Reduced fusion systems over 2-groups of sectional rank at most 4

Bob Oliver

Author address:

Universit Paris 13, Sorbonne Paris Cit, LAGA, UMR 7539 du CNRS, 99, Av. J.-B. Clment, 93430 Villetaneuse, France.

E-mail address: bobol@math.univ-paris13.fr

Contents

Introduction 1 Chapter 1. Background on fusion systems 5

1.1. Essential subgroups in fusion systems 7 1.2. Reduced fusion systems 8 1.3. The focal subgroup 9

Chapter 2. Normal dihedral and quaternion subgroups 13 Chapter 3. Essential subgroups in 2-groups of sectional rank at most 4 25

3.1. Essential subgroups of index 4 in their normalizer 26 3.2. Essential pairs of type (II) 29 3.3. Essential subgroups of index 2 in S 47

Chapter 4. Fusion systems over 2-groups of type G_{2}(q) 52 Chapter 5. Dihedral and semidihedral wreath products 57 Chapter 6. Fusion systems over extensions of UT_{3}(4) 69 Appendix A. Background results about groups 76 Appendix B. Subgroups of 2-groups of sectional rank 4 80 Appendix C. Some explicit 2-groups of sectional rank 4 83 Appendix D. Actions on 2-groups of sectional rank at most 4 89 Appendix. Bibliography 95

Abstract

We classify all reduced, indecomposable fusion systems over finite 2-groups of sectional rank at most 4. The resulting list is very similar to that by Gorenstein and Harada of all simple groups of sectional 2-rank at most 4. But our method of proof is very different from theirs, and is based on an analysis of the essential subgroups which can occur in the fusion systems.

2010 Mathematics Subject Classification. Primary 20D20. Secondary 20D05, 20E32, 20E45.

Key words and phrases. finite groups, finite simple groups, Sylow subgroups, fusion.

B. Oliver is partially supported by UMR 7539 of the CNRS, and by project ANR BLAN08-2 338236, HGRT.

Introduction

A saturated fusion system F over a finite p-group S is a category whose objects are the subgroups of S, whose morphisms are injective homomorphisms between subgroups, and where the morphism sets satisfy certain axioms first formulated by Puig and motivated by the properties of conjugacy relations among p-subgroups of a finite group. In particular, for each finite group G and each Sylow p-subgroup S ≤ G, the category F_{S}(G) whose objects are the subgroups of G and whose morphisms are those homomorphisms induced by conjugation in G is a saturated fusion system over S. We refer to Puig’s paper [Pg], and to [AKO] and [Cr], for more background details on saturated fusion systems.

A saturated fusion system F is reduced if it contains no nontrivial normal psubgroups, and no proper normal subsystems of p-power index or of index prime to p. All of these concepts are defined by analogy with finite groups; the precise definitions are given in Section 1.2. The class of reduced fusion systems is larger than that of simple fusion systems, although a reduced fusion system which is not simple has to be fairly large. We refer to main theorems in [AOV1] for the motivation for defining this class.

The sectional p-rank of a finite group G is the largest possible value of rk(P/Q), where Q ⊴ P ≤ G are p-subgroups and P/Q is elementary abelian. When G is a p-group, we just call this the sectional rank, and denote it r(G). In their book which appeared in 1974, Gorenstein and Harada [GH] gave a classification of all finite simple groups whose sectional 2-rank is at most 4.

A fusion system is indecomposable if it is not isomorphic to a product of fusion systems over smaller p-groups. The following theorem, where we list all reduced, indecomposable fusion systems over finite 2-groups of sectional rank at most 4, is the main result of this paper. We refer to the end of the introduction for the notation used for certain central products and semidirect products. When q is a prime power and n ≥ 2, UT_{n}(q) denotes the group of upper triangular matrices over 𝔽_{q} with 1’s on the diagonal. Also, we write L_{n}^{+}(q) = PSL_{n}(q) and L_{n}^{−}(q) = PSU_{n}(q).

A fusion system is simple if it contains no nontrivial proper normal subsystems. We refer to [AKO, Definition I.6.1] for the precise definition of a normal subsystem. Here, we just note that a reduced fusion system F over S is simple if S contains no nontrivial proper subgroup strongly closed in F.

Theorem A. Let F be a reduced, indecomposable fusion system over a nontrivial 2-group S of sectional rank at most 4. Then one of the following holds.

(1) S ≅ D_{2^{k}} for some k ≥ 3, and F is isomorphic to the fusion system of L_{2}^{+}(q) (when v_{2}(q ± 1) = k).

(2) S ≅ SD_{2^{k}} for some k ≥ 4, and F is isomorphic to the fusion system of L_{3}^{±}(q) (when v_{2}(q ± 1) = k − 2).

(3) S ≅ C_{2^{k}} ≀ C_{2} for some k ≥ 2, and F is isomorphic to the fusion system of L_{3}^{±}(q) (when v_{2}(q ∓ 1) = k).

(4) S ≅ (C_{2^{k}} × C_{2^{k}}) ⋊^{−1,t} C_{2}^{2} for some k ≥ 2, and F is isomorphic to the fusion system of G_{2}(q) (when v_{2}(q ± 1) = k), or of M_{12} (if k = 2).

(5) S ≅ (D_{2^{k}} ×_{C_{2}} D_{2^{k}}) ⋊^{t} C_{2} ≅ (Q_{2^{k}} ×_{C_{2}} Q_{2^{k}}) ⋊^{t} C_{2} for some k ≥ 3, and F

is isomorphic to the fusion system of PSp_{4}(q) (when v_{2}(q^{2} − 1) = k), or of GL_{4}(2) ≅ A_{8} (if k = 3).

(6) S ≅ D_{2^{k}} ≀ C_{2} for some k ≥ 3, and F is isomorphic to the fusion system of L_{4}^{±}(q) (when v_{2}(q ± 1) = k − 1), or of A_{10} (if k = 3).

(7) S ≅ SD_{2^{k}} ≀ C_{2} for some k ≥ 4, and F is isomorphic to the fusion system of L_{5}^{±}(q) (when v_{2}(q ± 1) = k − 2).

(8) S contains a normal subgroup T ≅ UT_{3}(4), where [S:T] ≤ 4 and Aut_{S}(T) is generated by field and/or graph automorphisms; and F is isomorphic to the fusion system of PSL_{3}(4), L_{4}^{±}(q) for q ≡ ±5 (mod 8), M_{22}, M_{23}, McL, J_{2}, J_{3}, or Ly.

Conversely, if G is any of the groups listed in (1)–(8) and S ∈ Syl_{2}(G), then F_{S}(G) is indecomposable and reduced, and is in fact simple.

Certain simple groups with sectional 2-rank 4, such as those with abelian Sylow 2-subgroup, do not appear in the above list because their fusion system is not reduced. (See Proposition 1.12(b) for more detail.) A few other simple groups, such as A_{7} and M_{11}, fail to appear because their fusion system is isomorphic to that of another simple group in the list.

It will be convenient to have names for some of the classes of 2-groups which appear in the statement of Theorem A. See the end of the introduction for an explanation of the notation used, especially that used for semidirect products.

Definition 0.1. Fix a finite 2-group S.

• S ∈ D if S ≅ D_{2^{n}} for some n ≥ 3.

• S ∈ Q if S ≅ Q_{2^{n}} for some n ≥ 3.

• S ∈ S if S ≅ SD_{2^{n}} for some n ≥ 4.

• S ∈ W if S ≅ C_{2^{n}} ≀ C_{2} for some n ≥ 2.

• S ∈ V if S ≅ Δ ≀ C_{2} or S ≅ (Δ ×_{C_{2}} Δ) ⋊^{t} C_{2} for some Δ ∈ D or Δ ∈ S.

• S ∈ G if S = (C_{2^{n}} × C_{2^{n}}) ⋊^{t,λ} C_{2}^{2}, for n ≥ 2, and for λ = −1 or λ = 2^{n−1} − 1

(the latter only if n ≥ 3). (When λ = −1, these are all of type G_{2}(q) for odd q.)

• S ∈ U if there is T ⊴ S such that T ≅ UT_{3}(4), and C_{S/Z(T)}(T/Z(T)) = T/Z(T).

• Juxtaposition of these symbols denotes union; e.g., DSQ is the family of 2groups which are (nonabelian) dihedral, semidihedral, or quaternion.

• If C is among the classes listed above, then S ∈ C×C if S = S_{1} × S_{2} where S_{1}, S_{2} ∈ C.

Note that UT_{4}(2) ≅ (D_{8} ×_{C_{2}} D_{8}) ⋊^{t} C_{2} ∈ V (Lemma C.4).

Proof of Theorem A. Fix a reduced, indecomposable fusion system F over a 2-group S of sectional rank at most 4. By the results of Section 3, summarized in Theorem 3.1, S ∈ DSWGUV or S has type Aut(M_{12}). If S has type Aut(M_{12}), then by Proposition 4.3, there are no reduced fusion systems over S.

By [BMO, Theorem A(d)], if F is the fusion system (at the prime 2) of P SU_{n}(q) for some odd prime power q, then it is also the fusion system of P SL_{n}(q′) for any q′ such that ⟨q′⟩ = ⟨−q⟩ as closed subgroups of ℤ_{2}^{×}. Hence for each statement in Theorem A about fusion systems of P SL_{n}^{±}(q), it suffices to handle the linear case.

When S ∈ DSW, F is as described in (1)–(3) by [AOV1, Propositions 4.3 & 4.4] and [AOV2, Proposition 3.1]. When S ∈ G, F is as in (4) by Proposition 4.2; and when S ∈ V (cases (5)–(7), and including the case S ≅ UT_{4}(2)) by Propositions 5.1, 5.5, and 5.6.

Assume S ∈ U: an extension of UT_{3}(4) as described above. The isomorphism classes in U are listed in Lemma 6.2(a). Reduced fusion systems over 2-groups of type M_{22} or J_{2} (denoted S_{ϕ} and S_{θ} in Lemma 6.2) are listed in [OV, Theorems 4.8 & 5.11]. The remaining cases are handled in Propositions 6.4, 6.5, and 6.6.

Conversely, assume G is one of the simple groups listed in (1)–(8), fix S ∈ Syl_{2}(G), and set F = F_{S}(G). Then O^{2}(F) = F and O_{2}(F) = 1 by Proposition 1.12(a,b). Also, by [Gd1, Theorem A] and [Ft, Theorem 1], S has no proper subgroups strongly closed in G. Hence F is indecomposable, and if it has any proper normal subsystems, they must be over S, and hence contain O^{2′}(F) by [AOV1, Lemma 1.26]. So F is reduced and simple if O^{2′}(F) = F.

If S ∈ DS or S ∈ W, then Aut(S) is a 2-group by point (b) or (a), respectively, in Corollary A.10. If S ∈ V and S ≅ U̸T_{4}(2), or if S has type Ly, then 𝒴 (S) ≠ ∅ (see Definition 2.1) and hence Aut(S) is a 2-group by Corollary 2.5. If S ∈ G, then Aut(S) is a 2-group by Proposition 4.2. Hence O^{2′}(F) = F in all of these cases by Proposition 1.12(c), and F is reduced and simple.

If S is of type M_{22} or J_{2}, then F is reduced by [AOV1, Proposition 4.5]. If S ≅ UT_{4}(2) or UT_{3}(4), then F is reduced by Proposition 5.1 or 6.4, respectively. □

The main idea behind our proof of Theorem A is to analyze and classify reduced fusion systems by studying their essential subgroups. These are subgroups whose automorphisms generate the fusion system (see Definition 1.1 and Proposition 1.6), and we refer to Theorem 3.1 for a brief summary of results in Section 3 describing them. The main tools used for handling essential subgroups are Bender’s classification of finite groups with strongly 2-embedded subgroups [Be, Satz 1], and Goldschmidt’s classification of amalgams of index (3, 3) [Gd2, Theorem A].

It is unclear to me whether or not this paper, when combined with the deep group theoretic results classifying finite simple groups having Sylow 2-subgroups in certain families, gives a shorter proof of the Gorenstein-Harada theorem than that in [GH]. In any case, that is not our goal here. Our proof of Theorem A is organized very differently from that by Gorenstein and Harada, by setting focus on the essential subgroups in the fusion systems rather than on the centralizers of involutions, and we hope that this approach can give some new insight into the classification of these groups.

The paper is organized as follows. Section 1 is mostly a review of background results on fusion systems. The properties of certain families of subgroups of 2groups are studied in Section 2, and this is applied in Section 3 to describe the (potential) essential subgroups and prove Theorem 3.1. This is then followed by three chapters dealing with fusion systems over the families G, V, and U, respectively. Fusion systems over groups in the families DSW were studied in the earlier papers [AOV1] and [AOV2]. Background results on groups and actions on groups are then collected in the appendices.

Notation and terminology: Most of the notation used here is standard among group theorists. For a prime p, “p-group” always means a finite p-group. For a group G, Z_{i}(G) denotes the i-th term in the upper central series for G; thus Z_{1}(G) = Z(G) and Z_{i+1}(G)/Z_{i}(G) = Z(G/Z_{i}(G)). Also, G^{#} = G∖{1}, and I(G) is the set of involutions in G (elements of order 2). When G and H are finite groups, and Z is identified as a subgroup of Z(G) and of Z(H), then G ×_{Z} H denotes the central product:

G ×_{Z} H = (G × H)/{(z, z^{−1}) | z ∈ Z}.

When G is a finite group and S is a 2-group, S is “of type G” if S is isomorphic to a Sylow 2-subgroup of G. Also, C_{n}, D_{n}, Q_{n}, and SD_{n} denote cyclic, dihedral, quaternion, and semidihedral groups of order n, and 2_{+}^{1+4} = Q_{8}×_{C_{2}} Q_{8} ≅ D_{8}×_{C_{2}} D_{8} and 2_{−}^{1+4} = D_{8} ×_{C_{2}} Q_{8}. When H ≤ G is a subgroup, we write

⟨H^{G}⟩ = ⟨H^{g} | g ∈ G⟩

for the normal closure of H in G.

As perhaps less standard notation, for a group G, we set

G^{ab} = G/[G, G] ,

the abelianization of G; and let

[α] = α·Inn(G) ∈ Out(G)

denote the class of α ∈ Aut(G).

When A is a finite abelian group, B is cyclic, and λ ∈ ℤ is prime to |A|, we let

A⋊^{λ}B denote the semidirect product in which a generator of B acts on A via a ↦ a^{λ}.

When A is any group and B is cyclic, then (A × A) ⋊^{t} B denotes the semidirect product where a generator of B exchanges the two factors A, and similarly for

(A ×_{Z} A) ⋊^{t} B when Z ≤ Z(A). Similarly, when A is abelian, (A × A) ⋊^{λ,t} C_{2}^{2} is the semidirect product where one generator of C_{2}^{2} acts via g ↦ g^{λ} and the other acts by exchanging the factors.

When q = 2^{k} and n ≥ 2, UT_{n}(q) ∈ Syl_{2}(SL_{n}(q)) denotes the subgroup of strict upper triangular natrices. For 1 ≤ i < j ≤ n and a ∈ 𝔽_{q}, e_{ij}^{a} ∈ UT_{n}(q) is the elementary matrix whose unique nonzero off-diagonal entry is a in position (i, j). When q = 2, we write e_{ij} = e_{ij}^{1}.

I would like very much to thank Andy Chermak for first telling me about Goldschmidt’s classification of amalgams. That was when I became convinced that this project should be possible. I would also like to thank the referee for going through the paper in great detail and making many very helpful suggestions.

CHAPTER 1 Background on fusion systems

A saturated fusion system over a p-group S is a category F whose objects are the subgroups of S, and where for each P, Q ≤ S, Mor_{F}(P, Q) is a set of injective homomorphisms from P to Q which includes all morphisms induced by conjugation in S, and which satisfies a set of axioms which are described, for example, in [AKO, § I.2], [BLO2, Definition 1.2], or [Cr, Definition 4.11]. We write Hom_{F}(P, Q) = Mor_{F}(P, Q) to emphasize that the morphisms are all homomorphisms.

The following terminology for subgroups in a fusion system will be used frequently. Recall that a subgroup H < G is strongly p-embedded if p||H|, and p ∤ |H ∩^{g}H| for g ∈ G∖H.

Definition 1.1. Fix a prime p, a p-group S, and a saturated fusion system F over S. Let P ≤ S be any subgroup.

• Let P^{F} denote the set of subgroups of S which are F-conjugate (isomorphic in F) to P. Similarly, g^{F} denotes the F-conjugacy class of an element g ∈ S.

• P is fully normalized in F (fully centralized in F) if |N_{S}(P)| ≥ |N_{S}(R)| (|C_{S}(P)| ≤ |C_{S}(R)|) for each R ∈ P^{F}.

• P is fully automized in F if Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)).

• P is F-centric if C_{S}(P′) = Z(P′) for all P′ which is F-conjugate to P.

• P is F-essential if P is F-centric and fully normalized in F, and Out_{F}(P) contains a strongly p-embedded subgroup. Let E_{F} denote the set of all F-essential subgroups of S.

• P is central in F if every morphism φ ∈ Hom_{F}(Q, R) in F extends to a morphism φ ∈ Hom_{F}(P Q, P R) such that φ|_{P} = Id_{P}.

• P is normal in F if every morphism φ ∈ Hom_{F}(Q, R) in F extends to a morphism φ ∈ Hom_{F}(P Q, P R) such that φ(P) = P.

• For any φ ∈ Aut(S), ^{φ}F denotes the fusion system over S defined by

Hom_{^{φ}F}(P, Q) = φ ∘ Hom_{F}(φ^{−1}(P), φ^{−1}(Q)) ∘ φ^{−1}  (all P, Q ≤ S)

By analogy with finite groups, the maximal normal p-subgroup of a saturated fusion system F is denoted O_{p}(F). Also, for any P ≤ S, N_{F}(P) ⊆ F is the largest fusion subsystem over N_{S}(P) in which P is normal. If P is fully normalized in F, then N_{F}(P) is a saturated fusion system by, e.g., [AKO, Theorem I.5.5].

Since we will have frequent need to refer to the “Sylow axiom” and the “extension axiom” for a saturated fusion system, we state them here in the form of a proposition. (These conditions are used to define saturation in [BLO2] and other papers.)

Proposition 1.2 ([AKO, Proposition I.2.5]). A fusion system F over a pgroup S is saturated if and only if the following two conditions hold.

(I) (Sylow axiom) If P ≤ S is fully normalized, then P is fully centralized and fully automized.

(II) (Extension axiom) For each P, Q ≤ S and φ ∈ Iso_{F}(P, Q) such that Q is fully centralized, if we set N_{φ} = {g ∈ N_{S}(P) |^{φ}c_{g} ∈ Aut_{S}(Q)}, then φ extends to some φ ∈ Hom_{F}(N_{φ}, S).

Proposition 1.3. Let F be a saturated fusion system over a p-group S.

(a) For each P ≤ S, and each R ∈ P^{F} which is fully normalized in F, there is φ ∈ Hom_{F}(N_{S}(P), N_{S}(R)) such that φ(P) = R.

(b) If Q < P ≤ S are such that Q is characteristic in P, Q is fully normalized in F, and P is fully normalized in N_{F}(Q), then P is fully normalized in F.

(c) Assume Q ⊴ P ≤ S, where P is fully normalized in F, N_{S}(Q) = N_{S}(P), and

N_{S}(φ(Q)) ∩ N_{S}(φ(N_{S}(P))) ≤ N_{S}(φ(P)) ∀ φ ∈ Hom_{F}(N_{S}(P), S).  (1.1)

Then Q is also fully normalized in F.

Proof. (a) See, e.g., [AKO, Lemma I.2.6(c)].

(b) By (a), there are φ ∈ Hom_{F}(N_{S}(P), S) and ψ ∈ Hom_{F}(N_{S}(φ(Q)), N_{S}(Q)) such that φ(P) is fully normalized in F and ψ(φ(Q)) = Q. Also, φ(N_{S}(P)) ≤ N_{S}(φ(P)) ≤ N_{S}(φ(Q)) since Q is characteristic in P.

Set χ = ψφ; then χ ∈ Hom_{N_{F} (Q)}(N_{S}(P), N_{S}(Q)) since χ(Q) = Q. Since P is fully normalized in N_{F}(Q) (and since N_{S}(P) ≤ N_{S}(Q)), χ(N_{S}(P)) = N_{S}(χ(P)). Since ψ(N_{S}(φ(P))) ≤ N_{S}(χ(P)), this proves that φ(N_{S}(P)) = N_{S}(φ(P)), so P is fully normalized in F since φ(P) is.

(c) By (a), there is φ ∈ Hom_{F}(N_{S}(Q), S) such that φ(Q) is fully normalized. If Q is not fully normalized, then N_{S}(φ(Q)) > φ(N_{S}(Q)) = φ(N_{S}(P)). Hence by Lemma A.1(a), N_{S}(φ(Q)) ∩ N_{S}(φ(N_{S}(P))) > φ(N_{S}(P)). Together with (1.1), this shows that N_{S}(φ(P)) > φ(N_{S}(P)), contradicting the assumption that P is fully normalized in F. □

The next theorem is a special (much weaker) version of the model theorem, first shown in [BCGLO1]. That theorem says that if F is a saturated fusion system and Q ⊴ F is normal and centric, then there is a unique “model” G for F: a unique group G which realizes the fusion system F and contains Q as normal centric subgroup.

Theorem 1.4 ([AKO, Proposition III.5.8(a)]). Let F be a saturated fusion system over a p-group S, and let Q ≤ S be an F-centric subgroup which is fully normalized in F. There is a finite group M such that N_{S}(Q) ∈ Syl_{p}(M), Q ⊴ M, C_{M}(Q) ≤ Q, and M/Q ≅ Out_{M}(Q) = Out_{F}(Q).

Proof. Since Q is F-centric, it is normal and centric in the normalizer fusion system N_{F}(Q). Hence N_{F}(Q) is constrained in the sense of [BCGLO1, § 4] or [AKO, § I.4]. So by the model theorem [BCGLO1, Proposition 4.3] or [AKO, Theorem I.4.9(a)], there is a finite group M (a “model” for N_{F}(Q)) which satisfies the above conditions (and also F_{N_{S}(Q)}(M) ≅ N_{F}(Q)). □

The following lemma on automorphisms will also be useful.

Lemma 1.5. Let F be a fusion system over a p-group S. Let Q ⊴ P ≤ S be a pair of subgroups both fully normalized in F, such that Q is F-centric and normalized by Aut_{F}(P). Set

Out(P, Q) = N_{Aut(P)}(Q)/Inn(P) = {α ∈ Aut(P) | α(Q) = Q}/Inn(P) ,

and let

R: Out(P, Q) ⟶ N_{Out(Q)}(Out_{P} (Q))/Out_{P} (Q)

be the homomorphism

R([α]) = [α|_{Q}]·Out_{P} (Q) .

Here, [α] ∈ Out(P) denotes the class of α ∈ Aut(P). Then the following hold.

(a) R sends Out_{F}(P) isomorphically to N_{Out_{F} (Q)}(Out_{P} (Q))/Out_{P} (Q).

(b) Assume that p = 2, and that either Z(Q) has exponent 2 and P/Q acts freely on some basis of Z(Q), or that |Z(Q)| = |P/Q| = 2. If Γ ≤ Out(P, Q) is any subgroup such that R(Γ) = N_{Out_{F} (Q)}(Out_{P} (Q))/Out_{P} (Q) and Out_{S}(P) ∈ Syl_{p}(Γ), then Γ = Out_{F}(P).

Proof. By [OV, Lemma 1.2], R is well defined and Ker(R) ≅ H^{1}(P/Q; Z(Q)). In particular, Ker(R) is a p-group since Z(Q) is a p-group. Also, Out_{F}(P) ≤ Out(P, Q) since Aut_{F}(P) normalizes Q.

(a) By the extension axiom (and since C_{S}(Q) ≤ Q and Q is fully normalized), R sends Out_{F}(P) onto N_{Out_{F} (Q)}(Out_{P} (Q))/Out_{P} (Q). Also, Out_{S}(P) ∈ Syl_{p}(Out_{F}(P)) since P is fully normalized, so Ker(R|_{Out_{F} (P)}) ≤ Out_{S}(P) since Ker(R) is a p-group. Hence if α ∈ Aut_{F}(P) and [α] ∈ Ker(R), then α = c_{g} for some g ∈ N_{S}(P), g ∈ P C_{S}(Q) = P since [α|_{Q}] ∈ Out_{P} (Q) and Q is F-centric, and thus [α] = 1 in Out_{F}(P). So R|_{Out_{F} (P)} is injective.

(b) If Z(Q) has exponent 2, and the conjugation action of P/Q permutes freely some basis for Z(Q), then R is injective by [OV, Corollary 1.3], and the result is immediate.

If |P/Q| = |Z(Q)| = 2, then each element in Ker(R) is represented by some α ∈ Aut(P) such that α|_{Q} = Id, and α(g) ∈ gZ(Q) for all g ∈ P ∖Q. Thus |Ker(R)| ≤ 2, and in particular, Ker(R) ≤ Z(Out(P, Q)). By (a), R sends Out_{F}(P) isomorphically onto N_{Out_{F} (Q)}(Aut_{P} (Q))/Out_{P} (Q). By a similar argument, for Γ ≤ Out(P, Q) as in (b), R sends Γ isomorphically onto N_{Out_{F} (Q)}(Aut_{P} (Q))/Out_{P} (Q). Since Ker(R) is central,

Out(P, Q) = Ker(R) × Out_{F}(P) = Ker(R) × Γ.

In particular, Out_{F}(P) and Γ have the same p′-elements. By assumption, Out_{S}(P) is a Sylow p-subgroup of both Out_{F}(P) and Γ, and hence Out_{F}(P) = Γ. □

1.1. Essential subgroups in fusion systems

Recall that E_{F} denotes the set of F-essential subgroups of a fusion system F. We begin with Alperin’s fusion theorem for fusion systems, in the form originally proven by Puig.

Proposition 1.6 ([AKO, Theorem I.3.5]). Let F be a saturated fusion system over a p-group S. Then each morphism in F is a composite of restrictions of automorphisms in Aut_{F}(S), and in O^{p′}(Aut_{F}(P)) for P ∈ E_{F}.

Lemma 1.7. Let F be a saturated fusion system over a p-group S, and assume P ∈ E_{F}. Then O_{p}(Out_{F}(P)) = 1, and Out_{F}(P) acts faithfully on P/Fr(P).

Proof. Since Out_{F}(P) has a strongly p-embedded subgroup, O_{p}(Out_{F}(P)) = 1 (cf. [AKO, Proposition A.7(c)]). The kernel of the action of Aut_{F}(P) on P/Fr(P) is a p-group by Lemma A.9, so Out_{F}(P) acts faithfully since O_{p}(Aut_{F}(P)) = Inn(P). □

The next two results give some necessary conditions for a subgroup to be essential. They were in fact proven in [OV] as conditions for a subgroup to be “critical”, but by [OV, Proposition 3.2], a subgroup of S which is F-essential for some saturated fusion system over S is a critical subgroup of S.

Lemma 1.8 ([OV, Lemma 3.4]). Let F be a saturated fusion system over a p-group S. Let P < S, let Θ be a characteristic subgroup in P, and assume there is g ∈ N_{S}(P)∖P such that

(i) [g, P] ≤ Θ·Fr(P), and

(ii) [g, Θ] ≤ Fr(P). Then c_{g} ∈ O_{p}(Aut(P)), and hence P ∉ E_{F}.

The proof of Lemma 1.8 is based on the fact that O_{p}(Out_{F}(P)) = 1 (Lemma 1.7). The next proposition is based on Bender’s classification [Be, Satz 1] of groups with strongly 2-embedded subgroups.

