A CHARACTERIZATION OF THE 2-FUSION SYSTEM OF L_{4}(q)

JUSTIN LYND

Abstract. We study a saturated fusion system F on a finite 2-group S having a Baumann component based on a dihedral 2-group. Assuming F = O^{2}(F), O_{2}(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 L_{4}(q_{1}) for some q_{1} ≡ 3 (mod 4). This should be viewed as a contribution to a program recently outlined by M. Aschbacher for the classification of simple fusion systems at the prime 2. The corresponding problem in the componenttype portion of the classification of finite simple groups (the L_{2}(q), A_{7} standard form problem) was one of the last to be completed, and was ultimately only resolved in an inductive context with heavy artillery. Thanks primarily to requiring the component to be Baumann, our main arguments by contrast require only 2-fusion analysis and transfer. We deduce a companion result in the category of groups.

Saturated fusion systems (or Frobenius categories) are categories, defined by Puig, codifying simultaneously the properties of G-conjugacy of p-subgroups in a finite group, and of Brauer pairs associated to a p-block of a group algebra for G. The study of such pfusion in finite groups began in the last decade of the 19th century with Burnside and Frobenius. In the latter half of the twentieth, the analysis of fusion was indispensable for the classification of finite simple groups. Abstract fusion theory has evolved in the last decade via the work of many to become the foundation for investigation of spaces which behave like p-completed classifying spaces of finite groups as well as a natural setting for studying the p-local structure of finite groups. In the latter setting, the structure theory of saturated fusion systems parallels that of finite groups. A saturated fusion system has appropriate analogues of O_{p}(G), O^{p}(G) and O^{p′}(G) [BCG^{+}07], a transfer map [BLO03], normal subgroups and quotients, simplicity, components, layer E(G), and the generalized Fitting subgroup [Asc08,Asc11].

With this groundwork in place, Aschbacher has proposed [AKO11, Section II.13-15] a program for the classification of simple fusion systems at the prime 2, and has begun to carry out substantial parts of it [Asc10, Asc13b, Asc13a, Asc13c]. See also work of Henke [Hen11] and Welz [Wel12]. One reason for doing this is to effect a directed search for new exotic 2-fusion systems other than the Solomon systems. A more central aim is to simplify portions of the classification of finite simple groups. Working in the category of fusion systems provides a clean separation of the analysis of fusion in a finite group from other considerations, such as the obstruction of the existence of cores of local subgroups (which vanishes in the fusion system setting), and of group recognition from local structure. This suggests such simplifications might occur most prominently in the component-type case.

Date: March 25, 2013.

2000 Mathematics Subject Classification. Primary 20D20, Secondary 20D05.

Early examples of this can be seen in Aschbacher’s E-balance theorem [Asc11, Theorem 7] and the Dichotomy Theorem [AKO11, II.14.3] for saturated fusion systems.

Denote by J(S) the Thompson subgroup of S generated by elementary abelian subgroups of S of maximum rank. The Baumann subgroup Baum(S) is C_{S}(Ω_{1}(ZJ(S))) [AS04, B.2.2]. Let ν_{2} be the 2-adic valuation. In this paper, we make a contribution to Aschbacher’s program in the (Baumann) component-type case by carrying out a standard form problem for the 2-fusion system of L_{2}(q) (q odd) resulting in the following characterization of the 2-fusion system of L_{4}(q_{1}) (q_{1} ≡ 3 (mod 4)).

Theorem A. Let F be a saturated fusion system on the 2-group S with O^{2}(F) = F and O_{2}(F) = 1. Let x ∈ S be a fully F-centralized involution, set C = C_{F}(x), T = C_{S}(x), and suppose K is a perfect normal subsystem of C. Assume

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

(2) Q := C_{T} (K) is cyclic, and

(3) Baum(S) ⩽ T. Then S ≅ D_{2^{k}} ≀ C_{2}, and F ≅ F_{S}(G) where G ≅ L_{4}(q_{1}) for some q_{1} ≡ 3 (mod 4) with ν_{2}(q_{1} + 1) = k − 1.

Proof. From Lemma 2.10, S is of 2-rank 3 or 4. Proposition 3.18 says that x does not lie in the center of S, while Theorems 4.1 and 5.1 show that S is of 2-rank 4 and Q is of order at least 4. Finally, Theorem 6.1 identifies F and completes the proof of the theorem. □

Here, K is determined as the unique simple saturated fusion system on a nonabelian dihedral group of the given order, i.e. as F_{2}(L_{2}(q)) for some (any) q ≡ ±1 (mod 8) with ν_{2}(q^{2} − 1) = k + 1. Hypothesis (2) means in particular that K should be regarded as a “standard subsystem” of F by analogy with [Asc75]. Currently there is no notion of a standard subsystem K of a fusion system, which presumably would say in particular that C_{F}(K) is tightly embedded in F. Aschbacher has defined an appropriate notion of tightly embedded subsystem [Asc13c]. The main obstruction seems to be what one means by the centralizer C_{F}(K), and more generally by the normalizer N_{F}(K) of a subsystem. In [Asc11], Aschbacher defines the centralizer of a normal subsystem, and we rely on this to talk about C_{T} (K) as in (2). Hypotheses (1) and (2) are equivalent to specifying the structure of the generalized Fitting subsystem of C as F^{∗}(C) = F_{Q}(Q) × K, and K should be standard subsystem of F under any appropriate definition. We note Matthew Welz considers the complementary situation of Theorem A, where C_{T} (K) has 2-rank at least 2, and assuming (1) (but not (3)); see [Wel12].

Let S be an arbitrary finite 2-group, F a saturated fusion system over S and W a weakly F-closed subgroup of S. Then F is of W -characteristic 2-type if N_{F}(P) is constrained for every fully F-normalized 1 ≠ P ⩽ S, and of W -component type if there is a fully Fcentralized involution x with the property that C_{F}(x) has a component. In the latter case, we say that a subsystem K is a W -component if it is a component in some C_{F}(x) with the property that W ⩽ C_{S}(x).

Hypothesis (3) is the statement that K is a Baum(S)-component and thus F is of Baum(S)-component type (alternatively, Baumann component type). The reason one might want to make this restriction in the context of a classification of simple 2-fusion systems is detailed in [AKO11, §II.14]. As regards our specific situation, (3) allows us to avoid building the Sylow 2-subgroups of certain larger groups in characteristic 2. Without (3) for instance, the fusion systems of Aut(L_{5}(2)), Aut(U_{5}(2)), and Aut(Sp_{4}(4)) appear, as these have involutory automorphisms fixing L_{2}(9) ≅ Ω_{5}(2) ≅ Sp_{4}(2)′. In the group case, these, HS, and Aut(He) are identified by Fritz [Fri77] and (independently) by Harris-Solomon [HS77] and Harris [Har77]. All of these almost simple groups are of component type. However, all but Aut(He) (which has a 2^{2}L_{3}(4) Baumann component) are of Baumann characteristic 2-type, as are their 2-fusion systems. More seriously, (3) permits us to avoid an inductive approach like that taken by Harris [Har81], where the solution to a large number of other standard form problems were required together with a K-group hypothesis in the midst of a still-lingering unbalanced group conjecture. It was here that the natural targets, L_{4}(q^{\frac{1}{2}} ) (q^{\frac{1}{2}} ≡ 3 (mod 4)) and U_{4}(q^{\frac{1}{2}} ) (q^{\frac{1}{2}} ≡ 1 (mod 4))) appeared (together with, e.g., Aut(Ω_{8}^{−}(q^{\frac{1}{4}} )) via an appeal to Aschbacher’s Classical Involution Theorem. We note that L_{4}(q_{1}) and U_{4}(q_{2}) have equivalent fusion systems at the prime 2 whenever their Sylow 2-subgroups are isomorphic; this follows from a more general theorem due to Broto, Møller, and Oliver [BMO12, Theorem 3.3].

For the heart of the arguments in this paper, only elementary 2-group analysis, fusion, and transfer are required. Use of transfer is made via the Thompson-Lyons transfer lemma for fusion systems [Lyn13]; see Subsection 1.7. At the beginning of the analysis, we encounter a difficulty unique to the fusion system setting in getting ahold on the structure of subsystems of C/C_{C}(K) containing K, which a priori contain among them exotic extensions of K. However, work of Andersen, Oliver, and Ventura [AOV12] shows that this is in fact not the case provided certain higher limits associated to K vanish; see Subsection 1.8. At the end, after determining the structure of S, we apply a piece of Oliver’s classification of fusion systems on 2-groups of sectional rank at most 4 [Oli] in order to identify F.

When combined with a theorem of David Mason [Mas73] and Glauberman’s Z^{∗}-theorem, we obtain the following companion of Theorem A in the category of groups. Recall Z^{∗}(G) is the preimage in G of Z(G/O_{2′}(G)).

Theorem B. Let G be a fusion simple finite group (i.e. with G = O^{2}(G) and Z^{∗}(G) = O_{2′}(G)), S ∈ Syl_{2}(G), and x ∈ Ω_{1}(ZJ(S)). Assume

(1) C = C_{G}(x) has a perfect normal subgroup K with K/O_{2′}(K) ≅ L_{2}(q) (q ≡ ±1 (mod 8)) or A_{7}, and

(2) C_{C}(K/O_{2′}(K)) has cyclic Sylow 2-subgroups.

Then K/O_{2′}(K) ≅ L_{2}(q), q is a square, and O^{2′}(G/O_{2′}(G)) ≅ L_{4}(q^{\frac{1}{2}} ) if q^{\frac{1}{2}} ≡ 3 (mod 4), while O^{2′}(G/O_{2′}(G)) ≅ U_{4}(q^{\frac{1}{2}} ) if q^{\frac{1}{2}} ≡ 1 (mod 4).

The proof of Theorem B, assuming Theorem A, is found near the beginning of Section 2.

Notation. Homomorphisms are applied on the right. We prefer to write conjugation-like maps in the exponent. For instance, the image of an element s ∈ S (or subgroup P ⩽ S) under a morphism φ in a fusion system is denoted s^{φ} (or P^{φ}).

• C_{n} is the cyclic group of order n, D_{2^{n}} (n ⩾ 2), Q_{2^{n}} (n ⩾ 3), SD_{2^{n}} (n ⩾ 4) are the dihedral, quaternion, and semidihedral groups, respectively, of order 2^{n} • G^{#} is the set of nonidentity elements of G

• I_{p}(G) is the set of elements of G of order p

• Ω_{n}(P) = ⟨x ∈ P | x^{p^{n}} = 1⟩

• ℧^{n}(P) = ⟨x^{p^{n}} | x ∈ P ⟩

• ℰ_{p^{n}}(P) is the set of elementary abelian subgroups of P of order p^{n}

• Nonstandardly, for P with Z(P) of order 2, P ≀_{∗} C_{2} is the quotient of P ≀ C_{2} by its center (a wreathed commuting product).

Acknowledgements. This work was carried out as part of the author’s Ph.D. thesis at Ohio State under the direction of Ron Solomon, to whom the author is indebted. Thanks to Richard Lyons for reading an early version of this paper and offering many probing questions and helpful comments, to Bob Oliver for his willingness in making available an early version of his sectional 2-rank 4 paper, and to Matthew Welz for stimulating discussions.

1. Background

We recall in this section (mostly) known results needed for the proof of Theorem A. Our main references for group theoretic material are [Gor80], [Suz82], and [GLS05]. For the background on fusion systems, we follow [AKO11] and [Cra11b].

1.1. Automorphism groups of p-groups. We list some results about automorphism groups of p-groups needed later.

Theorem 1.1. Let A be a p′-group of automorphisms of the p-group S which stabilizes a normal series 1 = S_{0} ⩽ S_{1} ⩽ ⋯ ⩽ S_{n} = S and acts trivially on each factor S_{i+1}/S_{i}. 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.

Proposition 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 the current statement is given in [Oli13, Proposition 3.1]. □

Next we describe the outer automorphism group of a nonabelian dihedral 2-group D, and list a couple of additional statements which 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.

Lemma 1.3. Let D be a 2-group isomorphic to D_{2^{k+1}} for some k ⩾ 2. Fix the presentation ⟨b, c | b^{2} = c^{2^{k}} = 1, b^{−1}cb = c^{−1}⟩ for D and let C = ⟨c⟩ 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 ≅ C_{2} and B ≅ C_{2^{k−2}} is the kernel of the action of Out(D) on the D-classes of four subgroups of D.

(b) [S, S] ⩽ C_{S}(C), and

(c) if S_{0} is the preimage of Ω_{1}(S/C_{S}(D)D) in S, then ℧^{1}(S_{0}) ⩽ C_{S}(D)C.

Proof. For (a), A may be generated by the class [η] of the automorphism η sending b ↦ c^{−1}b and inverting c, and B may be generated by the class [φ] of φ centralizing b and sending c ↦ c^{5}.

For (b), fix a 2-group S containing D as a normal subgroup. Then C ⊴ S as C is a characteristic subgroup of D. Hence (b) follows from the exact sequence 1 → C_{S}(C) → S → Aut_{S}(C) → 1 and the fact that Aut(C) is abelian.

Let S_{0} be the preimage of Ω_{1}(S/C_{S}(D)D) in S as in (c). Thus, S_{0} consists of the elements of S which square into C_{S}(D)D. Let s ∈ S_{0}. If s ∈ C_{S}(D)D we have that s squares into C_{S}(D)℧^{1}(D) = C_{S}(D)℧^{1}(C), so we may assume that s induces a nontrivial (involutory) outer automorphism of D. If s induces some element of the coset B[η] as in (a), then s centralizes no noncentral involution of D, and hence s^{2} ∈ C_{S}(D)C as claimed. So we may further assume that s induces an involutory outer automorphism in B. Then c^{s} = cz or c^{−1}z, and so s^{2} ∈ C_{C_{S}(D)D}(C) = C_{S}(D)C as claimed. □

Lemma 1.4. Suppose k ⩾ 3 and let D be a 2-group isomorphic to Q_{2^{k+1}}, SD_{2^{k+1}}, C_{2}×D_{2^{k}}, D_{2^{k}} × D_{2^{k}}, or D_{2^{k}} ≀ C_{2}. Then Aut(D) is a 2-group.

Proof. This follows from Lemma 1.3 and [Asc00, 23.3] after applying Theorem 1.1 to an appropriate normal series of D. □

1.2. Fusion systems. Let G be a group. Write c_{g}: x ↦ g^{−1}xg for the conjugation homomorphism induced by g ∈ G. For subgroups H and K denote by Hom_{G}(H, K) = {c_{g} | g^{−1}Hg ⩽ K} the set of group homomorphisms from H to K induced by conjugation by elements of the group G. Write Aut_{G}(H) for Hom_{G}(H, H). When φ: H → K is any isomorphism, we write the induced map from Aut(H) → Aut(K) as α ↦ α^{φ}.

Fix a prime p and a p-group S. If G is finite and S ∈ Syl_{p}(G), the fusion system of G at p is the category F_{S}(G) with objects the subgroups of S, and with morphisms Hom_{G}(P, Q) for P, Q ⩽ S.

A fusion system is a category F with objects the subgroups of S and with morphisms injective group homomorphisms between subgroups, containing Hom_{S}(P, Q) for each P, Q ⩽ S. In addition, each morphism in a fusion system is assumed to factor as an inclusion followed by an isomorphism.

