma follows. □

As we will be working in the normalizer of U for the remainder of this subsection, we may choose notation so that

U ∈ F^{f}. (8.26)

As U contains its centralizer in S, U is therefore F-centric. We fix a model H for N_{F}(U) for the remainder. Then Z(H) = Z(U) = ⟨t_{1}⟩, and we denote with bars the quotient by Z(U), as before.

Adopt the notation of Lemma 8.24, which gives a description of H̄ = Aut_{F}(U). Let Y be the preimage of Ȳ in S.

Lemma 8.27. The following hold.

(a) Z(O^{2}(H)) = Z(Y) = ⟨s_{1}, t_{1}⟩

(b) [Y, Y] = Z(Y).

Proof. By Lemma 8.24, Z(O^{2}(H)) ≤ Z(Y) ≤ ⟨s_{1}, t_{1}⟩. However, C_{H}(⟨s_{1}, t_{1}⟩) is of index 2 in H, so (a) holds. Now (b) holds because [Ȳ , Ȳ] = ⟨s̄_{1}⟩ and t_{1} ∈ [U, U] ≤ [Y, Y]. □

Let P_{i} be the preimage of P̄_{i} in S. Choose the complement B̄ of Lemma 8.24 so that B̄ ∩ \overline{N_{S}(U)} is a Sylow 2-subgroup of B̄. The preimage of B̄ in H splits over ⟨t_{1}⟩, since it does so in N_{C_{H}(x)}(E). We denote by B the commutator subgroup of this preimage, so that B ≅ A_{5}. We thus have that

O^{2}(H)⟨x⟩ = P_{1}P_{2}(⟨x⟩ × B),

with P_{1} and P_{2} interchanged by x and normal in O^{2}(H).

Lemma 8.28. For i = 1, 2, we have P_{i} = Y_{i} × ⟨t_{1}⟩ where Y_{i} ≅ D_{8} ∗ Q_{8} and (a) Y_{i} = [Y_{i}, B] = [P_{i}, B];

(b) Z(Y_{1}) = ⟨s_{1}⟩, Z(Y_{2}) = ⟨s_{1}t_{1}⟩, and Y_{1}^{x} = Y_{2} after suitable choice of notation; and (c) [Y_{1}, Y_{2}] ≤ ⟨t_{1}⟩.

Proof. We first show that

(8.29) [P_{1}, P_{1}] is of order 2.

Assume otherwise, so that P_{1} is special with [P_{1}, P_{1}] = ⟨s_{1}, t_{1}⟩ by Lemma 8.27. Then [P_{1}, B] covers P̄_{1}, and so [P_{1}, B] = P_{1}. Moreover, the same holds for a Sylow 5-subgroup F of B. The hypotheses of Lemma 5.26 of [Tho68] thus hold for P_{1}F, and so P_{1} is isomorphic to a Sylow 2-subgroup of U_{3}(4) by that lemma. Since the automorphism group of such a 2-group is solvable [GLS05, Lemma 2.2.9], this contradiction establishes (8.29).

Since t_{1} ∉ [P_{1}, P_{1}] and x interchanges P_{1} with P_{2} and also s_{1} with s_{1}t_{1} , we may choose notation so that

[P_{1}, P_{1}] = ⟨s_{1}⟩ and [P_{2}, P_{2}] = ⟨s_{1}t_{1}⟩.

Now [P_{1}, P_{1}]t_{1} is a fixed point for the action of B on the commutator quotient of P_{1}, and so since P_{1}/⟨s_{1}, t_{1}⟩ is a natural module for A_{5}, there is a B-invariant complement Y_{1} to ⟨t_{1}⟩ in P_{1}. Thus P_{1} = Y_{1} × ⟨t_{1}⟩ with Y_{1} ≅ D_{8} ∗ Q_{8}, and (a) and (b) hold after setting Y_{2} = Y_{1}^{x}. Finally, since Ȳ_{1} = P̄_{1} commutes with Ȳ_{2} = P̄_{2} in H̄, we see that (c) holds. □

We can now obtain a contradiction to the standing assumption that m_{2}(S) = m_{2}(T) = 5.

Lemma 8.30. m_{2}(N_{S}(U)) ≥ 6.

Proof. Each noncentral involution y ∈ Y_{1} is centralized by some four subgroup V_{y} of B, and so

if [y, y^{x}] = 1, then V_{y}Z(Y)⟨y, y^{x}⟩ ≅ E_{2^{6}}.

Hence we may assume that

[y, y^{x}] = t_{1} (8.31) for each such involution y by Lemma 8.28(c). On the other hand, we will show this cannot be the case. Let V = B ∩ S, a four group, and let N_{B}(V) = V ⟨b⟩ where b is of order 3. Write D_{1} := C_{Y_{1}}(b) = ⟨d, e⟩, a dihedral group generated by the involutions d and e, and write Q_{1} := [Y_{1}, b] = ⟨f, g⟩, a quaternion group commuting with D_{1}. Thus, d, e, def, deg, and defg constitute half of the noncentral involutions in Y_{1}. Now use bilinearity of the commutator, Lemma 8.28(c), and (8.31) to see that

[f, f^{x}] = [def, (def)^{x}] = t_{1};
[g, g^{x}] = [deg, (deg)^{x}] = t_{1}; and
[f, f^{x}][g, g^{x}] = [defg, (defg)^{x}] = t_{1}.

These three facts are incompatible. □

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Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854

  E-mail address: jlynd@math.rutgers.edu

Department of Mathematics and Computer Science, Saint Louis University, 220 North Grand Blvd., Saint Louis, MO 63103

  E-mail address: rainbolt@slu.edu