Proposition 1.9 ([OV, Proposition 3.3(c)]). Let F be a saturated fusion system over a 2-group S. Fix P ∈ E_{F}, and let k be such that |N_{S}(P)/P | = 2^{k}. Then rk(P/Fr(P)) ≥ 2k.

1.2. Reduced fusion systems

We now consider the class of reduced fusion systems, as defined in [AOV1]. First recall the following definitions from [BCGLO2].

Definition 1.10. Let F be a saturated fusion system over a p-group S.

(a) The focal subgroup of F is the subgroup

𝔣𝔬𝔠(F) =^{def} ⟨s^{−1}t | s, t ∈ S, t ∈ s^{F}⟩ = ⟨s^{−1}α(s) | s ∈ P ≤ S, α ∈ Aut_{F}(P)⟩.

(b) The hyperfocal subgroup of F is the subgroup

𝔥𝔶𝔭(F) = ⟨s^{−1}α(s) | s ∈ P ≤ S, α ∈ O^{p}(Aut_{F}(P))⟩.

For any saturated fusion subsystem F_{0} ⊆ F over a subgroup S_{0} ≤ S,

(c) F_{0} has p-power index in F if S_{0} ≥ 𝔥𝔶𝔭(F), and Aut_{F_{0}}(P) ≥ O^{p}(Aut_{F}(P)) for all P ≤ S_{0}; and

(d) F_{0} has index prime to p in F if S_{0} = S, and Aut_{F_{0}}(P) ≥ O^{p′}(Aut_{F}(P)) for all P ≤ S.

By [BCGLO2, Theorem 4.3], each fusion system F over a p-group S contains a unique minimal saturated fusion subsystem O^{p}(F) (over 𝔥𝔶𝔭(F)) of p-power index, and O^{p}(F) = F if and only if 𝔥𝔶𝔭(F) = S. By [BCGLO2, Theorem 5.4], each such F contains a unique minimal saturated fusion subsystem O^{p′}(F) (over S) of index prime to p, and O^{p′}(F) = F if and only if Aut_{O^{p′} (F)}(S) = Aut_{F}(S).

Definition 1.11. A reduced fusion system is a saturated fusion system F such that O_{p}(F) = 1, O^{p}(F) = F, and O^{p′}(F) = F.

For any saturated fusion system F, the reduction of F is the fusion system 𝔯𝔢𝔡(F) which is defined as follows: first set F_{0} = C_{F}(O_{p}(F))/Z(O_{p}(F)), and then let 𝔯𝔢𝔡(F) ⊆ F_{0} be the minimal subsystem which can be obtained by alternately taking O^{p}(−) and O^{p′}(−). A certain concept of “tameness” for fusion systems is defined in [AOV1], and the main results there state that a reduced fusion system is tame if and only if it is not the reduction of any exotic fusion system. Thus Theorem A, together with the result that all of the fusion systems listed in the theorem are tame (to be shown in later papers), imply that all fusion systems over 2-groups of sectional rank at most 4 are realizable.

In many, but not all cases, the 2-fusion system of a simple group is reduced. The following proposition, which is based on a theorem of Goldschmidt, is an attempt to make this statement more precise.

Proposition 1.12. Let G be a finite simple group. Fix S ∈ Syl_{2}(G), and set F = F_{S}(G). Then

(a) O^{2}(F) = F;

(b) O_{2}(F) = 1 if S is nonabelian and G is not isomorphic to a unitary group PSU_{3}(2^{n}) (n ≥ 2) nor to a Suzuki group Sz(2^{2n+1}) (n ≥ 1); and

(c) O^{2′}(F) = F if Out_{G}(S) = 1 (in particular, if Aut(S) is a 2-group). Thus F_{S}(G) is reduced whenever the assumptions in (b) and (c) hold.

Proof. (a) By the focal subgroup theorem for groups (cf. [G, Theorem 7.3.4] or [Sz2, Theorem 5.2.8]), 𝔣𝔬𝔠(F) = [G, G] ∩ S. Hence 𝔣𝔬𝔠(F) = S since [G, G] = G, so 𝔥𝔶𝔭(F) = S and hence O^{2}(F) = F by [AOV1, Theorem 1.22(a)] or [AKO, Corollary I.7.5]. (See also Proposition 1.14(b).)

(b) Assume O_{2}(F) ≠ 1, and set A = Z(O_{2}(F)) ≠ 1. Then A ⊴ F (cf. [AKO, Proposition I.4.4], or Lemma 1.15 below), and hence is strongly closed in S with respect to G. Since G is simple, G is the normal closure of A in G. By a theorem of Goldschmidt [Gd1, Theorem A], either S is abelian, or G ≅ PSU_{3}(2^{n}) or Sz(2^{2n+1}).

(c) See [BCGLO2, Theorem 5.4] or [AKO, Theorem I.7.7(a,b)]. □

Note that Out_{G}(S) = 1 whenever N_{G}(S) = S. Of course, O^{2′}(F_{S}(G)) = F_{S}(G) in many cases when Out_{G}(S) ≠ 1, but it seems to be very difficult to find more general conditions which imply this.

1.3. The focal subgroup

We now list some conditions on a 2-group S, or on a saturated fusion system F over S, which imply that F (or all saturated fusion systems over S) have proper subsystems of 2-power index. All of these are based on Proposition 1.14, which says that this is equivalent to showing that 𝔣𝔬𝔠(F) < S. So we need techniques for computing the focal subgroup, or for showing that it is properly contained in S. The following definitions are useful when doing this.

Definition 1.13. Let F be a saturated fusion system over a p-group S. For each P ≤ S, define

Aut_{F}^{∗}(P) = \begin{cases}O^{p}(Aut_{F}(P)) & if P = S \\ O^{p}(O^{p′}(Aut_{F}(P))) & if P < S. \end{cases}

Set 𝔣𝔬𝔠(F, P) = ⟨[Aut_{F}^{∗}(P), P]^{S}⟩: the normal closure in S of [Aut_{F}^{∗}(P), P].

For example, if P < S and Aut_{F}(P) ≅ Σ_{3} × C_{3}, then Aut_{F}^{∗}(P) ≅ C_{3}.

Recall that for any group P and any H ≤ Aut(P), [H, P] is normal in P (cf. [G, Theorem 2.2.1]). Thus 𝔣𝔬𝔠(F, S) = [Aut_{F}^{∗}(S), S], and [Aut_{F}^{∗}(P), P] ⊴ P for each P. So when P < S, 𝔣𝔬𝔠(F, P) is the subgroup generated by all [Aut_{F}^{∗}(Q), Q] for Q S-conjugate to P.

Proposition 1.14. The following hold for any saturated fusion system F over a p-group S.

(a) Each morphism in F is a composite of restrictions of morphisms in Inn(S) and in Aut_{F}^{∗}(P) for P = S or P ∈ E_{F}, and

𝔣𝔬𝔠(F) = ⟨[S, S], 𝔣𝔬𝔠(F, P) | P ∈ E_{F} ∪ {S}⟩ .

(b) O^{p}(F) = F ⟺ 𝔣𝔬𝔠(F) = S ⟺ S = ⟨𝔣𝔬𝔠(F, P) | P ∈ E_{F} ∪ {S}⟩ . In particular, these all hold if F is reduced.

(c) If P ≤ S, and Γ ≤ Aut_{F}^{∗}(P) is such that Aut_{F}^{∗}(P) ≤ Inn(P)Γ, then

[Aut_{F}^{∗}(P), P] = [Γ, P].

Proof. (a) By Proposition 1.6, each morphism in F is a composite of restrictions of automorphisms in Aut_{F}(S) = Aut_{F}^{∗}(S)Inn(S), and in O^{p′}(Aut_{F}(P)) = Aut_{F}^{∗}(P)Aut_{S}(P) for P ∈ E_{F}. Hence F is generated by restrictions of automorphisms in Inn(S) and in Aut_{F}^{∗}(P) for P ∈ E_{F} ∪ {S}, and

𝔣𝔬𝔠(F) = ⟨[S, S], [Aut_{F}^{∗}(P), P] | P ∈ E_{F} ∪ {S}⟩ .

Since this is clearly normal in S (each subgroup S-conjugate to an essential subgroup is essential), we can replace the commutators [Aut_{F}^{∗}(P), P] by their normal closures 𝔣𝔬𝔠(F, P).

(b) The first equivalence is shown in [AOV1, Theorem 1.22(a)] or [AKO, Corollary I.7.5]. The second follows from (a), since for U ≤ S, U[S, S] = S implies U = S (cf. [G, Theorems 5.1.1 & 5.1.3]). The last statement follows from the definition of a reduced fusion system.

(c) Assume Γ ≤ Aut_{F}^{∗}(P) ≤ Inn(P)Γ. Then

Aut_{F}^{∗}(P) = O^{p}(Inn(P)Γ) = ⟨^{α}Γ | α ∈ Inn(P)⟩:

by definition when P = S, and since Inn(P)Γ ≤ O^{p′}(Aut_{F}(P)) when P < S. Also, Inn(P)Γ normalizes [Γ, P] and hence acts on P/[Γ, P] (cf. [G, Theorem 2.2.1(iii)]), and Γ acts on P/[Γ, P] via the identity. Since Aut_{F}^{∗}(P) is the normal closure of Γ in Inn(P)Γ, it also acts trivially on P/[Γ, P], and so [Aut_{F}^{∗}(P), P] = [Γ, P]. □

We next note three consequences of Proposition 1.14. The first one provides a simple condition for showing that a subgroup is normal in a fusion system, and is a slightly strengthened version of [AKO, Proposition I.4.5].

Lemma 1.15. Let F be a saturated fusion system over a p-group S. Fix a normal subgroup Q ⊴ S. Then Q is normal in F if and only if for each P ∈ E_{F} ∪ {S}, P ≥ Q and Aut_{F}^{∗}(P) normalizes Q.

Proof. The condition is clearly necessary for Q to be normal. Conversely, if P ≥ Q and Aut_{F}^{∗}(P) normalizes Q for each Q ∈ E_{F} ∪ {S}, then by Proposition 1.14(a) (and since Q ⊴ S), each φ ∈ Hom_{F}(P_{1}, P_{2}) extends to some φ ∈ Hom_{F}(P_{1}Q, P_{2}Q) such that φ(Q) = Q, so Q ⊴ F. □

Lemma 1.16. Let F be a saturated fusion system over a p-group S.

(a) If P < S is not fully normalized in F, then there are R ∈ E_{F} and α ∈ Aut_{F}^{∗}(R), such that R ≥ N_{S}(P) and α(P) is not S-conjugate to P.

(b) Assume P < S and S_{0} ⊴ S are such that S_{0} ≥ [S, S] and [Aut_{F}(P), P] ≰ S_{0}. Then there is R ∈ E_{F} ∪ {S} such that R ≥ Q for some Q ∈ P^{F} and 𝔣𝔬𝔠(F, R) ≰ S_{0}.

Proof. (a) Assume P < S is not fully normalized. By Proposition 1.3(a), there is φ ∈ Hom_{F}(N_{S}(P), S) such that |N_{S}(φ(P))| > |N_{S}(P)|. By Proposition 1.14(a), there are a sequence of subgroups R_{1}, …, R_{m} ∈ E_{F} ∪ {S}, automorphisms α_{i} ∈ Aut_{F}(R_{i}), and restrictions β_{i} of α_{i}, such that φ = β_{m} ∘ ⋯ ∘ β_{1}. If α_{1}(P) is S-conjugate to P, then we can replace R_{1} by S and α_{1} by an element of Inn(S), without changing φ(P). If R_{1} = S, then we can define R_{i}^{∗} = α_{1}^{−1}(R_{i}) and α_{i}^{∗} = α_{1}^{−1}α_{i}α_{1}, and get a shorter sequence R_{2}^{∗}, …, R_{m}^{∗} without changing |N_{S}(φ(P))|. We can thus arrange that R_{1} ∈ E_{F} and α_{1}(P) not be S-conjugate to P. This proves (a), with (R, α) = (R_{1}, α_{1}).

(b) Choose β ∈ Aut_{F}(P) and g ∈ P such that β(g)g^{−1} ∉ S_{0}. By Proposition 1.14(a), there are F-essential subgroups R_{1}, …, R_{m} each of which contains a subgroup F-conjugate to P, and automorphisms γ_{i} ∈ Aut_{F}^{∗}(R_{i}) or (if R_{i} = S) γ_{i} ∈ Inn(S), such that β = γ′_{m} ∘ ⋯ ∘ γ′_{1} where γ′_{i} is a restriction of γ_{i}. Set g_{i} = γ′_{i} ∘ ⋯ ∘ γ′_{1}(g) (and g_{0} = g). Hence β(g)g^{−1} = g_{m}g_{0}^{−1} ∉ S_{0}. So there is 1 ≤ i ≤ m such that g_{i}g_{i−1}^{−1} ∉ S_{0}, and γ_{i} ∉ Inn(S) (γ_{i} ∈ Aut_{F}^{∗}(R_{i})) since [γ_{i}, R_{i}] ≰ [S, S]. Thus 𝔣𝔬𝔠(F, R_{i}) ≰ S_{0}. □

Lemma 1.17. Let S be a 2-group such that S/[S, S] ≅ C_{2^{n}} × A where A has exponent at most 2^{n−1}. Set S_{0} = {g ∈ S | g^{2^{n−1}} ∈ [S, S]}, and let F be a reduced fusion system over S. Then there are subgroups P ∈ E_{F} and Q ⊴ P such that P/Q ≅ C_{2^{n}} × C_{2^{n}} and P ≰ S_{0}. Furthermore, for any R ⊴ P such that R ≤ S_{0} and P/R ≅ C_{2^{n}}, there are g ∈ R and α ∈ Aut_{F}^{∗}(P) such that α(g) ∈ P ∖S_{0} and hence R⟨α(g)⟩ = P.

Proof. By definition, S_{0} is characteristic in S. Also, [S:S_{0}] = 2 since by hypothesis, S/[S, S] contains no subgroup C_{2^{n}} × C_{2^{n}}. Hence

𝔣𝔬𝔠(F, S) = ⟨[Aut_{F}^{∗}(S), S]^{S}⟩ ≤ S_{0}.

By Proposition 1.14(b) (and since F is reduced), there is a subgroup P ∈ E_{F} such that 𝔣𝔬𝔠(F, P) ≰ S_{0}. Set

P_{0} = {g ∈ P | g^{2^{n−1}} ∈ [P, P]} ≤ S_{0} .

Thus P_{0} is characteristic in P, and P_{0} ≤ S_{0}. Since Aut_{F}^{∗}(P) = O^{2}(O^{2′}(Aut_{F}(P))) is generated by automorphisms of odd order, there is α ∈ Aut_{F}^{∗}(P) of odd order such that [α, P] ≰ S_{0}, and hence such that α acts nontrivially on P/P_{0}. So P/P_{0} must be noncyclic (Corollary A.10(a)). If a, b ∈ P are elements whose classes are distinct of order 2 in P/P_{0}, then the classes of a^{2^{n−1}} and b^{2^{n−1}} are distinct of order 2 in P/[P, P]. Thus P/[P, P] has a subgroup ⟨[a], [b]⟩ ≅ C_{2^{n}} × C_{2^{n}}, and hence a quotient group isomorphic to C_{2^{n}} × C_{2^{n}}.

Now assume R ≤ P ∩S_{0} is such that R ⊴ P and P/R ≅ C_{2^{n}}. Since [α, P] ≰ S_{0}, there is h ∈ P ∩ S_{0} such that α(h) ∉ S_{0}. Since P/R is cyclic and [S:S_{0}] = 2, P = R⟨α(h)⟩, and P ∩S_{0} = R⟨α(h^{2})⟩. Thus there is m ∈ ℤ such that g =^{def} hα(h^{2m}) ∈ R, and α(g) ∈ α(h)Fr(P) ⊆ P ∖S_{0}. So P = R⟨α(g)⟩. □

As examples of how Lemma 1.17 can be applied, there are no reduced fusion systems over either of the groups C_{2}^{4} ⋊ C_{4} (where C_{4} acts freely on a basis) or (C_{4} × C_{4}) ⋊ C_{4} (where C_{4} acts via the matrix ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ∈ GL_{2}(ℤ/4)): neither group has a subquotient isomorphic to C_{4} × C_{4}.

The next proposition gives another way to handle the focal subgroup of a fusion system.

Proposition 1.18. Let F be a saturated fusion system over a 2-group S.

(a) Set S_{0} = Ω_{1}(Z(S)) and S_{1} = S_{0} ∩ [S, S], and assume |S_{0}/S_{1}| = 2. Then for g ∈ S_{0}∖S_{1}, g ∉ 𝔣𝔬𝔠(F).

(b) Let U ⊴ S be such that Aut_{F}(S) normalizes U, and U ≤ [P, P] for each P < S of index 2. Assume g ∈ S∖[S, S] is such that [g, S] ≤ U, g^{2} ∈ U, and each α ∈ Aut_{F}(S) sends the coset g[S, S] to itself. Then g ∉ 𝔣𝔬𝔠(F).

(c) Let U ⊴ S be such that Aut_{F}(S) normalizes U, and U ≤ Fr(P) for each P < S of index 2. Assume g ∈ S∖Fr(S) is such that [g, S] ≤ U, g^{2} ∈ U, and each α ∈ Aut_{F}(S) sends the coset gFr(S) to itself. Then g ∉ 𝔣𝔬𝔠(F).

In any of these cases, F is not reduced.

Proof. Point (a) is shown in [AKO, Corollary I.8.5].

To prove (b), we refer to [AKO, § I.8] for some of the properties of the transfer homomorphism trf_{F}: S ⟶ S/[S, S] for a saturated fusion system F over S. In particular, Ker(trf_{F}) ≥ 𝔣𝔬𝔠(F). Let g ∈ S be as above, and let [g] ∈ S^{ab} = S/[S, S] be its class. By assumption, [g] ≠ 1.

For P < S, let trf_{P}^{S} : S^{ab} ⟶ P^{ab} be the usual transfer homomorphism (cf. [AKO, Lemma I.8.1(b)]). If [S:P] = 2, then trf_{P}^{S} ([g]) = [gxgx^{−1}] for any choice of x ∈ S∖P: this follows from the construction in [AKO] upon taking coset representatives {1, x}. Since gxgx^{−1} ∈ g^{2}[g, S] ⊆ U ≤ [P, P] by assumption, trf_{P}^{S} ([g]) = 1. Since this holds for each P < S of index 2, trf_{P}^{S} ([g]) = 1 for each P < S since transfers compose (cf. [AKO, Lemma I.8.1(d)]).

By assumption, for each α ∈ Aut_{F}(S), α([g]) = [g]. So by [AKO, Proposition I.8.4(a)], trf_{F}(g) = [g]^{k} ≠ 1, where k = |Out_{F}(S)| is odd. Thus trf_{F}(g) ≠ 1, so g ∉ 𝔣𝔬𝔠(F) since Ker(trf_{F}) ≥ 𝔣𝔬𝔠(F), and 𝔣𝔬𝔠(F) < S. By Proposition 1.14(b), F is not reduced.

The proof of (c) is similar, but carried out by regarding trf_{F} as a homomorphism to S/Fr(S), and replacing P^{ab} by P/Fr(P) for each P ≤ S. □

CHAPTER 2 Normal dihedral and quaternion subgroups

The definitions and results in this chapter will be applied in Section 3, when analyzing certain essential subgroups (those of index 2 in their normalizer). Recall that r(S) denotes the sectional rank of a 2-group S.

Definition 2.1. Let S be a 2-group with r(S) ≤ 4.

(a) A (nonabelian) dihedral or quaternion subgroup Q ≤ S will be called strongly automized if two of the three subgroups of index 2 in Q are N_{S}(Q)-conjugate.

(b) 𝒳 (S) = {Q ⊴ S | Q ∈ DQ and is strongly automized}.

(c) 𝒴_{0}(S) is the set of all Y_{0} ≤ S such that Y_{0} ≅ C_{2}^{4}, 2_{±}^{1+4}, or Q_{8} × Q_{8}, and N_{S}(Y_{0})/Fr(Y_{0}) ≅ D_{8} ≀ C_{2}.

(d) 𝒴 (S) = {⟨(Y_{0})^{S}⟩ | Y_{0} ∈ 𝒴_{0}(S)}: the set of all normal closures in S of subgroups in 𝒴_{0}(S).

The sets 𝒳 (S) and 𝒴 (S) will play a central role in the next chapter (see Theorem 3.1 and Proposition 3.9), when identifying and characterizing essential subgroups. Most of this chapter is aimed at describing 2-groups for which one of these sets is nonempty. The next definition will be used later in this chapter, but is placed here for easier reference.

Definition 2.2. Let S be a 2-group such that 𝒴 (S) ≠ ∅ (and hence 𝒴_{0}(S) ≠ ∅). Fix a subgoup Y_{0} ∈ 𝒴_{0}(S).

(a) Set

𝒜_{S}^{+}(Y_{0}) = {Γ ≤ Out(Y_{0}) | Γ ≥ Aut_{S}(Y_{0}) and Γ ≅ SO_{4}^{+}(2) ≅ Σ_{3} ≀ C_{2}}
𝒜_{S}^{−}(Y_{0}) = {Γ ≤ Out(Y_{0}) | Γ ≥ Aut_{S}(Y_{0}) and Γ ≅ SO_{4}^{−}(2) ≅ Σ_{5}}
𝒜_{S}(Y_{0}) = 𝒜_{S}^{+}(Y_{0}) ∪ 𝒜_{S}^{−}(Y_{0}) .

(b) Let 𝒰_{S}(Y_{0}) be the set of unordered pairs {U_{1}, U_{2}} of subgroups of Y_{0} such that

• for i = 1, 2, U_{i} ⊴ Y_{0}, and U_{i} ≅ C_{2}^{2} or Q_{8};

• [U_{1}, U_{2}] ≤ U_{1} ∩ U_{2} ≤ Fr(U_{1}) and Y_{0} = U_{1}U_{2}; and

• each element of Aut_{S}(Y_{0}) either normalizes U_{1} and U_{2} or exchanges them.

(c) Elements Γ ∈ 𝒜_{S}(Y_{0}) and {U_{1}, U_{2}} ∈ 𝒰_{S}(Y_{0}) are compatible if each α ∈ Aut(U_{i}) (i = 1, 2) extends to some α ∈ Aut(Y_{0}) such that [α] ∈ Γ.

When Y_{0} ∈ 𝒴_{0}(S), and F is a reduced fusion system over S, we will show that Out_{F}(Y_{0}) ∈ 𝒜_{S}(Y_{0}) (Propositions 3.9(a) and 3.11(b.1)). Thus this set contains the “candidates” for Out_{F}(Y_{0}). The sets 𝒰_{S}(Y_{0}) and the compatibility relation will be used to help identify Out_{F}(Y_{0}) among the elements of 𝒜_{S}(Y_{0}) (Proposition 3.11(b.2)), and also when determining the list of all essential subgroups in F (see Proposition 3.11 and Lemma 5.2). We will see in Lemma 2.9 that this compatibility relation defines a bijection between 𝒰_{S}(Y_{0}) and 𝒜_{S}^{±}(Y_{0}).

Lemma 2.3. Let P < S be 2-groups, where P ≅ D_{8} ≀ C_{2} and r(S) ≤ 4. Then Z(S) = Z(P) ≅ C_{2}, |N_{S}(P)/P | = 2, and N_{S}(P)/Z(S) ≅ D_{8} ≀ C_{2}. If V ⊴ P and V ≅ C_{2}^{4}, then N_{S}(V) = P.

Proof. Let Q < P be the unique subgroup isomorphic to 2_{+}^{1+4} (see Lemma C.5(a)). Then N_{S}(P) ≤ N_{S}(Q). Since r(S) ≤ 4 = r(Q/Z(Q)), C_{S}(Q) ≤ Q by Lemma A.6(a). Thus Z(S) = Z(Q) = Z(P), and the homomorphism

cj: N_{S}(Q)/Z(S) = N_{S}(Q)/Z(Q) ⟶ Aut(Q) ≅ Aut(Q_{8}) ≀ C_{2} ≅ Σ_{4} ≀ C_{2}

induced by conjugation is injective. Also, |N_{S}(P)/Z(S)| > |P/Z(P)| = 2^{6} since N_{S}(P) > P by Lemma A.1(a). Thus N_{S}(P)/Z(S) ≅ D_{8} ≀ C_{2}, a Sylow 2-subgroup of Aut(Q). Also, N_{S}(Q) = N_{S}(P), and |N_{S}(P)/P | = 2.

Assume V ⊴ P and V ≅ C_{2}^{4}. Then V ∩Q is one of six subgroups of Q isomorphic to C_{2}^{3} (Lemma C.5(a) again), these subgroups are permuted transitively by Aut(Q), so none is normalized by a Sylow 2-subgroup of Aut(Q). Hence V ∩ Q ⋬ N_{S}(P), so V ⋬ N_{S}(P). If N_{S}(V) > P, then N_{N_{S}(P)}(V) > P by Lemma A.1(a), V ⊴ N_{S}(P) since |N_{S}(P)/P | = 2, and we just saw this is impossible. So N_{S}(V) = P. □

Recall that Z_{i}(G) denotes the i-th term in the upper central series for G. Thus Z_{0}(G) = 1, Z_{1}(G) = Z(G), and Z_{i}(G)/Z_{i−1}(G) = Z(G/Z_{i−1}(G)).

Lemma 2.4. Let S be a 2-group such that r(S) ≤ 4 and 𝒴 (S) ≠ ∅. Let m be such that |S| = 2^{m}. Then the following hold.

(a) For each 0 ≤ i ≤ m − 5, |Z_{i}(S)| = 2^{i}. Also, S/Z_{m−7}(S) ≅ D_{8} ≀ C_{2}, S/[S, S] ≅ C_{2}^{3}, and

P ⊴ S ⟹ P ≥ Z_{m−5}(S) or P = Z_{i}(S) for some 0 ≤ i ≤ m − 6. (2.1)

For each Y_{0} ∈ 𝒴_{0}(S), Fr(Y_{0}) ⊴ S.

(b) For each Y ∈ 𝒴 (S), Y ≥ Z_{m−5}(S), the image of Y in S/[S, S] has order 2, and Y/Z_{m−7}(S) ≅ C_{2}^{4} or 2_{+}^{1+4}. If Y is the normal closure of Y_{0} ∈ 𝒴_{0}(S), then [Y:Y_{0}] = 2^{k} for even k. There are at most two subgroups in 𝒴 (S) of index 8 in S and at most one subgroup of index 4 in S. If Y_{1}, Y_{2} ∈ 𝒴 (S) and Y_{1} ≠ Y_{2}, then Y_{1} ≰ Y_{2}.

(c) For each 0 ≤ i ≤ m − 7, Z_{i+2}(S)/Z_{i}(S) ≅ C_{2}^{2}, and for each 0 ≤ i ≤ m − 8, Z_{i+3}(S)/Z_{i}(S) ≅ C_{4} × C_{2}.

(d) If Y ∈ 𝒴 (S) has index 4 in S, then m ≥ 8 and Y/Z_{m−8}(S) ≅ D_{8} × D_{8}.

Proof. Fix a subgroup Y_{0} ∈ 𝒴_{0}(S), and let Y ∈ 𝒴 (S) be its normal closure in S. Let j = 0, 1, 2 be such that 2^{j} = |Fr(Y_{0})|. We first claim that Fr(Y_{0}) = Z_{j}(S). This is clear when j = 0, and follows from Lemma A.6(a) when j = 1.