Fix a fusion system F on S. 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. Elements x and y are F-conjugate if ⟨x⟩ and ⟨y⟩ are F-conjugate by a morphism which sends x to y. Note that Aut_{P} (P) = Inn(P) ⊴ Aut_{F}(P) ⩽ Aut(P). Write Out_{F}(P) = Aut_{F}(P)/ Aut_{P} (P) for the F-outer automorphism group of P.

Definition 1.5. Let F be a fusion system on S. A subgroup P of S is said to be • fully F-centralized if |C_{S}(P)| ⩾ |C_{S}(Q)| for every Q ∈ P^{F},

• fully F-normalized if |N_{S}(P)| ⩾ |N_{S}(Q)| for every Q ∈ P^{F},

• fully F-automized if Aut_{S}(P) is a Sylow p-subgroup of Aut_{F}(P).

• F-centric if C_{S}(Q) ⩽ Q for every Q ∈ P^{F}, and

• F-radical if O_{p}(Out_{F}(P)) = 1.

• weakly F-closed if P^{φ} = P for every φ ∈ Hom_{F}(P, S), and

• strongly F-closed if R^{φ} ⩽ P for every R ⩽ P and φ ∈ Hom_{F}(R, S).

We refer to [AKO11, I.2.5] for the definition of a saturated fusion system in the form used in this paper. Thus F is saturated if it satisfies the Sylow and extension axioms. The extension axiom says that given an isomorphism φ ∈ Hom_{F}(P, Q) with Q fully Fcentralized, there is a morphism φ̃ ∈ Hom_{F}(N_{φ}, S), such that φ̃|_{P} = φ. Here,

N_{φ} = {s ∈ N_{S}(P) | (c_{s})^{φ} ∈ Aut_{S}(Q)}.

Note C_{S}(P)P ⩽ N_{φ} for any such isomorphism φ, so every map P → Q with Q fully Fcentralized must extend to C_{S}(P)P; we will apply the extension axiom in this special case quite often. A fusion system of a finite group is saturated [AKO11, Theorem 2.3].

We will often say an element x ∈ S is fully F-centralized if ⟨x⟩ is fully F-centralized, especially when x is an involution. 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. Concatenation in the superscript denotes the intersection of the relevant sets. For example, F^{cr} is the set of subgroups which are both F-centric and F-radical. 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 for saturated fusion systems [BLO03, A.10].

The most important weakly closed subgroups for our purposes are the Thompson subgroup J(S) generated by elementary abelian subgroups of maximum rank in S, and the Baumann subgroup Baum(S) = C_{S}(Ω_{1}(Z(J(S)))) [AS04, B.2.2]. Each of these are weakly closed in any fusion system over S.

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. For a subgroup T of S, the normalizer N_{F}(T) is the fusion subsystem on N_{S}(T) with morphism sets Hom_{N_{F} (T)}(P, Q) consisting of those φ ∈ Hom_{F}(P, Q) having an extension φ̃ ∈ Hom_{F}(T P, T Q) with T φ̃ = T. The centralizer C_{F}(T) is the fusion subsystem on C_{S}(T) with morphism sets Hom_{C_{F} (T)}(P, Q) consisting of those φ ∈ Hom_{F}(P, Q) having an extension φ̃ ∈ Hom_{F}(T P, T Q) with φ̃|_{T} = id_{T}. The subgroup T is normal in F if F = N_{F}(T), and central if F = C_{F}(T). The normalizer (resp. centralizer) is saturated if T is fully F-normalized (resp. fully F-centralized). We will often use without comment that C_{F_{S}(G)}(T) = F_{C_{S}(T)}(C_{G}(T)) in a finite group, as is easily observed (and the same holds for normalizers). Write O_{p}(F) for the largest normal subgroup of F, and Z(F) for the center of F, the largest central subgroup.

We need a version of Burnside’s fusion theorem, which will often be applied in the case T = J(S).

Lemma 1.6 (Burnside’s Fusion Theorem). Let F be a saturated fusion system on the pgroup 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 N_{F}(T).

Proof. Suppose P and Q are subgroups of Z(T), and let φ ∈ Hom_{F}(P, Q). Let ψ ∈ Iso_{F}(Q, Q′) with Q′ fully F-centralized. By the extension axiom, φψ and ψ have extensions to C_{S}(P) and C_{S}(Q) respectively, and these subgroups both contain T. Restricting these extensions to T and using the fact that T is weakly F-closed, we get automorphisms α, β ∈ Aut_{F}(T) such that (αβ^{−1})|_{P} = φ, which is what was to be shown. □

1.3. Normal subsystems and quotients. Next we consider notions of normality for fusion subsystems.

Definition 1.7. 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.

• E is weakly normal if E is saturated, and whenever P ⩽ Q ⩽ T, φ ∈ Hom_{E}(P, Q), and ψ ∈ Hom_{F}(Q, S), then φ^{ψ} ∈ Hom_{E}(P^{ψ}, Q^{ψ}).

• E is normal in F, written E ⊴ F, if E is weakly normal and every element α ∈ Aut_{E}(T) has an extension α̃ ∈ Aut_{F}(C_{S}(T)T) such that [C_{S}(T)T, α] ⩽ T.

• F is simple if it contains no normal subsystems other than F_{1}(1) and F.

Craven has shown that the two notions of normality give equivalent definitions of simplicity for saturated fusion systems; see [Cra11a].

Morphisms of saturated fusion systems are defined in [AKO11, II.2]. The quotient of a saturated fusion system by a strongly closed subgroup is defined in [AKO11, II.4,5]. See also [Cra11b, Chapter 5].

Lemma 1.8 ( [Asc08, Lemmas 8.9, 8.10]). Assume that θ: F → F^{+} is a surjective morphism of saturated fusion systems, and S_{0} is the kernel of θ: S → S^{+}. Then the map T ↦ T θ is a bijection between the strongly F-closed overgroups of S_{0} in S and the set of strongly F^{+}-closed subgroups of S^{+}.

A theorem of Craven gives analogues of the second isomorphism theorem.

Theorem 1.9 ( [Cra10, Theorem E]). Let F be a saturated fusion system on S, and suppose T_{1} ⩽ T_{2} are strongly F-closed subgroups of S. Then F/T_{2} ≅ (F/T_{1})/(T_{2}/T_{1}).

1.4. The hyperfocal and residual subsystems. We now look at analogues of O^{p}(G) and O^{p′}(G) in a saturated fusion system.

Definition 1.10. 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) = ⟨[s, φ] | φ ∈ Hom_{F}(⟨s⟩, S)⟩
= ⟨[s, φ] | s ∈ P ⩽ S and φ ∈ Aut_{F}(P)⟩.

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

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

Here [s, φ] := s^{−1}s^{φ} for s ∈ S and φ ∈ Hom(⟨s⟩, S). The equivalence between the two descriptions of 𝔣𝔬𝔠(F) in the definition can be seen via Alperin’s fusion theorem.

Lemma 1.11. Let F be a saturated fusion system on the p-group S. Then (a) 𝔥𝔶𝔭(F) ⩽ 𝔣𝔬𝔠(F) and both subgroups are strongly F-closed,

(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

(c) if S/𝔣𝔬𝔠(F) is cyclic, then 𝔣𝔬𝔠(F) = 𝔥𝔶𝔭(F).

Proof. Part (a) is straightforward. We present a concise proof for (b) due to Craven. By Alperin’s fusion theorem, F/T is the fusion system of the p-group S/T if and only if there are no p′-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 P of S, each p′- automorphism α of P, we have [P, α] ⩽ T. Since O^{p}(Aut_{F}(P)) is generated by the p′ elements in Aut_{F}(P), we conclude that F/T is the fusion system of S/T if and only if T ⩾ 𝔥𝔶𝔭(F) = ⟨[P, O^{p}(Aut_{F}(P))] | P ⩽ S⟩. 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 1.9) 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. □

Let F be a saturated fusion system on the p-group S. A fusion subsystem F_{0} on the subgroup S_{0} ⩽ S has p-power index in F if S_{0} ⩾ 𝔥𝔶𝔭(F), and Aut_{F_{0}}(P) ⩾ O^{p}(Aut_{F}(P)) for every P ⩽ S_{0}, and p-prime index in F if S_{0} = S and Aut_{F_{0}}(P) ⩾ O^{p′}(Aut_{F}(P)) for every P ⩽ S_{0}.

The saturated subsystems of p-power index in F are in one-to-one correspondence with the overgroups T of 𝔥𝔶𝔭(F) in S [BCG^{+}07, Theorem 4.3]. Such a subsystem is normal if and only if T is normal in S [Asc11, Section 7]. Hence there is a unique minimal normal subsystem of p-power index, the hyperfocal subsystem O^{p}(F) based on 𝔥𝔶𝔭(F).

We will require the following theorem of Aschbacher, which allows one to consider the product of a p-group with a normal subsystem. See also [Hen13] for a simplification of Aschbacher’s construction and proof of saturation.

Theorem 1.12 ( [Asc11, 8.21]). Let F be a saturated fusion system on the p-group S and let S_{0} and T be strongly F-closed subgroups of S with S_{0} ⩽ T. Suppose F_{0} is a normal subsystem of F on S_{0}. Then there exists a saturated fusion subsystem F_{0}T of F with the following properties.

(a) F_{0} ⊴ F_{0}T,

(b) F_{0}T/S_{0} ≅ F_{T^{+}}(T^{+}) where T^{+} = T/S_{0}, and

(c) the map X ↦ F_{0}X is a bijection between the set of subgroups X ⩽ T containing S_{0} and the set of saturated subsystems of F_{0}T containing F_{0}.

Note the particular case of the preceding theorem: if F_{0} = O^{p}(F_{0}), then as 𝔥𝔶𝔭(F_{0}) ⩽ 𝔥𝔶𝔭(F_{0}T) ⩽ S_{0} by (b), we have F_{0} = O^{p}(F_{0}T).

Broto, Castellana, Grodal, Levi, and Oliver [BCG^{+}07, Theorem 5.4] (see also [Pui06, Theorem 6.11]) give a description of the fusion subsystems of index prime to p: they are in one-to-one correspondence with overgroups of a subgroup Γ ⩽ Aut_{F}(S) containing Aut_{S}(S), and hence there is a unique minimal one O^{p′}(F), the residual subsystem of F. The following corollary to this result will suffice for our purposes.

Proposition 1.13. Let S be a finite p-group with automorphism group a p-group. Then F = O^{p′}(F) for every saturated fusion system F on S.

The following lists the relationship between the hyperfocal and residual subsystems, surjective morphisms, and direct products we will need. (For the definition of a direct product of two fusion systems, we refer to [AKO11, I.6.5].)

Lemma 1.14. 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(O^{p}(F)) = O^{p}(F^{+}) and θ(O^{p′}(F)) = O^{p′}(F^{+}).

(b) If F = F_{1} × F_{2} with each F_{i} saturated, then O^{p}(F) = O^{p}(F_{1}) × O^{p}(F_{2}) and O^{p′}(F) = O^{p′}(F_{1}) × O^{p′}(F_{2}).

Proof. As a surjective morphism of fusion systems is surjective on morphisms, part (a) follows from Alperin’s fusion theorem and the fact that O^{p}(G) and O^{p′}(G) are fully invariant subgroups. Part (b) is [AOV12, Proposition 3.4]. □

Later in Section 6, we will make use of the following special case of a more general theorem of Oliver.

Theorem 1.15 ([Oli13, Theorem C]). Let F be a perfect saturated fusion system on a direct product D_{1} × D_{2} of two nonabelian dihedral 2-groups of the same order. Then F = F_{1} × F_{2} where F_{i} is a perfect fusion system on D_{i}.

The hypotheses of Oliver’s theorem require that F = O^{2′}(F), but this holds in the case stated above by Lemma 1.4 and Proposition 1.13. Also, the F_{i} must be perfect by Lemma 1.14(b).

1.5. Centralizers. To date an appropriate notion of the normalizer or centralizer of an arbitrary saturated fusion subsystem has been elusive. Aschbacher has shown that in the case that a subsystem E is normal in F, one can define the centralizer C_{F}(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 1.16 ([Asc11, (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 ⩽ C_{S}(T) for which C_{F}(X) contains E. Then X has a unique maximal element, denoted by C_{S}(E) and called the centralizer in S of E. Moreover, C_{S}(E) is strongly F-closed, and there is a normal subsystem C_{F}(E) of F based on C_{S}(E).

Most of the time we will use the characterization of C_{S}(E) in Theorem 1.16, but for the proof of Theorem B, it is more natural to use Aschbacher’s direct construction of C_{S}(E) as follows.

First we recall some terminology, and some ideas from [Asc08, Section 4]. A saturated fusion system F over S is constrained if C_{S}(O_{p}(F)) ⩽ O_{p}(F). If F is constrained, then it has a model G [AKO11, III.5.10]. That is, G is a finite group with S ∈ Syl_{p}(G), O_{p′}(G) = 1 and C_{G}(O_{p}(G)) ⩽ O_{p}(G) such that F = F_{S}(G). Any two models for F are isomorphic by an isomorphism which is the identity on S; we refer to this as the strong uniqueness of models.

Now let E a normal subsystem of F on T. Let U ∈ E^{c} ∩ F^{f}. Then U ∈ E^{fc} and E(U) := N_{E}(U) is saturated and constrained system on N_{S}(U). Also UC_{S}(U) ∈ F^{fc}, and so D(U) := N_{F}(UC_{S}(U)) is saturated and constrained system on N_{T} (U). Furthermore E(U) ⊴ D(U). Let G(U) be a model for D(U), and H(U) be the unique normal subgroup of G(U) which is a model for E(U) [Asc08, Theorem 1]. Then C_{N_{S}(U)}(H(U)) is a well-defined subgroup of S by strong uniqueness of G(U). Aschbacher defines

I = ⋂_{U∈E^{c}∩F^{f}} C_{N_{S}(U)}(H(U))

and

C_{S}(E) = ⋂_{φ∈Aut_{F} (T C_{S}(T))} I^{φ}.

This is motivated by the fact that E = ⟨Aut_{E}(U)^{φ} | U ∈ E^{c} ∩ F^{f} | φ ∈ Aut_{F}(T)⟩ [Asc08, Theorem 3] and that the restriction map Aut_{F}(T C_{S}(T)) → Aut_{F}(T) is surjective.

The model version of the construction makes clear the situation in a finite group, as in the following.

Lemma 1.17. Let G be a finite group and N a normal subgroup of G. Let S ∈ Syl_{p}(G), T = S ∩ N, F = F_{S}(G), and E = F_{T} (N). Let H(N) = {N_{N}(V) | V ∈ E^{fc}}. Then

(a) C_{S}(E) is the unique largest subgroup of Y of S for which [H, Y] ⩽ O_{p′}(H) for every H ∈ H(N),

(b) C_{S}(N/O_{p′}(N)) ⩽ C_{S}(E), and

(c) C_{S}(E) = C_{S}(N/O_{p′}(N)) if Aut(N/O_{p′}(N)) contains no element of order p centralizing H/O_{p′}(H) for every H ∈ H(N/O_{p′}(N)).

Proof. For each U ∈ E^{fc}, set H_{U} = N_{N}(U) for brevity. Let Y ⩽ S and assume [H_{U}, Y] ⩽ O_{p′}(H_{U}) for each such U. Note then that Y centralizes T because [T, Y] ⩽ T ∩O_{p′}(H_{T}) = 1. Fix U ∈ E^{c} ∩ F^{f} ⊆ E^{fc} and set G_{U} = N_{G}(UC_{S}(U)). Then G(U) := G_{U}/O_{p′}(G_{U}) is a model for N_{F}(UC_{S}(U)) with Sylow p-subgroup N_{S}(U) ⩾ Y. Since H_{U} ⊴ G_{U}, we have O_{p′}(H_{U}) = O_{p′}(G_{U}) ∩ H_{U}. Hence H(U) := H_{U}/O_{p′}(H_{U}) is a normal subgroup of G(U) while being a model for N_{E}(U), and so Y ⩽ I with I as above. But Aut_{F}(T C_{S}(T)) acts on E^{fc}, and so Y ⩽ C_{S}(E).

Now set Y = C_{S}(E). Then Y is strongly F-closed, and [H_{V} , Y] ⩽ O_{p′}(H_{V}) whenever V ∈ (E^{c} ∩ F^{f})^{N_{G}(T C_{S}(T))} by reversing the argument in the last paragraph. Let U ∈ E^{fc}. By a Frattini argument, there is V ∈ (E^{c} ∩ F^{f})^{N_{G}(T C_{S}(T))} ∩ U^{E}. Fix such a V, and let h ∈ N with V^{h} = U. By Theorem 1.16, there is g ∈ C_{G}(Y) and t ∈ C_{G}(V) with h = tg. Hence V^{g} = U and [H_{U}, Y] = [H_{V}^{g} , Y^{g}] = [H_{V} , Y]^{g} ⩽ O_{p′}(H_{V})^{g} = O_{p′}(H_{U}) as required. This completes the proof of (a).

Let X = C_{S}(N/O_{p′}(N)). Then [T, X] ⩽ T ∩ O_{p′}(N) = 1. Also, [H, X] ⩽ H ∩ O_{p′}(N) ⩽ O_{p′}(H) for any subgroup H of N normalized by X, so (b) is immediate from (a).

For (c), assume that X < Y. Then Y/X acts on N/O_{p′}(N) and centralizes (the image of) H/O_{p′}(H) for each H ∈ H(N). Since O_{p′}(N) ∩ H ⩽ O_{p′}(H), the same is true for each \overline{H} ∈ H(N/O_{p′}(N)). Now (c) is clear. □

We will need a lemma examining in a special case how centralizers behave under quotienting by a strongly closed subgroup. First, the following shows that for the purposes of computing the centralizer in S of a normal subsystem E of F, we may restrict to the subsystem ES of Theorem 1.12.

Lemma 1.18. Suppose F is a saturated fusion system on S and E is a normal subsystem of F on T. Let Q ⩽ C_{S}(T). Then C_{F}(Q) ⩾ E if and only if C_{ES}(Q) ⩾ E.

Proof. Let φ ∈ Hom_{E}(U, V) for subgroups U and V of T. If φ lies in C_{ES}(Q), then it clearly lies in C_{F}(Q). Suppose φ ∈ C_{F}(Q). Then φ extends to a morphism φ̃ ∈ Hom_{F}(QU, QV) with φ̃|_{Q} = id_{Q}, and it suffices to show that φ̃ ∈ ES. By Alperin’s fusion theorem applied in E, it is enough to show this when U = V and U ∈ E^{fc}. But then φ̃ ∈ Aut_{N (C_{S}(U)U)}(U) with N (C_{S}(U)U) in the sense of [Asc11, Notation 8.4], and ES is generated by such automorphism groups. □

Lemma 1.19. 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 = C_{S}(E), and assume that Q ∩ T = 1. Then C_{S/Q}(ES/Q) = 1.

Proof. Without loss of generality we will assume that F = ES by Lemma 1.18. By Theorem 1.16, 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] ⩽ Q ∩ T = 1. Also, for every U, V ⩽ T,

θ_{U,V} : Hom_{F}(U, V) ⟶^{≃} Hom\overline{F}(\overline{U}, \overline{V}). (1.20)

is a bijection since Q ∩ T = 1.

Suppose the proposition is false and let \overline{Q}_{1} = C\overline{S}(\overline{E}) ≠ 1, where Q_{1} > Q is the preimage of \overline{Q}_{1} under θ. By Theorem 1.16, \overline{Q}_{1} is strongly \overline{F}-closed. By Lemma 1.8(a), Q_{1} is strongly F-closed. Since T ⊴ S, we have [Q_{1}, T] ⩽ T. But [Q_{1}, T] ⩽ Q as well because [\overline{Q}_{1}, \overline{T}] = 1. Therefore, [Q_{1}, T] ⩽ Q ∩ T = 1.

We will show that E ⩽ C_{F}(Q_{1}), supplying a contradiction. Let U be a fully E-normalized, E-centric subgroup of T and let φ ∈ Aut_{E}(U). Since U is fully E-normalized, we have that φ = φ′c_{t} for some φ′ ∈ Aut_{E}(U) of p′-order and some t ∈ T. If φ′ ∈ C_{F}(Q_{1}), then so is φ since Q_{1} centralizes T, and so we may assume that φ has order prime to p.

Let \overline{φ} = (φ)θ. Then by definition of \overline{Q}_{1}, \overline{φ} extends to \overline{φ}_{1} ∈ Aut\overline{F}(\overline{Q}_{1}\overline{U}) such that \overline{φ}_{1}|_{\overline{Q}_{1}} = id_{\overline{Q}_{1}}. Let φ_{1} be a morphism in Aut_{F}(Q_{1}U) such that (φ_{1})θ = \overline{φ}_{1}. Then by (1.20), φ_{1} restricts to φ on U. Let φ̃ be the p′-part of φ_{1}. As φ has order prime to p, φ̃ still restricts to φ on U. Furthermore, φ_{1}|_{Q_{1}} stabilizes the series 1 ⩽ Q ⩽ Q_{1} and centralizes Q_{1}/Q, so the same is true for φ̃.

Recall that we have assumed as we may that F = ES. Quotienting now by T and applying (1.20) with the roles of Q and T interchanged, we have that Aut_{F}(Q) ≅ Aut_{ES/T} (QT/T) ≅ Aut_{S/T} (Q) is a p-group, since ES/T is the fusion system of the p-group S/T by Theorem 1.12(b). It follows that φ̃|_{Q} = id_{Q}. By Theorem 1.1, φ̃ = id_{Q_{1}}. We have thus produced for arbitrary U ∈ E^{fc} and φ ∈ Aut_{E}(U), an extension φ̃ ∈ Aut_{F}(Q_{1}U) of φ which restricts to the identity on Q_{1}. Therefore, E ⩽ C_{F}(Q_{1}) by Alperin’s fusion theorem, contradicting the maximality of Q. □

1.6. Components and generalized Fitting subsystem. A saturated subsystem E of F is subnormal if there is a sequence E = E_{0} ⊆ E_{1} ⊆ ⋯ ⊆ E_{n} = F with E_{i} normal in E_{i+1} for each i. A saturated fusion system E is quasisimple if E = O^{p}(E) and E/Z(E) is simple. E is a component of F if E is a subnormal, quasisimple subsystem of F.

The intersection of two normal subsystems E_{i} on T_{i} need not be normal. However, in [Asc11, Theorem 1], Aschbacher locates a normal subsystem E := E_{1} ∧ E_{2} ⊴ F on T := T_{1} ∩ T_{2} which is contained in E_{1} ∩ E_{2}. Then E is the largest subsystem of E_{1} ∩ E_{2} on T which is normal in E_{1} and E_{2}. This allows one to speak of the normal subsystem generated by a collection of subsystems.

Definition 1.21. 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) = O_{p}(F)E(F).

Theorem 1.22 ( [Asc11, Theorem 6]). The layer E(F) is a central product of the components of F. Furthermore, F^{∗}(F) is a central product of O_{p}(F) and E(F), and C_{F}(F^{∗}(F)) = Z(F^{∗}(F)).

1.7. Thompson-Lyons transfer lemma. The main technical tool in the proof of Theorem A is the Thompson-Lyons transfer lemma, proved in [Lyn13]. We recall two consequences of this result for easy reference.

Proposition 1.23. Let F be a perfect saturated fusion system on a finite 2-group S. 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. Then u has a fully F-centralized F-conjugate in T.

Proposition 1.24. Let F be a perfect saturated fusion system on a finite 2-group S. 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 Ω_{1}(S/T). Then each involution u ∈ S − T has a fully F-centralized F-conjugate in T.

Since these are used so often, we illustrate how Proposition 1.23 can be applied together with Alperin’s fusion theorem to describe the perfect fusion systems on nonabelian 2-groups of maximal class.

Lemma 1.25. Let F be a saturated fusion system on the 2-group S with F = O^{2}(F). If S is nonabelian of maximal class, then F is uniquely determined by S up to isomorphism and one of the following holds.

(a) S ≅ D_{2^{k}} with k ⩾ 3, and for any odd prime power q with ν_{2}(q^{2} − 1) = k + 1, we have F ≅ F_{S}(G) with G ≅ L_{2}(q),

(b) S ≅ SD_{2^{k}} with k ⩾ 4, and for any odd prime power q ≡ 3 (mod 4) with ν_{2}(q + 1) = k − 2, we have F ≅ F_{S}(G) with G ≅ L_{3}(q).

(c) S ≅ Q_{2^{k}} with k ⩾ 3, and for any odd prime power q with ν_{2}(q^{2} − 1) = k, we have F ≅ F_{S}(G) with G ≅ SL_{2}(q).

Proof. If P is a subgroup of S, then P is cyclic, dihedral, semidihedral, or quaternion. Hence P has automorphism group a 2-group unless P ≅ C_{2} × C_{2} or Q_{8}. Hence, if P is a proper F-radical subgroup of S, then P ≅ C_{2} × C_{2} or Q_{8} with Aut_{F}(P) isomorphic to S_{3} or S_{4}, respectively.

Let C = ⟨c⟩ be the cyclic maximal subgroup of S, and let Z(S) = ⟨z⟩ ⩽ C. Let P ∈ F^{cr} and suppose that P ≅ C_{2} × C_{2}. Let u ∈ P − Z(S). Then u lies outside the cyclic maximal subgroup T of S. By Proposition 1.23, 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 Aut_{F}(P)-conjugate to z by a morphism of order 3, and Aut_{F}(P) ≅ S_{3}.

Let P ∈ F^{cr} with P ≅ Q_{8}. Then S is semidihedral or quaternion, and P = ⟨u, z_{1}⟩ with z_{1} ∈ P ∩ C. If S is semidihedral, let T = Ω_{1}(S), the dihedral maximal subgroup of S. If S is quaternion, let T = C. In either case, u is of least order outside T, and the only elements of T of order 4 are z_{1} and z_{1}^{−1}. Hence by Proposition 1.23, u is F-conjugate to z_{1}. As in the previous paragraph, it follows that Aut_{F}(P) ≅ S_{4} unless S = P, in which case Aut_{F}(P) ≅ A_{4}.

We have determined the automorphism groups Aut_{F}(P) for P ∈ F^{cr}. Therefore, F is uniquely determined by Alperin’s fusion theorem, and as is described in (a)-(c). □

1.8. Tameness and p-power extensions. In this subsection we address the problem of determining the structure of extensions of fusion systems, which is resolved via recent work of Andersen, Oliver, and Ventura [AOV12]. Under the conditions of Theorem A, this problem manifests itself in the determination of an involution centralizer from the description of its generalized Fitting subsystem. For suppose given a saturated fusion system on a 2-group and assume C is an involution centralizer on the subgroup T with F^{∗}(C) = O_{2}(C)E where E is quasisimple. One is confronted with the possibility that \overline{C} = C/O_{2}(C) is exotic extension of \overline{E} = E/Z(E) even when the latter is realizable by a simple group.

Let F be a saturated fusion system on S and F_{0} = O^{p}(F) on S_{0}. Assume O_{p}(F_{0}) = 1. Theorem A of [AOV12] says that if F_{0} is strongly tamely realizable by G_{0} (with O_{p′}(G) = 1), then F is realizable as well. Here we indicate how to follow the proof of that theorem to obtain the additional information that G may be chosen so that F = F_{S}(G) and Inn(G_{0}) ⩽ G ⩽ Aut(G_{0}), provided C_{S}(F_{0}) = 1.

Roughly, F_{0} is tame if F_{0} is realizable by a finite group G_{0} such that every outer automorphism of the canonical linking system of G_{0} is induced by an outer automorphism of G_{0}. An example of a fusion system F_{0} which is tame, realizable by G_{0}, but not tamely realizable by G_{0} is obtained with G_{0} = A_{7}, which is missing the “diagonal automorphism” present on F_{2}(L_{2}(q)), (q ≡ ±7 (mod 16)). See Proposition 2.6.

The fusion system F_{0} is strongly tame if, in addition, certain higher limits of functors associated to G_{0} vanish; this is expressed by saying G_{0} lies in a certain class 𝔊(p) of finite groups. We refer to [AKO11, II.§ 3,4] for background on linking systems. We also point to [AOV12] for details on tameness and the precise meaning of 𝔊(p).

Proposition 1.26. Let F be a saturated fusion system on the p-group S. Suppose that F_{0} = O^{p}(F), C_{S}(F_{0}) = 1, and that F_{0} is strongly tamely realized by G_{0} with O_{p′}(G_{0}) = 1. Then F is realized by a finite group G such that Inn(G_{0}) ⩽ G ⩽ Aut(G_{0}).

Proof. A tame fusion system can always be realized by a finite group with no nontrivial normal p′ 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 G_{0} ∈ 𝔊(p), F_{0} has a unique centric linking system L_{0}. In addition, there is a unique centric linking system L associated to F by [AOV12, Proposition 2.12(a)].

Next observe that Z(F) = Z(F_{0}) = 1, since C_{S}(F_{0}) = 1. As O_{p′}(G_{0}) = 1, any central subgroup of G_{0} must be a p-subgroup of S central in F_{0}, so it follows that Z(G_{0}) = Z(F_{0}) = 1. Now [AOV12, Proposition 1.31(a)] and [AOV12, Proposition 2.16] apply together to give that F is tamely realized by a finite group G such that G_{0} ⊴ G. Now C_{S}(G_{0}) = 1 again follows from C_{S}(F_{0}) = 1. Thus G is isomorphic to a subgroup of Aut(G_{0}) containing Inn(G_{0}) ≅ G_{0}. □

We will apply Proposition 1.26 in the case where F_{0} ≅ F_{2}(L_{2}(q)) for appropriate q. Hence we need to know

Proposition 1.27. Let K ≅ L_{2}(q) for q ≡ ±1 (mod 8). Then K ∈ 𝔊(2).

Proof. It is shown in [Oli06, Proposition 7.5] that classical groups in odd characteristic lie in 𝔊(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 𝔊(2) from general considerations. □

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 1.26, and fixing notation.

First we record some information about Aut(L_{2}(q)).

Lemma 2.1. Let K be a finite group isomorphic to L_{2}(q) and P ∈ Syl_{2}(K). Let H = Aut(K), and T ∈ Syl_{2}(H). Then

(a) H = K⟨h⟩F with h ∈ I_{2}(T), F is cyclic, and Out(K) ≅ ⟨h⟩ × F,

(b) P is dihedral of order 2^{k} where ν_{2}(q^{2} − 1) = k + 1,

(c) K⟨h⟩ ≅ P GL_{2}(q) and P ⟨h⟩ is dihedral,

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

(e) if F_{T} := T ∩ F, then P ⟨h⟩ ∩ F_{T} = 1, and |F_{T} | ⩽ 2^{k−2},

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

(g) if f is an involution in F_{T} := T ∩ F, then P ⟨f⟩ = P × ⟨f⟩, q is a square, C_{K}(f) ≅ P GL_{2}(q^{1/2}), and P ∩ O^{2}(C_{K}(f)) is dihedral of order 2^{k−1}.