Assume j = 2, and hence Y_{0} ≅ Q_{8} × Q_{8}. Set Y_{0} = Y_{0}/Z(Y_{0}), and X = XZ(Y_{0})/Z(Y_{0}) for X ≤ Y_{0}. Fix U_{1}, U_{2} ⊴ Y_{0} such that U_{i} ≅ Q_{8} and Y_{0} = U_{1} × U_{2}. Let z_{i} ∈ Z(U_{i}) be a generator, and set z = z_{1}z_{2}. For g = gZ(Y_{0}) ∈ (Y_{0})^{#} and i = 1, 2, g^{2} = z_{i} if and only if g ∈ U_{i}. Hence under the (faithful) action of Out_{S}(Y_{0}) ≅ D_{8} on Y_{0}, each element either normalizes the U_{i} or exchanges them, and there is some g ∈ N_{S}(Y_{0}) which exchanges them. Hence ^{g}z_{1} = z_{2}, and Z(N_{S}(Y_{0})) = ⟨z⟩ < Z(Y_{0}). So Fr(Y_{0}) = Z_{2}(S) by Lemma A.6(b).

Write Z_{i} = Z_{i}(S) for short (for all i ≥ 0). Let

N_{S}(Y_{0}) = N_{0} < N_{1} < N_{2} < ⋯ < N_{r} = S

be such that N_{i} = N_{S}(N_{i−1}) for i > 0.

(a) We just showed that Fr(Y_{0}) = Z_{j}(S) ⊴ S.

By definition of 𝒴_{0}(S), N_{0}/Z_{j} ≅ D_{8} ≀ C_{2}. If N_{0} < S, then by Lemma 2.3 (applied to the inclusion N_{0}/Z_{j} < S/Z_{j}), |Z_{j+1}/Z_{j}| = |Z(S/Z_{j})| = 2, |N_{1}/N_{0}| = 2, and N_{1}/Z_{j+1} ≅ D_{8}≀C_{2}. Upon repeating this procedure, we see that for all 1 ≤ i ≤ r,

|Z_{j+i}/Z_{j+i−1}| = 2, |N_{i}/N_{i−1}| = 2, and N_{i}/Z_{j+i} ≅ D_{8} ≀ C_{2}.  (2.2)

Since |S| = |N_{r}| = 2^{m}, |Z_{j+r}| = 2^{m−7} = 2^{j+r}, and thus

j + r = m − 7.  (2.3)

In particular, S/Z_{m−7} ≅ D_{8} ≀ C_{2}. Since |Z_{2}(D_{8} ≀ C_{2})| = 4, |Z_{m−5}| = 2^{m−5}. Point (2.1) now follows from Lemma A.2. In particular, [S, S] > Z_{m−7}, so S/[S, S] ≅ C_{2}^{3} (the abelianization of D_{8} ≀ C_{2}).

(c) By Lemma A.6(a), C_{S/Z_{i}}(N_{0}/Z_{i}) ≤ N_{0}/Z_{i} for all i ≤ j + 2 (i.e., all i such that r(N_{0}/Z_{i}) = 4). Since Z_{2}(D_{8} ≀ C_{2}) ≅ C_{2}^{2}, |Z_{i}(N_{0})| = 2^{i} = |Z_{i}| for such i, and thus

Z_{i}(N_{0}) = Z_{i} for all i ≤ j + 2.

If Y_{0} ≅ Q_{8} × Q_{8}, then Z_{1}(N_{0}) ≅ C_{2}, Z_{2} = Z_{2}(N_{0}) = Z(Y_{0}) ≅ C_{2}^{2}, and Z_{3} = Z_{3}(N_{0}) ≅ C_{4} × C_{2} since all elements of order 2 in Y_{0} are in its center.

If Y_{0} ≅ 2_{−}^{1+4}, then Y_{0}/Z(Y_{0}) = Y_{0}/Z_{1} has 5 involutions which lift to involutions in Y_{0} (Lemma C.2(a)). Four of these are permuted by Out_{N_{0}}(Y_{0}) ≅ D_{8} while the fifth is fixed. Hence Z_{2} = Z_{2}(N_{0}) ≅ C_{2}^{2}, and Z_{3} = Z_{3}(N_{0}) ≅ C_{4} × C_{2} since there are no involutions in Z_{3}∖Z_{2}.

If Y_{0} ≅ 2_{+}^{1+4}, then Y_{0}/Z(Y_{0}) = Y_{0}/Z_{1} has a basis {a_{1}, a_{2}, a_{3}, a_{4}} such that each of the subgroups ⟨a_{1}, a_{2}⟩ and ⟨a_{3}, a_{4}⟩ both lifts to a quaternion subgroup of Y_{0}, and such that Aut_{N_{0}}(Y_{0}/Z_{1}) ≅ D_{8} is generated by the permutations (1 2), (3 4), and (1 3)(2 4) (with respect to this indexing). Thus Z_{2}/Z_{1} = ⟨a_{1}a_{2}a_{3}a_{4}⟩, so Z_{2} ≅ C_{2}^{2}; and Z_{3}/Z_{1} = ⟨a_{1}a_{2}, a_{3}a_{4}⟩, so Z_{3} ≅ C_{4} × C_{2}.

Now assume Y_{0} ≅ C_{2}^{4}. Since N_{1}/Z_{1} ≅ D_{8} ≀ C_{2} and r((N_{1}/Z_{2})/Z(N_{1}/Z_{2})) = 4, C_{S/Z_{2}}(N_{1}/Z_{2}) ≤ N_{1}/Z_{2} by Lemma A.6(a). Hence Z_{3} ≤ N_{1}. Also, N_{0} ≅ D_{8} ≀ C_{2}, so Z_{2} = Z_{2}(N_{0}) ≅ C_{2}^{2}. If N_{0} < S (if m ≥ 8), then for x ∈ N_{1}∖N_{0}, c_{x} exchanges Y_{0} with the other normal subgroup in N_{0} isomorphic to C_{2}^{4} (see Lemma C.5(a), and recall that N_{S}(Y_{0}) = N_{0} by Lemma 2.3). Hence c_{x} acts on Z_{3}(N_{0}) ≅ C_{2} × D_{8} by exchanging the two subgroups C_{2}^{3}, and so Z_{3} = Z_{3}(N_{1}) ≅ C_{4} × C_{2}.

Thus Z_{2} ≅ C_{2}^{2} in all cases, and Z_{3} ≅ C_{4} × C_{2} if m ≥ 8. For each 1 ≤ i ≤ m − 8, Z_{i+2}/Z_{i} ≅ C_{2}^{2} and Z_{i+3}/Z_{i} ≅ C_{4} × C_{2} by a similar argument applied to N_{i−j}/Z_{i} < S/Z_{i} if i ≥ j (recall N_{i−j}/Z_{i} ≅ D_{8} ≀ C_{2} by (2.2) and i − j = r + (i + 7 − m) < r by (2.3)), or to N_{0}/Z_{1} < S/Z_{1} if i = 1 and j = 2. If i = m − 7, then S/Z_{i} ≅ D_{8} ≀ C_{2} by (a), and Z_{i+2}/Z_{i} ≅ Z_{2}(D_{8} ≀ C_{2}) ≅ C_{2}^{2}.

(b) Let Y_{i} be the normal closure of Y_{0} in N_{i}, and set Y = Y_{r}: the normal closure of Y_{0} in S. We claim that for each i ≤ m − 7 − j,

Y_{i} > Z_{j+i}  and  Y_{i}/Z_{j+i} ≅ \begin{cases}C_{2}^{4} & if i is even \\ 2_{+}^{1+4} & if i is odd. \end{cases}  (2.4)

This holds by definition when i = 0. If i is even and Y_{i}/Z_{j+i} ≅ C_{2}^{4}, then by (2.2), N_{i}/Z_{j+i} ≅ D_{8} ≀ C_{2} is the normalizer of Y_{i}/Z_{j+i} in S/Z_{j+1} by the last statement in Lemma 2.3, so Y_{i}/Z_{j+i} is N_{i+1}-conjugate to the other normal subgroup C_{2}^{4} in N_{i}/Z_{j+i} (see Lemma C.5(a)). Thus Y_{i+1}/Z_{j+i} ≅ D_{8} × D_{8}, and Y_{i+1}/Z_{j+i+1} ≅ D_{8} ×_{C_{2}} D_{8} ≅ 2_{+}^{1+4}. If i is odd and Y_{i}/Z_{j+i} ≅ 2_{+}^{1+4}, then since this is the only subgroup of N_{i}/Z_{j+i} of this isomorphism type (Lemma C.5(a) again), Y_{i} ⊴ N_{i+1}. Hence in this case, Y_{i+1} = Y_{i} and Y_{i+1}/Z_{i+j+1} ≅ C_{2}^{4}. This proves (2.4).

In particular, [Y:Y_{0}] = |Y_{r}|/2^{4+j} = |Y_{r}/Z_{r+j}|·2^{r−4} is always an even power of 2.

When i = r, so N_{i} = S, and i = m−7−j by (2.3), (2.4) implies that Y > Z_{m−7}, and that Y/Z_{m−7} ≅ C_{2}^{4} or 2_{+}^{1+4}. Since S/Z_{m−7} ≅ D_{8} ≀ C_{2} contains exactly two normal subgroups isomorphic to C_{2}^{4} and one isomorphic to 2_{+}^{1+4} (Lemma C.5(a)), 𝒴 (S) contains at most two subgroups of index 8 and at most one of index 4 (and none of any other index). Also, since none of these three subgroups of D_{8} ≀ C_{2} is contained in any other by Lemma C.5(a), no member of 𝒴 (S) is contained in any other member.

(d) If Y ∈ 𝒴 (S) and [S:Y] = 4, then Y > Y_{0} since [S:Y_{0}] ≥ [N_{0}:Y_{0}] = 8. So N_{0} = N_{S}(Y_{0}) < S, and 2^{m} = |S| ≥ 2·|N_{0}| = 2^{8+j}. Also, N_{m−8−j}/Z_{m−8} ≅ D_{8} ≀ C_{2}, as seen in the proof of (a). Let Y < N_{m−8−j} be such that Y /Z_{m−8} ≅ D_{8} × D_{8}. Then Y /Z_{m−7} ≅ D_{8} ×_{C_{2}} D_{8} ≅ 2_{+}^{1+4}, and Y = Y since there is a unique such subgroup in S/Z_{m−7}. Thus Y/Z_{m−8} ≅ D_{8} × D_{8}. □

As one example, set S = ⟨a_{1}, b_{1}, a_{2}, b_{2}, t⟩ ≅ D_{2^{n}} ≀ C_{2}, with the presentation of Notation 5.4. Then 𝒴 (S) = {Y_{1}, Y_{2}, Y_{3}}, where Y_{1} = ⟨a_{1}^{2}, b_{1}, a_{2}^{2}, b_{2}⟩ ≅ D_{2^{n−1}} × D_{2^{n−1}}, Y_{2} = ⟨a_{1}^{2}, a_{1}b_{1}, a_{2}^{2}, a_{2}b_{2}⟩ ≅ D_{2^{n−1}} × D_{2^{n−1}}, and Y_{3} = ⟨a_{1}a_{2}, a_{1}^{2}, b_{1}b_{2}, t⟩ ≅ Q_{2^{n}} ×_{C_{2}} Q_{2^{n}}.

Corollary 2.5. If S is a 2-group such that r(S) ≤ 4 and 𝒴 (S) ≠ ∅, then 𝒳 (S) = ∅ and Aut(S) is a 2-group.

Proof. Let m be such that 2^{m} = |S|. By Lemma 2.4(a), there is a sequence of subgroups 1 < Z_{1} < Z_{2} < … < Z_{m−7} < S characteristic in S such that |Z_{i}| = 2^{i} for each i and S/Z_{m−7} ≅ D_{8} ≀ C_{2}. Since Aut(D_{8} ≀ C_{2}) is a 2-group by Corollary A.10(c), Aut(S) is a 2-group by Lemma A.9.

If R ∈ 𝒳 (S), then by definition, R ⊴ S, is dihedral or quaternion of order at least 8, and is strongly automized in S. By Lemma 2.4(a), either R = Z_{i}(S) for some i ≤ m−5, or R ≥ Z_{m−5}(S). If m ≥ 8, this is impossible since Z_{3}(S) ≅ C_{4}×C_{2} by Lemma 2.4(c).

If m = 7, then R contains Z_{2}(S) ≅ C_{2}^{2} (Lemma 2.4(c) again) as a normal subgroup. Hence R ≅ D_{8}. Since Z_{2}(S) < R is normal in S, this contradicts the assumption that R is strongly automized. □

We next look at conditions which imply that a subgroup Y ⊴ S lies in 𝒴 (S).

Lemma 2.6. Let S be a 2-group with r(S) ≤ 4. Assume Y = Θ_{1}Θ_{2} ⊴ S, where {Θ_{1}, Θ_{2}} is an S-conjugacy class, Θ_{i} ≅ D_{2^{k}} (k ≥ 3) or Q_{2^{k}} (k ≥ 4) and is strongly automized in S for i = 1, 2, [Θ_{1}, Θ_{2}] ≤ Θ_{1} ∩Θ_{2} ≤ Z(S), and Θ_{1} ∩Θ_{2} = 1 if Θ_{i} ∈ D. Then the conjugation action of S/Y on Y/[Y, Y] ≅ C_{2}^{4} permutes transitively a basis, and one of the following holds.

(a) If S/Y ≅ C_{2}^{2} or D_{8}, then Y ∈ 𝒴 (S), and

{Y_{0} ∈ 𝒴_{0}(S) | Y_{0} < Y } = {U_{1}U_{2} | U_{i} < Θ_{i}, U_{i} ≅ C_{2}^{2} or Q_{8}} ,  (2.5)

where this set consists of one S-conjugacy class if S/Y ≅ D_{8} and two classes if S/Y ≅ C_{2}^{2}. Also, if Y_{0} = U_{1}U_{2} ∈ 𝒴_{0}(S) as in (2.5), then {U_{1}, U_{2}} ∈ 𝒰_{S}(Y_{0}).

(b) If S/Y ≅ C_{4}, then S/[S, S] ≅ C_{4} × C_{2} and 𝒴 (S) = ∅.

Proof. By the 3-subgroup lemma [G, Theorem 2.2.3], and since [Θ_{1}, Θ_{2}] ≤ Z(S),

[[Θ_{1}, Θ_{1}], Θ_{2}] = 1  and  [[Θ_{2}, Θ_{2}], Θ_{1}] = 1.  (2.6)

Set Z_{∗} = Z(Θ_{1})Z(Θ_{2}). Then Y/Z_{∗} ≅ (Θ_{1}/Z(Θ_{1})) × (Θ_{2}/Z(Θ_{2})), so Z(Y) ≤ Z_{2}(Θ_{1})Z_{2}(Θ_{2}). If z_{1}z_{2} ∈ Z(Y), where z_{i} ∈ Z_{2}(Θ_{i}) ≤ [Θ_{i}, Θ_{i}], then [z_{i}, Θ_{i}] = [z_{1}z_{2}, Θ_{i}] = 1 (i = 1, 2) by (2.6), so z_{1}z_{2} ∈ Z(Θ_{1})Z(Θ_{2}). Thus Z(Y) = Z_{∗}, and |Z(Y)| ≤ 4. Also, Z(S) ≤ Z(Y) by Lemma A.6(a) and since r(Y/Z(Y)) = 4. If |Z(Y)| = 4, then Z(Y) = Z(Θ_{1}) × Z(Θ_{2}), and Z(S) < Z(Y) since Θ_{1} and Θ_{2} are S-conjugate. Thus |Z(S)| = 2 in all cases, and Z(S) = Z(Y) = Z(Θ_{1}) = Z(Θ_{2}) if Θ_{1} ∩ Θ_{2} ≠ 1.

We first check that

U_{1} ≤ Θ_{1}, U_{2} ≤ Θ_{2}, U_{i} ≅ C_{2}^{2} (if Θ_{i} ∈ D) or Q_{8} (if Θ_{i} ∈ Q) ⟹

U_{1}U_{2} ≅ C_{2}^{4}, 2_{+}^{1+4}, 2_{−}^{1+4}, or Q_{8} × Q_{8}. (2.7)

This is clear whenever [U_{1}, U_{2}] = 1 (recall that Θ_{1} ∩ Θ_{2} = 1 if Θ_{i} ∈ D). If [Θ_{1}, Θ_{2}] = [U_{1}, U_{2}] = Z(S), then U_{i} ≅ Q_{8}, |C_{U_{1}}(U_{2})| ≥ |U_{1} ∩ [Θ_{1}, Θ_{1}]| = 4 by (2.6), and so U_{1}U_{2} ≅ 2_{−}^{1+4} by Lemma C.2(a).

For each i = 1, 2, let Q_{i1}, Q_{i2} < Θ_{i} be the two noncyclic subgroups of index 2. Set Q = {Q_{ij} | i, j = 1, 2}. We first claim that the conjugation action of S/Y on Q is faithful. Assume otherwise: then there is x ∈ S∖Y such that x ∈ N_{S}(Q_{ij}) for each i, j. Fix U_{i} ≤ Q_{i1} as in (2.7). Each subgroup of Q_{i1} which is isomorphic to U_{i} is Θ_{i}-conjugate to U_{i}, and Aut_{S}(U_{i}) = Aut_{Θ_{i}}(U_{i}) ∈ Syl_{2}(Aut(U_{i})). Hence there are elements x_{i} ∈ Θ_{i} (i = 1, 2) such that c_{x}|_{U_{i}} = c_{x_{i}}|_{U_{i}}. Upon replacing x by xx_{1}^{−1}x_{2}^{−1}, we can assume that [x, U_{1}U_{2}] ≤ [Θ_{1}, Θ_{2}]. If [Θ_{1}, Θ_{2}] = Z(S), then U_{i} ≅ Q_{8}, and U_{1}U_{2} is extraspecial of order 2^{5} by (2.7). So c_{x}|_{U_{1}U_{2}} ∈ Inn(U_{1}U_{2}) in all cases. But this is impossible, since U_{1}U_{2} is centric in S by Lemma A.6(a) (and since r(U_{1}U_{2}) = 4 by (2.7) again).

Thus S/Y acts faithfully on Q. Also, Q_{i1} is S-conjugate to Q_{i2} (Θ_{i} is strongly automized) and Θ_{1} is S-conjugate to Θ_{2}. Hence S/Y acts transitively on Q, and S/Y ≅ C_{2}^{2}, C_{4}, or D_{8}. Furthermore, each Q_{ij} has image in Y^{ab} of order 2, the involutions in these images form a basis for Y^{ab} ≅ Θ_{1}^{ab} × Θ_{2}^{ab} ≅ C_{2}^{4}, and thus S/Y permutes this basis transitively.

(a) Assume S/Y ≅ C_{2}^{2} or D_{8}. Fix indices j_{1}, j_{2} ∈ {1, 2}. There are elements g, h ∈ S∖Y such that ^{g}Q_{1j_{1}} = Q_{2j_{2}} and ^{h}Q_{i1} = Q_{i2} for i = 1 and i = 2, and such that g^{2}, h^{2} ∈ Y. In particular, c_{g} exchanges Θ_{1} and Θ_{2}, and ⟨c_{g}, c_{h}⟩ acts freely and transitively on Q. Set g^{2} = d_{1}d_{2} where d_{i} ∈ Θ_{i}. Then d_{2} ≡ gd_{1}g^{−1} (mod Θ_{1} ∩ Θ_{2}) since [g, d_{1}d_{2}] = 1, so

(gd_{1}^{−1})^{2} = gd_{1}^{−1}gd_{1}^{−1} ≡ d_{2}^{−1}g^{2}d_{1}^{−1} ≡ 1 (mod Θ_{1} ∩ Θ_{2}) .

Upon replacing g by gd_{1}^{−1}, we can arrange that g^{2} ∈ Θ_{1} ∩ Θ_{2} ≤ Z(Y).

Choose U_{1}^{∗} ≤ Q_{1j_{1}} such that U_{1}^{∗} ≅ C_{2}^{2} if Θ_{1} ∈ D or U_{1}^{∗} ≅ Q_{8} if Θ_{1} ∈ Q. Set U_{2}^{∗} = ^{g}U_{1}^{∗} ≤ Q_{2j_{2}} and Y_{0}^{∗} = U_{1}^{∗}U_{2}^{∗}. Since g^{2} ∈ Z(Y), g ∈ N_{S}(Y_{0}^{∗}). By (2.7), Y_{0}^{∗} ≅ C_{2}^{4}, 2_{+}^{1+4}, 2_{−}^{1+4}, or Q_{8} × Q_{8}.

Set N_{i} = N_{Θ_{i}}(U_{i}^{∗}), so that N_{1}N_{2} = N_{Y} (Y_{0}^{∗}) and N_{1}N_{2}/Fr(Y_{0}^{∗}) ≅ D_{8} × D_{8}. For each x ∈ N_{S}(Θ_{1})∖Y, c_{x} exchanges Q_{i1} with Q_{i2} for either or both i = 1, 2, and hence cannot normalize Y_{0}^{∗}. Thus N_{S}(Y_{0}^{∗}) = N_{1}N_{2}⟨g⟩ for g as above. Also, gN_{i} = N_{3−i} and g^{2} ∈ Fr(Y_{0}^{∗}), so N_{S}(Y_{0}^{∗})/Fr(Y_{0}^{∗}) ≅ D_{8} ≀ C_{2}. Hence Y_{0}^{∗} ∈ 𝒴_{0}(S) in this case, and Y ∈ 𝒴 (S) since it is the normal closure of Y_{0}^{∗} in S. Moreover, for any U_{1} < Q_{1j_{1}} and U_{2} < Q_{2j_{2}} isomorphic to U_{1}^{∗} and U_{2}^{∗}, U_{1}U_{2} ∈ 𝒴_{0}(S) since it is Y -conjugate to Y_{0}^{∗}. Since j_{1}, j_{2} ∈ {1, 2} were arbitrary, this proves that the right hand side in (2.5) is contained in the left hand side.

Set 𝒴_{0} = {Y_{0} ∈ 𝒴_{0}(S) | Y_{0} < Y }. Since no subgroup in 𝒴 (S) is contained in any other (Lemma 2.4(b)), Y is the normal closure of each Y_{0} ∈ 𝒴_{0}. For each Y_{0} ∈ 𝒴_{0}, [Y:Y_{0}] is an even power of 2 by Lemma 2.4(b). So if |Y | = 2^{m} for even m, then Θ_{1} ∩ Θ_{2} = 1, and Y_{0} ≅ C_{2}^{4} or Q_{8} × Q_{8}. If Y_{0} ≅ C_{2}^{4}, then its images under projection to each Θ_{i} have order at most 4, hence have order exactly 4, so U_{i} = Y_{0} ∩ Θ_{i} ≅ C_{2}^{2} for i = 1, 2 (and Θ_{i} ∈ D). If Y_{0} ≅ Q_{8} × Q_{8}, then a similar argument shows that U_{i} = Y_{0} ∩ Θ_{i} ≅ Q_{8} for i = 1, 2 and Θ_{i} ∈ Q. So (2.5) holds in this case. Also, Y_{0} = U_{1} × U_{2}, each element of Aut_{S}(Y_{0}) normalizes U_{1} and U_{2} or exchanges them since each element of S normalizes or exchanges the Θ_{i}, and so {U_{1}, U_{2}} ∈ 𝒰_{S}(Y_{0}) (Definition 2.1(e)).

If |Y | is an odd power of 2, then Z(S) = Z(Y_{0}) = Z(Y) = Θ_{1} ∩Θ_{2}, Θ_{i} ∈ Q, and the hypotheses of the lemma hold after replacing S, Y, and Θ_{i} by S/Z(S), Y/Z(S), and Θ_{i}/Z(S). For each Y_{0} ∈ 𝒴_{0}(S) contained in Y, |Y_{0}| = 2^{5} since [Y:Y_{0}] is an even power of 2, so Y_{0}/Z(S) ≅ C_{2}^{4}, Y_{0}/Z(S) ∈ 𝒴_{0}(S/Z(S)), and Y_{0} = U_{1}U_{2} for some U_{i} < Θ_{i} with U_{i} ≅ Q_{8}. This finishes the proof of (2.5), and {U_{1}, U_{2}} ∈ 𝒰_{S}(Y_{0}) by an argument similar to that used in the last paragraph.

By (2.5), there are exactly four Y -conjugacy classes 𝒴_{0}^{ij} ⊆ 𝒴_{0} (i, j ∈ {1, 2}), where 𝒴_{0}^{ij} is the set of those U_{1}U_{2} such that U_{1} < Q_{1i} and U_{2} < Q_{2j}. If S/Y ≅ C_{2}^{2}, then S = Y ⟨g, h⟩ where g and h are as defined above, and 𝒴_{0}^{11}∪𝒴_{0}^{22} and 𝒴_{0}^{12}∪𝒴_{0}^{21} are the two S-conjugacy classes in 𝒴_{0}. If S/Y ≅ D_{8}, then there is also a ∈ S such that ^{a}Q_{11} = Q_{12} and ^{a}Q_{21} = Q_{21}, so these two sets are S-conjugate.

(b) Assume S/Y ≅ C_{4}. Since S/Y permutes the basis B transitively, Y/[S, S] = Y/[S, Y] ≅ C_{2}, and S/[S, S] ≅ C_{4} × C_{2} since if S/[S, S] were cyclic then S would be cyclic. Thus 𝒴 (S) = ∅ by Lemma 2.4(a). □

The next lemma can be regarded as a converse to Lemma 2.6(a), but with the extra (necessary) hypothesis (2.8) added.

Lemma 2.7. Let S be a 2-group with r(S) ≤ 4 and 𝒴 (S) ≠ ∅. Choose Y_{0} ∈ 𝒴_{0}(S), and let Y ∈ 𝒴 (S) be its normal closure in S. Fix {U_{1}, U_{2}} ∈ 𝒰_{S}(Y_{0}), and assume that

U_{1} is not S-conjugate to any other subgroup of U_{1}Z_{2}(S).  (2.8)

Then there is an S-conjugacy class {Θ_{1}, Θ_{2}} of subgroups such that U_{i} ≤ Θ_{i}, Y = Θ_{1}Θ_{2}, and

Y_{0} ≅ C_{2}^{4} ⟹ Θ_{i} ∈ D or Θ ≅ C_{2}^{2}, and Y = Θ_{1} × Θ_{2},
Y_{0} ≅ 2_{±}^{1+4} ⟹ Θ_{i} ∈ Q and [Θ_{1}, Θ_{2}] ≤ Θ_{1} ∩ Θ_{2} = Z(S),
Y_{0} ≅ Q_{8} × Q_{8} ⟹ Θ_{i} ∈ Q and Y = Θ_{1} × Θ_{2}.

Proof. If Y_{0} = Y, then the lemma holds (with Θ_{i} = U_{i}) by definition of 𝒰_{S}(Y_{0}). So assume Y > Y_{0}. Case 1: Assume that Y_{0} ≅ C_{2}^{4} or Q_{8} × Q_{8}, and thus that Y_{0} = U_{1} × U_{2}. Set S_{0} = C_{S}(Z_{2}(S)). Since |Z_{2}(S)| = 4 by Lemma 2.4(a), [S:S_{0}] = 2. We prove the lemma in this case by induction on |S|.