(h) if f is an involution in F_{T} := T ∩ F and ⟨z⟩ = Z(P), then [f, h] = (fh)^{2} = z, P ⟨fh⟩ is semidihedral, and all involutions of T lie in P ∪ P h ∪ P f.

Proof. This is taken from Chapter 10, Lemma 1.2 of [GLS05]. □

We are now in a position to prove Theorem B assuming Theorem A.

Proof of Theorem B. Let F = F_{S}(G). We may assume ⟨x⟩ is fully F-centralized by choosing S appropriately. Then T := C_{S}(x) is a Sylow 2-subgroup of C = C_{G}(x). Set P = T ∩ K ∈ Syl_{2}(K), a dihedral group of order 2^{k} with k ⩾ 3, and set K = F_{P} (K). Put Q = C_{T} (K), \overline{K} = K/O_{2′}(K) and Q_{0} = C_{T} (K/O(K)). Then Q_{0} is a Sylow 2-subgroup of C_{C}(K/O(K)), which is cyclic by hypothesis.

We check the hypotheses of Theorem A for F. First we verify that

(2.2) Q = Q_{0} is cyclic.

By Lemma 1.17(b,c), it suffices to show that no involution in H := Aut(\overline{K}) centralizes mod core the normalizer of \overline{P} and of each four subgroup of \overline{P}. Assume otherwise that w ∈ I_{2}(Aut(\overline{K})) is such an involution. Also suppose K = \overline{K} to lighten notation. Since w centralizes N_{K}(P) mod core, it centralizes P. If w ∈ Z(P), then each four group V ⩽ P contains w and N_{K}(V) permutes transitively the involutions in V. So we may assume w induces a nontrivial field automorphism of K by Lemma 2.1(h,e,c) in case K ≅ L_{2}(q), or w is a transposition or a product of three transpositions in case K ≅ A_{7}. Suppose K ≅ L_{2}(q). If V is a four subgroup of C_{K}(w), then Lemma 2.1(g) implies that N_{C_{K}(w)}(V) permutes transitively the three involutions of V w − {w}. Hence zw does not centralize N_{K}(V) mod core. So w = (zw)^{h} does not centralize N_{K}(V^{h}) mod core, and this is the final contradiction. The argument in the case K ≅ A_{7} is as straightforward. Hence, (2.2) implies (2) of Theorem A is satisfied.

Now we show:

Q = O_{2}(C). (2.3)

By (2.2), C_{C}(K) is the fusion system of the 2-group Q. By [Asc11, 9.8.2], K = E(C). Now [Asc11, 9.12] applies to give that O_{2}(C) = F^{∗}(C_{C}(K)) = Q, as desired.

Next, we show

O_{2}(F) = 1. (2.4)

Let Y = O_{2}(F) and suppose that Y ≠ 1. Then Ω_{1}(Z(Y)) ⊴ F. Hence Ω_{1}(Z(Y)) ∩ T is (nontrivial) normal in C = C_{F}(x); this follows for instance from the Aschbacher-Stancu characterization of normality [Cra11b, Proposition 4.62,Theorem 5.29]. So Ω_{1}(Z(Y))∩T ⩽ Ω_{1}(Q) = ⟨x⟩. If T < S, this is a contradiction. So S = T and Ω_{1}(Z(Y)) = ⟨x⟩ is strongly F-closed. This contradicts the Z^{∗}-theorem and hence Y = 1.

As F = O^{2}(F) by Puig’s hyperfocal subgroup theorem [Pui00], and Baum(S) ⩽ C_{S}(x) = T by assumption on x, F satisfies the hypotheses of Theorem A. Hence S ≅ D_{2^{k}} ≀ C_{2} and F ≅ F_{2}(L_{4}(q_{1})) for some q_{1} ≡ 3 (mod 4). Now a Theorem of David Mason [Mas73, Theorems 1.1, 3.14] shows that O^{2′}(G/O_{2′}(G)) is isomorphic with L_{4}(q_{2}) for some q_{2} ≡ 3 (mod 4) or with U_{4}(q_{3}) for some q_{3} ≡ 1 (mod 4). From the structure of involution centralizers in L_{4}(q_{2}) and U_{4}(q_{3}) (see [Suz86, 6.5.2, 6.5.15]) it follows that q_{2}^{2} = q and q_{3}^{2} = q in the respective cases. This completes the proof of Theorem B. □

We now begin work on the proof of Theorem A. The following hypothesis simply extracts those of Theorem A and fixes some notation.

Hypothesis 2.5. Suppose that F is a saturated fusion system on the 2-group S with F = O^{2}(F) and O_{2}(F) = 1. Assume x is a fully F-centralized involution in S with Baum(S) ⩽ C_{S}(x). Write T = C_{S}(x) and C = C_{F}(x), and assume that K = E(C) is a fusion system on a dihedral group P of order 2^{k}. Assume Q = C_{T} (K) is cyclic. Set R = QP = Q × P.

Unless otherwise specified, we assume for the remainder of this paper that F is a fusion system satisyfing Hypothesis 2.5, adopting the notation there. The next Proposition allows us to choose a suitable realization of K.

Proposition 2.6. Let P be a nonabelian dihedral group of order 2^{k}. There is a unique perfect saturated fusion system K on P. Fix a prime p ≡ 5 (mod 8), let q = p^{2^{k−2}}, and set K = L_{2}(q). Then K is strongly tamely realized by K.

Proof. Lemma 1.25(a) shows there is a unique fusion system K on P with K = O^{2}(K). That K is tamely realized by K is the content of [AOV12, Proposition 4.3]. Strong tameness follows from Proposition 1.27. □

For the remainder, we fix an odd prime power q ≡ ±1 (mod 8) with ν_{2}(q^{2} − 1) = k + 1 and K tamely realizing K as in Proposition 2.6.

Since K = E(C) is a normal subsystem of C, for each T_{1} ⩽ T with P ⩽ T_{1} we may form the product KT_{1} as in Theorem 1.12. Then O^{2}(KT_{1}) = O^{2}(K) = K.

Proposition 2.7. For each T_{1} ⩽ S with R ⩽ T_{1}, the quotient KT_{1}/Q is isomorphic to the 2-fusion system of a subgroup of Aut(K) containing K.

Proof. We verify the hypotheses of Proposition 1.26 with F = KT_{1}/Q and F_{0} = KQ/Q ≅ K. Denote quotients by Q with bars. By Proposition 2.6, K is strongly tamely realized by K and O_{2′}(K) = 1. By Lemma 1.14, we have that \overline{K} = O^{2}(\overline{KT_{1}}). Since Q ∩ P = 1, C_{\overline{T}_{1}}(\overline{KT_{1}}) = 1 by Lemma 1.19. Therefore KT_{1}/Q is the 2-fusion system of a subgroup of Aut(K) containing Inn(K) ≅ K. □

Lemma 2.8. Let f ∈ T be an involution. If f ∈ C_{T} (P) but f ∉ R, then

(a) KP ⟨f⟩ is the fusion system of K⟨f⟩ where f is an involutory field automorphism of K, and

(b) if V is a four subgroup of P, there exists a unique i ∈ {0, 1} such that C_{C}(fz^{i}) contains Aut_{K}(V). In this case, if V′ is another four subgroup of P not P -conjugate to V, then C_{C}(fz^{1−i}) contains Aut_{K}(V′).

Proof. By Theorem 1.12(c) the set of saturated subsystems of KT is in one-to-one correspondence with the set of subgroups of T containing P via the bijection X ↦ KX. Factoring by Q induces an isomorphism of KP ⟨f⟩ with the fusion system of an extension of K (by Proposition 2.7) containing an involution outside K centralizing a Sylow 2-subgroup of K. This is unique and the required extension by Lemma 2.1, proving (a).

Now by Lemma 2.1(g), C_{KP ⟨f⟩}(f) contains Aut_{K}(U) for each U in some unique P -class of four-subgroups in P. Furthermore, there is an (abstract) KP ⟨f⟩-fusion preserving isomorphism c_{h} of P ⟨f⟩ which swaps the P -classes of four-subgroups of P and interchanges f and fz by Lemma 2.1(c,h). So either Aut_{K}(V) or Aut(V′) is contained in C_{KP ⟨f⟩}(f), and the other K-automorphism group is contained in C_{KP ⟨f⟩}(fz). □

We write K⟨f⟩ in place of KP ⟨f⟩, for brevity. In view of Lemma 2.8, we also make the following definition in the situation of Hypothesis 2.5.

Definition 2.9. We say that an involution f ∈ T is an f-element on K if f ∈ C_{T} (P) but f ∉ R.

Viewing \overline{T} = T/Q as a subgroup of the Sylow 2-subgroup of Aut(K), let \overline{F} denote the intersection of \overline{T} with group of field automorphisms of Aut(K), and let F be the preimage in T of \overline{F}. Also, let F_{1} be the preimage in T of Ω_{1}(\overline{F}). Hence, F_{1} ∩ R = Q, |F_{1} : Q| = 1 or 2, and [F_{1}, P] ⩽ Q ∩ P = 1 by Lemma 2.1(g). So C_{T} (P) = F_{1}Z(P) by Lemma 2.1(c).

Let R_{d} denote the largest subgroup of T containing R for which R_{d}/Q is dihedral of order 2^{k+1}. Thus R_{d} contains R with index 1 or 2. An argument as in part (a) of Lemma 2.8 applies to give that KR_{d}/Q is the 2-fusion system of L_{2}(q) or P GL_{2}(q), respectively.

Lemma 2.10. Assume Hypothesis 2.5. If T contains an f-element on K, then T (and hence S) has 2-rank 4 and J(S) = J(RF_{1}). Otherwise T (and hence S) has 2-rank 3.

Proof. As remarked previously, C_{T} (P) = F_{1}Z(P). Thus, there is an f-element on K in T if and only if F_{1} splits over Q. If h_{1} ∈ I_{2}(T − RF), then C_{R}(h_{1}) is of 2-rank 2 by Lemma 2.1(c). Lemma 2.1(h) shows that h_{1} centralizes no involution in RF − R. Hence, C_{T} (h_{1}) is of 2-rank 3. Suppose f is an f-element of T. Then as f ∈ C_{T} (P) the 2-rank of RF is 4. Hence, J(T) = J(RF) = J(RF_{1}) 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 J(S) ⩽ Baum(S) ⩽ T, we have J(S) = J(T), and the 2-rank of S is the 2-rank of T. □

3. The 2-central case

We begin now the heart of the analysis of a fusion system F satsifying Hypothesis 2.5. 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 3.18, we will reach the conclusion that there is no such F. 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 2 and in particular of Hypothesis 2.5. Thus C = C_{F}(x), K is a component of C on the dihedral group P, T = C_{S}(x) and Q = C_{T} (K). Recall the definition of f-element from Definition 2.9 and the definitions of R_{d} and F. Also set Z(P) = ⟨z⟩, T_{0} = Ω_{1}(T), and Z = Ω_{1}(Z(T_{0})). Directly from the definition of the centralizer and the fact that K has a single class of involutions, we have

(3.1) all involutions of P^{#}x are C-conjugate

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

Lemma 3.2. z is weakly F-closed in Z.

Proof. As T_{0} = Ω_{1}(T) is weakly F-closed in T, Aut_{F}(T_{0}) controls fusion in Z = Ω_{1}Z(T_{0}) by Lemma 1.6. Suppose z is not weakly F-closed in Z, and let φ ∈ Aut_{F}(T_{0}) such that z ≠ z^{φ} ∈ Z. Since z ∈ Z(T_{0}), we may take φ to be of odd order. Set P_{i} = P^{φ^{i}} for i ∈ Z_{⩾0}. As P ⊴ T and P_{i} = Ω_{1}(P_{i}) for all i, we have P_{i} ⊴ T_{0} for all i. Then

Z(P_{i}) ∩ Z(P_{j}) = 1 for each i ≠ j ∈ {0, 1, 2} (3.3)

because φ has odd order and z^{φ} ≠ z. Furthermore, Z(P_{i}) ⩽ Z for all i.

Let i ≠ j ∈ {0, 1, 2}, and suppose that P_{i} ∩ P_{j} ≠ 1. As P_{i} ∩ P_{j} ⊴ P_{j}, Z(P_{j}) ⩽ P_{i} ∩ P_{j}. Hence Z(P_{j}) ⩽ P_{i} ∩ Z = Z(P_{i}), contrary to (3.3). As [P_{i}, P_{j}] ⩽ P_{i} ∩ P_{j}, we have P_{0}P_{1}P_{2} ≅ P_{0} × P_{1} × P_{2}. But T has 2-rank at most 4 by Lemma 2.10, so this is a contradiction. □

Lemma 3.4. If φ ∈ Hom_{F}(P, T), then z^{φ} = z.

Proof. First recall that P ⊴ T. Let C be the cyclic maximal subgroup of P. Then C ⊴ T. Now [T, T] ⩽ R from Proposition 2.7, and in fact [T, T] ⩽ C_{R}(C) = QC by Lemma 1.3(b). So T/QC is abelian. Thus

Ω_{1}([T, T]) ⩽ Ω_{1}(QC) = ⟨x, z⟩ ⩽ Z. (3.5)

Let φ ∈ Hom_{F}(P, T). Then [u_{1}, u_{2}] = z for a pair of involutions u_{1} and u_{2} of P, and so z^{φ} = [u_{1}^{φ}, u_{2}^{φ}] ⩽ Ω_{1}([T, T]) ⩽ Z by (3.5). Now Lemma 3.2 shows that z^{φ} = z. □

Proposition 3.6. Suppose F satisfies Hypothesis 2.5 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.

Lemma 3.7. z^{F} ∩ R⟨f⟩ ⊆ P.

Proof. We first show that

z^{F} ∩ C_{T} (P) = ⟨z⟩. (3.8)

Let y ∈ z^{F} ∩ C_{T} (P) and choose φ ∈ F with y^{φ} = z. Since z ∈ Z(T), φ extends to a morphism on P ⩽ C_{T} (y). Therefore y = z by Lemma 3.4, and (3.8) holds.

Now we suppose the lemma fails and let y ∈ z^{F} ∩ (R⟨f⟩ − P ∪ C_{T} (P)) be arbitrary. We claim that

(3.9) y is C-conjugate to an element of C_{T} (P).

Together with (3.8) and the fact that P is strongly C-closed, this will yield a contradiction. Recalling that R⟨f⟩ = Q⟨f⟩ × P, write y = uf_{0}v with u ∈ Q, f_{0} ∈ ⟨f⟩, and v ∈ P. Since y is an involution outside P ∪ C_{T} (P), we have uf_{0} ≠ 1 = (uf_{0})^{2}, v ∉ Z(P), and v^{2} = 1. Let V be the four subgroup of P containing v. Then Aut_{K}(V) ⩽ C_{K⟨f⟩}(f_{0}z^{i}) for some i ∈ {0, 1} by Lemma 2.8, and so there exists a morphism φ ∈ Aut_{K⟨f⟩}(V ⟨f⟩) with (f_{0}v)^{φ} = (f_{0}z^{i})^{φ}(z^{i}v)^{φ} = f_{0}z^{i} · z ∈ ⟨f, z⟩. As Q ⊴ C, this φ extends to a morphism φ̃ ∈ C fixing Q, and hence yφ̃ = (uf_{0}v)φ̃ ∈ Q⟨f, z⟩ = C_{T} (P) confirming (3.9). □

Lemma 3.10. P is weakly F-closed.

Proof. Suppose not. Choose by Alperin’s fusion theorem a fully F-normalized subgroup D ⩽ T containing P and an automorphism φ ∈ Aut_{F}(D) with P^{φ} ≠ P. Since P is normal in T, we can choose such a φ of odd order. Set P_{i} = P^{φ^{i}} for each i. Thus the subgroups P_{0} = P, P_{1} and P_{2} are distinct by choice of φ, whereas Z(P_{i}) = Z(P_{0}) = ⟨z⟩ for all i by Lemma 3.4.

Now we examine the images of the P_{i} in \overline{T} = T/Q. Since \overline{z} ≠ 1, we have that \overline{P_{i}} ≅ D_{2^{k}} for all i. Furthermore, P_{i} = Ω_{1}(P_{i}) ⩽ ⟨z^{F}⟩ as all involutions of P are F-conjugate. Thus there exists h ∈ R_{d} − R squaring into Q, and

\overline{P_{i}} ⩽ \overline{P} ⟨\overline{h}⟩

for all i by Lemma 3.7 and the fact (Lemma 2.1(h)) that there are no involutions in \overline{P} \overline{h}\overline{f}.

Suppose that \overline{P} = \overline{P_{i}} for some i. Then P Q ⩾ P_{i}, and so I_{2}(P_{i}) ⊆ R. By Lemma 3.7 then, P_{i} ⩽ ⟨z^{F} ∩ P_{i}⟩ ⩽ P. So P = P_{i}. This shows that \overline{P_{0}} ≠ \overline{P_{1}} and \overline{P_{0}} ≠ \overline{P_{2}}. But \overline{P} ⟨\overline{h}⟩ is dihedral and the \overline{P_{i}} are among the two dihedral maximal subgroups of \overline{P} ⟨\overline{h}⟩ so

\overline{P_{1}} = \overline{P_{2}}. (3.11)

Set S_{0} = P_{0}P_{1} and S_{1} = P_{1}P_{2}, so that S_{0}^{φ} = S_{1}. Then \overline{S_{0}} = \overline{P} ⟨\overline{h}⟩ ≅ D_{2^{k+1}} but

\overline{S_{1}} ≅ D_{2^{k}} (3.12)

from (3.11).

As \overline{P_{0}} ≠ \overline{P_{1}} are dihedral maximal subgroups of \overline{P} ⟨\overline{h}⟩, we have [\overline{P_{0}}, \overline{P_{1}}] is the cyclic maximal subgroup of \overline{P_{0}}. So [P_{0}, P_{1}] is the cyclic maximal subgroup of P_{0}. But [P_{0}, P_{1}] ⩽ P_{1} because the P_{i} normalize each other, and hence [P_{0}, P_{1}] is the cyclic maximal subgroup of P_{1} as well. It follows that P_{0} ∩ P_{1} has index 2 in P_{0} and P_{1}, and |S_{0}| = 2|P_{0}| = 2^{k+1}. Hence, S_{0} ≅ \overline{S_{0}} ≅ D_{2^{k+1}} and S_{1} = S_{0}^{φ} is also isomorphic to D_{2^{k+1}} with center ⟨z⟩^{φ} = ⟨z⟩. As \overline{z} ≠ 1, \overline{S_{1}} ≅ D_{2^{k+1}}. This contradicts (3.12) and completes the proof. □

Lemma 3.13. Let u be an involution in C_{T} (P). If u is fully F-centralized, then so is uz.

Proof. Let φ ∈ Hom_{F}(⟨uz⟩, T) with (uz)^{φ} fully F-centralized. Then φ extends to a morphism φ̃ on C_{T} (uz) = C_{T} (u) ⩾ P, and zφ̃ = z by Lemma 3.4. Since u is fully F-centralized, we have

|C_{T} (uz)| = |C_{T} (u)| ⩾ |C_{T} (uφ̃)| = |C_{T} (uφ̃z)| = |C_{T} ((uz)φ̃)|

and so uz is fully F-centralized as well. □

Lemma 3.14. There exists f_{0} ∈ f⟨z⟩ and φ ∈ N_{F}(P) such that f_{0}^{φ} = x.

Proof. We have R ⊴ T with T/R abelian. By Lemma 2.1(h), all involutions in T − R lie in Rh ∪ Rf. As F = O^{2}(F), there exists φ ∈ Hom_{F}(⟨f⟩, R) with f^{φ} fully F-centralized by Proposition 1.24. Then φ extends (by the extension axiom) to a morphism φ̃ on C_{T} (f) ⩾ P normalizing P by Lemma 3.10. Thus, φ̃ ∈ N_{F}(P) and so f φ̃ ∈ Ω_{1}(C_{R}(P)) − ⟨z⟩ = x⟨z⟩ as [P, f] = 1. Since one of f φ̃ or (fz)φ̃ = f φ̃z equals x, we are finished. □

Now the next two lemmas give a contradiction in the proof of Proposition 3.6.

Lemma 3.15. xz ∈ x^{F}.

Proof. Suppose not. Since x is not F-conjugate to z by Lemma 3.7, and all involutions of P^{#}x are F-conjugate by Lemma 3.1, we have that

(3.16) ⟨x⟩ is weakly F-closed in ⟨x⟩ × P.

Replacing f by fz if necessary, there exists a subgroup D ⩽ T with P ⩽ D and a morphism φ ∈ Hom_{F}(D, T) with x^{φ} = f and P^{φ} = P by Lemma 3.14. By Lemma 2.8(b), ⟨f⟩ is not weakly closed in K⟨f⟩ and so there exists ψ ∈ Hom_{F}(⟨f⟩, ⟨f⟩ × P) with f^{ψ} ∈ P f − {f}. Then x^{φψφ^{−1}} ∈ P x − {x}, which contradicts (3.16) and completes the proof. □

Lemma 3.17. xz ∉ x^{F}.

Proof. Let φ ∈ F with x^{φ} = xz. Then as x and z lie in Z(T) we may assume φ ∈ Aut_{F}(T) is of odd order by the extension and Sylow axioms. Then z^{φ} = z by Lemma 3.10, and so φ induces an automorphism of ⟨x, z⟩ of order 2, a contradiction. □

This completes the proof of Proposition 3.6. We now can prove the main result of this section.

Proposition 3.18. Suppose F satisfies Hypothesis 2.5. Then T < S.

Proof. Suppose to the contrary that T = S. From Proposition 3.6, there is no involutory f-element in T, and hence there are no involutions in Rf. By Lemma 2.1(h), it follows that

T_{0} ⩽ R_{d} (3.19)

and hence Z = ⟨x, z⟩ ⩽ Z(T) in the present case. Furthermore,

(3.20) T is of 2-rank 3

by Lemma 2.10.

In view of Lemma 3.2, we know z is not F-conjugate to x or xz. Since fusion in Z(T) is controlled in Aut_{F}(T),

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

Our assumption that O_{2}(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 ∈ Aut_{F}(D) with x^{α} ∉ Z. Set h = x^{α}.

Note that h ∉ R, as otherwise h would lie in Ω_{1}(R) = ⟨x⟩ × P. Because all involutions of P ∪ P^{#}x are F-conjugate to xz or z, (3.21) would yield h = x, contrary to the choice of h. Therefore, by (3.19):

(3.22) h ∈ R_{d} − R and P ⟨h⟩ is dihedral.

Since D is F-centric, it contains Z = ⟨x, z⟩ ⩽ Ω_{1}(Z(T)), and hence Ω_{1}(Z(D)) = Z⟨h⟩ as T is of 2-rank 3. Set A = Z⟨h⟩. Then h is N_{P} (A)-conjugate to hz by (3.22). If h is Aut_{F}(A)-conjugate to hx or hxz then it is Aut_{F}(A)-conjugate to both, so x has exactly five conjugates under Aut_{F}(A) by (3.21), which is not the case. So

h^{Aut_{F} (A)} = {x, h, hz}. (3.23)

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

Ω_{1}(T) = Ω_{1}(R_{d}) = ⟨x⟩ × P ⟨h⟩. (3.24)

We claim that

(3.25) P ⟨h⟩ is normal in T.

If this does not hold, then from (3.24) and the fact that P is normal in T, there exists t ∈ T with h^{t} ∈ P hx ∪ P hxz. But all involutions in P h are P -conjugate (3.22), and hence multiplying t by a suitable element of P, we have that h is N_{T} (A)-conjugate to hx or to hxz, contradicting (3.23).

We now complete the proof via transfer arguments. Note that F is cyclic or quaternion by Proposition 3.6. If F is cyclic, then as it covers T/P ⟨h⟩, we can apply Proposition 1.23 to get that x is F-conjugate to z, an immediate contradiction to (3.21). So F is quaternion and consequently, |F : Q| = 2 by an argument analogous to that of Lemma 1.3(c). Let w ∈ F − Q of order 4, so that w^{2} = x. In the present situation, R_{d} = Q × P ⟨h⟩, w is of least order in T − R_{d} by (3.24), and T/R_{d} is cyclic of order 2. So Proposition 1.23 yields a morphism φ ∈ F with w^{φ} ∈ R_{d}, and hence (w^{φ})^{2} ∈ Z by the structure of R_{d}. As w^{2} = x, x^{φ} = x by (3.21). Thus φ ∈ C and so φ extends to a morphism φ̃ on a subgroup of T containing Q because Q ⊴ C. This forces F φ̃ ⩽ ⟨Q, wφ̃⟩ to be abelian, a final contradiction. □

4. The 2-rank 3 case

Continuing the notation from Section 3, we prove here the following reduction.

Theorem 4.1. Let F be a fusion system on S satifying Hypothesis 2.5. Then T is of 2-rank 4.

Throughout this section, assume to the contrary that T is of 2-rank 3. By Hypothesis 2.5, S is also of 2-rank 3. From Proposition 3.18, we may assume that

x ∉ Z(S). (4.2)

Recall F_{1} contains Q with index 1 or 2 and F_{1}/Q induces field automorphisms on K. Then

C_{T} (P) = F_{1} × ⟨z⟩. (4.3)

By Lemma 2.10, there exists no involution in T which is an f-element. Set J = J(S) = J(T) for short. We have the inclusions P ⩽ J ⩽ Ω_{1}(T) ⩽ R_{d}. This shows that Z = Ω_{1}(Z(Ω_{1}(T))) ⩽ Ω_{1}(C_{T} (P)) = Ω_{1}(F_{1} × ⟨z⟩) = ⟨x, z⟩. So Z = ⟨x, z⟩ and Z coincides with Ω_{1}(Z(J)). Therefore, since Baum(S) ⩽ T and (4.2), we have that T = Baum(S) and

(4.4) T is of index 2 in S.

Fix a ∈ S − T.

Lemma 4.5. z ∉ x^{F} ∪ (xz)^{F}.

Proof. Since x is fully F-centralized and not central in S, we need only show that z ∈ Z(S). Suppose that z^{a} ≠ z and hence P^{a} ≠ P. Since P and P^{a} are normal in T, we have [P^{a}, P] ⩽ P^{a} ∩ P is normal in both P and P^{a}. Furthermore, [P^{a}, P] ≠ 1 since otherwise P^{a}P = 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 x^{a} = xz.

Lemma 4.6. Q = ⟨x⟩ and F is cyclic.

Proof. Suppose that Q > ⟨x⟩ and let u ∈ Q with u^{2} = x. Then ⟨u^{a}⟩ is a normal subgroup of T. Since (u^{a})^{2} = xz, we have [⟨u^{a}⟩, T] ⩽ ⟨xz⟩. But [⟨u^{a}⟩, P] ⩽ P as P is normal, and it follows that u^{a} ∈ C_{T} (P) = F_{1} ×⟨z⟩ by (4.3). Therefore, xz ∈ ℧^{1}(⟨u^{a}⟩) ⩽ ℧^{1}(F_{1} ×⟨z⟩) ⩽ Q, which is absurd. So Q = ⟨x⟩ and consequently F is cyclic (since F/Q is cyclic). □

Lemma 4.7. Let V be a four subgroup of P and set E = ⟨x⟩ × V. Then E^{a} is not T -conjugate to E.

Proof. Suppose that E^{a} is T -conjugate to E. Modifying a if necessary, we may assume that a normalizes E. Now the subgroup N = ⟨c_{a}, Aut_{C}(E)⟩ of Aut_{F}(E) lies in L_{3}(2) and does not act transitively on I_{2}(E) by Lemma 4.5. As x^{a} = xz, N does not stabilize a point of E. So N must fix a line, which is then V. It follows that N = Aut_{F}(E) ≅ S_{4}. Now | Aut_{T} (E)| = 2, and we can obtain a contradiction to (4.4) by showing that | Aut_{S}(E)| = 8, i.e. that E is fully automized in F.

Suppose that E is not fully F-automized. Either J = ⟨x⟩×P or there exists an involution h ∈ R_{d} − R and J = ⟨x⟩ × P ⟨h⟩. In either case, there are exactly two S-classes of elementary abelian subgroups of order 8. Moreover, if E_{1} ∈ E^{F} is fully F-automized, then E_{1} ∉ E^{S}, and so ⟨E, E_{1}⟩ = J. By Alperin’s fusion theorem, there is a subgroup D ∈ F^{fc} and an automorphism α ∈ Aut_{F}(D) of odd order such that E_{1} := E^{α} is fully F-automized. But then J = ⟨E, E_{1}⟩ ⩽ 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 1.4, a contradiction. □

Lemma 4.8. R_{d} > R.

Proof. Suppose on the contrary that R_{d} = R. If |T: P F | = 2, then J = ⟨x⟩ × P, and T acts transitively on ℰ_{2^{3}}(T) contrary to Lemma 4.7. So T = P F. If |F | > 2, then Z(T) = Ω_{2}(F) × ⟨z⟩ and so ℧^{1}(Z(T)) = ⟨x⟩ is normal in S, at odds with (4.2). So T = R = J = ⟨x⟩ × 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 = ⟨z^{F} ∩ T ⟩ is normal in S by Lemma 4.5. Moreover, the quotient S/P is abelian. If b ∈ x^{F} is fully F-centralized, then C_{S}(b) has 2-rank 3, whence b ∈ T ∩ x^{F} ⊆ P x. Proposition 1.24 now says that x is F-conjugate into P, contradicting Lemma 4.5 and completing the proof. □

As a consequence of the previous lemma, T is transitive on ℰ_{2^{3}}(T). Fix a four subgroup V of P. By Lemma 4.7 and the preceding remark, V^{a} ∈ R_{d} − R. Fix an involution h ∈ V^{a} − P. Then P_{1} := P ⟨h⟩ is dihedral of order 2|P |, and therefore is generated by F-conjugates of z. We have at this point that J = ⟨x⟩ × P_{1} = R_{d}. As every involution in P hx is P -conjugate to hx, and hence S-conjugate into P x, it follows that

P_{1} = ⟨z^{F} ∩ P_{1}⟩ = ⟨z^{F} ∩ T ⟩. (4.9)

So

(4.10) P_{1} is normal in S.

Recall now that C_{T} (P) = Ω_{2}(F)×⟨z⟩ from (4.3) and Lemma 4.6. Moreover, if Ω_{2}(F) > ⟨x⟩, then [Ω_{2}(F), h] = ⟨z⟩ by (4.10) and Lemma 2.1(h). Consequently C_{S}(P_{1}) = Z and so

(4.11) S/P_{1}Z is abelian

by Lemma 1.3 because P_{1} is nonabelian dihedral.

Notice F P_{1}/P_{1} is a cyclic normal subgroup of S/P_{1} of index 2. So S/P_{1} is either abelian or modular, or else |F | = 4 by (4.11) and S/P_{1} is dihedral or quaternion. We rule out each of these cases in turn.