Set N_{0} = N_{S}(Y_{0}) and Ŝ = Y N_{0}. By Lemma 2.3 and since Y > Y_{0}, the two normal subgroups of N_{0}/Fr(Y_{0}) ≅ D_{8} ≀ C_{2} isomorphic to C_{2}^{4} are N_{S}(N_{0})-conjugate and hence are both contained in Y/Fr(Y_{0}). Hence [N_{0}:Y ∩ N_{0}] ≤ 2. Also, Y = ⟨(Y_{0})^{S}⟩ ≤ S_{0} since Y_{0} ≤ S_{0} ⊴ S, and N_{0} ≰ S_{0} since U_{1} and U_{2} are N_{0}-conjugate (by definition of 𝒰_{S}(Y_{0})) but not S_{0}-conjugate. Thus [Ŝ:Y] = [N_{0}:Y ∩N_{0}] = 2, and Y = Ŝ∩S_{0} = CŜ(Z_{2}(Ŝ)) (where Z_{2}(Ŝ) = Z_{2}(S) by Lemma 2.4(a) again). Let Ŷ be the normal closure of Y_{0} in Ŝ (hence that in Y). Then NŜ(Y_{0}) = N_{0}, so Y_{0} ∈ 𝒴_{0}(Ŝ) and Ŷ ∈ 𝒴 (Ŝ). Also, {U_{1}, U_{2}} ∈ 𝒰Ŝ(Y_{0}), and (2.8) holds in Ŝ.

Now, Ŝ < S, since [S:Y] ≥ 4 by Lemma 2.4(b) while [Ŝ:Y] = 2. So by the induction hypothesis, Ŷ = Θ̂_{1} × Θ̂_{2}, where {Θ̂_{1}, Θ̂_{2}} is an Ŝ-conjugacy class, U_{i} ≤ Θ̂_{i} ⊴ Y, and Θ̂_{i} ∈ DQ or Θ̂_{i} = U_{i} ≅ C_{2}^{2}.

Let 𝒫_{i} be the S_{0}-conjugacy class of Θ̂_{i}, and set 𝒫 = 𝒫_{1}∪𝒫_{2}. Then Y = ⟨𝒫⟩, since Y is the normal closure in S of Y_{0} = U_{1}U_{2} and hence that of Θ̂_{1}Θ̂_{2} ≥ Y_{0}. Since Θ̂_{i} ⊴ Y, the hypotheses of Lemma B.6 hold with Y in the role of S. By that lemma, there are subgroups Θ_{i} ⊴ Y such that Θ_{i} ≤ ⟨𝒫_{i} ⟩ ≤ Θ_{i}Z_{2}(S), Θ_{i} ∈ DSQ, and Y = Θ_{1} × Θ_{2}.

Set P_{1} = ⟨𝒫_{1} ⟩ for short. If Θ_{1} = P_{1}, then Θ_{1} ⊴ S_{0}, and hence {Θ_{1}, Θ_{2}} is an S-conjugacy class which satisfies the conditions in the lemma. So assume otherwise: assume P_{1} = Θ_{1}Z_{2}(S) = Θ_{1} × Z(S). Then Θ̂_{1} ⊴ P_{1} with index 4 (recall Θ̂_{1} ⊴ Y), so Θ̂_{1} ≥ [P_{1}, P_{1}] = Fr(P_{1}) with index 2. Hence for each Q ∈ 𝒫_{1}, the image of Q in P_{1}/Fr(P_{1}) ≅ C_{2}^{3} has order 2. So |𝒫_{1}| ≥ 3, and |𝒫_{1}| = 4 since |𝒫_{1}| = [S_{0}:N_{S}(Θ̂_{i})] ≤ [S_{0}:Y] = 4.

The image in P_{1}/Z(S) of each Q ∈ 𝒫_{1} is one of the two noncyclic subgroups of index 2. Hence there is Q ∈ 𝒫_{1} such that Q ≠ Θ̂_{1} and QZ_{2}(S) = Θ̂_{1}Z_{2}(S). Let g ∈ S_{0} be such that Q = ^{g}Θ̂_{1}. Since Θ̂_{1} is the normal closure of U_{1} in P_{1} = Θ_{1} × Z(S), there is x ∈ Θ_{1} such that ^{g}(^{x}U_{1}) ≰ Θ̂_{1}. Let y ∈ Θ_{1} be such that gxU_{1} ≤ (^{y}U_{1})Z_{2}(S). Then U_{1} ≠ ^{y^{−1}gx}U_{1} ≤ U_{1}Z_{2}(S). Since this contradicts (2.8), we now conclude that Θ_{1} = P_{1} = ⟨𝒫_{1} ⟩. Thus Θ_{1} ∈ D or Θ_{1} ∈ Q (depending on whether U_{1} ≅ C_{2}^{2} or Q_{8}), and the lemma holds in this case.

Case 2: Now assume that Y_{0} ≅ 2_{±}^{1+4}. Set Z = Z(S), S = S/Z, and X = XZ/Z for each X ≤ S. The hypotheses of the lemma hold for Y_{0} < Y < S, where {U_{1}, U_{2}} ∈ 𝒰_{S}(Y_{0}) and U_{i} ≅ C_{2}^{2}. In particular, (2.8) holds since U_{1}Z_{2}(S) =

U_{1}Z(S) = U_{1}Z_{2}(S). So by Case 1, there is an S-conjugacy class {Θ_{1}, Θ_{2}} such that Y = Θ_{1} × Θ_{2} and Θ_{i} ∈ D. Let Θ_{i} ⊴ Y be the preimage of Θ_{i} ⊴ Y.

By construction, Y = Θ_{1}Θ_{2}, and [Θ_{1}, Θ_{2}] ≤ Θ_{1} ∩ Θ_{2} ≤ Z. So the only thing to check is that Θ_{i} ∈ Q for i = 1, 2. Since Θ_{i} > U_{i} ≅ Q_{8} and Z = [U_{i}, U_{i}], (Θ_{i})^{ab} ≅ (Θ_{i})^{ab} ≅ C_{2}^{2}. So Θ_{i} ∈ DSQ by Proposition B.2, and Θ_{i} ∈ Q since it is generated by subgroups conjugate to U_{i} ∈ Q (recall Y = ⟨(U_{1}U_{2})^{S}⟩ = ⟨(U_{i})^{S}⟩). □

The following example helps illustrate why condition (2.8) is needed in the last lemma. It also shows that 𝒰_{S}(Y_{0}) can be empty for Y_{0} ∈ 𝒴_{0}(S) when Y_{0} ≅ Q_{8}×Q_{8}.

Example 2.8. Fix n ≥ 4. Set S = ⟨a_{1}, b_{1}, a_{2}, b_{2}, t⟩ = Δ_{1}Δ_{2}⟨t⟩, where for i = 1, 2, Δ_{i} = ⟨a_{i}, b_{i}⟩ ≅ Q_{2^{n}}, |a_{i}| = 2^{n−1}, and |b_{i}| = 4. Also, [b_{1}, b_{2}] = [a_{1}, a_{2}] = 1, [a_{1}, b_{2}] = [a_{2}, b_{1}] = b_{1}^{2}b_{2}^{2} ∈ Z(S), t^{2} = 1, ^{t}a_{1} = a_{2}, and ^{t}b_{1} = b_{2}. In particular, S/⟨b_{1}^{2}, b_{2}^{2}⟩ ≅ D_{2^{n−1}} ≀ C_{2}.

Set U_{i} = ⟨a_{i}^{2^{n−3}}, b_{i}⟩ ≅ Q_{8} (i = 1, 2), and set Y_{0} = U_{1}U_{2} ≅ Q_{8} × Q_{8}. Then N_{S}(Y_{0})/⟨b_{1}^{2}, b_{2}^{2}⟩ ≅ D_{8} ≀ C_{2}, so Y_{0} ∈ 𝒴_{0}(S). Also, ⟨(Y_{0})^{S}⟩ = Θ_{1} × Θ_{2} ∈ 𝒴 (S), where Θ_{i} = ⟨a_{i}^{2}, b_{i}⟩ ≅ Q_{2^{n−1}} for i = 1, 2. If n ≥ 5, then {U_{1}, U_{2}} ∈ 𝒰_{S}(Y_{0}), but (2.8) does not hold in this case, and the conclusion of Lemma 2.7 also fails to hold since {Θ_{1}, Θ_{2}} is not an S-conjugacy class. If n = 4 (so Y = Y_{0}), then 𝒰_{S}(Y_{0}) = ∅.

We are now ready to look at the sets 𝒜_{S}(Y_{0}) of Definition 2.2, and the compatibility relation defined there between elements of 𝒜_{S}(Y_{0}) and 𝒰_{S}(Y_{0}).

Lemma 2.9. Let S be a 2-group such that r(S) ≤ 4 and 𝒴 (S) ≠ ∅. Then the following hold for each Y_{0} ∈ 𝒴_{0}(S).

(a) Y_{0} ≅ C_{2}^{4}, 2_{±}^{1+4}, or Q_{8} × Q_{8}, Out_{S}(Y_{0}) ≅ D_{8}, and the action of Out_{S}(Y_{0}) on Y_{0}/Fr(Y_{0}) ≅ C_{2}^{4} is faithful and permutes a basis.

(b) For each Γ ∈ 𝒜_{S}(Y_{0}), there is a unique pair {U_{1}, U_{2}} ∈ 𝒰_{S}(Y_{0}) which is compatible with Γ. If, furthermore, U ⊴ Y_{0} is such that U ≅ C_{2}^{2} or Q_{8}, |N_{Aut_{S}(Y_{0})}(U)| = 4, and each α ∈ Aut(U) extends to α ∈ Aut(Y_{0}) such that [α] ∈ Γ, then U ∈ {U_{1}, U_{2}}.

(c) If Y_{0} ≅ C_{2}^{4}, 2_{+}^{1+4}, or Q_{8} × Q_{8}, then for each {U_{1}, U_{2}} ∈ 𝒰_{S}(Y_{0}), there is a unique subgroup Γ ∈ 𝒜_{S}^{+}(Y_{0}) which is compatible with {U_{1}, U_{2}}. If Y_{0} ≅ C_{2}^{4} or 2_{−}^{1+4}, then for each {U_{1}, U_{2}} ∈ 𝒰_{S}(Y_{0}), there is a unique subgroup Γ ∈ 𝒜_{S}^{−}(Y_{0}) which is compatible with {U_{1}, U_{2}}.

Proof. (a) By Definition 2.1, N_{S}(Y_{0})/Fr(Y_{0}) ≅ D_{8} ≀ C_{2}, where Y/Fr(Y_{0}) ≅ C_{2}^{4}. Hence Out_{S}(Y_{0}) ≅ N_{S}(Y_{0})/Y_{0} ≅ D_{8}, and its action on Y_{0}/Fr(Y_{0}) is faithful and permutes a basis. (b,c) We consider separately the cases Γ ≅ Σ_{3} ≀ C_{2} and Γ ≅ Σ_{5}. Case 1: Γ ≅ SO _{4}^{+}(2) ≅ Σ_{3} ≀ C_{2}. If Γ ≤ Out(Y_{0}) and Γ ≅ Σ_{3} ≀ C_{2}, then there are exactly two subgroups H_{1}, H_{2} < Γ of order 3 with rk([H_{i}, Y_{0}/Fr(Y_{0})]) = 2. (If H < Γ is any other subgroup of order 3, then rk([H_{i}, Y_{0}/Fr(Y_{0})]) = 4.) Let α_{i} ∈ Aut(Y_{0}) be of order 3 such that [α_{i}] ∈ Out(Y_{0}) generates H_{i}, and set U_{i} = [α_{i}, Y_{0}]. Then {U_{1}, U_{2}} ∈ 𝒰_{S}(Y_{0}) and is compatible with Γ.

If U ≤ Y_{0} is as in point (b), then the condition on extending automorphisms implies that U = [α, Y_{0}] for some α ∈ Aut(Y_{0}) of order 3 such that [α] ∈ Γ. Since U_{1} and U_{2} are independent of the choice of ⟨α_{i}⟩ (any two choices are conjugate by an element of Inn(Y_{0})), U ∈ {U_{1}, U_{2}}.

Conversely, if {U_{1}, U_{2}} ∈ 𝒰_{S}(Y_{0}), then let Γ ≤ Out(Y_{0}) be the group of (classes of) all automorphisms of Y_{0} which either normalize the U_{i} or exchange them. (Note that [U_{1}, U_{2}] = 1 since Y_{0} ≅ 2̸_{−}^{1+4}.) Then Γ ≅ Σ_{3} ≀ C_{2}, Γ ≥ Out_{S}(Y_{0}), and hence Γ ∈ 𝒜_{S}^{+}(Y_{0}) is the unique element which is compatible with {U_{1}, U_{2}}. This proves (c).

Case 2: Γ ≅ SO _{4}^{−}(2) ≅ Σ_{5}. Assume first that Y_{0} ≅ C_{2}^{4}. Fix a basis B = {e_{1}, e_{2}, e_{3}, e_{4}} for Y_{0} which is permuted transitively by Aut_{S}(Y_{0}), ordered so that Aut_{S}(Y_{0}) contains the transpositions (e_{1} e_{2}) and (e_{3} e_{4}). Set z = e_{1}e_{2}e_{3}e_{4}, the generator of C_{V} (Aut_{S}(Y_{0})), and set e′_{i} = e_{i}z for i = 1, 2, 3, 4. Since Aut_{S}(Y_{0}) permutes the fifteen involutions in V in orbits of length 4, 4, 4, 2, 1, and the orbit {e_{1}e_{3}, e_{1}e_{4}, e_{2}e_{3}, e_{2}e_{4}} is not a basis, B′ = {e′_{1}, e′_{2}, e′_{3}, e′_{4}} is the only other basis which is permuted transitively by Aut_{S}(Y_{0}). Let Γ, Γ′ ≤ Aut(Y_{0}) be the subgroups of automorphisms which permute the sets B ∪ {z} and B′ ∪ {z}, respectively. Set e_{0} = e′_{0} = z for convenience.

For any Δ ∈ 𝒜_{S}^{−}(Y_{0}), by Proposition D.1(d), Y_{0} is the orthogonal module for Δ, since otherwise it cannot contain Aut_{S}(Y_{0}). Thus Δ acts on Y_{0}^{#} with an orbit of length 5 the product of whose elements is the identity. This orbit contains four elements permuted transitively by Aut_{S}(Y_{0}) and which generate Y_{0} (since the Δ-action is irreducible) and one which is fixed. Thus Δ permutes one of the sets B ∪ {z} or B′ ∪ {z}. It follows that 𝒜_{S}^{−}(Y_{0}) = {Γ, Γ′}.

Assume U < Y_{0} is such that U ≅ C_{2}^{2}, |N_{Aut_{S}(Y_{0})}(U)| = 4, and each α ∈ Aut(U) of order 3 extends to α ∈ Γ < Aut(Y_{0}) of order 3. Since rk([β, Y_{0}]) = 2 for each β ∈ Γ ≅ Σ_{5} of order 3, U = [α, Y_{0}]. Also, α permutes cyclically e_{i}, e_{j}, e_{k} for some triple i, j, k ∈ {0, 1, 2, 3, 4} of distinct indices, and hence U = ⟨e_{i}e_{j}, e_{i}e_{k}⟩. The only such triples of indices which are normalized by a subgroup of index 2 in Aut_{S}(Y_{0}) are {e_{0}, e_{1}, e_{2}} and {e_{0}, e_{3}, e_{4}}, so U ∈ {U_{1}, U_{2}} where U_{1} = ⟨e′_{1}, e′_{2}⟩ and U_{2} = ⟨e′_{3}, e′_{4}⟩. Thus {U_{1}, U_{2}} is the only pair in 𝒰_{S}(Y_{0}) which is compatible with Γ, and the U_{i} are the only subgroups which satisfy the hypotheses in the second statement in (b). Similarly, {⟨e_{1}, e_{2}⟩, ⟨e_{3}, e_{4}⟩}, is the only pair in 𝒰_{S}(Y_{0}) which is compatible with Γ′. This proves (b) and (c) when Y_{0} ≅ C_{2}^{4}.

Now assume that Y_{0} ≅ 2_{−}^{1+4}, and set Z = Z(Y_{0}) for short. Then Γ = Out(Y_{0}) is the unique element of 𝒜_{S}(Y_{0}). Let a_{0}, a_{1}, …, a_{4} ∈ Y_{0} be such that {a_{i}Z | 0 ≤ i ≤ 4} are the five cosets of noncentral involutions in Y_{0} (Lemma C.2(a)). Then a_{0}a_{1}a_{2}a_{3}a_{4} ∈ Z. Each element of order 4 in Y_{0} lies in a_{i}a_{j}Z for some unique pair of distinct indices i, j. So each quaternion subgroup has the form U = ⟨a_{i}a_{j}, a_{k}a_{ℓ}⟩ for indices i ≠ j and k ≠ ℓ, and {i, j} ∩ {k, ℓ} ≠ ∅ since otherwise a_{i}a_{j}a_{k}a_{ℓ} has order 2. Thus U = ⟨a_{i}a_{j}, a_{i}a_{k}⟩ for some triple of distinct indices i, j, k, and U = [α, Y_{0}] for any α ∈ Aut(Y_{0}) of order 3 which permutes cyclically the cosets a_{i}Z, a_{j}Z, a_{k}Z. Thus there are exactly ten quaternion subgroups in Y_{0}. Since Out_{S}(Y_{0}) ≅ D_{8} fixes one of the cosets a_{i}Z and permutes the other four transitively, it permutes the ten quaternion subgroups in two orbits of length 4 and one of length 2. Thus there is a unique pair {U_{1}, U_{2}} ∈ 𝒰_{S}(Y_{0}) (the orbit of length 2). Since U_{i} = [α_{i}, Y_{0}] (i = 1, 2) for some α_{1}, α_{2} ∈ Aut(Y_{0}) of order 3, this pair is compatible with Γ. □

We now look at 2-groups S for which 𝒳 (S) ≠ ∅.

Lemma 2.10. Fix a 2-group S and a subgroup Δ ∈ 𝒳 (S). Let A ⊴ Δ be the cyclic subgroup of index 2 (the one which is normal in S if Δ ≅ Q_{8}), fix a generator a ∈ A, and set A_{0} = ⟨a^{2}⟩. Let Δ_{0} ≤ Δ be dihedral or quaternion of order 8, and set T = C_{S}(Δ_{0}).

(a) [S:T Δ] = 2, and for all g ∈ S∖T Δ, ^{g}b = a^{j}b for some odd j. Also, T A_{0} ⊴ S, S/T A_{0} ≅ D_{8}, and T A/T A_{0} = Z(S/T A_{0}).

(b) There is x ∈ S∖T Δ such that ^{x}a = a^{1+4ℓ} for some ℓ ∈ ℤ, ^{x}b = ab, and x^{2} = a^{i}t for t ∈ T and i odd. Also, A = [S, Δ].

(c) If y ∈ T Ax, where x is as in (b), then either |y| > |a|, or |y| = |a|, Z(Δ) ≤ Fr(T), and ⟨y⟩ ∩ A = 1.

Proof. (a) Most of this was shown in [AOV2, Lemma B.3], but we give a slightly different argument here. Recall that by Definition 2.1, Δ ∈ 𝒳 (S) implies that Δ ⊴ S, Δ ∈ DQ, and Δ is strongly automized in S. Also, A ⊴ S: by assumption if Δ ≅ Q_{8}, and since A is characteristic in Δ if Δ ≅ Q̸_{8}.

Set B = ⟨a^{4}⟩. Then Δ/B ≅ D_{8}, and Aut(Δ/B) = ⟨α, β⟩ ≅ D_{8} where

α: \{ \begin{matrix}a ↦ a \\ b ↦ ab \end{matrix}  and  β: \{ \begin{matrix}a ↦ a^{−1} \\ b ↦ b \end{matrix}

(Here, g ∈ Δ/B denotes the class of g ∈ Δ). Consider the homomorphism

ψ: S ⟶ Aut(Δ/B) ≅ D_{8}

induced by conjugation. For t ∈ T = C_{S}(Δ_{0}), ^{t}b = b and ^{t}a = a^{4i+1} for some i, so t ∈ Ker(ψ). Thus T A_{0} ≤ Ker(ψ). Conversely, if g ∈ Ker(ψ), then ^{g}b = a^{4j}b and ^{g}a = a^{4k+1} for some j, k ∈ ℤ, so [ga^{2m}, b] = 1 whenever m ∈ ℤ is such that m(4k + 1) ≡ −j (mod |a|). Also, [ga^{2m}, a] ∈ B, so ga^{2m} ∈ T and g ∈ T A_{0}. This proves that Ker(ψ) = T A_{0} and hence that T A_{0} ⊴ S.

Since ψ(Δ) = ψ(T Δ) = Inn(Δ/B) = ⟨α^{2}, β⟩, and since there is x ∈ S such that ψ(x) ∉ Inn(Δ/B) (Δ is strongly automized), ψ is onto. Thus S/T A_{0} ≅ Aut(Δ/B) ≅ D_{8}. Also, [S:T Δ] = |Out(Δ/B)| = 2, and Z(S/T A_{0}) = T A/T A_{0}. For g ∈ S∖T Δ, ψ(g) ∉ Inn(Δ/B) and hence ^{g}b = a^{j}b for odd j.

(b) Choose x ∈ ψ^{−1}(α). Thus x ∈ S∖T Δ, ^{x}b = a^{m}b for some m ≡ 1 (mod 4), and upon replacing x by an appropriate element of xA_{0}, we can arrange that ^{x}b = ab (and still ψ(x) = α). Also, ψ(x^{2}) = α^{2} = ψ(a), so x^{2} ∈ aT A_{0}. Since α acts via the identity on A/B, we have ^{x}a = a^{1+4ℓ} for some ℓ ∈ ℤ.

Thus A = ⟨[x, b]⟩ ≤ [S, Δ]. Conversely, [S, Δ] ≤ A since Δ/A ≤ Z(S/A).

(c) Assume y ∈ T Ax. Then T A_{0}y has order 4 in S/T A_{0} ≅ D_{8} since y ∈ T A_{0}x or y ∈ T A_{0}ax, so y^{2} = a^{k}t for t ∈ T and k odd. Set 2^{n} = |a|. Since t ∈ C_{S}(Δ_{0}), ta = a^{1+4m} for some m. Then y^{4} = a^{k}(^{t}a)^{k}t^{2} = a^{2k(1+2m)}t^{2}. Upon iterating this procedure, we get that y^{2^{n−1}} = a^{±2^{n−2}}t^{2^{n−2}} ≠ 1, and so y^{2^{n}} = a^{2^{n−1}}t^{2^{n−1}}. Thus either |y| > 2^{n} = |x|, or |y| = 2^{n}, t^{2^{n−1}} = a^{2^{n−1}}, and thus Z(Δ) = ⟨a^{2^{n−1}}⟩ ≤ Fr(T). In the latter case, y^{2^{n−1}} ∉ A since t^{2^{n−2}} ∈ T ∖Z(Δ), so ⟨y⟩ ∩ A = 1. □

The next lemma provides a necessary condition on certain 2-groups S with 𝒳 (S) ≠ ∅ for there to be a nontrivial automorphism of odd order.

Lemma 2.11. Fix a 2-group S with r(S) ≤ 4, and a normal dihedral subgroup Δ ⊴ S with |Δ| ≥ 8. Set Z = Z(Δ), let Δ_{0} ≤ Δ be dihedral of order 8, and assume Z is a direct factor of C_{S}(Δ_{0}). Let 1 ≠ G ≤ Aut(S) be a subgroup of odd order, and set T̂ = [G, S]. Then

(a) T̂ ⊴ S, [T̂, Δ] = T̂ ∩ Δ = 1, and [S:T̂Δ] ≤ 2; and

(b) |G| = 3, and T̂ ≅ Q_{8} or C_{2^{k}} × C_{2^{k}} for some k ≥ 1.

Proof. Let A ⊴ Δ be the cyclic subgroup of index 2, let a ∈ A be a generator, and fix b ∈ Δ_{0}∖A. Set A_{0} = ⟨a^{2}⟩ and Z = Z(Δ). Set T = C_{S}(Δ_{0}). By assumption, there is T_{0} < T such that T = T_{0}Z and T_{0} ∩ Z = 1. For each t ∈ T, since T centralizes the subgroup of order 4 in A, ^{t}a = a^{1+4k} for some k. Thus [T, A] ≤ ⟨a^{4}⟩. Step 1: We first show that for each α ∈ G,

α(A_{0}) = A_{0}  and  α(T Δ) = T Δ .  (2.9)

By Lemma B.7, α(Z) = Z. If A_{0} > Z (if |A| > 4), then by Lemma B.7 applied to S/Z, α sends the subgroup of order 4 in A to itself. Upon iterating this procedure, we get that α(A_{0}) = A_{0}.

If Δ is not strongly automized, then for each g ∈ S, [g, b] ∈ A_{0}, so [ga^{i}, b] = 1 for some i, and ga^{i} ∈ T or ga^{i}b ∈ T. Thus S = T Δ, and α(T Δ) = T Δ trivially.

Now assume that Δ is strongly automized in S. By Lemma 2.10(a,b), [S:T Δ] = 2, and there is x ∈ S∖T Δ such that ^{x}a = a^{1+4ℓ} for some ℓ and ^{x}b = ab. If |Δ| ≥ 16, then T A⟨x⟩ is the centralizer of Δ_{0} ∩ A_{0} (the subgroup of order 4 in A_{0}), so α(T A⟨x⟩) = T A⟨x⟩.

If |Δ| = 8, then T = C_{S}(Δ) ⊴ S and T Δ ≅ T_{0} × Δ. Since A ≤ [S, S] ≤ T A by Lemma 2.10(b), α(a) ∈ T A, and hence

[T A, α(a)] ≤ [T A, T A] ∩ α([S, a]) ≤ T_{0} ∩ A_{0} = 1 .

Thus T A ≤ C_{S}(α(a)) = α(C_{S}(a)) = α(T A⟨x⟩). Let t ∈ T and i ∈ ℤ be such that α(a) = ta^{i}. Then a^{2} = α(a^{2}) = t^{2}a^{2i}, so i is odd, and b ∉ C_{S}(α(a)). Thus α(T A⟨x⟩) = T A⟨x⟩ or T A⟨bx⟩, the same holds for α^{i}(T A⟨x⟩) for all i, and since |α| is odd, we get α(T A⟨x⟩) = T A⟨x⟩.

Thus α(T A⟨x⟩) = T A⟨x⟩, independently of |Δ|. Hence

[α(T A), S] = [α(T A), α(T A⟨x⟩)][α(T A), α(b)]
≤ [T A⟨x⟩, T A⟨x⟩]α([T A, b]) ≤ (T A_{0})α(A_{0}) = T A_{0}.

Since Z(S/T A_{0}) = T A/T A_{0} by Lemma 2.10(a) again, this proves that α(T A) = T A. Also, α induces the identity on S/T A ≅ C_{2}^{2} since it has odd order and sends the class of x to itself, and hence α(T Δ) = T Δ. This finishes the proof of (2.9). Step 2: Now, T Δ/⟨a^{4}⟩ ≅ T_{0} × D_{8} (recall that T_{0} ∩ Δ = 1 and [T_{0}, Δ] = [T, Δ] ≤ ⟨a^{4}⟩). By (2.9), each α ∈ G^{#} induces an automorphism of T Δ/⟨a^{4}⟩ which sends A_{0}/⟨a^{4}⟩ = Z(Δ/⟨a^{4}⟩) to itself. So by the Krull-Schmidt theorem (Theorem A.8(b)), α(T A_{0}/⟨a^{4}⟩) = T A_{0}/⟨a^{4}⟩. Thus α(T A_{0}) = T A_{0}. Also, α(T A) = T A since A_{0} is in the Frattini subgroup of T A but not those of T A_{0}⟨b⟩ or T A_{0}⟨ab⟩. To summarize,

α(T A_{0}) = T A_{0}, α(T A) = T A, and α(T Δ) = T Δ .  (2.10)

Now, α|_{A_{0}} = Id since Aut(A_{0}) is a 2-group by Corollary A.10(a). Since α ≠ Id and |α| is odd, (2.10) together with Lemma A.9 imply that the automorphism of T A_{0}/A_{0} induced by α is nontrivial. Since r(T_{0}) ≤ r(S) − r(D_{8}) ≤ 2 (recall T_{0} × Δ_{0} ≤ S), T_{0} ≅ T A_{0}/A_{0} is metacyclic by Lemma B.1(a). By Lemma B.1(c), either T_{0} ≅ C_{2^{n}} × C_{2^{n}} for some n ≥ 1, or T_{0} ≅ Q_{8}.