Lemma 4.12. S/P_{1} is not abelian.

Proof. Suppose S/P_{1} is abelian. For any b ∈ x^{F} which is fully F-centralized, C_{S}(b) is of 2-rank 3, and so b ∈ J ∩ x^{F} ⊆ P_{1}x by (4.9) and Lemma 4.5. Now x has an F-conjugate in P_{1} by Proposition 1.24, and this contradicts Lemma 4.5. □

The next lemma shows that S/P_{1} is not quaternion.

Lemma 4.13. There exists an involution b in S − T.

Proof. Let b_{1} ∈ S − T. Modifying b_{1} by an element of F, we may assume b_{1}^{2} ∈ P_{1}Z because S/P_{1} is not cyclic (by the previous lemma). But then P_{1}Z⟨b_{1}⟩/⟨x⟩ is dihedral or semidihedral because b_{1} swaps the P_{1}-classes of four-subgroups of P_{1} by Lemma 4.7. Modifying b_{1} by an element of P_{1} then, we may assume b_{1}^{2} ∈ Z. Now C_{Z}(b_{1}) = ⟨z⟩ so b_{1}^{2} ∈ ⟨z⟩. Set b = b_{1} if b_{1} is an involution, and set b = xb_{1} otherwise. Then b is an involution. □

Lemma 4.14. S/P_{1} is not modular.

Proof. Suppose it is. Then S/P_{1} has a unique four subgroup, covered by ⟨x, b⟩. Hence, S_{0} := Ω_{1}(S) = P_{1}Z⟨b⟩ and S/S_{0} is cyclic. Let w ∈ F with w^{2} = x. Then w is of least order outside S_{0} and centralizes F P, whence |S: C_{S}(w)| ⩽ 4. Apply Proposition 1.23 to obtain a morphism φ in F with w^{φ} in S_{0} and fully F-centralized.

Any element of S_{0} − P_{1}Z interchanges the two classes of four-subgroups of P_{1}. Hence if b_{1} ∈ S_{0} − P_{1}Z is of order 4, then b_{1}^{2} ∈ Z and b_{1} induces an involutory automorphism of P_{1} interchanging the two classes of four-subgroups of P_{1}. So C_{P_{1}}(b_{1}) = ⟨z⟩ and |S: C_{S}(b_{1})| ⩾ |P_{1} : C_{P_{1}}(b_{1})| ⩾ 8 as |P_{1}| ⩾ 16.

As w^{φ} is fully F-centralized, the preceding paragraph implies w^{φ} ∈ P_{1}Z = ⟨x⟩ × P_{1}, and consequently w^{φ} = x_{0}v for some x_{0} ∈ ⟨x⟩ and v ∈ P_{1} of order 4. Now x^{φ} = (w^{φ})^{2} = z, contrary to Lemma 4.5. □

Therefore by the previous three lemmas, |F | = 4 and S/P_{1} is dihedral of order 8. We now obtain the final contradiction, completing the proof of Theorem 4.1.

Lemma 4.15. S/P_{1} is not dihedral.

Proof. Suppose it is. Again let w ∈ F with w^{2} = x. Then F = ⟨w⟩. Since C_{T} (F) = R is of index 2 and [F, h] = ⟨z⟩, we have F ⩽ Z_{2}(T) in the present situation. Moreover, T/Z = F/Z × P_{1}Z/Z with the second factor dihedral of order at least 8, and so Z_{2}(T) = F × V where V is cyclic of order 4 in P_{1}. Now b inverts P_{1}w by assumption; hence ww^{b} ∈ Z_{2}(T) ∩ P_{1} = V. As [w, w^{b}] = 1, we have on the one hand that ww^{b} ∈ C_{V} (b) = ⟨z⟩, because b^{2} = 1, and on the other (ww^{b})^{2} = w^{2}(w^{b})^{2} = w^{2}(w^{2})^{b} = x(xz) = z. These two facts are incompatible, and the proof is complete. □

5. The 2-rank 4 case: |Q| = 2

For a fusion system F on S satisfying Hypothesis 2.5, T has 2-rank 3 or 4 by Lemma 2.10. By Theorem 4.1, 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.

Theorem 5.1. Assume that F satisfies Hypothesis 2.5 with T of 2-rank 4. Then |Q| > 2.

The notation follows that begun in Section 2, in particular that of Hypothesis 2.5. For instance, K is the unique component of the involution centralizer C = C_{F}(x), and is a fusion system of a finite group K isomorphic with L_{2}(q) for suitable q ≡ ±1 (mod 8) (as chosen once and for all after Proposition 2.6). The Sylow subgroup of K is denoted by P, a dihedral group of order 2^{k} (k = ν_{2}(q^{2} − 1) − 1 ⩾ 3). Consistent with Sections 3 and 4, we also set Z(P) = ⟨z⟩. Denote by C the cyclic maximal subgroup of P. By Hypothesis 2.5, the Thompson subgroup J(T) = J(S) and so this common subgroup is denoted simply by J.

By Lemma 2.10,

(5.2) there exists an involutory f-element f ∈ C_{T} (P).

We fix such an involution f.

Before beginning the proof of Theorem 5.1, we collect some facts seen before, and which hold throughout 2-rank 4 case. In particular,

J ⩽ R⟨f⟩ = Q⟨f⟩ × P (5.3)

from Lemma 2.8, and

C_{T} (P) = Q⟨f⟩ × ⟨z⟩ (5.4)

from Lemma 2.7 and the structure of Aut(K) in Lemma 2.1. Finally,

T < S (5.5)

by Proposition 3.18.

Assume for the remainder of this section that Q = ⟨x⟩ is of order 2, as we prove Theorem 5.1 by way of contradiction in a series of lemmas. Thus, J = ⟨x, f⟩ × P by (5.3) and Z(J) = ⟨x, f, z⟩ is elementary abelian of order 8.

Lemma 5.6. z^{F} ∩ Z(J) = ⟨z⟩.

Proof. The Thompson subgroup J = ⟨x, f⟩ × P is weakly F-closed. By Lemma 1.6 fusion in Z(J) is controlled in Aut_{F}(J). But ℧^{1}(J) ∩ Z(J) = ⟨z⟩ is characteristic in J, so the statement follows. □

Lemma 5.7. Let y ∈ Z(J). Then each involution of yP is C-conjugate to y or to yz. In particular, z^{F} ∩ J = z^{F} ∩ P.

Proof. Since K has one class of involutions and x ∈ C_{T} (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 ∉ ⟨x, z⟩. 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 = ⟨t, z⟩, and let φ ∈ Aut_{K}(U) of order 3 such that t^{φ} = z. Then φ extends to φ̃ ∈ C on U⟨y⟩ and centralizes either y or yz by Lemma 2.8. In the former case, (yt)φ̃ = yz, and in the latter, (yt)^{φ̃^{2}} = (yztz)^{φ̃^{2}} = y. This completes the proof of the first statement. The second statement now follows from Lemma 5.6. □

Lemma 5.8. The following hold.

(a) P is normal in S,

(b) there exists a fully F-centralized four subgroup of P,

(c) [S, S] ⩽ C_{S}(C), and

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

Proof. Let s ∈ S. Then z^{s} ∈ Z(J), and so z^{s} = z. But all involutions of P^{s} are K^{s}- conjugate by Lemma 2.1(f). Hence P^{s} = Ω_{1}(P^{s}) = ⟨z^{K^{s}}⟩ ⩽ P by Lemma 5.7 and (a) holds.

Let U be a four subgroup of P, and let ψ ∈ Hom_{F}(U, S) such that U^{ψ} is fully Fcentralized. By the extension axiom, ψ extends to C_{S}(U), which contains an elementary abelian subgroup of maximal rank. Thus, U^{ψ} ⩽ J and the nonidentity elements of U^{ψ} consist of F-conjugates of z. It follows that U^{ψ} ⩽ P by Lemma 5.7, proving (b).

Part (c) is Lemma 1.3(b). For (d), suppose that s ∈ S with s^{2} ∈ J − Z(J)C. Then s^{2} must lie in C_{S}(P)C by Lemma 1.3(c). Thus, s^{2} ∈ C_{J}(P)C = Z(J)C. □

Lemma 5.9. J = P C_{S}(P).

Proof. Suppose the lemma is false and choose a ∈ C_{S}(P) − J with a^{2} ∈ J. Let Z_{a} = C_{Z(J)}(a), which contains z and is of order 4. Then a^{2} ∈ Z(J) ∩ C_{S}(a) = Z_{a} since a centralizes P. By (5.4) and (5.3), J = P C_{T} (P), and so a does not centralize x.

Fix a fully F-centralized four subgroup U of P guaranteed by Lemma 5.8(b), a Kautomorphism φ of U of order 3, and an extension φ̃ ∈ Hom_{F}(C_{S}(U), S) (by the extension axiom). Observe that

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

because each element of Z(J)a lies outside J and centralizes the subgroup Z_{a}U, which is of 2-rank 3. Also, φ̃ is defined on a; it follows that

a^{2} ≠ z, (5.11)

since otherwise aφ̃ is an element of S squaring to a noncentral involution of P, contrary to Lemma 5.8(d).

If [x, a] = z, then [xφ̃, aφ̃] = z^{φ} is a noncentral involution of P, contradicting Lemma 5.8(c). Finally, we consider the case in which [x, a] = y ≠ z. Here, a^{2} ∈ Z_{a} = ⟨y, z⟩. If a^{2} = y, then x inverts a, and so (xa)^{2} = 1 contrary to (5.10). Hence a^{2} = yz by (5.11). In this case, we may replace a by xa to obtain a^{2} = z, again contradicting (5.11), and completing the proof. □

Let Ω be the two element set consisting of the P -classes of four-subgroups of P. Let N be kernel of the action of S on Ω. Then J ⩽ N. By the previous two lemmas S/J embeds into Out(P). Thus by Lemma 1.3(a), S/J has a cyclic subgroup B with index 1 or 2 and with B = N/J cyclic of order dividing 2^{k−3}. Thus, N is of index 1 or 2 in S.

Lemma 5.12. We have J < N. In particular, |P | ⩾ 16.

Proof. Suppose to the contrary that J = N. Then N ⩽ T and T < S by (5.5). Since |S : N| ⩽ 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) = ⟨y, z⟩ for some y ∈ Z(J) − ⟨x, z⟩. 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) − ⟨z⟩ is not F-conjugate to z by Lemma 5.6. But then Proposition 1.24, applied with C representing the T of that Proposition, forces z ∈ y^{F} 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

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

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 ⟨a⟩ because Z(J) ∩ P = ⟨z⟩ and P ⟨a⟩ is transitive on the involutions in P − C. So |C_{S}(x)| > |C_{S}(s)|, and (5.13) holds.

Thus, (5.13) 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 Proposition 1.24, with Z(S)C = [S, S] in the role of T, to obtain an F-conjugate of x in Ω_{1}([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 four-subgroups of P. Thus N induces inner automorphisms on P, i.e. N = P C_{S}(P) = J, contrary to J < N. Therefore, |P | ⩾ 16. □

By an earlier remark and Lemma 5.12, 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 four subgroup U = ⟨e, z⟩ of P. Replacing w by ew if necessary, we may assume also that

(5.14) w centralizes C/℧^{2}(C).

Let f_{1} ∈ ⟨w⟩ such that f_{1} ∉ J but f_{1}^{2} ∈ J. Since J = P C_{S}(P) by Lemma 5.9, we have f_{1}^{2} ∈ C_{S}(C) by Lemma 1.3(d) applied with D = P there. Then f_{1} takes a generator c of C to cz by choice of f_{1} ∈ ⟨w⟩ and (5.14), and hence f_{1} centralizes ℧^{1}(C). As |P | ⩾ 16 from Lemma 5.12, it follows that

(5.15) C_{P} (f_{1}) is the nonabelian dihedral subgroup P_{1} := ⟨U, ℧^{1}(C)⟩ of P.

So f_{1}^{2} ∈ C_{S}(P_{1}) ∩ J = Z(J), and f_{1} is of order at most 4. But f_{1} is not an involution, otherwise ⟨f_{1}, C_{Z(J)}(f_{1}), U⟩ is an elementary 16 outside J, so f_{1} is of order 4. In fact, it is shown below that there are no involutions in Jf_{1}. For this, we will need that

(5.16) f_{1} does not square to z.

Assume to the contrary that f_{1}^{2} = z. Then as [℧^{1}(C), f_{1}] = 1 and |P | ⩾ 16 by Lemma 5.12, there exists an element v ∈ C_{P} (f_{1}) with (vf_{1})^{2} = 1. But then C_{P} (vf_{1}) contains the four subgroup ⟨ce, z⟩ of P, and so vf_{1} ∈ J, yielding the same contradiction as before and thus confirming (5.16). We now show

Lemma 5.17. There is no involution in Jf_{1}.

Proof. By (5.16), we may assume that f_{1}^{2} ∈ Z(J)−⟨z⟩. If [Z(J), f_{1}] ⩽ ⟨z⟩, then J⟨f_{1}⟩/P ≅ C_{2} × C_{4} with P f_{1} of order 4. In this case, every element of order 2 in J⟨f_{1}⟩ lies in J as claimed. Hence we may assume that [Z(J), f_{1}] = ⟨y⟩ ≠ ⟨z⟩. Then Z(J)P_{1}⟨f_{1}⟩ = ⟨y_{1}, f_{1}⟩ × P_{1} with D_{1} := ⟨y_{1}, f_{1}⟩ ≅ D_{8}. If E_{1} is the other four subgroup of D_{1}, then E_{1} × P_{1} has 2-rank 4 and so E_{1} ⩽ J. But then D_{1} ⩽ C_{S}(P_{1}) ∩ J = Z(J), a contradiction. □

In addition, we let h_{1} ∈ S − N be an element such that h_{1}^{2} ∈ J or set h_{1} = 1 if such an element does not exist. Note that if h_{1} = 1, then S/J is cyclic by the structure of Out(P) (Lemma 1.3(a)). In any case, S/J⟨h_{1}⟩ is cyclic.

Assume that h_{1} ≠ 1. Then

(5.18) both h_{1} and h_{1}f_{1} square into J.

Let s ∈ Jh_{1} ∪ Jh_{1}f_{1}. Then for any e_{1} ∈ P − C, we have e_{1}^{s} = e_{1}c for some generator c of C because s ∉ N. Therefore,

(5.19) [P, s] = C and C_{J}(s) ⩽ Z(J)C is abelian.

Furthermore, as e_{1}^{s^{2}} = e_{1}cc^{s}, we have

(5.20) if s^{2} ∈ C_{S}(P) then s inverts C.

With this setup, the next two lemmas contradict each other and complete the proof of Theorem 5.1.

Lemma 5.21. h_{1} ≠ 1 and both cosets Jh_{1} and Jh_{1}f_{1} contain involutions.

Proof. Suppose either that h_{1} = 1 or that there are no involutions in Jh_{1}f_{1}. The argument is the same in case h_{1} ≠ 1 and Jh_{1} contains no involutions. (Alternatively, swap the roles of h_{1} and h_{1}f_{1} in this extra case.) By (5.17) and assumption, Ω_{1}(S) ⩽ J⟨h_{1}⟩. Also S/J⟨h_{1}⟩ is cyclic, and f_{1} is of least order outside J⟨h_{1}⟩. By Proposition 1.23, there exists a morphism φ ∈ F such that f_{1}^{φ} ∈ J⟨h_{1}⟩ is fully F-centralized, and C_{S}(f_{1})^{φ} ⩽ C_{S}(f_{1}^{φ}).

Now if h_{1} ≠ 1, then f_{1}^{φ} cannot lie in the coset Jh_{1}. This is because Ω_{1}(C_{P} (f_{1})) is nonabelian dihedral by (5.15), whereas Ω_{1}(C_{S}(s)) is abelian for every s ∈ Jh_{1} by (5.19). So f_{1}^{φ} ∈ J whether or not h_{1} = 1. Since f_{1} is of order 4 and Ω_{1}(℧^{1}(J)) = ⟨z⟩, we have that (f_{1}^{2})^{φ} = z. But z is weakly F-closed in Z(J) by Lemma 5.6 and so f_{1}^{2} = z, contrary to (5.16). □

Lemma 5.22. Jh_{1}f_{1} contains no involution.

Proof. We may assume h_{1} ≠ 1 = h_{1}^{2} by Lemma 5.21, and then h_{1} inverts C by (5.20). So

(5.23) h_{1}f_{1} sends a generator c of C to c^{−1}z.

In particular, C_{C}(h_{1}f_{1}) = ⟨z⟩ and so C_{Z(J)C}(h_{1}f_{1}) ⩽ Z(J)Ω_{2}(C). As (h_{1}f_{1})^{2} ∈ J from (5.18), we have (h_{1}f_{1})^{2} ∈ C_{J}(h_{1}f_{1}) = C_{Z(J)C}(h_{1}f_{1}) ⩽ Z(J)Ω_{2}(C) with the equality by (5.19). But (h_{1}f_{1})^{2} does not lie in C_{S}(P) = Z(J) by (5.20), since h_{1}f_{1} does not invert C.

Set M = J⟨h_{1}f_{1}⟩ and \overline{M} = M/Z(J). Then \overline{M} contains the dihedral group \overline{P} as a maximal subgroup, which is nonabelian as |P | ⩾ 16. Furthermore \overline{M} is of maximal class by (5.19) and (5.23). As \overline{h_{1}f_{1}} is of order 4 squaring into the center of \overline{M}, we know \overline{M} is semidihedral. But then \overline{M} contains no involutions outside its dihedral maximal subgroup \overline{P}. It follows that M = J⟨h_{1}f_{1}⟩ contains no involutions outside J, which is what was to be shown. □

6. The 2-rank 4 case: |Q| > 2

For this final section, we continue to assume F is a saturated fusion system on the 2group S satisfying Hypothesis 2.5. By the main results of the previous three sections, we are reduced to the following situation in describing F.

(1) T = C_{S}(x) is a proper subgroup of S (Proposition 3.18),

(2) S is of 2-rank 4 (Theorem 4.1), and

(3) Q = C_{T} (K) is of order at least 4 (Theorem 5.1).

Theorem 6.1. Let F be a saturated fusion system on the 2-group S. Assume F satisfies Hypothesis 2.5 and, in addition, the above three items. Then S ≅ D_{2^{k}} ≀ C_{2}, and F is the fusion system of L_{4}(q) for some q ≡ 3 (mod 4) with ν_{2}(q + 1) = k − 1.

Adopt the notation of Hypothesis 2.5 and the set up at the beginning of Section 5. By Lemma 2.10,

there exists an involutory f-element f ∈ C_{T} (P).

We continue to fix such an involution f. As T a proper subgroup of S, we also

fix a ∈ N_{S}(T) − T with a^{2} ∈ T.

As usual, we prove Theorem 6.1 in a sequence of lemmas. It will emerge quickly (after Lemma 6.8) that J = R⟨f⟩ is the product of two dihedral groups Q⟨f⟩ 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 2.7), this means that either T = J = R⟨f⟩, or T = R⟨h, f⟩ for some 1 ≠ h ∈ T such that h^{2} ∈ Q and R⟨h⟩/Q is dihedral of order 2|P |. Thus R⟨h⟩K/Q is uniquely determined as the fusion system of P GL_{2}(q) (see the description of Out(K) in Lemma 2.1). In anticipation of this we let

(6.2)  h ∈ T − R such that h^{2} ∈ Q and R⟨h⟩/Q is dihedral, or
h = 1 if such an element does not exist.

Note in the case h ≠ 1,

[Qh, Qf] = Qz (6.3)

by Lemma 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 Q⟨h⟩ splits over Q. Together with the target F = F_{S}(L_{4}(q)) appearing within the case T = J, the following table lists the fusion systems (of finite groups) which nearly satisfy the conditions of Theorem 6.1, and why they are eventually ruled out.

Scenario  T  S  Group  Contradiction
h ≠ 1 and Q⟨h⟩ is dihedral J⟨h⟩ Q_{2^{k+1}} ≀_{∗} C_{2} P Sp_{4}(q) C_{T} (K) ≅ D_{2^{k}}
h ≠ 1 and Q⟨h⟩ is cyclic J⟨h⟩ SD_{2^{k+1}} ≀_{∗} C_{2} P GL_{4}(q)  O^{2}(F) < F
h = 1  J  D_{8} ≀ C_{2}  A_{10}  C_{T} (K) ≅ C_{2} × C_{2}
h = 1  J  D_{2^{k}} ≀ C_{2}  L_{4}(q)

The 2-group and transfer analysis is carried out through Lemma 6.23, where it is shown that S is of type P Sp_{4}(q) or P GL_{4}(q) when h ≠ 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 GL_{4}(q)-case. Analyzing the resulting fusion information allows us to conclude that that h = 1, and S is then isomorphic to D_{2^{k}} ≀ C_{2}. Lastly, we appeal to a result of Oliver [Oli] to identify F as the fusion system of L_{4}(q).

We begin by pinning down the structure of J in the next few lemmas.

Lemma 6.4. The following hold.

(a) Q⟨f⟩ is dihedral or semidihedral,

(b) J = Ω_{1}(Q⟨f⟩) × P,

(c) no element of Rf is a square in T, and

(d) T = R⟨h, f⟩.

Proof. We claim that Z(Q⟨f⟩) = ⟨x⟩. 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 = ⟨x, f⟩ × P by (5.3). Then D = C_{T} (J) is normal in S, and D is equal to Q × ⟨f, z⟩ if f centralizes Q and to ℧^{1}(Q) × ⟨f, z⟩ otherwise. In any case, x is the only involution which is a square in D, so x ∈ Z(S). This contradicts T < S. Thus Q⟨f⟩ is of maximal class and f is an involution outside Q, so (a) holds. Now (b) follows by (5.3).

Suppose some element f_{1} ∈ T squares into Rf. Write T̃ = T/P and denote images modulo P similarly. Then D_{1} := Q̃⟨f̃_{1}⟩ contains D := Q̃⟨f̃⟩ as a normal subgroup. Moreover as Q⟨f⟩ ∩ P = 1, D ≅ Q⟨f⟩. If Q⟨f⟩ is dihedral in (a), then Lemma 1.3(d), with D_{1} in the role of S there, gives a contradiction. If Q⟨f⟩ is semidihedral, then considering the images of D_{1} and D modulo ⟨x̃⟩, we obtain the same contradiction to Lemma 1.3(d). Therefore,

(c) holds. By the structure of Out(K) and (c), T/R is a four group covered by ⟨h, f⟩, yielding (d). □

Lemma 6.5. x^{S} = {x, z}. Consequently, |S : T | = 2.

Proof. By Lemma 6.4, S normalizes Z(J) = ⟨x, z⟩ and J = J(Q⟨f⟩) × P = Ω_{1}(Q⟨f⟩) × P is the product of two nonabelian dihedral groups. By Proposition 1.2, S must permute the commutator subgroups of these factors. Therefore, x^{S} ⊆ {x, z} and as T = C_{S}(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 ⊴ S = T ⟨a⟩ and a swaps x and z. So Z(S) = ⟨xz⟩, and

(6.6) x is not F-conjugate to xz

because x ∉ Z(S) is fully F-centralized.

Lemma 6.7. [P^{a}, P] = P^{a} ∩ P = 1.

Proof. Note first that both P and P^{a} are normal in T, So [P, P^{a}] ⩽ P ∩ P^{a}. Suppose that Z_{0} := P ∩ P^{a} ≠ 1. Then Z_{0} is nontrivial normal in P^{a}, and so ⟨x⟩ = Z(P^{a}) ⩽ Z_{0} ⩽ P, a contradiction. □

Lemma 6.8. Q⟨f⟩ is dihedral of the same order as P.

Proof. Recall from Lemma 6.4 that Ω_{1}(Q⟨f⟩) is nonabelian dihedral and J = Ω_{1}(Q⟨f⟩)×P. Let C_{1} and C_{2} be the cyclic maximal subgroups of Ω_{1}(Q⟨f⟩) and P, respectively. Because x^{a} = z, we have C_{1}^{a} is a cyclic subgroup of J with z as its unique involution, and so |C_{1}| ⩽ |C_{2}| by the structure of J. We conclude similarly that |C_{2}| ⩽ |C_{1}| by considering C_{2}^{a}, and hence C_{1} and C_{2} are of the same order. Therefore either Q⟨f⟩ is either dihedral of the same order as P, or Q⟨f⟩ is semidihedral with |Q⟨f⟩| = 2|P |.

Suppose Q⟨f⟩ is semidihedral. Then Ω_{1}(Q⟨f⟩) ≅ P. Set S_{1} = (Q⟨f⟩)^{a} for short. Then S_{1} centralizes P^{a}. But P also centralizes P^{a} by Lemma 6.7. In fact,

P^{a} × ⟨z⟩ ⩽ Ω_{1}(C_{T} (P)) = Ω_{1}(Q⟨f⟩) × ⟨z⟩

so the above three subgroups are equal, as the outside two are of the same order. Taking centralizers, we get that

S_{1} ⩽ C_{T} (P^{a}) = C_{T} (Ω_{1}(Q⟨f⟩)) ⩽ C_{T} (f) (6.9)

Now f centralizes no element in the coset Rh when h ≠ 1 by (6.3). So C_{T} (f) = C_{R⟨f⟩}(f) = P × ⟨f, x⟩ by Lemma 6.4(d,a), and (6.9) is a contradiction because P × ⟨f, x⟩ contains no semidihedral subgroup. □

In view of the previous lemma, it is now determined that

J = Q⟨f⟩ × P = P^{a} × P (6.10)

with a interchanging P and P^{a}. Since P^{a} ⩽ C_{T} (P) = Q⟨f⟩ × ⟨z⟩, we may replace f by fz and assume that

f ∈ P^{a}. (6.11)

We fix notation for the maximal cyclic subgroup of P, calling it C. Then

C^{a} ⩽ Q⟨z⟩ and Q^{a} ⩽ C⟨x⟩ (6.12)

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 6.25 to rule out the P Sp_{4}(q)-case mentioned above and determine that in fact h = 1.

Lemma 6.13. The following hold.

(a) If C = ⟨c⟩, then ff^{a} is not F-conjugate to f(cf^{a}).

(b) There exists an involution in S − T.

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 (6.11) we have chosen f ∈ P^{a}, so f^{a} ∈ P. Thus, U_{0} = ⟨f^{a}, z⟩ and U_{1} = ⟨cf^{a}, z⟩ are four-subgroups of P which are not P -conjugate. Since f is an f-element on K (Definition 2.9), C_{C}(f) contains Aut_{K}(U_{j}) for some j, and C_{C}(fz) contains Aut_{K}(U_{1−j}) by Lemma 2.8(b). Thus, there is an element φ ∈ Aut_{C}(⟨f⟩U_{j}) of order 3 with (f · c^{j}f^{a})^{φ} = f^{φ}(c^{j}f^{a})^{φ} = fz. On the other hand, there is a similar element ψ ∈ Aut_{C}(⟨f⟩U_{1−j}) with (fz · zc^{1−j}f^{a})^{ψ} = fz · z = f. Since f is K^{a}-conjugate to x in C_{F}(z), and so fz is C_{F}(z)-conjugate to xz, this contradicts (6.6).

For (b), suppose that a^{2} ∈ J. It will be shown first that (b) holds in this situation. Write a^{2} = ts^{−1} with t ∈ P^{a} and s ∈ P. Let a_{0} = as. Then P^{a_{0}} = P^{a}, and a_{0}^{2} = a^{2}s^{a}s = ts^{a} as [P, P^{a}] = 1. So a_{0}^{2} ∈ P^{a} and centralizes a_{0}. Therefore a_{0}^{2} = 1, as claimed.

So it remains to prove that a^{2} ∈ J. If a does not square into J, then h ≠ 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 ⟨x⟩ ∪ P^{a}⟨z⟩. Thus 𝒥 has cardinality 4, and for any generator c of C, the set {ff^{a}, (c^{a}f)f^{a}, f(cf^{a}), c^{a}f(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 ⟨a⟩ 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 a^{2} = 1. We narrow down the structure of T to two possibilities in the next lemma, depending on whether Q⟨h⟩ splits over Q or not, as described in the introduction to this section.

Lemma 6.14. Suppose h ≠ 1. Then one of the following holds.

(a) h^{2} = 1 and Q⟨h⟩ is dihedral, or

(b) Q = ⟨h^{2}⟩.

Proof. Recall that Q ⊴ T and h^{2} ∈ Q by the choice of h. Since T = R⟨h, f⟩ and J = R⟨f⟩, it follows that h^{a} ∈ Jh. The coset P^{a}h^{a} 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^{a}h^{a} is an involution in T/P^{a} − J/P^{a}, or P^{a}h^{a} squares to a generator of the cyclic maximal subgroup CP^{a}/P^{a} of J/P^{a}.

Suppose that P^{a}h^{a} is an involution in T/P^{a} − J/P^{a}. Then P^{a}h^{a} inverts CP^{a}/P^{a} ≅ C, and so h^{a} inverts C. It follows that h inverts C^{a^{−1}} ⩽ Q × ⟨z⟩. Since h normalizes Q, h must invert Q. As P^{a}h^{a} is an involution, we have (h^{a})^{2} ∈ P^{a}. So h^{2} ∈ P. But h^{2} ∈ Q by choice of h. Therefore, h^{2} ∈ Q ∩ P = 1 giving (a).

Suppose that P^{a}h^{a} squares to a generator of CP^{a}/P^{a}. Then h^{a} and hence h has order at least 2|C|. But h^{2} ∈ Q and |Q| = |C|, so we must have Q = ⟨h^{2}⟩, giving (b). □

Set J_{0} = QC = Q × C = C^{a} × C, a homocyclic normal subgroup of S. It will be helpful for what follows to call attention to the action of T on J_{0}, and describe what this means for the structure of the quotient S/J_{0}. Recall that C^{a} ⩽ Q⟨z⟩ from (6.12), and so the action of an element of T on C^{a} is the same as on Q. From Lemma 6.8 and the two possibilities in Lemma 6.14, each element in T centralizes C or inverts it, and the same holds for C^{a} in place of C. Conjugation by an element in S − T swaps the actions. Moreover, T/J_{0} is elementary abelian of order 8 when h ≠ 1, and a induces an automorphism of T/J_{0} fixing pointwise a four group. For instance, from the actions of h, f, and f^{a} on J_{0}, and since J ⊴ S,

(6.15) if h inverts Q, then ⟨h, ff^{a}⟩ covers C_{T/J_{0}}(a).

and

(6.16) if h centralizes Q, then ⟨fh, ff^{a}⟩ covers C_{T/J_{0}}(a),

Lastly,

[S, S] = J_{0}⟨ff^{a}⟩. (6.17)

Suppose h is involution as in Lemma 6.14(a) from now through the next lemma. From (6.15), we have [h, a] ∈ J_{0} = C^{a}C and we may arrange to have [h, a] ∈ C^{a} by replacing h by an appropriate element in C^{a}h ⩽ Q⟨z⟩h. Then

[h, a] = (ha)^{2} ∈ C^{a} ∩ C_{J_{0}}(ha) = 1,

and h still squares to the identity. Fix this choice for h now through the next lemma. Thus, S is a split extension of J by the four group ⟨h, a⟩.