By Lemma A.9 and since T_{0}/Fr(T_{0}) ≅ C_{2}^{2}, |Aut(T_{0})| = 3·2^{m} for some m. So |G| = 3, since G acts faithfully on T A_{0}/A_{0} ≅ T_{0}, and G acts via the identity on S/T A_{0} by (2.10) and Lemma A.9. So T̂ =^{def} [G, S] ≤ T A_{0}, and hence T̂ = [G, T A_{0}] (see [G, Theorem 5.3.6]). Also, T̂A_{0} = T A_{0}, since [G, T A_{0}/A_{0}] = T A_{0}/A_{0} in either case (T A_{0}/A_{0} ≅ C_{2^{n}} × C_{2^{n}} or Q_{8}).

Since G centralizes A_{0}, T̂ = [G, S] also centralizes A_{0}. Thus [T A_{0}, A_{0}] = [T̂A_{0}, A_{0}] = 1. Also,

T̂ ∩ A_{0} = [G, T A_{0}] ∩ C_{T A_{0}}(G) ∩ A_{0} ≤ [T A_{0}, T A_{0}] ∩ A_{0} ≤ [T, T] ∩ A_{0} = 1 ,

where the second relation holds by [G, Theorem 5.2.3]. Finally, T̂ ⊴ S (cf. [G, Theorem 2.2.1(iii)]), and hence

[T̂, Δ] ≤ T̂ ∩ Δ = (T̂ ∩ T A_{0}) ∩ Δ
= T̂ ∩ ((T ∩ Δ)A_{0}) = T̂ ∩ (C_{Δ}(Δ_{0})A_{0}) = T̂ ∩ A_{0} = 1.  □

CHAPTER 3 Essential subgroups in 2-groups of sectional rank at most 4

We now analyze the different possibilities for F-essential subgroups when F is a saturated fusion system over a 2-group S with r(S) ≤ 4. It will be convenient to use the following shorthand to refer to the different “types” of F-essential subgroups R < S which can occur. We say that R has

type (I) when |N_{S}(R)/R| ≥ 4,

type (II) when |N_{S}(R)/R| = 2 and R is not normal in S, or

type (III) when |S/R| = 2.

We let E_{F}^{(I)}, E_{F}^{(II)}, and E_{F}^{(III)} denote the sets of F-essential subgroups of types (I), (II), and (III), respectively, so that E_{F} = E_{F}^{(I)} ∪ E_{F}^{(II)} ∪ E_{F}^{(III)}. For each Y ⊴ S,

E_{F}(Y) = {P ∈ E_{F} | 𝔣𝔬𝔠(F, P) = Y } ,

E_{F}^{(I)}(Y) = E_{F}^{(I)} ∩ E_{F}(Y), etc.

The results in this chapter are summarized in the following theorem, formulated in terms of the sets 𝒳 (S) and 𝒴 (S) of Definition 2.1.

Theorem 3.1. Let F be a reduced, indecomposable fusion system over a 2-group S such that r(S) ≤ 4.

(a) If R ∈ E_{F}^{(I)}, then R ≅ C_{2}^{4} or 2_{−}^{1+4}, and S ≅ UT_{4}(2) ∈ V or S ∈ U.

(b) If R ∈ E_{F}^{(II)}, then either

(b.1) 𝔣𝔬𝔠(F, R) ∈ 𝒳 (S), R is as in Lemma 3.8(a), and S ∈ DSWG; or

(b.2) 𝔣𝔬𝔠(F, R) ∈ 𝒴 (S) and S ∈ UV; or

(b.3) 𝔣𝔬𝔠(F, R) ∉ 𝒴 (S), 𝒳 (S) = ∅, 𝔣𝔬𝔠(F, R) ≅ C_{2}^{4} or UT_{3}(4), R is as in Lemma 3.7(b), and S ∈ U.

(c) If E_{F} = E_{F}^{(III)} (i.e., E_{F}^{(I)} = E_{F}^{(II)} = ∅), then S ≅ D_{8}, C_{4} ≀ C_{2}, or UT_{4}(2), or S has type M_{12} or Aut(M_{12}), or S ∈ U.

Proof. (a) If R ∈ E_{F}^{(I)}, then by Lemma 3.3, |N_{S}(R)/R| = 4. By Proposition 3.5, N_{S}(R)/R ≅ C̸_{4}. The result thus follows from Proposition 3.4.

(b) Assume R ∈ E_{F}^{(II)}. If 𝔣𝔬𝔠(F, R) ∈ 𝒴 (S), then S ∈ UV by Proposition 3.12, and (b.2) holds.

If 𝔣𝔬𝔠(F, R) ∉ 𝒴 (S), then by Lemma 3.6(b), we are in one of the following two situations:

• If 3.6(b.i) holds, then we are in the situation of Lemma 3.7(b): S ∈ U, 𝒳 (S) = ∅, 𝔣𝔬𝔠(F, R) ≅ C_{2}^{4} or UT_{3}(4), and (b.3) holds.

• If 3.6(b.ii) holds, then (since F is reduced and 𝔣𝔬𝔠(F, R) ∉ 𝒴 (S)) we are in the situation of Lemma 3.8(a). In particular, 𝔣𝔬𝔠(F, R) ∈ 𝒳 (S), so S ∈ DSWG by Proposition 3.14, and (b.1) holds.

(c) This is shown in Proposition 3.15. □

In this chapter, in addition to proving Theorem 3.1, we also collect more detailed information about the essential subgroups: information which will be useful in later chapters when analyzing the fusion systems themselves.

The following lemma limits the possibilities for F-automorphism groups of essential subgroups of type (II) or (III).

Lemma 3.2. Fix a 2-group S with r(S) ≤ 4, a saturated fusion system F over S, and a subgroup R ∈ E_{F}. Assume that either R ∈ E_{F}^{(II)}, or R ∈ E_{F}^{(III)} and

|E_{F}^{(III)}| ≥ 2. Then Out_{F}(R) ≅ Σ_{3}, Σ_{3} × C_{3}, or (C_{3} × C_{3}) ⋊^{−1} C_{2}.

Proof. Assume otherwise. By Lemma 1.7, Out_{F}(R) acts faithfully on the quotient R/Fr(R), and hence is isomorphic to a subgroup of GL_{4}(2) ≅ A_{8}. By the Sylow axiom, |Out_{F}(R)| = 2m for some odd m, so |O_{2′}(Out_{F}(R))| = m by Burnside’s normal p-complement theorem [G, Theorem 7.4.3]. By Proposition D.1(a), and since Out_{F}(R) is not isomorphic to one of the groups listed above, Out_{F}(R) is

isomorphic to a subgroup of C_{15} ⋊^{2}C_{4} or (C_{3} ×A_{5})⋊C_{2}, and in particular, contains a subgroup Aut_{F}^{0}(R)/Inn(R) ≅ D_{10}. By Lemma D.8, Fr(R) ≤ Z(R).

If R ∈ E_{F}^{(II)}, set S_{0} = N_{S}(R), fix x ∈ N_{S}(S_{0})∖S_{0} such that x^{2} ∈ S_{0}, and set Q = ^{x}R and Aut_{F}^{0}(Q) = c_{x}Aut_{F}^{0}(R)c_{x}^{−1}. Then S_{0} = RQ since [S_{0}:R] = 2. If R ∈ E_{F}^{(III)} (thus [S:R] = 2) and |E_{F}^{(III)}| ≥ 2, choose Q ∈ E_{F}^{(III)} different from R, and choose Aut_{F}^{0}(Q) ≤ Aut_{F}(Q) such that Aut_{F}^{0}(Q)/Inn(Q) ≅ Σ_{3} or D_{10}. In either case, let T ⊴ R ∩ Q < R be the largest subgroup which is normalized by Aut_{F}^{0}(R) and by Aut_{F}^{0}(Q). Since T < R and Out_{F}^{0}(R) ≅ D_{10} acts irreducibly on R/Fr(R) ≅ C_{2}^{4}, T ≤ Fr(R) ≤ Z(R), and hence C_{S}(T) ≥ R.

Thus T is not centric in S. This situation is impossible by [AOV2, Theorem 4.5 or 4.6(a)]: Out_{F}(R) cannot contain D_{10} when T (as defined here) is not centric in S. □

3.1. Essential subgroups of index 4 in their normalizer

We begin with a very general lemma on essential subgroups of index 4 in their normalizer, and then make it more explicit in two propositions.

Lemma 3.3. Let F be a saturated fusion system over a 2-group S. Assume R ≤ S is an F-essential subgroup with |N_{S}(R)/R| ≥ 4 and rk(R/Fr(R)) ≤ 4. Then rk(R/Fr(R)) = 4, Out_{F}(R) acts faithfully on R/Fr(R), and one of the following holds: either

(a) N_{S}(R)/R ≅ C_{4} permutes freely some basis of R/Fr(R); or

(b) N_{S}(R)/R ≅ C_{2}^{2} permutes freely some basis of R/Fr(R); or

(c) N_{S}(R)/R ≅ C_{2}^{2} acts on R/Fr(R) with rk(C_{R/Fr(R)}(N_{S}(R)/R)) = 2.

Also, Out_{F}(R) ≅ C_{15} ⋊^{2} C_{4}, C_{5} ⋊^{2} C_{4}, or (C_{3} × C_{3}) ⋊ C_{4} in case (a); Out_{F}(R) ≅ A_{5} and R/Fr(R) is its orthogonal module in case (b); and Out_{F}(R) ≅ A_{5} or C_{3} × A_{5} and R/Fr(R) is its L_{2}(4)-module in case (c).

Proof. By Proposition 1.9, rk(R/Fr(R)) ≥ 4. Since the opposite inequality holds by assumption, rk(R/Fr(R)) = 4. Also, Out_{S}(R) ≅ N_{S}(R)/R ≅ C_{2}^{2} or C_{4}. By Lemma 1.7, Out_{F}(R) acts faithfully on R/Fr(R), and hence is isomorphic to a subgroup of GL_{4}(2) ≅ A_{8}.

Set Γ = Out_{F}(R). If Out_{S}(R) ≅ C_{4}, then |Γ/O_{2′}(Γ)| = 4 by Burnside’s normal p-complement theorem (cf. [G, Theorem 7.4.3]). The involution in Out_{S}(R) is not

central in Γ since Γ has a strongly 2-embedded subgroup, so Γ ≅ C_{15} ⋊^{2}C_{4}, C_{5} ⋊^{2}C_{4}, or (C_{3} × C_{3}) ⋊ C_{4} by Proposition D.1(a). In either of the first two cases, Out_{S}(R) acts on R/Fr(R) ≅ C_{2}^{4} via the Galois action on 𝔽_{16}, hence permutes a basis by the Hilbert normal basis theorem. If O_{2′}(Γ) ≅ C_{3} × C_{3}, then Out_{S}(R) acts by exchanging the two irreducible factors in R/Fr(R), and acts on each by exchanging the elements in a basis.

If Out_{S}(R) ≅ C_{2}^{2}, then Γ/O_{2′}(Γ) ≅ A_{5} by Bender’s theorem on groups with strongly 2-embedded subgroups [Be, Satz 1] and since |Out(A_{5})| = 2. Hence by Proposition D.1(a,d), Γ ≅ A_{5} or A_{5} × C_{3}, and R/Fr(R) is its orthogonal module (in case (b)) or L_{2}(4)-module (case (c)). □

We now deal separately with the cases where N_{S}(R)/R ≅ C_{2}^{2} or C_{4}. As in Proposition D.1(d), when V ≅ 𝔽_{2}^{4} is an A_{5}- or Σ_{5}-module, we call it the “L_{2}(4)- module” if V |_{A_{5}} is the natural module for SL_{2}(4) ≅ A_{5}, and the “orthogonal module” if it is the natural module for Ω_{4}^{−}(2) ≅ A_{5}.

Proposition 3.4. Let F be a saturated fusion system over a 2-group S with r(S) ≤ 4. Assume R ∈ E_{F} is such that N_{S}(R)/R ≅ C_{2}^{2}. Then R ≅ C_{2}^{4} or 2_{−}^{1+4}, |S| ≤ 2^{7}, and either R ⊴ S and S/Fr(R) ≅ UT_{4}(2), or R ≅ C_{2}^{4} and N_{S}(R) ≅ UT_{3}(4). If F is reduced, then S ≅ UT_{4}(2) or S ∈ U.

Proof. By Lemma 3.3, Out_{F}(R) ≅ A_{5} or C_{3} × A_{5} and acts faithfully on R/Fr(R) ≅ C_{2}^{4}. By Theorem 1.4, there is a finite group G such that N_{S}(R) ∈ Syl_{2}(G), R ⊴ G, C_{G}(R) ≤ R, and G/R ≅ Out_{G}(R) = O^{3}(Out_{F}(R)) ≅ A_{5}. Then R ≅ C_{2}^{4} or 2_{−}^{1+4} by Lemma D.4.

Since any nontrivial extension of C_{2}^{4} by A_{5} splits by [GH, Lemma II.2.6], G/Fr(R) splits over R/Fr(R). Hence by Lemmas C.4(a) and C.7, N_{S}(R)/Fr(R) is isomorphic to C_{2} ≀ C_{2}^{2} = C_{2}^{4} ⋊ C_{2}^{2} ≅ UT_{4}(2) (if R/Fr(R) is the orthogonal module for A_{5}), or to UT_{3}(4) (if R/Fr(R) is the L_{2}(4)-module). Case 1: Assume N_{S}(R)/Fr(R) ≅ UT_{3}(4). If R ≅ 2_{−}^{1+4}, then Out(R) ≅ Σ_{5}, and this group acts on R/Fr(R) as the orthogonal module, contradicting our assumption. Thus R ≅ C_{2}^{4}. If R ⊴ S, then S ≅ UT_{3}(4) ∈ U, so assume R ⋬ S.

Set T = N_{S}(R). Since T ≅ UT_{3}(4) contains exactly two subgroups Q, R isomorphic to C_{2}^{4}, each element of N_{S}(T)∖T exchanges them, and thus T has index 2 in its normalizer. Set S_{0} = N_{S}(T). Then S_{0} ∈ U (see Definition 0.1), so T is characteristic in S_{0} by Lemma C.9. Hence S = S_{0} ∈ U by Lemma A.1(b), and |S| = 2^{7}. Case 2: Assume N_{S}(R)/Fr(R) ≅ UT_{4}(2). Since Fr(R) = 1 or Z(N_{S}(R)), Fr(R) is characteristic in N_{S}(R). Also, R/Fr(R) ≅ C_{2}^{4} is the unique abelian subgroup of rank 4 in N_{S}(R)/Fr(R) (Lemma C.4(a)), so R is characteristic in N_{S}(R), and S = N_{S}(R) by Lemma A.1(b). Thus R ⊴ S, and S/Fr(R) ≅ UT_{4}(2).

Now assume F is reduced; we must show that S ≅ UT_{4}(2) or S ∈ U. If R ≅ C_{2}^{4}, then S ≅ S/Fr(R) ≅ UT_{4}(2), so assume R ≅ 2_{−}^{1+4}. Set Z = ⟨z⟩ = Z(R) = Fr(R). Let U < S be such that U/Z is the unique subgroup of S/Z ≅ UT_{4}(2) isomorphic to 2_{+}^{1+4} (Lemma C.4(b)). Then Z(U) = [U, U] ≅ C_{2}^{2} by Lemma D.3 (or by explicit computations), so U is special of type 2^{2+4}. Also, there is α ∈ Aut_{G}(S) of order 3 since N_{G}(S)/R ≅ A_{4}, α acts nontrivially on U/(R ∩ U) and on R ∩ U, and trivially on Z(U) since it fixes Z = Z(S). Thus C_{U/Z(U)}(α) = 1.

By Lemma D.2, either U ≅ UT_{3}(4), or U ≅ Q_{8} × Q_{8}, or U/⟨x⟩ ≅ 2_{+}^{1+4} for exactly two of the involutions x ∈ Z(U). Fix z′ ∈ Z(U)∖Z. Since U/⟨z⟩ ≅ 2_{+}^{1+4} and U/⟨z′⟩ ≅ U/⟨zz′⟩ (z′ and zz′ are S-conjugate), the last case is impossible.

If U ≅ Q_{8} ×Q_{8}, then I(U) ⊆ Z(U) ≤ R. By Lemma C.4(c), I(S/Z) ⊆ (R/Z)∪ (U/Z), so I(S) ⊆ R. All noncentral involutions in R ≅ 2_{−}^{1+4} are F-conjugate to each other since Aut_{F}(R) ≅ A_{5} (see Lemma C.2(a)). Since Z(F) = 1, they are all F-conjugate to z, and so z′ ∈ z^{F}. By the extension axiom and since C_{S}(z′) = U, there is φ ∈ Hom_{F}(U, S) such that φ(z′) = z. Then φ(U)/Z ≅ U/⟨z′⟩ ≅ Q_{8} × C_{2}^{2} is a subgroup of index 2 in S/Z ≅ UT_{4}(2), which is impossible by Lemma C.4(c).

Thus if F is reduced, then U ≅ UT_{3}(4), and S ∈ U. □

We next consider essential subgroups R such that N_{S}(R)/R ≅ C_{4}.

Proposition 3.5. Let F be a saturated fusion system over a 2-group S with r(S) ≤ 4. Assume R ∈ E_{F} with N_{S}(R)/R ≅ C_{4}. Then R ⊴ S (so S/R ≅ C_{4}), |R| ≤ 2^{6}, and F is not reduced.

Proof. By Theorem 1.4, there is a finite group G such that N_{S}(R) ∈ Syl_{2}(G), R ⊴ G, C_{G}(R) ≤ R, and G/R ≅ Out_{G}(R) = Out_{F}(R). By Lemma 3.3, G/R ≅

Out_{F}(R) acts faithfully on R/Fr(R) ≅ C_{2}^{4}, and is isomorphic to C_{5} ⋊^{2} C_{4}, C_{15} ⋊^{2} C_{4}, or (C_{3} × C_{3}) ⋊ C_{4}. So by Lemma D.8 (if G/R contains a subgroup isomorphic to C_{5} ⋊ C_{4}) or Lemma D.7 (if G/R ≅ (C_{3} × C_{3}) ⋊ C_{4}), we have

R ≅ C_{2}^{4}, 2_{±}^{1+4}, Q_{8} × Q_{8}, or is of type PSU_{3}(4).  (3.1)

Set S_{0} = N_{S}(R) for short. By Lemma A.6(a,b), either R ≅ C_{2}^{4}; or Z(R) = Z(S_{0}); or Z(R) > Z(S_{0}), |Z(R)| = 4, R ≅ Q_{8} × Q_{8} or is of type PSU_{3}(4), and Z(R) = Z_{2}(S_{0}) since all involutions in R are central (cf. [Sz2, Lemma 6.4.27(iii)] when R is of type PSU_{3}(4)). Thus Fr(R) is characteristic in S_{0} in all cases. By Lemma 3.3(a), S_{0}/Fr(R) ≅ C_{2} ≀ C_{4} = C_{2}^{4} ⋊ C_{4} where S_{0}/R ≅ C_{4} permutes freely some basis of R/Fr(R). So by Lemma A.4(b), R/Fr(R) is the only abelian subgroup of rank 4 in S_{0}/Fr(R), and hence R is characteristic in S_{0} = N_{S}(R). Thus R ⊴ S (S_{0} = S) by Lemma A.1(b). In particular, S^{ab} ≅ (C_{2} ≀ C_{4})^{ab} ≅ C_{2} × C_{4}.

It remains to show that F is not reduced. Assume otherwise. By Lemma 1.17, there are subgroups Q ⊴ P ≤ S such that S = RP and P/Q ≅ C_{4} × C_{4}. Furthermore, by the same lemma, there are elements g, h ∈ P and α ∈ Aut_{F}^{∗}(P) such that h = α(g), g ∈ R, and S = R⟨h⟩. In particular, g^{2} ∉ [P, P] since h^{2} ∉ R ≥ [S, S].

Now, 4 ≤ |h| = |g| ≤ 4, so |h| = |g| = 4 and R has exponent 4. Thus R is nonabelian. Set T = [h^{2}, R]. For x ∈ R, [x, h^{2}]^{−1} = [h^{2}, x] = ^{h^{2}}[x, h^{2}] since h^{4} = 1. Thus c_{h^{2}} inverts T, and T is abelian. Also, T ⊴ R by [G, Theorem 2.1(iii)], and its image in R/Fr(R) has rank 2 (since c_{h} permutes freely a basis for R/Fr(R)). By inspection of the list in (3.1), T ≥ Fr(R).

If g ∈ T, then g^{2} = [g, h^{2}] ∈ [P, P], which is impossible by the above remarks. Thus g ∉ T. Upon regarding R/Z(R) as an 𝔽_{2}[⟨h⟩] ≅ 𝔽_{2}[C_{4}]-module, we have

R/Z(R) ≅ 𝔽_{2}[h]/(h^{4} − 1) = 𝔽_{2}[h]/(h − 1)^{4}.

Since the polynomial ring 𝔽_{2}[X] is a PID, R/Z(R) contains a unique 𝔽_{2}[⟨h⟩]- submodule of rank k for each 0 ≤ k ≤ 4 (generated by (h − 1)^{4−k} under the above identification). Since g ∉ T, the subgroup generated by g and iterated conjugates with h has rank at least 3 in R/Z(R), so the image of Q in R/Z(R) has rank at least 2, and Q ≥ T. But this is impossible, since Q ≥ Z(R) and g^{2} ∉ Q. We conclude that F is not reduced. □

3.2. Essential pairs of type (II)

Let F be a saturated fusion system over a 2-group S. An F-essential pair of type (II) in F is a pair of subgroups (R_{1}, R_{2}) such that

• R_{1}, R_{2} ∈ E_{F}^{(II)},

• N_{S}(R_{1}) = N_{S}(R_{2}) = R_{1}R_{2} < S, and

• R_{2} = ^{x}R_{1} for some x ∈ N_{S}(R_{1}R_{2})∖R_{1}R_{2} where x^{2} ∈ R_{1}R_{2}.

The F-essential pairs play a key role when describing fusion systems containing essential subgroups of type (II).

We first show that each F-essential subgroup of type (II) lies in an F-essential pair of type (II), and prove some of the basic properties of such pairs.

Lemma 3.6. Let F be a saturated fusion system over a 2-group S with r(S) ≤ 4, and assume R_{1} ∈ E_{F} is of type (II). Then there is a subgroup R_{2} ∈ E_{F} of type (II) such that (R_{1}, R_{2}) is an F-essential pair of type (II). Set R = R_{1}R_{2} = N_{S}(R_{1}) = N_{S}(R_{2}), and let x ∈ N_{S}(R)∖R be such that x^{2} ∈ R and ^{x}R_{1} = R_{2}.

(a) For i = 1, 2, O^{2′}(Out_{F}(R_{i})) ≅ Σ_{3} or (C_{3} × C_{3}) ⋊ C_{2}. There are subgroups

Aut_{F}^{0}(R_{i}) ≤ Aut_{F}(R_{i}) and Out_{F}^{0}(R_{i}) = Aut_{F}^{0}(R_{i})/Inn(R_{i}) ≤ Out_{F}(R_{i})

such that Out_{S}(R_{i}) ≤ Out_{F}^{0}(R_{i}) ≅ Σ_{3} and c_{x}Aut_{F}^{0}(R_{i})c_{x}^{−1} = Aut_{F}^{0}(R_{3−i}).

(b) For Aut_{F}^{0}(R_{i}) as in (a), let T ≤ R_{1} ∩ R_{2} be the largest subgroup normalized by Aut_{F}^{0}(R_{1}) and by Aut_{F}^{0}(R_{2}). Then x ∈ N_{S}(T), and either

(b.i) T = R_{1} ∩ R_{2}, C_{S}(T) ≤ T, and N_{S}(T) = R⟨x⟩; or

(b.ii) T < R_{1} ∩ R_{2} and C_{R}(T) ≰ T.

(c) For Out_{F}^{0}(R_{i}) as in (a), there are groups G_{1} > R < G_{2} and an isomorphism β ∈ Iso(G_{1}, G_{2}), such that [G_{i}:R] = 3, R_{i} ⊴ G_{i}, Out_{F}^{0}(R_{i}) = Out_{G_{i}}(R_{i}) ≅ G_{i}/R_{i} (i = 1, 2), and β|_{R} = c_{x}|_{R}.

Proof. Set R = N_{S}(R_{1}), choose x ∈ N_{S}(R)∖R such that x^{2} ∈ R, and set R_{2} = ^{x}R_{1}. Then R_{2} ∈ E_{F}^{(II)}, N_{S}(R_{2}) = ^{x}R = R, and R = R_{1}R_{2} since the R_{i} are distinct of index 2 in R. Thus (R_{1}, R_{2}) is an F-essential pair of type (II).

(a,c) The first statement in (a) follows from Lemma 3.2, and the others immediately from that. Point (c) follows from the model theorem (Theorem 1.4); we refer to [AOV2, Theorem 4.6] for more details. The uniqueness in the choice of T implies that x ∈ N_{S}(T).

(b) If C_{S}(T) ≰ T, then T < R_{1} ∩R_{2} by [AOV2, Theorem 4.6(a.ii)]. So it remains to prove that

C_{R}(T) ≤ T ⟹ T = R_{1} ∩ R_{2} and N_{S}(T) = R⟨x⟩.  (3.2)

If C_{R}(T) ≤ T, then by [AOV2, Theorem 4.6(b)], Out_{S}(T) ≅ N_{S}(T)/T acts faithfully on T/Fr(T). Since rk(T/Fr(T)) ≤ 4, N_{S}(T)/T is isomorphic to a subgroup of GL_{4}(2) ≅ A_{8}. In particular, N_{S}(T)/T contains no elements of order 8.

Set R_{12} = R_{1} ∩ R_{2} for short. Then R/R_{12} = (R_{1}/R_{12}) × (R_{2}/R_{12}) ≅ C_{2}^{2}, c_{x} exchanges the two factors R_{i}/R_{12}, and hence R⟨x⟩/R_{12} ≅ D_{8}. If N_{S}(T) > R⟨x⟩, then there is g ∈ N_{N_{S}(T)}(R⟨x⟩)∖R⟨x⟩ such that g^{2} ∈ R⟨x⟩ (Lemma A.1). Since g ∉ R = N_{S}(R_{i}) (i = 1, 2), g ∉ N_{S}(R), and hence ^{g}(R/R_{12}) ≠ R/R_{12}. In other words, R⟨x⟩/R_{12} ≅ D_{8} is strongly automized in R⟨x, g⟩/R_{12}. By Lemma 2.10(c), applied with R⟨x, g⟩/R_{12} and R⟨x⟩/R_{12} in the role of S and Δ = T Δ, R⟨x, g⟩/R_{12} contains an element of order 8, which is a contradiction. (In fact, R⟨x, g⟩/R_{12} ≅ D_{16} or SD_{16}.)

Thus N_{S}(T) = R⟨x⟩. Now assume T < R_{12}. By the maximality of T, the amalgam (G_{1}/T > R_{12}/T < G_{2}/T) of (c) is primitive of index (3, 3) as defined in [Gd2]. Hence it is one of those in Goldschmidt’s list (Table 1 and Theorem A in [Gd2]). Since G_{1}/T ≅ G_{2}/T and |R_{i}/T | ≥ 4, we have G_{i}/T ≅ Σ_{4} × C_{2}^{k}, R/T ≅ D_{8} × C_{2}^{k}, and R_{i}/T ≅ C_{2}^{2+k} for some k = 0, 1.

Choose elements y_{i} ∈ R_{i}∖R_{12} (i = 1, 2). Thus y_{i}^{2} ∈ T (since R_{i}/T ≅ C_{2}^{2+k}), and (xy_{1})^{2} ≡ y_{1}y_{2} (mod R_{12}). So (y_{1}y_{2})^{2} ≡ [y_{1}, y_{2}] ≢ 1 (mod T), and xy_{1}T has order 8 in R⟨x⟩/T, which is impossible. This finishes the proof of (3.2). □

Recall (Definition 1.13) that for a fusion system F over a p-group S, we set Aut_{F}^{∗}(R) = O^{p}(O^{p′}(Aut_{F}(R))) if R < S and Aut_{F}^{∗}(S) = O^{p}(Aut_{F}(S)), and let 𝔣𝔬𝔠(F, R) be the normal closure in S of [Aut_{F}^{∗}(R), R].

In the next two lemmas, we examine F-essential pairs of the two types described in points (b.i) and (b.ii) of Lemma 3.6. A priori, this type depends on the choice of subgroups Aut_{F}^{0}(R_{i}) ≤ Aut_{F}(R_{i}), but we will see in each case that Aut_{F}(R_{i}) ≅ Σ_{3} or C_{3} × Σ_{3}, and hence that this choice is unique.

Lemma 3.7. Let F be a saturated fusion system over a 2-group S with r(S) ≤ 4. Let (R_{1}, R_{2}) be an F-essential pair of type (II), choose subgroups Σ_{3} ≅ Out_{F}^{0}(R_{i}) ≤ Out_{F}(R_{i}) as in Lemma 3.6(a), and assume we are in the situation of Lemma 3.6(b.i). Thus T = R_{1} ∩ R_{2} in the notation of that lemma. Then one of the following holds.

(a) If 𝔣𝔬𝔠(F, R_{i}) ∈ 𝒴 (S), then T ∈ 𝒴_{0}(S), [Aut_{F}^{∗}(R_{i}), R_{i}] ≤ T ≤ 𝔣𝔬𝔠(F, R_{i}), Aut_{F}(T) ∈ 𝒜_{S}(T) (Definition 2.2), and either

• Out_{F}(T) ≅ Σ_{3} ≀ C_{2}, and T ≅ C_{2}^{4}, 2_{+}^{1+4}, or Q_{8} × Q_{8}; or

• Out_{F}(T) ≅ Σ_{5}, T ≅ C_{2}^{4} or 2_{−}^{1+4}, and T/Fr(T) is the orthogonal module

for Out_{F}(T).

(b) If 𝔣𝔬𝔠(F, R_{1}) ∉ 𝒴 (S), then T ≅ C_{2}^{4} and is the L_{2}(4)-module for Out_{F}(T) ≅ Σ_{5} or (A_{5} × C_{3}) ⋊ C_{2}. Also, S ∈ U, [Aut_{F}^{∗}(R_{1}), R_{1}] = T, and 𝔣𝔬𝔠(F, R_{1}) = T or 𝔣𝔬𝔠(F, R_{1}) ≅ UT_{3}(4). If 𝒴 (S) ≠ ∅, then |S| = 2^{8}, S/Z(S) ≅ D_{8} ≀ C_{2}, and 𝔣𝔬𝔠(F, R_{1})/Z(S) ≅ 2_{+}^{1+4}.

Also, in all cases, 𝒳 (S) = ∅, and for i = 1, 2, Out_{R_{i}}(T) ≰ O^{2}(Out_{F}(T)), and Out_{F}(R_{i}) ≅ Σ_{3}.

Proof. Set R = R_{1}R_{2}, let (G_{1} > R < G_{2}) be an amalgam as in Lemma 3.6(c), and set Out_{F}^{0}(R_{i}) = Out_{G_{i}}(R_{i}) ≤ Out_{F}(R_{i}). Thus R = N_{S}(R_{1}) = N_{S}(R_{2}), [G_{i}:R] = 3 and Out_{F}^{0}(R_{i}) ≅ Σ_{3} i = 1, 2). Let x ∈ N_{S}(R)∖R be such that x^{2} ∈ R and R_{2} = ^{x}R_{1}. By assumption, T = R_{1} ∩ R_{2} is normal in both G_{1} and G_{2}.

By Lemma 3.6(b), T is centric in S and Out_{S}(T) ≅ N_{S}(T) = R⟨x⟩. If T^{∗} ∈ T^{F} is fully normalized, then there is φ ∈ Hom_{F}(N_{S}(T), N_{S}(T^{∗})) such that φ(T) = T^{∗} (Proposition 1.3(a)), N_{S}(T^{∗}) = φ(N_{S}(T)) by the same lemma applied to φ(R_{i}), φ(R), φ(x), etc. Thus T is fully normalized in F. So by Lemma 1.5(a),

Out_{F}(R_{i}) ≅ N_{Out_{F} (T)}(Out_{R_{i}}(T))/Out_{R_{i}}(T) for i = 1, 2.  (3.3)
If Out_{R_{i}}(T) ≤ O^{2}(Out_{F}(T)) for some i = 1, 2, then
Out_{R_{i}}(T) ≤ [Out_{F}(T), Out_{F}(T)],

and by the focal subgroup theorem for groups [G, Theorem 7.3.4], Out_{R_{i}}(T) is conjugate in Out_{F}(T) to the center of Out_{S}(T) ≅ D_{8}. Hence R_{i} is F-conjugate to the third subgroup R_{3} < R of index 2 containing T. Since R_{3} ⊴ R⟨x⟩, this would contradict the assumption that R_{1} is F-essential (hence fully normalized). Thus Out_{R_{i}}(T) ≰ O^{2}(Out_{F}(T)).

Let x ∈ Out(T) be such that ⟨x⟩ = Out_{R_{1}}(T). Thus x ∉ O^{2}(Out_{F}(T)). By (3.3) and since O_{2}(Out_{F}(R_{i})) = 1 (Lemma 1.7), O_{2}(C_{Out_{F} (T)}(x)) = ⟨x⟩. Hence by Proposition D.1(f), Out_{F}(T) ≅ Σ_{3} ≀ C_{2}, Σ_{5}, or ΓL_{2}(4) ≅ (A_{5} × C_{3}) ⋊ C_{2}. So in all cases, by (3.3) and since Out_{F}(R_{i}) is not a 2-group, we have Out_{F}(R_{i}) ≅ Σ_{3}.

By Theorem 1.4, there is a finite group G such that N_{S}(T) ∈ Syl_{2}(G), T ⊴ G, C_{G}(T) ≤ T, and Out_{F}(T) = Out_{G}(T) ≅ G/T. If Out_{F}(T) ≅ Σ_{3} ≀ C_{2}, then by Lemma D.7, T ≅ C_{2}^{4}, 2_{+}^{1+4}, or Q_{8} × Q_{8}. If Out_{F}(T) ≅ Σ_{5} or ΓL_{2}(4), then by Lemma D.4, T ≅ C_{2}^{4} or 2_{−}^{1+4}. Finally, if T ≅ C_{2}^{4} and Out_{F}(T) ≅ Σ_{5}, then Out_{F}(T) acts on C_{2}^{4} as either the L_{2}(4)-module or the orthogonal module.

In all cases, G/Fr(T) splits as a semidirect product

G/Fr(T) ≅ (T/Fr(T)) ⋊ Out_{F}(T) :  (3.4)

by [GH, Lemma II.2.6] when Out_{F}(T) ≅ Σ_{5} or ΓL_{2}(4), and by [AOV2, Lemma A.8] when Out_{F}(T) ≅ Σ_{3} ≀ C_{2}.

We now consider the individual cases.

(a) Assume that either Out_{F}(T) ≅ Σ_{3} ≀ C_{2}, or Out_{F}(T) ≅ Σ_{5} and T/Fr(T) is the orthogonal module for Out_{F}(T). We will show that 𝔣𝔬𝔠(F, R_{i}) ∈ 𝒴 (S), that 𝒳 (S) = ∅, and prove the other claims in (a) which have not already been shown.

If Out_{F}(T) ≅ Σ_{5}, then by Proposition D.1(d), T/Fr(T) is generated by an Out_{F}(T)-orbit of length 5, and hence Out_{S}(T) permutes a basis. If Out_{F}(T) ≅ Σ_{3} ≀ C_{2}, let V_{1}, V_{2} < T/Fr(T) be the irreducible components for the action of the Sylow 3-subgroup, choose bases for each V_{i} permuted by N_{Out_{S}(T)}(V_{i}), and the union of these two bases is a basis for T/Fr(T) permuted by Out_{S}(T). In either case, N_{S}(T)/Fr(T) splits over T/Fr(T) by (3.4), and hence N_{S}(T)/Fr(T) ≅ D_{8} ≀C_{2}.

Since T ≅ C_{2}^{4}, 2_{+}^{1+4}, 2_{−}^{1+4}, or Q_{8}×Q_{8}, this proves that T ∈ 𝒴_{0}(S). Also, Out_{F}(T) ∈ 𝒜_{S}(T) since it is isomorphic to Σ_{5} or Σ_{3} ≀ C_{2}.

If Out_{F}(T) ≅ Σ_{3} ≀ C_{2}, then the Out_{R_{i}}(T) are distinct noncentral subgroups of order 2 which are conjugate in Out_{F}(T). Choose α_{i} ∈ Aut_{F}(T) of order 3 such that [[α_{i}], Out_{R_{i}}(T)] = 1 in Out_{F}(T). By the extension axiom (and since T is fully normalized), each α_{i} extends to α_{i} ∈ Aut_{F}^{∗}(R_{i}). Then [α_{1}] ≠ [α_{2}] in Out_{F}(T), so

T ≥ [Aut_{F}^{∗}(R_{1}), R_{1}][Aut_{F}^{∗}(R_{2}), R_{2}] ≥ [α_{1}, T][α_{2}, T] = [⟨α_{1}, α_{2}⟩, T] = T.

Hence [Aut_{F}^{∗}(R_{i}), R_{i}] ≤ T, and 𝔣𝔬𝔠(F, R_{i}) = ⟨T^{S}⟩ ∈ 𝒴 (S).

Now assume that Out_{F}(T) ≅ Σ_{5}. Identify these groups in such a way that Out_{R_{1}}(T) = ⟨(1 2)⟩ and Out_{R_{2}}(T) = ⟨(3 4)⟩ (recall that Out_{R_{i}}(T) ∉ O^{2}(Out_{F}(T))). Let α_{1}, α_{2} ∈ Aut_{F}(T) be elements of order 3 such that [α_{1}] = (3 4 5) and [α_{2}] = (1 2 5). Then ⟨α_{1}, α_{2}⟩ ≅ A_{5}, so [α_{1}, T][α_{2}, T] = T.

For i = 1, 2, α_{i} normalizes Aut_{R_{i}}(T), so by the extension axiom (and since T is fully normalized), α_{i} extends to α_{i} ∈ Aut_{F}(R_{i}). In particular, α_{i} ∈ Aut_{F}^{∗}(R_{i}), so

T ≥ [Aut_{F}^{∗}(R_{1}), R_{1}][Aut_{F}^{∗}(R_{2}), R_{2}] ≥ [α_{1}, T][α_{2}, T] = T.

Hence [Aut_{F}^{∗}(R_{i}), R_{i}] ≤ T, and 𝔣𝔬𝔠(F, R_{i}) = ⟨T^{S}⟩ ∈ 𝒴 (S).

In either case, 𝒳 (S) = ∅ by Corollary 2.5 and since 𝒴 (S) ≠ ∅.

(b) Assume T ≅ C_{2}^{4} is the L_{2}(4)-module for Out_{F}(T) = ΣL_{2}(4) or ΓL_{2}(4). We must show that 𝔣𝔬𝔠(F, R_{1}) ∉ 𝒴 (S), and prove the other claims in (b).

Let G ≥ N_{S}(T) be as in (3.4), let G_{0} ⊴ G be such that G_{0} > T and G_{0}/T ≅ SL_{2}(4), and let T̂ ≤ N_{S}(T) be such that T̂ ∈ Syl_{2}(G_{0}). Thus T̂ is a semidirect product of C_{2}^{4} by C_{2}^{2} where C_{2}^{2} acts as UT_{2}(4). So T̂ ≅ UT_{3}(4) by Lemma C.7(a), T̂/Z(T̂) is centric in N_{S}(T)/Z(T̂), and hence N_{S}(T) ∈ U.

Set S_{0} = N_{S}(T) = R⟨x⟩ for short. By (3.4) again, S_{0} splits as a semidirect product of C_{2}^{4} by D_{8}. If S > S_{0}, then Out_{S}(S_{0}) exchanges the two subgroups of T̂ ≅ UT_{3}(4) isomorphic to C_{2}^{4}, and hence [N_{S}(S_{0}):S_{0}] = 2 and N_{1} =^{def} N_{S}(S_{0}) ∈ U. Hence T̂ is characteristic in N_{1} by Lemma C.9, and S = N_{1} ∈ U by Lemma A.1(b). Thus |S| ≤ 2^{8}. Also, [Aut_{F}^{∗}(R_{i}), R_{i}] = T, and so 𝔣𝔬𝔠(F, R_{i}) = T (if T ⊴ S) or T̂.

If 𝒴 (S) ≠ ∅, then S > S_{0} since |S_{0}| = 2^{7} = |D_{8} ≀ C_{2}| and S_{0} ≅ D̸_{8} ≀ C_{2}. Hence |S| = 2^{8}, and S/Z(S) ≅ D_{8} ≀ C_{2} by Lemma 2.4(a). Also, T ∉ 𝒴_{0}(S) (hence T ∉ 𝒴 (S)) since S_{0} = N_{S}(T) ≅ D̸_{8} ≀ C_{2}. Also, T̂ ∉ 𝒴_{0}(S) since |T̂| = 2^{6} and T̂ ≅ Q̸_{8} × Q_{8}, and for any Q < T̂ of index 2, |Z(Q)| ≥ 4 and hence Q is not extraspecial. Thus no subgroup of T̂ lies in 𝒴_{0}(S), and T̂ ∉ 𝒴 (S). Also, T̂/Z(S) ≅ 2_{+}^{1+4} (since UT_{3}(4)/Z ≅ 2_{+}^{1+4} for each Z < Z(UT_{3}(4)) of order 2).

It remains to show that 𝒳 (S) = ∅. Assume otherwise: assume Q ∈ 𝒳 (S). Thus Q ∈ DQ, Q ⊴ S, and Q is strongly automized in S. Since Z_{2}(S) = Z_{2}(S_{0}) = Z(T̂) has order 4, Q ≥ Z(T̂) by Lemma A.2(b). Thus C_{2}^{2} ≅ Z(T̂) ⊴ Q, so Q ≅ D_{8}, and (since Z(T̂) ⊴ S) Q is not strongly automized in S, a contradiction. □

We next look at essential pairs of type (II) where T < R_{1} ∩ R_{2} in the notation of Lemma 3.6(b). The starting point for doing this is the description in [AOV2, Theorem 4.6].

Lemma 3.8. Let F be a saturated fusion system over a 2-group S with r(S) ≤ 4. Let (R_{1}, R_{2}) be an F-essential pair of type (II), choose subgroups Σ_{3} ≅ Out_{F}^{0}(R_{i}) ≤ Out_{F}(R_{i}) as in Lemma 3.6(a), and assume we are in the situation of Lemma 3.6(b.ii). Thus T < R_{1} ∩ R_{2} in the notation of Lemma 3.6(b). Set R = R_{1}R_{2} = N_{S}(R_{1}) = N_{S}(R_{2}).

Set U_{i} = [Aut_{F}^{∗}(R_{i}), R_{i}] ⊴ R and U = U_{1}U_{2}. Set W = Fr(U), S_{∗} = N_{S}(W), and let Δ be the normal closure of U in S_{∗}. Then either

(a) S_{∗} = S and 𝔣𝔬𝔠(F, R_{1}) = Δ ∈ 𝒳 (S); or

(b) S_{∗} < S, N_{S}(Δ) = S^{∗}, [S:S_{∗}] = 2, 𝔣𝔬𝔠(F, R_{1}) = ΔΔ^{∗} where Δ^{∗} ≠ Δ, {Δ, Δ^{∗}} is an S-conjugacy class, [Δ, Δ^{∗}] ≤ Δ ∩ Δ^{∗} ≤ Z(S), and Δ ∩ Δ^{∗} = 1 if Δ ∈ D. In this case, if F is reduced or 𝒴 (S) ≠ ∅, then 𝔣𝔬𝔠(F, R_{1}) ∈ 𝒴 (S).

Also, the following hold in both cases.

(c) Either (c.1) U_{1} ≅ U_{2} ≅ C_{2}^{2}, U ≅ D_{8}, Δ ∈ D, and W = Z(U) = Z(Δ) ≅ C_{2}; or

(c.2) U_{1} ≅ U_{2} ≅ Q_{8}, U ≅ Q_{16}, Δ ∈ Q, and W = Z_{2}(U) = Z_{2}(Δ) ≅ C_{4}.

(d) For i = 1, 2, U_{i} is fully normalized in F, and R_{i} = U_{i}C_{S}(U_{i}). If U_{i} ≅ C_{2}^{2}, then it is a direct factor of R_{i}.

(e) For i = 1, 2, Out_{F}(R_{i}) ≅ Σ_{3} or C_{3} × Σ_{3}.

Proof. Let G_{1} > R < G_{2} be as in Lemma 3.6(c). Thus R_{i} ⊴ G_{i} and Out_{F}^{0}(R_{i}) = Out_{G_{i}}(R_{i}) ≅ Σ_{3} for i = 1, 2. Set Û_{i} = R∩O^{2}(G_{i}) ⊴ R and Û = Û_{1}Û_{2}. We first prove (e), and then use that to show that Û_{i} = U_{i} and Û = U.

(e) By [AOV2, Theorem 4.6(a)], there is a subgroup T^{•} < R_{1} ∩ R_{2} such that R_{i} = Û_{i}T^{•} and R = ÛT^{•}, and such that either

Û_{i} ≅ C_{2}^{2}, Û ≅ D_{8}, and [T^{•}, Û] = T^{•} ∩ Û = 1; or  (3.5)
Û_{i} ≅ Q_{8}, Û ≅ Q_{16}, and [T^{•}, Û] ≤ T^{•} ∩ Û = Z(Û).  (3.6)

In particular, since [T^{•}, Û] ≤ [Û_{i}, Û_{i}] in both cases,

[R, R_{i}] = [ÛT^{•}, Û_{i}T^{•}] = [Û, Û_{i}][T^{•}, T^{•}] ≤ [Û, Û_{i}]Fr(R_{i}) .

So rk([R, R_{i}/Fr(R_{i})]) ≤ rk([Û, Û_{i}/Fr(Û_{i})]) = 1. Hence Out_{F}(R_{i}) ≅ (C_{3}×C_{3}) ⋊^{−1}C_{2}, and by Lemma 3.2, Out_{F}(R_{i}) ≅ Σ_{3} or C_{3} × Σ_{3}.

In particular, O^{2′}(Aut_{F}(R_{i})) = Aut_{G_{i}}(R_{i}), and hence

Aut_{F}^{∗}(R_{i}) = O^{2}(Aut_{G_{i}}(R_{i})) = Aut_{O^{2}(G_{i})}(R_{i}).

For any Q ∈ Syl_{3}(G_{i}), Q ≤ O^{2}(G_{i}) ≤ R_{i}Q (recall R_{i} ⊴ G_{i} and G_{i}/R_{i} ≅ Σ_{3}). Thus Aut_{Q}(R_{i}) ≤ Aut_{F}^{∗}(R_{i}) ≤ Inn(R_{i})Aut_{Q}(R_{i}). So

Û_{i} = R ∩ O^{2}(G_{i}) = R ∩ (Q[Q, R_{i}]) = [Q, R_{i}] = [Aut_{F}^{∗}(R_{i}), R_{i}] = U_{i} ,

where the fourth equality holds by Proposition 1.14(c) (applied with Aut_{Q}(R_{i}) in the role of Γ).

(c) By Lemma 3.6(b), T is not centric in S. So points (c.1) and (c.2) follow from [AOV2, Theorem 4.6(a)] and since U_{i} = Û_{i}. Note that since Δ ⊴ S_{∗} and [S:S_{∗}] = 2, the S-conjugacy class of Δ has order at most 2, and hence is {Δ, Δ^{∗}}.

(d) By [AOV2, Theorem 4.6(a)] again, T^{•} ≤ C_{S_{∗}}(U_{1}) ≤ T^{•}U_{1}. So R_{1} = T^{•}U_{1} = U_{1}C_{S_{∗}}(U_{1}). Since W = Fr(U) ≤ U_{1}, C_{S}(U_{1}) ≤ N_{S}(W) = S_{∗}, so C_{S}(U_{1}) = C_{S_{∗}}(U_{1}), and R_{1} = U_{1}C_{S}(U_{1}). By Lemma 3.6(c), there are β ∈ Iso(G_{1}, G_{2}) and x ∈ N_{S}(R) such that β(R) = R, β|_{R} = c_{x}|_{R}, and β(R_{1}) = ^{x}R_{1} = R_{2}. Then xU_{1} = β(U_{1}) = U_{2}, and so R_{2} = ^{x}R_{1} = U_{2}C_{S}(U_{2}). If U_{i} ≅ C_{2}^{2} (i = 1, 2), then U ∩ T^{•} = 1 by (3.5), and hence U_{i} is a direct factor of R_{i} = T^{•}U_{i}.

For i = 1, 2, R = UT^{•} = UR_{i} ≤ N_{S}(U_{i}) ≤ N_{S}(U_{i}C_{S}(U_{i})) = N_{S}(R_{i}) = R, so N_{S}(U_{i}) = N_{S}(R_{i}). Since R_{i} is fully normalized, U_{i} is also fully normalized by Proposition 1.3(c), applied with U_{i} ⊴ R_{i} in the role of Q ⊴ P. Note that since R = N_{S}(R_{i}), (1.1) takes the form N_{S}(φ(U_{i})) ∩ N_{S}(φ(R)) ≤ N_{S}(φ(R_{i})) for each φ ∈ Hom_{F}(R, S), and this holds since R_{i} = U_{i}C_{R}(U_{i}).

(a,b) By (e), 𝔣𝔬𝔠(F, R_{1}) is the normal closure of U_{1} in S. Also, U_{1} = R ∩ O^{2}(G_{1}) and U_{2} = R ∩ O^{2}(G_{2}) are S-conjugate by the conditions on G_{1} and G_{2} in Lemma 3.6(c). Since Δ is the normal closure of U = U_{1}U_{2} in S_{∗}, this shows that 𝔣𝔬𝔠(F, R_{1}) is the normal closure of Δ in S.

If S_{∗} = S, then Δ ⊴ S, so 𝔣𝔬𝔠(F, R_{1}) = Δ. Also, Δ is fully automized in S since it is the normal closure of U_{1} < Δ, so Δ ∈ 𝒳 (S).

Assume for the rest of the proof that S_{∗} < S. Thus Δ ⋬ S. Also, S_{∗} ≤ N_{S}(Δ) ≤ N_{S}(W) = S_{∗} since Δ ⊴ S_{∗} by construction and W is characteristic in Δ. Thus S_{∗} = N_{S}(W). If Δ ∈ D, then S_{∗} = C_{S}(Z(Δ)), and [S:S_{∗}] = 2 by Lemma B.4.

Now assume Δ ∈ Q, and set S_{0} = C_{S}(Z(Δ)). Thus S_{∗} ≤ S_{0} ≤ S, and [S_{0}:S_{∗}] ≤ 2 by Lemma B.4 applied to Δ/Z(Δ) < S_{0}/Z(Δ). If S_{0} = S_{∗}, then upon applying the lemma again, we get that [S:S_{∗}] = 2. If [S_{0}:S_{∗}] = 2, and Δ̂ is the normal closure of Δ in S_{0}, then Δ̂/Z(Δ) ≅ (Δ/Z(Δ)) × (Δ/Z(Δ)) by Lemma B.3, so r(Δ̂) = 4, and Lemma B.4 applied to Δ̂ < S implies that S = S_{0} and hence [S:S_{∗}] = 2.

Thus in both cases, [S:N_{S}(Δ)] = 2, so 𝔣𝔬𝔠(F, R_{1}) = ΔΔ^{∗} where Δ^{∗} = ^{g}Δ for any g ∈ S∖S_{∗}. Since S_{∗} = N_{S}(W), Z(Δ) ≠ Z(Δ^{∗}) if Δ ∈ D, and Z_{2}(Δ) ≠ Z_{2}(Δ^{∗}) if Δ ∈ Q. So [Δ, Δ^{∗}] ≤ Δ ∩ Δ^{∗} ≤ Z(Δ) by Lemma B.3 (applied with Δ/Z(Δ) and Δ^{∗}/Z(Δ) in the role of P and Q if Δ ∈ Q and Z(Δ) = Z(Δ^{∗})), and Δ ∩ Δ^{∗} = 1 if Δ ∈ D.

Assume ΔΔ^{∗} ∉ 𝒴 (S); we must show that 𝒴 (S) = ∅ and F is not reduced. Set Y = ΔΔ^{∗}. By Lemma 2.6 and since Y ∉ 𝒴 (S), 𝒴 (S) = ∅, S^{ab} ≅ C_{4} × C_{2}, S/Y ≅ C_{4}, and the action of S/Y on Y^{ab} ≅ C_{2}^{4} permutes freely a basis. Also, Y < S_{∗} < S since S_{∗} ≥ ΔΔ^{∗} and [S:S_{∗}] = 2.

Assume F is reduced; we must find a contradiction. By Lemma 1.17, there are subgroups Q ⊴ P ∈ E_{F} such that P/Q ≅ C_{4} × C_{4}, and such that [Aut_{F}^{∗}(P), P] surjects onto S/Y ≅ C_{4}. Then |N_{S}(P)/P | ≤ 4 by Lemma 3.3, N_{S}(P)/P ≅ C̸_{2}^{2} by Proposition 3.4 and since P^{ab} is not elementary abelian, and N_{S}(P)/P ≅ C̸_{4} by Proposition 3.5 and since F is reduced. Thus P ∉ E_{F}^{(I)}. Since 𝒴 (S) = ∅ by Lemma 2.4(a) (S^{ab} ≅ C̸_{2}^{3}) and P ≅ C̸_{2}^{4}, P is not in a pair of the type described in Lemma 3.7. By (e) and since [Aut_{F}^{∗}(P), P] surjects on to S/Y, P is not in a pair of the type described in this lemma. Thus P ∉ E_{F}^{(II)}, and hence [S:P] = 2.

Fix g ∈ P ∖S_{∗}, choose a_{1}, b_{1} ∈ Δ such that [Δ:⟨a_{1}⟩] = 2 and b_{1} ∈ Δ∖⟨a_{1}⟩, and set a_{2} = ^{g}a_{1}, b_{2} = ^{g}b_{1} ∈ Δ^{∗}. Then ^{g}b_{2} = ^{g^{2}}b_{1} = a_{1}^{i}b_{1}, where i is odd since c_{g} permutes freely a basis of Y^{ab}. Also, b_{1}b_{2}^{−1} ∈ [S, S], so b_{1}b_{2} ∈ Fr(S) ≤ P, and g(b_{1}b_{2})(b_{1}b_{2})^{−1} = (b_{2}a_{1}^{i}b_{1})(b_{1}b_{2})^{−1} ≡ a_{1}^{i} (mod [Δ, Δ^{∗}] ≤ Z(Δ)). Thus a_{1}, a_{2} ∈ [P, P], so P/[P, P] = ⟨[g], [b_{1}b_{2}]⟩ ≅ C_{4} × C_{2} contains no subgroup C_{4} × C_{4}, which contradicts our original assumption. Hence this situation is impossible. □

We now summarize the results in Section 3.1 and in the last two lemmas, as they apply when 𝒴 (S) ≠ ∅.