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 6.18. Suppose h is an involution. Then S has presentation

⟨ d, c, f, e, h, a | d^{2^{k−1}} = c^{2^{k−1}} = f^{2} = e^{2} = a^{2} = 1,
[d, c] = [f, e] = 1,
d^{f} = d^{−1}, c^{e} = c^{−1}, c^{a} = d, e^{a} = f,
h^{2} = 1, e^{h} = ec, h^{a} = h ⟩

with notation consistent with that fixed. Here, P = ⟨c, e⟩, P^{a} = ⟨d, f⟩, J = P^{a}P, J_{0} = ⟨d, c⟩, x = d^{2^{k−2}}, z = c^{2^{k−2}}, Z(S) = ⟨xz⟩, and T = ⟨d, c, f, e, h⟩. Furthermore, the following hold.

(a) J⟨a⟩ ≅ D_{2^{k}} ≀ C_{2}.

(b) Q = ⟨dz⟩.

(c) h inverts J_{0} and all involutions of Jh are J-conjugate.

(d) C_{S}(h) = ⟨h⟩ × B_{0} where B_{0} = ⟨x, a⟩ is dihedral of order 8.

(e) C_{S}(a) = ⟨a⟩ × B_{a} where B_{a} = ⟨ff^{a}h, h⟩ is dihedral of order 2^{k+1}.

(f) C_{S}(ha) = ⟨ha⟩ × B_{ha} where B_{ha} = ⟨[h, f]ff^{a}h, h⟩ is dihedral of order 2^{k+1}.

(g) All involutions of Ja are J-conjugate as are all involutions of Jha.

(h) C_{S}(ff^{a}) = ⟨f, x, a⟩ ≅ (C_{2} × C_{2}) ≀ C_{2} and all elements of J_{0}ff^{a} are S-conjugate.

(i) D_{1} := ⟨ff^{a}h, xff^{a}a⟩ and D_{2} := ⟨[h, f]ff^{a}h, h⟩ are quaternion of order 2^{k+1} with [D_{1}, D_{2}] = 1, D_{1} ∩ D_{2} = ⟨xz⟩, D_{1}^{f} = D_{2}, D := D_{1}D_{2} = J_{0}⟨ff^{a}, h, a⟩ = [S, S]⟨h, a⟩,

and S = D⟨f⟩ ≅ Q_{2^{k+1}} ≀_{∗} C_{2} is of type P Sp_{4}(q).

From now through the next lemma, assume Q = ⟨h^{2}⟩ as in Lemma 6.14(b). We adjust h slightly as follows. As a consequence of (6.17), we have that h^{a} ∈ J_{0}ff^{a}h = CC^{a}ff^{a}h. Since a^{2} = 1, we may write h^{a} = c_{1}^{a}c_{1}ff^{a}h for some c_{1} ∈ C. Replacing h by (c_{1}^{a})^{−1}h, which lies in Q⟨z⟩h by (6.12), we arrange that

h^{a} = ff^{a}h, (6.19)

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. (6.20)

Now fix the generator c = [f^{a}, h] of C and set d = c^{a}. Then

d = c^{a} = [f, h^{a}] = [f, ff^{a}h] = [f, h]. (6.21)

From (6.20) and the fact that (fh)^{2} ∈ Qz in (6.3), we have (fh)^{2} = xz, and so xz = fhfh = h^{−2}[f, h] = h^{−2}d as f inverts Q. Hence,

h^{2} = dxz. (6.22)

Lastly, from (fh)^{2} = xz = (xa)^{2} and (6.20), we have

xfha ∈ Jha is an involution.

We can now write down a presentation for S in case Q = ⟨h^{2}⟩. The rest of the following lemma is similarly verified by direct computation.

Lemma 6.23. Suppose Q = ⟨h^{2}⟩. Then S has presentation

⟨ d, c, f, e, h, a | d^{2^{k−1}} = c^{2^{k−1}} = f^{2} = e^{2} = a^{2} = 1,
[d, c] = [f, e] = 1,
d^{f} = d^{−1}, c^{e} = c^{−1}, c^{a} = d, e^{a} = f
h^{2} = dd^{2^{k−2}}c^{2^{k−2}}, e^{h} = ec, h^{a} = feh ⟩

with notation consistent with that fixed. Here P = ⟨c, e⟩, P^{a} = ⟨d, f⟩, J = P P^{a}, J_{0} = ⟨d, c⟩, x = d^{2^{k−2}}, z = c^{2^{k−2}}, Z(S) = ⟨xz⟩, and T = ⟨d, c, f, e, h⟩. Furthermore, the following hold.

(a) J⟨a⟩ ≅ D_{2^{k}} ≀ C_{2}.

(b) Q = ⟨dxz⟩ = ⟨dz⟩.

(c) There are no involutions in Jh.

(d) fh inverts J_{0}; all elements of J_{0}fh square to xz and are J-conjugate.

(e) C_{S}(a) = ⟨a⟩×B_{a} where B_{a} = ⟨f^{a}h, ff^{a}⟩ is semidihedral of order 2^{k+1} with Z(B_{a}) = ⟨xz⟩.

(f) Set b_{1} = xfha ∈ Jha. Then b_{1}^{2} = 1 and C_{S}(b_{1}) = ⟨b_{1}⟩ × B_{ha} where B_{ha} = ⟨d^{−1}f^{a}h, xa⟩ is semidihedral of order 2^{k+1} with Z(B_{ha}) = ⟨xz⟩.

(g) All involutions of Ja are J-conjugate as are all involutions of Jha.

(h) C_{S}(ff^{a}) = ⟨x, f, a⟩ ≅ (C_{2} × C_{2}) ≀ C_{2} and all elements of J_{0}ff^{a} are S-conjugate.

(i) D_{1} = ⟨f^{a}h, ff^{a}a⟩ and D_{2} = ⟨d^{−1}f^{a}h, a⟩ are semidihedral of order 2^{k+1} with [D_{1}, D_{2}] = 1, D_{1} ∩ D_{2} = ⟨xz⟩, D_{1}^{f} = D_{2}, D := D_{1}D_{2} = J_{0}⟨ff^{a}, fh, a⟩ = [S, S]⟨fh, a⟩ and S = D⟨f⟩ ≅ SD_{2^{k+1}}≀_{∗}C_{2} is of type P GL_{4}(q) for q ≡ 3 (mod 4) with ν_{2}(q+1) = k−1.

Armed with this data, the centralizer of the central involution is computed next.

Lemma 6.24. Suppose h ≠ 1. Then C_{F}(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 SL_{2}(q) ∗ SL_{2}(q) of index 2 with f interchanging the two SL_{2}(q) factors. In particular, S ≅ Q_{2^{k+1}} ≀_{∗} C_{2} and h is an involution.

Proof. Assume that h ≠ 1. The two possibilities for S in Lemmas 6.18 and 6.23 will be treated simultaneously. Fix t ∈ J such that tha is an involution as follows. When in the case of Lemma 6.18, we take t = 1. In the other case, we take t = xf as in Lemma 6.23(f). In either case th commutes with a and inverts J_{0}. Let b be one of a or tha. Then C_{S}(b) = ⟨b⟩ × B where B is dihedral or semidihedral of order 2^{k+1} with Z(C_{S}(b)) ∩ [C_{S}(b), C_{S}(b)] = Z(S) by Lemma 6.18(e,f) and Lemma 6.23(e,f). Moreover, all involutions of Jb are J-conjugate by Lemma 6.18(g) and Lemma 6.23(g).

When h is an involution, all involutions in the coset Jh are J-conjugate by Lemma 6.18(c). From Lemma 6.18(d), C_{S}(h) = ⟨h⟩ × B_{0} where B_{0} is dihedral of order 8 < 2^{k+1}, whence |C_{S}(h)| < |C_{S}(b)|. In the case where Q = ⟨h^{2}⟩, there are no involutions in Jh by Lemma 6.23(c). This shows that in either case the set of fully F-centralized F-conjugates of b outside J lies in Ja ∪ Jha. By Proposition 1.24, there exists a morphism φ ∈ F such that b^{φ} ∈ J is fully F-centralized, and C_{S}(b)^{φ} ⩽ C_{S}(b^{φ}). Since C_{S}(b) has nilpotence class k ⩾ 3 and [S, S, S] ⩽ J_{0} from (6.17), we have (xz)^{φ} ∈ Ω_{1}(J_{0}) = ⟨x, z⟩. It follows that (xz)^{φ} = xz as xz is not F-conjugate to x or to z. Composing with c_{a} if necessary, we may assume that b^{φ} = z. Thus, we have shown there exist φ_{a}, φ_{ha} ∈ C_{F}(xz) such that a^{φ_{a}} = (tha)^{φ_{ha}} = z.

Set N = C_{F}(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 ⟨[a, φ_{a}]^{S}, [tha, φ_{ha}]^{S}⟩ is the commuting product D := D_{1} ∗ D_{2} of two quaternion or two semidihedral subgroups of order 2^{k+1} as in Lemma 6.18(i) or Lemma 6.23(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.

Set D_{0} = ⟨[a, φ_{a}]^{S}, [tha, φ_{ha}]^{S}⟩ = ⟨(xa)^{S}, (xtha)^{S}⟩. Taking products, D_{0} contains th inverting J_{0}. All elements of J_{0}th are J-conjugate by Lemma 6.18(c) and Lemma 6.23(d), so D_{0} contains J_{0}. This shows that [S, S] = J_{0}⟨ff^{a}⟩ = J_{0}⟨[f, a]⟩ ⩽ D_{0}. But D_{0} is a proper normal subgroup of S contained in D = [S, S]⟨th, a⟩, and so D_{0} = D. We conclude that 𝔣𝔬𝔠(N) ⩾ D is of index 1 or 2 in S.

As f interchanges D_{1} and D_{2}, fz ∉ D. Suppose that 𝔣𝔬𝔠(N) = S. Then by Proposition 1.23, there exists a morphism η ∈ N such that (fz)^{η} ∈ D is fully N -centralized and C_{S}(fz)^{η} ⩽ C_{S}((fz)^{η}). Since C_{S}(fz) = ⟨x, f⟩ × P is of 2-rank 4 we have that (fz)^{η} ∈ J ∩ D = J_{0}⟨ff^{a}⟩. Suppose (fz)^{η} ∈ J_{0}ff^{a}. By Lemma 6.18(h) and Lemma 6.23(h) then, C_{S}((fz)^{η}) ≅ (C_{2} × C_{2}) ≀ C_{2} is of order 2^{5}, forcing |P | = 8 and η|_{C_{S}(fz)} to be an isomorphism C_{S}(fz) → C_{S}((fz)^{η}). But |Z(C_{S}(fz))| = 8 whereas |Z(C_{S}((fz)^{η}))| = 4, a contradiction. Therefore (fz)^{η} ∈ Ω_{1}(J_{0}) = ⟨x, z⟩ and (fz)^{η} = x or z because η ∈ N. But fz is C_{F}(z)-conjugate to xz, another 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 1.11(c).

Let M = O^{2}(N), a saturated fusion system on D. Set M^{+} = M/⟨xz⟩, 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 D_{1}^{+} × D_{2}^{+} of dihedral groups each of order 2^{k}, and with M^{+} = O^{2}(M^{+}) by Lemma 1.14. Since k ⩾ 3, Aut(D^{+}) is a 2-group by Lemma 1.4, and so it follows that M^{+} = O^{2′}(M^{+}) as well, by Proposition 1.13. The hypotheses of Theorem 1.15 are now satisfied and therefore M^{+} ≅ M_{1}^{+} × M_{2}^{+}, by that theorem, for some pair M_{i}^{+} of saturated fusion systems on D_{i}^{+}. Note then that the M_{i}^{+} = O^{2}(M_{i}^{+}) are determined as the unique perfect 2-fusion system on the dihedral group D_{i}^{+} (Lemmas 1.14 and 1.25), i.e. as F_{D_{i}^{+}}(M_{i}^{+}) with M_{i}^{+} ≅ L_{2}(q).

Let M_{i} be the full preimage of M_{i}^{+} under τ; M_{i} is a saturated fusion system on D_{i}. Then M_{i} = O^{2}(M_{i}) since Z(S) = Z(M_{i}) ⩽ [D_{i}, D_{i}], and M_{i}/Z(M_{i}) = M_{i}^{+} for each i = 1, 2. As there are no perfect fusion systems on a semidihedral group with nontrivial center by Lemma 1.25(b), each of the D_{i} is quaternion, and hence M_{i} is the 2-fusion system of SL_{2}(q) by (c) of the same lemma. Furthermore as f interchanges D_{1} and D_{2}, f interchanges M_{1} and M_{2}.

In particular, we conclude that D is a commuting product of quaternion groups of the same order, and S ≅ Q_{2^{k+1}} ≀_{∗} C_{2}. Thus S is not isomorphic to SD_{2^{k+1}} ≀_{∗} C_{2} as the latter has an involution with centralizer isomorphic to C_{2} × SD_{2^{k+1}}, whereas the former does not. By Lemma 6.18(i) and Lemma 6.23(i), h is an involution. □

We now extract fusion information from the description of the centralizer of the central involution in Lemma 6.24 to show

Lemma 6.25. h = 1.

Proof. Suppose h ≠ 1. Then the structure of S is that of Lemma 6.18 and N = C_{F}(xz) is given by Lemma 6.24. Let N^{+} = N /⟨xz⟩ as before, and denote passage to the quotient by pluses. Recall S = D_{i} ≀_{∗} C_{2}, with f a wreathing element and with each D_{i} quaternion and given as in Lemma 6.18(h). We claim

(6.26) every element of J_{0}ff^{a} is N -conjugate to x,

and once shown, this contradicts Lemma 6.13(a).

To see (6.26), note that Z(S^{+}) = ⟨x^{+}⟩, and the image of each element of J_{0}ff^{a} = J_{0}a^{f}a = J_{0}(xa)^{f}(xa) in S^{+} is an involution which is not contained in either of the D_{i}^{+} 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 morphism N → N^{+}, it follows that each element of J_{0}ff^{a} is N -conjugate into Z(T) = ⟨x, z⟩, and hence N -conjugate to x. This finishes the proof of (6.26) and the lemma. □

Lemma 6.27. F is the fusion system of L_{4}(q_{1}) for some q_{1} ≡ 3 (mod 4) with ν_{2}(q_{1} + 1) = k − 1.

Proof. By Lemma 6.25, S = J⟨a⟩ is isomorphic to D_{2^{k}} ≀ C_{2}. By [Oli, Proposition 5.5(a)], either k = 3 and F is the fusion system of A_{10}, or F is the fusion system of L_{4}(q_{1}) for q_{1} ≡ 3 (mod 4) with ν_{2}(q_{1}+1) = k−1. In the case of A_{10}, x is a product of two transpositions with centralizer having a unique component K isomorphic to the fusion system of A_{6} ≅ L_{2}(9). But then Q = C_{T} (K) is a four group, contrary to hypothesis. □

Lemma 6.27 completes the identification of F and the proof of Theorem 6.1.

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

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