Proposition 3.9. Let F be a saturated fusion system over a 2-group S such that r(S) ≤ 4 and 𝒴 (S) ≠ ∅. Let Z_{∗} ⊴ S be the unique normal subgroup such that S/Z_{∗} ≅ D_{8} ≀ C_{2}. Let Y_{1}, Y_{2}, Y_{3} ⊴ S be the distinct normal subgroups such that Y_{1}/Z_{∗} ≅ Y_{2}/Z_{∗} ≅ C_{2}^{4} and Y_{3}/Z_{∗} ≅ 2_{+}^{1+4}.

(a) Out_{F}(S) = 1, and E_{F} = E_{F}(Y_{1}) ∪ E_{F}(Y_{2}) ∪ E_{F}(Y_{3}). If F is reduced, then E_{F}(Y_{i}) ≠ ∅ for each i = 1, 2, 3.

(b) If Y_{i} ∈ 𝒴 (S) (some i = 1, 2, 3), then E_{F}(Y_{i}) ⊆ E_{F}^{(II)}. Also, each R ∈ E_{F}(Y_{i}) is in an F-essential pair of the form described in Lemma 3.7(a) or 3.8(b).

(c) If Y_{i} ∉ 𝒴 (S) (some i = 1, 2, 3) and R ∈ E_{F}(Y_{i}), then i = 3, and either

(c.1) R ∈ E_{F}^{(II)}, |R| = 2^{5}, Y_{3} ≅ UT_{3}(4), and R is in an F-essential pair of the form described in Lemma 3.7(b); or

(c.2) R ∈ E_{F}^{(III)}, R > Y_{3}, and Y_{3} ≅ 2_{+}^{1+4}, Q_{8} × Q_{8}, or UT_{3}(4).

Proof. By definition of 𝒴 (S), |S| ≥ 2^{7}, with equality only if S ≅ D_{8}≀C_{2}. Also, (D_{8}≀C_{2})^{ab} ≅ C_{2}^{3}, D_{8}≀C_{2} contains no subgroup isomorphic to 2_{−}^{1+4} by Lemma C.5(a), and contains none isomorphic to UT_{3}(4) by Lemma C.5(b). So by Propositions 3.4 and 3.5, E_{F}^{(I)} = ∅. By Lemma 2.4(b) and Corollary 2.5, 𝒴 (S) ⊆ {Y_{1}, Y_{2}, Y_{3}}, and Aut(S) is a 2-group (hence Aut_{F}^{∗}(S) = 1).

By Corollary 2.5, 𝒳 (S) = ∅. So if R ∈ E_{F}^{(II)}, then by Lemma 3.6(b.1,b.2), R is in an F-essential pair of the form described in Lemma 3.7(a) or 3.8(b) (if 𝔣𝔬𝔠(F, R) ∈ 𝒴 (S)), or in Lemma 3.7(b). In the latter case, |R| = 2^{5}, 𝔣𝔬𝔠(F, R) = Y_{3} ∉ 𝒴 (S), Y_{3} ≅ UT_{3}(4), and we are in the situation of (c.1).

We claim that

R ∈ E_{F} of type (III) ⟹ R > Y_{3}, 𝔣𝔬𝔠(F, R) = Y_{3}, Y_{3} ∉ 𝒴 (S), and

Y_{3} ≅ 2_{+}^{1+4}, Q_{8} × Q_{8}, or UT_{3}(4). (3.7)

Once this has been shown, points (b) and (c) then follow. Also, for each R ∈ E_{F}, 𝔣𝔬𝔠(F, R) = Y_{i} for some i = 1, 2, 3, and hence E_{F} = ⋃_{i=1}^{3} E_{F}(Y_{i}). Finally, if F is reduced, then S = ⟨𝔣𝔬𝔠(F, R) | R ∈ E_{F} ∪{S}⟩ by Proposition 1.14(b), 𝔣𝔬𝔠(F, S) = 1 since Aut_{F}^{∗}(S) = 1, and so E_{F}(Y_{i}) ≠ ∅ for each i = 1, 2, 3 since no two of the Y_{i} generate S.

It remains to prove (3.7). Assume R ∈ E_{F}^{(III)}. In particular, [S:R] = 2 and Aut(R) is not a 2-group. Set |S| = 2^{m}, and for each 0 ≤ i ≤ m − 5, set Z_{i} = Z_{i}(S) for short. Let 𝒞 h(R) be the set of subgroups characteristic in R.

We first show, for each 1 ≤ i ≤ m − 6, that

(i) either Z_{i} ∈ 𝒞 h(R) or Z_{i−1} ∈ 𝒞 h(R); and

(ii) if Z_{i}, Z_{i−1} ∈ 𝒞 h(R), then Z_{j} ∈ 𝒞 h(R) for each j ≤ i. To prove (i), assume Z_{j} ∈ 𝒞 h(R) for some 0 ≤ j ≤ m − 8, and let P ⊴ S be such that P/Z_{j} = Ω_{1}(Z(R/Z_{j})). Thus P ∈ 𝒞 h(R). By Lemma 2.4(a), and since |P | ≥ 2^{j+1}, either P = Z_{k} for k = j+1, j+2, or P ≥ Z_{j+3}. Since P/Z_{j} is elementary abelian, and Z_{j+3}/Z_{j} ≅ C_{4}×C_{2} by Lemma 2.4(c), this last case is impossible. Thus Z_{j+1} ∈ 𝒞 h(R) or Z_{j+2} ∈ 𝒞 h(R), and (i) now follows by induction on i.

If i ≥ 2, Z_{i} and Z_{i−1} are both characteristic, and Z_{i−2} is not, then i ≥ 3 (Z_{0} ∈ 𝒞 h(R)), Z_{i−3} ∈ 𝒞 h(R) by (i), Z_{i}/Z_{i−2} ≅ C_{2}^{2} and Z_{i}/Z_{i−3} ≅ C_{4} × C_{2} by Lemma 2.4(c), so Z_{i−2}/Z_{i−3} = Fr(Z_{i}/Z_{i−3}) is characteristic in R/Z_{i−3}, which contradicts our assumption. Point (ii) now follows by induction on i.

By (i), either Z_{∗} = Z_{m−7} ∈ 𝒞 h(R) or Z_{m−6} ∈ 𝒞 h(R) (or both). Since Fr(R) ⊴ S, and [S:Fr(R)] ≤ 2^{5} since r(R) ≤ 4, Fr(R) ≥ Z_{m−5}(S) > Z_{∗} by Lemma 2.4(a) again. If Z_{∗} ∈ 𝒞 h(R), then Aut(R/Z_{∗}) is not a 2-group by Lemma A.9 and since Aut(R) is not a 2-group, so R/Z_{∗} ≅ (2_{+}^{1+4}) ⋊ C_{2} by Lemma C.5(b). Since Y_{3} < S is the unique subgroup such that Y_{3}/Z_{∗} ≅ 2_{+}^{1+4} (Lemma C.5(a)), R > Y_{3}.

If Z_{∗} is not characteristic in R, then m ≥ 8 since Z_{∗} ≠ 1, Z_{m−8}, Z_{m−6} ∈ 𝒞 h(R)

by (i), and so Z_{m−5} ∉ 𝒞 h(R) by (ii). Thus R/Z_{m−6} < S/Z_{m−6} ≅ (D_{8} ×_{C_{2}} D_{8}) ⋊^{t}

C_{2} ≅ UT_{4}(2) with index 2, R/Z_{m−6} ≅ 2̸_{+}^{1+4} (that would imply Z_{m−5} = Fr(R) ∈ 𝒞 h(R)), and Aut(R/Z_{m−6}) is not a 2-group by Lemma A.9 (recall Z_{m−6} ≤ Fr(R)). So R > Y_{3} by Lemma C.4(c).

We have now shown that R > Y_{3}. Also, Y_{3} ∈ 𝒞 h(R), since Y_{3}/Z_{∗} < S/Z_{∗} and Y_{3}/Z_{m−6} < S/Z_{m−6} are the unique subgroups of their isomorphism type (and Z_{∗} or Z_{m−6} is in 𝒞 h(R)). Since

S/Fr(Y_{3}) = S/Z_{m−6} ≅ (D_{8} ≀ C_{2})/Z(D_{8} ≀ C_{2})

≅ (D_{8} ×_{C_{2}} D_{8}) ⋊^{t} C_{2} ≅ UT_{4}(2) ≅ C_{2} ≀ C_{2}^{2}

(see Lemma C.4(a,b)), Out_{S}(Y_{3}) ≅ S/Y_{3} ≅ C_{2}^{2} acts faithfully on Y_{3}/Fr(Y_{3}) ≅ C_{2}^{4}, permuting a basis freely. Hence Out_{F}(Y_{3}) also acts faithfully on Y_{3}/Fr(Y_{3}) (Lemma A.9). Fix Id ≠ α ∈ Aut_{F}(R) of odd order, and set α_{0} = α|_{Y_{3}} ∈ Aut_{F}(Y_{3}). Then [[α_{0}], Out_{R}(Y_{3})] = 1 in Out_{F}(Y_{3}), so these commute as automorphisms of Y_{3}/Fr(Y_{3}) ≅ C_{2}^{4}, which implies that |α_{0}| = 3 and α_{0} acts on Y_{3}/Fr(Y_{3}) with trivial fixed component. Thus 𝔣𝔬𝔠(F, R)Fr(Y_{3}) = Y_{3}, so 𝔣𝔬𝔠(F, R) = Y_{3} (cf. [G, Theorem 5.1.1]).

Now, Z_{m−6} = Fr(Y_{3}) ∈ 𝒞 h(R), and hence Z_{m−j} ∈ 𝒞 h(R) for each even 6 ≤ j ≤ m by (i) and (ii). If m = 7, then Y_{3} ≅ 2_{+}^{1+4}. If m ≥ 8, then Z(Y_{3}/Z_{m−8}) ≅ C_{2}^{2} by Lemma D.3, so by Lemma D.2, Y_{3}/Z_{m−8} ≅ Q_{8}×Q_{8}, UT_{3}(4), or a certain special 2-group of type 2^{2+4} which contains a characteristic subgroup P/Z_{m−8} ≅ C_{2}^{4}. This last case is impossible, since then P > Z_{m−5} by Lemma 2.4(a), while Z_{m−5}/Z_{m−8} ≅ C_{4} × C_{2} by Lemma 2.4(c). If m ≥ 9, then Z(Y_{3}/Z_{m−9}) = Z_{m−6}/Z_{m−9} ≅ C_{2}^{3} by Lemma D.3, which again contradicts Lemma 2.4(c). Thus m ≤ 8, and Y_{3} is as described in (3.7). Finally, Y_{3} ∉ 𝒴 (S) by Lemma 2.4(d), since Y_{3}/Z_{m−8} ≅ D_{8} × D_{8}. □

Recall Definition 2.1(e): for a 2-group S and Y_{0} ∈ 𝒴_{0}(S), 𝒰_{S}(Y_{0}) is the set of all pairs {U_{1}, U_{2}} such that U_{i} ≅ C_{2}^{2} or Q_{8}, [U_{1}, U_{2}] ≤ U_{1}∩U_{2} ≤ Fr(U_{1}), Y_{0} = U_{1}U_{2}, and each element of Out_{S}(Y_{0}) ≅ D_{8} either normalizes the U_{i} or exchanges them.

Lemma 3.10. Let F be a saturated fusion system over a 2-group S such that r(S) ≤ 4 and 𝒴 (S) ≠ ∅. Fix Y_{0} ∈ 𝒴_{0}(S). Then Y_{0} is fully normalized in F, and for each {U_{1}, U_{2}} ∈ 𝒰_{S}(Y_{0}), U_{1} and U_{2} are fully normalized in F.

Proof. Set Z = Fr(Y_{0}) for short. By Lemma 2.4(a), Z ⊴ S.

We first prove that Y_{0} is fully normalized. By Proposition 1.3(a), there is φ ∈ Hom_{F}(N_{S}(Y_{0}), S) such that φ(Y_{0}) is fully normalized. Set Y_{1} = φ(Y_{0}) and Z_{1} = φ(Z) = Fr(Y_{1}) ⊴ N_{S}(Y_{1}). Then φ(N_{S}(Y_{0}))/Z_{1} ≅ N_{S}(Y_{0})/Z ≅ D_{8} ≀ C_{2}. By Lemma 2.3, applied with N_{S}(Z_{1})/Z_{1}, φ(N_{S}(Y_{0}))/Z_{1}, and Y_{1}/Z_{1} in the role of S, P, and V, N_{S}(Y_{1})/Z_{1} = φ(N_{S}(Y_{0}))/Z_{1}, and hence Y_{0} is also fully normalized.

Now fix {U_{1}, U_{2}} ∈ 𝒰_{S}(Y_{0}). It remains to show, for i = 1, 2, that U_{i} is fully normalized. Assume otherwise. Set M = N_{N_{S}(Y_{0})}(U_{i}). By Lemma 1.16(a), there is an F-essential subgroup R ≥ N_{S}(U_{i}) ≥ M. We will show that this is impossible.

By Definition 2.1(e), M/Z has index 2 in N_{S}(Y_{0})/Z ≅ D_{8} ≀ C_{2}, and normalizes the two complementary subgroups U_{1}Z/Z ≅ U_{2}Z/Z ≅ C_{2}^{2} in Y_{0}/Z ≅ C_{2}^{4}. Hence M = Y_{0}⟨g_{1}, g_{2}⟩ where rk([g_{i}, Y_{0}/Z]) = 1, so M/Z ≅ D_{8} × D_{8}.

By Proposition 3.9 and since |R| ≥ 2^{6}, either 𝔣𝔬𝔠(F, R) ∈ 𝒴 (S) and R is as described in 3.9(b), or R ∈ E_{F}^{(III)} and is as described in 3.9(c). In the former case, R is in an essential pair of the type described in Lemma 3.7(a) or 3.8(b). We consider these three cases individually.

If R is in a pair as in 3.7(a), then it contains a subgroup T < R of index 2, where T ∈ 𝒴_{0}(S) and hence T ≅ C_{2}^{4}, 2_{+}^{1+4}, 2_{−}^{1+4}, or Q_{8} × Q_{8}. Thus

T/Z < R/Z ≥ M/Z ≅ D_{8} × D_{8}

and [R/Z:T/Z] = 2. If |T/Z| ≤ 2^{5}, then R = M, and T/Z is isomorphic to a subgroup of index 2 in D_{8} × D_{8}. Hence T surjects onto D_{8}, which is impossible in all cases. If |T/Z| ≥ 2^{6}, then Z = 1 and T ≅ Q_{8} × Q_{8}, which is also impossible since Q_{8} × Q_{8} contains only three involutions while each subgroup of index 2 in D_{8} × D_{8} contains more.

If R ∈ E_{F}^{(III)}, then a similar argument applies using Proposition 3.9(c), except that we could have T ≅ UT_{3}(4). In this case, |R| = 2^{7} and |S| = 2^{8}. By Lemma 2.4(a), |Z(S)| = 2 and S/Z(S) ≅ D_{8} ≀ C_{2}. By Proposition 3.9(c) again, T/Z(S) is the unique subgroup of S/Z(S) isomorphic to 2_{+}^{1+4}.

If Z = 1, then Y_{0} ≅ C_{2}^{4}, M ≅ D_{8} × D_{8}, and N_{S}(Y_{0}) ≅ D_{8} ≀ C_{2}. Thus M/Z(S) = M/Z(N_{S}(Y_{0})) ≅ 2_{+}^{1+4}. Hence M/Z(S) = T/Z(S), which is impossible since M ≅ T̸.

If Z ≠ 1, then |M| = 2^{7} = |R|, so M = R and |Z| = 2. Then R/Z ≅ D_{8} × D_{8}, T/Z = T/Z(S) ≅ 2_{+}^{1+4}, R ≥ T, and this is impossible.

If R is in a pair as in 3.8(b), then there is V ⊴ R such that either V ≅ C_{2}^{2} and is a direct factor of R, or V ≅ Q_{8} and R = V C_{S}(V). By Lemma A.6(c) (applied with R, V, and M or Y_{0} in the role of S, U, and Q), V ≤ M, and V ≤ Y_{0} if Z ≠ 1. If Z = 1, this is impossible since M ≅ D_{8} × D_{8} contains no subgroup isomorphic to Q_{8}, and no direct factor isomorphic to C_{2}^{2}.

If Z ≠ 1, then V ⊴ Y_{0}, so V Z/Z ≅ C_{2}^{i} for i ≤ 2, and V Z/Z is a direct factor in M/Z ≅ D_{8} × D_{8}. Hence i = 0, V ≤ Z, so V ≅ C_{2}^{2} and is a direct factor in M, which is impossible. □

We now make a more precise analysis, for a fusion system F over S, of essential subgroups R such that 𝔣𝔬𝔠(F, R) ∈ 𝒴 (S).

Recall that D denotes the class of nonabelian dihedral 2-groups. It will be useful — in the following proposition only — to let D̂ be the extended class of dihedral 2-groups including the group D_{4} = C_{2}^{2}. Thus S ∈ D̂ if S ∈ D or S ≅ C_{2}^{2}. By extension of Definition 2.1, a subgroup P ≅ C_{2}^{2} is strongly automized in S > P if Aut_{S}(P) ≠ 1.

Proposition 3.11. Let F be a saturated fusion system over a 2-group S such that r(S) ≤ 4 and 𝒴 (S) ≠ ∅. Fix Y ∈ 𝒴 (S), and assume that E_{F}(Y) ≠ ∅. Let 𝒴_{0} be the set of all Y_{0} ∈ 𝒴_{0}(S) whose normal closure is Y; equivalently, those which are contained in Y. Then the following hold.

(a) There is a pair of subgroups Θ_{1}, Θ_{1} ⊴ Y such that (a.1) {Θ_{1}, Θ_{2}} is an S-conjugacy class and Θ_{i} ∈ D̂Q;

(a.2) [Θ_{1}, Θ_{2}] ≤ Θ_{1} ∩ Θ_{2} ≤ Z(S), Θ_{1} ∩ Θ_{2} = 1 if Θ_{i} ∈ D̂, and Y = Θ_{1}Θ_{2}; and

(a.3) for each U ≤ Θ_{1} or U ≤ Θ_{2} such that U ≅ C_{2}^{2} or Q_{8}, Aut_{F}(U) =

Aut(U). Furthermore, Θ_{1} and Θ_{2} are strongly automized in S.

(b) Let Θ_{1}, Θ_{2} ⊴ Y be as in (a). Let 𝒰_{F}(Y) be the set of all U such that U ≤ Θ_{1} or U ≤ Θ_{2}, and U ≅ C_{2}^{2} or Q_{8}. Then all subgroups in 𝒰_{F}(Y) are S-conjugate to each other, and

𝒰_{F}(Y) = {Y_{0} ∩ Θ_{i} | Y_{0} ∈ 𝒴_{0}, i = 1, 2}.  (3.8)

For each Y_{0} ∈ 𝒴_{0},

(b.1) Out_{F}(Y_{0}) ∈ 𝒜_{S}(Y_{0}), so Out_{F}(Y_{0}) ≅ Σ_{3} ≀ C_{2} or Σ_{5}; and

(b.2) {Y_{0} ∩ Θ_{1}, Y_{0} ∩ Θ_{2}} ∈ 𝒰_{S}(Y_{0}) and is the unique element of its isomorphism type compatible with Out_{F}(Y_{0}) in the sense of Definition 2.2(b).

(c) Let 𝒰_{F}(Y) be as in (b). For each U ∈ 𝒰_{F}(Y), if we set R = UC_{S}(U), then (c.1) U is fully normalized in F,

(c.2) R ∈ E_{F}^{(II)}(Y),

(c.3) [Aut_{F}^{∗}(R), R] = U, and

(c.4) Aut_{F}^{∗}(R) = O^{2}(Inn(R)⟨α⟩) for some α ∈ Aut_{F}^{∗}(R) of order 3 which normalizes U and induces the identity on R/U, and such that α|_{C_{S}(U)} = Id if U ≅ Q_{8}.

Proof. In Step 1, we prove that there are subgroups U_{1}, U_{2}, T, R ≤ S such that

T ∈ 𝒴_{0}, {U_{1}, U_{2}} ∈ 𝒰_{S}(T), Aut_{F}(U_{1}) = Aut(U_{1}); and  (3.9)
R = U_{1}C_{S}(U_{1}) ∈ E_{F}^{(II)}(Y) and [Aut_{F}^{∗}(R), R] = U_{1}.  (3.10)

Then, in Step 2, we apply this to prove the proposition.

Step 1: By assumption, E_{F}(Y) ≠ ∅. By Proposition 3.9, each F-essential pair (P_{1}, P_{2}) of subgroups in E_{F}(Y) = E_{F}^{(II)}(Y) has the form described in Lemma 3.7(a) or 3.8(b).

Assume first that there is a pair (P_{1}, P_{2}) is as described in Lemma 3.8(b). Set R = P_{1} and U_{1} = [Aut_{F}^{∗}(P_{1}), P_{1}]. Then (3.10) holds by Lemma 3.8(b), and hence Aut_{F}^{∗}(U_{1}) = Aut(U_{1}). By the same lemma, there is an S-conjugacy class {Δ, Δ^{∗}} such that Δ ∈ DQ, |Δ| ≥ 16 if Δ ∈ Q, Y = 𝔣𝔬𝔠(F, P_{1}) = ΔΔ^{∗}, and [Δ, Δ^{∗}] ≤ Δ ∩ Δ^{∗} ≤ Z(S). The hypotheses of Lemma 2.6(a) thus hold. By that lemma, for any U_{2} < Δ^{∗} with U_{2} ≅ U_{1}, U_{1}U_{2} ∈ 𝒴_{0}, and {U_{1}, U_{2}} ∈ 𝒰_{S}(U_{1}U_{2}). Thus (3.9) also holds in this case.

Now assume (for the rest of Step 1) that there is a pair (P_{1}, P_{2}) is as described in Lemma 3.7(a), and none as described in 3.8(b). Set T = P_{1} ∩ P_{2}. By Lemma 3.7(a), T ∈ 𝒴_{0}, Y = 𝔣𝔬𝔠(F, P_{1}) is its normal closure in S, and Out_{F}(T) ≅ Σ_{3} ≀ C_{2} or Σ_{5}. In particular, Out_{F}(T) ∈ 𝒜_{S}(T).

Let {U_{1}, U_{2}} ∈ 𝒰_{S}(T) be the unique element compatible with Out_{F}(T) ∈ 𝒜_{S}(T). Fix β ∈ Aut(U_{1}) of order 3. Since {U_{1}, U_{2}} is compatible with Out_{F}(T) (Definition 2.2(b)), β extends to an automorphism in Aut_{F}(T), and in particular, β ∈ Aut_{F}(U_{1}). This proves (3.9).

By Lemma 2.4(a,b), there is a unique normal subgroup Z_{∗} ⊴ S of index 2^{7}, and there is a unique triple of subgroups Y_{1}, Y_{2}, Y_{3} ⊴ S such that Y_{1}/Z_{∗} ≅ Y_{2}/Z_{∗} ≅ C_{2}^{4} and Y_{3}/Z_{∗} ≅ 2_{+}^{1+4}. By the same lemma, Y ∈ 𝒴 (S) ⊆ {Y_{1}, Y_{2}, Y_{3}}.

Set R = U_{1}C_{S}(U_{1}). Set 𝒴 ′ = {Y_{1}, Y_{2}, Y_{3}}∖{Y } and S_{0} = Fr(S)⟨𝒴 ′ ⟩. Since the images in S/Fr(S) ≅ C_{2}^{3} of the subgroups Y_{1}, Y_{2}, Y_{3} have rank 1 and are independent (Lemma C.5), [S:S_{0}] = 2 and Y ∩ S_{0} ≤ Fr(S).

Since U_{1} is fully normalized by Lemma 3.10 and R = U_{1}C_{S}(U_{1}), β ∈ Aut_{F}(U_{1}) extends to β̂ ∈ Aut_{F}(R). Since the normal closure of U_{1} in S is the normal closure of U_{1}U_{2} = T and hence equal to Y, there is h ∈ U_{1}∖Fr(S) ⊆ Y ∖Fr(S) such that β(h) ∈ Fr(S). Thus [Aut_{F}(R), R] ≰ S_{0}, so by Lemma 1.16(b), there is Q ∈ E_{F} such that Q ≥ R^{∗} for some R^{∗} ∈ R^{F}, and 𝔣𝔬𝔠(F, Q) ≰ S_{0}. Hence Q ∈ E_{F}(Y).

Assume first that |Q| ≥ 4·|T |. If Q is in a pair of type (3.7a), then it contains a subgroup T̂ ∈ 𝒴_{0}(S) with index 2. Also, |T̂| > |T | and T ∈ 𝒴_{0}(S). By Lemma 2.4(b), [Y:T̂] and [Y:T] are both even powers of 2, and hence T̂ ≅ Q_{8} × Q_{8} and T ≅ C_{2}^{4}. By construction, Q ≥ R^{∗} ∈ R^{F} where R = U_{1}C_{S}(U_{1}) > T ≅ C_{2}^{4}. Since [Q:T̂] = 2, T̂ contains a subgroup isomorphic to C_{2}^{3}, which is impossible. Thus Q is in an F-essential pair of type (3.8b), contradicting our assumption that there is no such pair.

Thus |Q| = 2·|T |. Then Q ∈ R^{F} and [R:T] = 2. Since Q ∈ E_{F}(Y) ⊆ E_{F}^{(II)} (Proposition 3.9(b)), |N_{S}(Q)/Q| = 2, so R is fully normalized in F and hence also F-essential. In particular, Out_{F}(R) ≅ Σ_{3} by Lemma 3.7. Upon replacing β̂ by an appropriate power, we can assume that |β̂| = 3 in Aut_{F}(R), and hence that

β̂ ∈ Aut_{F}^{∗}(R) = O^{2}(O^{2′}(Aut_{F}(R))) ≤ Inn(R)⟨β̂⟩.

By Proposition 1.14(c), [Aut_{F}^{∗}(R), R] = [β̂, R]. Also, β̂(T) = T by the condition defining T in Lemma 3.6(b), T/U_{1} ≅ U_{2}/(U_{1} ∩ U_{2}) ≅ C_{2}^{2} or Q_{8}, and R/T ≅ C_{2} acts on U_{2} as a subgroup of order 2 in Out(U_{2}) ≅ Σ_{3}. Thus no automorphism of T/U_{1} of order 3 extends to R/U_{1}, and β̂ induces the identity on R/U_{1} by Lemma A.9. This proves that [Aut_{F}^{∗}(R), R] = [β̂, R] = U_{1}. Finally, 𝔣𝔬𝔠(F, R) = Y since Y is the normal closure in S of T and hence of U_{1}.

This finishes the proof of (3.10).

Step 2: Let U_{1}, U_{2}, T = U_{1}U_{2}, and R = U_{1}C_{S}(U_{1}) be as in (3.9) and (3.10). Since {U_{1}, U_{2}} ∈ 𝒰_{S}(T), U_{1} is fully normalized in F by Lemma 3.10. By (3.9), there is β ∈ Aut_{F}(U_{1}) of order 3. By the extension axiom, β extends to some β̂ ∈ Aut_{F}(R).

Set W = U_{1}Z_{2}(S). We first check that condition (2.8) in Lemma 2.7 holds. Assume otherwise: then there is g ∈ S such that U_{1} ≠ ^{g}U_{1} ≤ W. In particular, since W = U_{1}Z_{2}(S) = ^{g}U_{1}Z_{2}(S), g ∈ N_{S}(W). By (3.10), U_{1} = [Aut_{F}^{∗}(R), R] ⊴ N_{S}(R).

If U_{1} ∩ U_{2} = 1, then W = U_{1}Z(S) since U_{1} ∩ Z_{2}(S) ≠ 1, so W C_{S}(W) = U_{1}C_{S}(U_{1}) = R. Hence g ∈ N_{S}(R) and g ∉ N_{S}(U_{1}), a contradiction.

If U_{1} ∩ U_{2} = Z(S), set Z = Z(S), and S = S/Z, U_{1} = U_{1}/Z, etc. Then R = U_{1}C_{S}(U_{1}) since each element of S which centralizes U_{1} = U_{1}/Z acts on U_{1} ≅ Q_{8} via an inner automorphism. Since W = U_{1}Z(S), this shows that W C_{S}(W) = R, so g ∈ N_{S}(R)∖N_{S}(U_{1}), which again is a contradiction.

Thus condition (2.8) in Lemma 2.7 holds. Let {Θ_{1}, Θ_{2}} be as in that lemma. Let 𝒰_{F}(Y) be the set of subgroups of the Θ_{i} isomorphic to C_{2}^{2} or Q_{8}.

(a) By Lemma 2.7, (a.1) and (a.2) hold, and Θ_{i} ≥ U_{i}.

If Y = T, then Θ_{i} = U_{i} for i = 1, 2. By definition of 𝒰_{S}(Y), there is a subgroup Λ < Out_{S}(Y) ≅ D_{8} of order 4 which normalizes U_{1} and U_{2}. If Λ induces the identity on the images of U_{1} and U_{2} in Y/Fr(Y), then (since T = U_{1}U_{2}) it induces the identity on T/Fr(T), which is impossible since Out_{S}(T) acts faithfully on this quotient. Thus for each i, there is g ∈ N_{S}(U_{i}) such that c_{g} acts nontrivially on U_{i}/Fr(U_{i}) ≅ C_{2}^{2}, so Θ_{i} = U_{i} is strongly automized in S.

If Y > T, then it is the normal closure of T and hence of U_{1} or U_{2}. Since {Θ_{1}, Θ_{2}} is an S-conjugacy class, 𝒰_{F}(Y) contains the S-conjugacy class of U_{1}, and each Θ_{i} is generated by subgroups in that class. Since Θ_{i} contains two Θ_{i}-conjugacy classes of subgroups isomorphic to C_{2}^{2} or Q_{8}, neither of which generates Θ_{i}, both conjugacy classes must be S-conjugate to U_{1}. Thus 𝒰_{F}(Y) is the S-conjugacy class of U_{1}, and Θ_{1} and Θ_{2} are strongly automized in S. Since Aut_{F}(U_{1}) = Aut(U_{1}), this also proves (a.3).

(c) We just showed that 𝒰_{F}(Y) is the S-conjugacy class of U_{1}. So it suffices to prove (c) when U = U_{1}. Points (c.2) and (c.3) hold for U_{1} by (3.10), and we already saw that U_{1} is fully normalized.

The image of Aut_{F}^{∗}(R) = O^{2}(O^{2′}(Aut_{F}(R))) in Out_{F}(R) has order 3 since Out_{F}(R) ≅ Σ_{3} or Σ_{3}×C_{3} (Lemmas 3.7 and 3.8(e)). Hence Aut_{F}^{∗}(R) ≤ Inn(R)⟨α⟩ ≤ O^{2′}(Aut_{F}(R)) for some α ∈ Aut_{F}^{∗}(R) of order 3, so Aut_{F}^{∗}(R) = O^{2}(Inn(R)⟨α⟩). Also, α normalizes U_{1} and is the identity on R/U_{1} by (c.3). If U_{1} ≅ Q_{8}, then α induces the identity on Z(U_{1}) and on C_{S}(U_{1})/Z(U_{1}), and hence on C_{S}(U_{1}) by Lemma A.9. This proves (c.4).

(b) Fix Y_{0} ∈ 𝒴_{0}. We have already seen that 𝒰_{F}(Y) is the S-conjugacy class of U_{1}. Set V_{i} = Y_{0} ∩ Θ_{i}.

If Y = T = Y_{0} ∈ 𝒴_{0}, then {Θ_{1}, Θ_{2}} = {U_{1}, U_{2}} ∈ 𝒰_{S}(Y) by assumption. So assume Y > T. Then Θ_{1} > U_{1}, so Θ_{i} ≅ D_{2^{n}} for n ≥ 3 or Q_{2^{n}} for n ≥ 4. Hence the hypotheses of Lemma 2.6 hold by (a); and (3.8) (𝒰_{F}(Y) is the set of all Y_{0} ∩ Θ_{i} for i = 1, 2 and Y_{0} ∈ 𝒴_{0}) follows from Lemma 2.6(a). By the same lemma, Y_{0} = V_{1}V_{2} and {V_{1}, V_{2}} ∈ 𝒰_{S}(Y_{0}).

It remains to prove (b.1) and the compatibility statement in (b.2). Recall (Lemma 2.9(a)) that there is a basis of Y_{0}/Fr(Y_{0}) which is permuted by Out_{S}(Y_{0}) ≅ D_{8}. Let B < Out_{S}(Y_{0}) be the subgroup generated by products of two disjoint transpositions, and let γ_{1}, γ_{2} ∈ Out_{S}(Y_{0}) be the two classes which act as transpositions. (Thus rk([γ_{i}, Y_{0}/Fr(Y_{0})]) = 1, while rk([β, Y_{0}/Fr(Y_{0})]) = 2 for β ∈ B^{#}.) In particular, no element in Out_{S}(Y_{0})∖B is Out_{F}(Y_{0})-conjugate to any element in B. By the focal subgroup theorem for groups (see [G, Theorem 7.3.4]), Out_{S}(Y_{0}) ∩ [Out_{F}(Y_{0}), Out_{F}(Y_{0})] ≤ B, and thus γ_{1}, γ_{2} ∉ O^{2}(Out_{F}(Y_{0})).

Now, ⟨γ_{1}, γ_{2}⟩ is the normalizer in Out_{S}(Y_{0}) of V_{1} and of V_{2}, and we can index them such that γ_{i} acts nontrivially on V_{i}/Fr(V_{i}) and trivially on V_{3−i}/Fr(V_{3−i}). Set R^{∗} = V_{1}C_{S}(V_{1}). By (c), and since V_{1} ∈ 𝒰_{F}(Y), R^{∗} ∈ E_{F}(Y), and there is β^{∗} ∈ Aut_{F}(R^{∗}) of order 3 which normalizes V_{1} and acts trivially on R^{∗}/V_{1}. Thus [β^{∗}|_{Y_{0}}] ∈ Out_{F}(Y_{0}) is inverted by γ_{1}, and normalizes (hence centralizes) γ_{2} ∈ Out_{R^{∗}}(Y_{0}). Since C_{Out_{S}(Y_{0})}(γ_{2}) = ⟨γ_{1}, γ_{2}⟩, this gives O_{2}(C_{Out_{F} (Y_{0})}(γ_{2})) = ⟨γ_{2}⟩.

Since Out_{S}(Y_{0}) acts faithfully on Y_{0}/Fr(Y_{0}), Out_{F}(Y_{0}) acts faithfully by Lemma A.9. Also, Out_{F}(Y_{0}) ≅ Γ̸L_{2}(4) since Out_{S}(Y_{0}) ≅ D_{8} permutes a basis of Y_{0}/Fr(Y_{0}). So by Proposition D.1(f), applied with γ_{2} in the role of x, Out_{F}(Y_{0}) ≅ Σ_{3} ≀ C_{2} or Σ_{5}. Thus Out_{F}(Y_{0}) ∈ 𝒜_{S}(Y_{0}). By (c.4), Out_{F}(Y_{0}) is compatible, in the sense of Definition 2.2(b), with {Y_{0} ∩ Θ_{1}, Y_{0} ∩ Θ_{2}} ∈ 𝒰_{S}(Y_{0}). □

Proposition 3.12. Let F be a reduced fusion system over a 2-group S such that r(S) ≤ 4 and 𝒴 (S) ≠ ∅. Then S ∈ UV.

Proof. Assume S ∉ U and S ≅ D̸_{8} ≀ C_{2}. By Lemma 2.4(a), |S| ≥ 2^{8}, there is a unique normal subgroup 1 ≠ Z_{∗} ⊴ S of index 2^{7}, and S/Z_{∗} ≅ D_{8} ≀ C_{2}. Let Y_{1}, Y_{2}, Y_{3} ⊴ S be the three distinct normal subgroups such that Y_{1}/Z_{∗} ≅ Y_{2}/Z_{∗} ≅ C_{2}^{4} and Y_{3}/Z_{∗} ≅ 2_{+}^{1+4}. By Lemma 2.4(b), 𝒴 (S) ⊆ {Y_{1}, Y_{2}, Y_{3}}.

By Proposition 3.9(a), E_{F} = E_{F}(Y_{1}) ∪ E_{F}(Y_{2}) ∪ E_{F}(Y_{3}), and E_{F}(Y_{i}) ≠ ∅ for each i = 1, 2, 3. Also, Y_{3} ≅ U̸T_{3}(4) since S ∉ U, and |Y_{3}| ≥ 2^{6} since |S| ≥ 2^{8}. So by Proposition 3.9(b,c), for each i = 1, 2, 3 and each R ∈ E_{F}(R_{i}), either

• Y_{i} ∈ 𝒴 (S) and E_{F}(Y_{i}) ⊆ E_{F}^{(II)}, and R is in an F-essential pair

as described in Lemma 3.7(a) or 3.8(b); or

• i = 3, Y_{3} ∉ 𝒴 (S), E_{F}(Y_{3}) = E_{F}^{(III)}, Y_{3} ≅ Q_{8}×Q_{8}, and [R:Y_{3}] = 2.

(3.11)

Together with Proposition 3.11(a) (applied when Y_{i} ∈ 𝒴 (S)), this proves that for each i = 1, 2, 3, there are subgroups Θ_{i1}, Θ_{i2} ⊴ Y_{i} where

|Y_{i}| = 2^{m} for m even ⟹ Θ_{ij} ∈ DQ, Y_{i} = Θ_{i1} × Θ_{i2}  (3.12)
|Y_{i}| = 2^{m} for m odd ⟹ Θ_{ij} ∈ Q, [Θ_{i1}, Θ_{i2}] ≤ Θ_{i1} ∩ Θ_{i2} = Z(S) . (3.13)

(Note that Y_{i} ≅ C̸_{2}^{4} since |S| ≥ 2^{8} and [S:Y_{i}] ≤ 2^{3}.) Also, by Proposition 3.11(a.1), {Θ_{i1}, Θ_{i2}} is an S-conjugacy class if Y_{i} ∈ 𝒴 (S), and in particular, if i = 1, 2.

We claim that

for some i = 1, 2, 3, Y_{i} ∈ D × D.  (3.14)

To see this, note first that since F is reduced, Z(S) ⋬ F, so by Lemma 1.15, there is R ∈ E_{F} and α ∈ Aut_{F}^{∗}(R) such that α(Z(S)) ≠ Z(S). Let i be such that R ∈ E_{F}(Y_{i}).

Let R_{0} ≤ R be any subgroup such that r(R_{0}/Z(R_{0})) = 4. By Lemma A.6(a) and since Aut_{F}^{∗}(R) does not normalize Z(S), Z(S) < Z(R) ≤ Z(R_{0}). Thus |Z(R_{0})| ≥ 4. If R_{0} ≅ Q_{8} × Q_{8}, then there is a unique element z ∈ Z(R_{0})^{#} such that z = g^{2} for 9 classes gZ(R_{0}) ∈ (R_{0}/Z(R_{0}))^{#}, so ⟨z⟩ is characteristic in R_{0}. If Aut_{F}^{∗}(R) normalizes R_{0}, then it also normalizes ⟨z⟩, so ⟨z⟩ ≠ Z(S). Each element of odd order in Aut_{F}^{∗}(R) centralizes the two elements in Z(R)∖⟨z⟩, and each element in Aut_{S}(R) centralizes ⟨z⟩ and Z(S). Thus Aut_{F}^{∗}(R) centralizes Z(R_{0}) > Z(S), which is impossible. We conclude that

there is no R_{0} ≤ R such that r(R_{0}/Z(R_{0})) = 4, and such that either |Z(R_{0})| = 2, or R_{0} ≅ Q_{8} × Q_{8} and is normalized by Aut_{F}^{∗}(R).

(3.15)

If R ∈ E_{F}^{(II)}(Y_{i}) is in an essential pair as in Lemma 3.7(a), then there is a subgroup T < R such that T ∈ 𝒴_{0}(S) and Aut_{F}^{∗}(R) normalizes T. By (3.15), T ≅ 2̸_{±}^{1+4} and T ≅ Q̸_{8} ×Q_{8}. Thus T ≅ C_{2}^{4}. If Θ_{i1}, Θ_{i2} ∈ Q, then T ∩Θ_{i1} ≤ Z(Θ_{i1}), so the image of T under projection to Y_{i}/Θ_{i1} has rank at least 3, which is impossible since r(Y_{i}/Θ_{i1}) ≤ r(Θ_{i2}) = 2. Thus Θ_{i1}, Θ_{i2} ∈ D, and so Y_{i} ∈ D × D.

If R is in a pair as in Lemma 3.8(b), then R = UC_{S}(U) for some U ≅ C_{2}^{2} or Q_{8}, and [Aut_{F}^{∗}(R), R] = U. By Lemma 3.8(b,c), U ≤ Δ for some Δ ∈ DQ in a conjugacy class {Δ, Δ^{∗}} such that [Δ, Δ^{∗}] ≤ Δ ∩ Δ^{∗} ≤ Z(S), |Δ| ≥ 16 if Δ ∈ Q, and Y_{i} = Δ × Δ^{∗} if Δ ∈ D. If Δ, Δ^{∗} ∈ Q, then U ≅ Q_{8}, and R ≥ UΔ^{∗} since Aut_{Δ^{∗}}(U) ≤ Inn(U). Choose U^{∗} < Δ^{∗} with U^{∗} ≅ Q_{8} and set R_{0} = UU^{∗}. Then Aut_{F}^{∗}(R) normalizes R_{0} since [Aut_{F}^{∗}(R), R] = U < R_{0}. Either [U, U^{∗}] = 1 and R_{0} ≅ 2_{+}^{1+4} or Q_{8} × Q_{8}; or [U, U^{∗}] ≠ 1, [Fr(Δ), U^{∗}] = 1, and hence R_{0} ≅ 2_{−}^{1+4} by Lemma C.2(a) (with U, U^{∗} in the role of Δ_{1}, Δ_{2}). Since this contradicts (3.15), we have Δ, Δ^{∗} ∈ D, and hence Y_{i} = Δ × Δ^{∗} ∈ D × D by (3.12) and (3.13).

Finally, if R ∈ E_{F}^{(III)}(Y_{3}), then Y_{3} ≅ Q_{8} ×Q_{8}, and Aut_{F}^{∗}(R) normalizes Y_{3} since 𝔣𝔬𝔠(F, R) = Y_{3}. This again contradicts (3.15), and finishes the proof of (3.14). Case 1: Assume |S| is an odd power of 2. Then |Y_{1}| = |Y_{2}| = 2^{m} and |Y_{3}| = 2^{m+1} for some even m. So Y_{i} = Θ_{i1} × Θ_{i2} for i = 1, 2 by (3.12), and Y_{3} ∉ D × D by (3.13). Hence by (3.14), for i = 1 or i = 2, Θ_{i1} ≅ Θ_{i2} ∈ D.

Set Z_{ij} = Z(Θ_{ij}). Set S_{0} = C_{S}(Z_{2}(S)), where Z(S) < Z_{2}(S) ≅ C_{2}^{2} by Lemma 2.4(c). Hence [S:S_{0}] = 2. For i = 1, 2, {Θ_{i1}, Θ_{i2}} is an S-conjugacy class by Proposition 3.11(a.1) (and since Y_{i} ∈ 𝒴 (S) by (3.11)), so Z_{i1}Z_{i2} ⊴ S, and Z_{i1}Z_{i2} = Z_{2}(S) ≤ Z(S_{0}) by Lemma 2.4(a). Thus no element of Inn(S_{0}) can exchange Θ_{i1} with Θ_{i2}, so Θ_{ij} ⊴ S for all i, j = 1, 2.

Let z_{1}, z_{2} ∈ Z_{2}(S) be the two elements not in Z(S). After renumbering, if necessary, we can assume that Z_{ij} = ⟨z_{j}⟩ for i, j = 1, 2. By Lemma B.6, applied with S_{0} and {Θ_{ij} | i, j = 1, 2} in the role of S and 𝒫, S_{0} = Γ_{1} × Γ_{2}, where for each j = 1, 2, Γ_{j} ∈ DSQ, Z(Γ_{j}) = ⟨z_{j}⟩, and Γ_{j} ≤ Θ_{1j}Θ_{2j} ≤ Γ_{j}Z(Γ_{3−j}). Then Γ_{1} ≅ Γ_{2}, and Γ_{j} ∈ DS since Θ_{1j} or Θ_{2j} is dihedral. Also, for any g ∈ S∖S_{0}, S_{0} = Γ_{1} × ^{g}Γ_{1} by Lemma B.3 (Z(^{g}Γ_{1}) = ⟨^{g}z_{1}⟩ = ⟨z_{2}⟩), so upon replacing Γ_{2} by gΓ_{1}, we can assume that Γ_{1} and Γ_{2} are S-conjugate.

Fix g ∈ S∖S_{0}, and let r, s ∈ Γ_{1} be such that g^{2} = r·^{g}s = ^{g}s·r. Then s = r and [g^{2}, r] = 1 since [g, g^{2}] = 1, so

(r^{−1}g)^{2} = r^{−1}(^{g}r^{−1})g^{2} = r^{−1}(^{g}r^{−1})(^{g}r)r = 1 .

Set h = r^{−1}g; then h^{2} = 1, and so S ≅ Γ_{1} ≀ C_{2} ∈ V in this case. Case 2: Now assume |S| is an even power of 2. Set Z = ⟨z⟩ = Z(S). Then |Y_{1}| = |Y_{2}| = 2^{m} for odd m, so by (3.13), for i = 1, 2, Θ_{i1} ∩ Θ_{i2} = Z(S) and Θ_{i1}, Θ_{i2} ∈ Q. An argument similar to that used in Case 1, applied to S/Z, Y_{i}/Z, and Θ_{ij}/Z, shows that S/Z(S) ≅ D ≀ C_{2} for some D ∈ D. Here, D is dihedral since it is generated by two of the subgroups Θ_{ij}/Z, which are dihedral or C_{2}^{2}. Thus S = ⟨a_{1}, a_{2}, b_{1}, b_{2}, t⟩, where D_{i} = ⟨a_{i}, b_{i}⟩ ∈ DSQ for i = 1, 2, |a_{i}| ≥ 8, [D_{i}:⟨a_{i}⟩] = 2, [D_{1}, D_{2}] ≤ D_{1} ∩ D_{2} = Z, and t^{2} ∈ Z. Upon replacing b_{2} by b_{2}z or a_{2} by a_{2}z, if necessary, we can arrange that ^{t}a_{i} = a_{3−i} and ^{t}b_{i} = b_{3−i} for i = 1, 2.

Choose w_{i} ∈ ⟨a_{i}^{2}⟩ of order 4. Since [a_{i}, D_{3−i}] ≤ Z, [w_{i}, D_{3−i}] ≤ [a_{i}^{2}, D_{3−i}] = 1. If [a_{1}, b_{2}] = z, then [a_{1}w_{2}, b_{2}] = 1. So after replacing a_{1} by a_{1}w_{2} and a_{2} by a_{2}w_{1} if necessary, we can arrange that [a_{1}, b_{2}] = [a_{2}, b_{1}] = 1. Also, upon replacing b_{i} by b_{i}w_{3−i} if necessary, we can arrange that b_{1}^{2} = b_{2}^{2} = 1.

To get more information about relations between these generators, we now look more closely at Y_{3}. We have Z_{∗} = ⟨a_{1}^{4}, a_{2}^{4}⟩ since it is the unique normal subgroup of S such that S/Z_{∗} ≅ D_{8} ≀ C_{2}. Also, Y_{3}/Z_{∗} ≅ 2_{+}^{1+4} by definition of Y_{3}, and hence

Y_{3} = ⟨a_{1}a_{2}^{−1}, t, a_{1}a_{2}, b_{1}b_{2}⟩.

By (3.14) (and since Y_{1}, Y_{2} ∉ D × D), Y_{3} ∈ D × D. Set A = ⟨a_{1}a_{2}, a_{1}a_{2}^{−1}⟩ = ⟨a_{1}a_{2}, a_{1}^{2}⟩. Thus A is abelian by the above remarks ([a_{1}^{2}, a_{2}] = 1). By Lemma C.1, A is the unique subgroup of Y_{3} ∈ D ×D which is abelian of rank 2 and index 4, and of the three involutions in A, exactly two are squares of elements in Y_{3}∖A. Since w_{1}w_{2} and w_{1}w_{2}^{−1} are S-conjugate, z is not the square of any g ∈ Y_{3}∖A. Hence t^{2} = 1, [b_{1}, b_{2}] = (b_{1}b_{2})^{2} = 1, and [a_{1}, a_{2}] = [a_{1}b_{1}, a_{2}b_{2}] = (a_{1}b_{1}a_{2}b_{2})^{2} = 1 (note that (a_{2}b_{2})^{2} = ^{t}((a_{1}b_{1})^{2}) = (a_{1}b_{1})^{2} ∈ Z). Thus D_{i} = ⟨a_{i}, b_{i}⟩ ∈ DS (recall b_{i}^{2} = 1), [D_{1}, D_{2}] = 1, D_{1} ∩ D_{2} = ⟨z⟩, ^{t}D_{1} = D_{2}, and so S = (D_{1} ×_{⟨z⟩} D_{2}) ⋊ ⟨t⟩ ∈ V. □

It remains to look more closely at essential subgroups of the type described in Lemma 3.8(a).

Lemma 3.13. Fix a 2-group S with r(S) ≤ 4, and a reduced fusion system F over S. Assume the following:

(i) There is a normal subgroup Δ ⊴ S which is quaternion of order at least 16. Let A ≤ Δ be the cyclic subgroup of index 2, and fix b ∈ Δ∖A.

(ii) There is a subgroup P ⊴ S of index 2, and an automorphism σ ∈ Aut(P) of odd order, such that A < P, b ∉ P, c_{b}σc_{b}^{−1} ≡ σ^{−1} ( mod Inn(P)), σ(A) ∩ A = 1, and such that conjugation by a generator of σ(A) exchanges the two noncyclic subgroups of index 2 in Δ.

Then S ∈ GW.

Proof. Set Â = A·α(A). We will prove the following statements:

(a) Â ⊴ S, Â = A × σ(A), α(Â) = Â, |σ| = 3, CÂ(σ) = 1;

(b) P = ÂV where V = C_{P} (σ) and Â ∩ V = 1;

(c) there is t ∈ Âb such that c_{t}σc_{t}^{−1} = σ^{−1} ∈ Aut(P), [V, t] = 1, t^{2} = 1, [t, Â] = A, and Â⟨t⟩ ≅ A ≀ C_{2};

(d) V ≅ P/Â is cyclic, and there is μ ∈ Hom(V, (ℤ/2^{n})^{×}) (where 2^{n} = |A|) such that ^{x}g = g^{μ(v)} for all x ∈ V, g ∈ Â; and

(e) μ is injective, |V | ≤ 2, and 2^{n−1} + 1 ∉ Im(μ).

Then by (c) and (e), either V = 1 and S ≅ A ≀ C_{2} ∈ W, or |V | = 2, S = Â ⋊ ⟨t, V ⟩ where V acts on Â via (g ↦ g^{λ}) for λ = −1 or 2^{n−1} − 1, and S ∈ G.

(a) Let Δ_{1}, Δ_{2} < Δ be the noncyclic subgroups of index 2 in Δ, and set S_{0} = N_{S}(Δ_{1}) = N_{S}(Δ_{2}). Fix a generator a ∈ A, and set y = σ(a). Then y ∉ S_{0} by assumption. Since A and σ(A) are both normal in P, [A, σ(A)] ≤ A ∩ σ(A), where A ∩ σ(A) = 1 by assumption. Thus Â ≅ A × A. Since [Â, b] ≤ A, Â ⊴ P ⟨b⟩ = S.

By Lemma A.11 (applied to the action of ⟨[σ], [c_{b}]⟩ on P^{ab}), and since [b, P] ≤ [b, S] ≤ A is cyclic, ⟨[σ], [c_{b}]⟩ ≅ Σ_{3} as a subgroup of Aut(P^{ab}). Also, ⟨σ⟩ acts faithfully on P^{ab} by Lemma A.9, so σ has order 3 in Aut(P).

Since ⟨σ⟩ ∈ Syl_{3}(Inn(P)⟨σ⟩), there is h ∈ P such that c_{hb}σc_{hb}^{−1} = σ^{−1} in Aut(P). Set t̂= hb for short. Then σ(y) = σ^{−1}(a) ∈ t̂(σ(A)) ≤ Â since A ⊴ S and Â ⊴ S. Thus σ(Â) = Â. Also, CÂ(σ) = 1 since |σ| = 3 and σ acts nontrivially on Â/Fr(Â).

(b) Since [b, P] ≤ A ≤ Â, c_{b} induces the identity on P/Â. Since |σ| is odd and [σ] is inverted by [c_{b}] in Out(P), σ induces the identity on (P/Â)^{ab}, and hence on P/Â by Lemma A.9. Since CÂ(σ) = 1, each coset gÂ in P/Â contains a unique element fixed by σ. Thus P = ÂV = V Â, where V = C_{P} (σ), and Â ∩ V = 1.

(c) Recall that t̂ ∈ Qb = V Âb. Let u ∈ V and t ∈ Âb be such that t̂ = ut. Then σc_{u}σ^{−1} = c_{σ(u)} = c_{u}, so c_{t}σc_{t}^{−1} = c_{u^{−1}t̂}σc_{u^{−1}t̂}^{−1} = c_{u}^{−1}σ^{−1}c_{u} = σ^{−1} in Aut(P). Also, c_{t}|Â = c_{b}|Â, and c_{t} induces the identity on P/Â ≅ V since [b, P] ≤ A. For each v ∈ V, since c_{t} normalizes ⟨σ⟩ in Aut(P), c_{t} sends C_{vÂ}(σ) = {v} to itself. So [V, t] = 1. Finally, [c_{t^{2}}, σ] = 1 in Aut(P) since c_{t} inverts σ, so t^{2} ∈ CÂ(σ) = 1.

In particular, c_{b}σc_{b}^{−1}|Â = σ^{−1}|Â. Also, aσ(a)σ^{2}(a) = ayσ(y) ∈ CÂ(σ) = 1, so σ(y) = a^{−1}y^{−1}. Since c_{b}(a) = a^{−1}, c_{b} sends the σ-orbit {a, y, a^{−1}y^{−1}} to the orbit {a^{−1}, ay, y^{−1}}, and thus ^{b}y = ay. Hence [b, Â] = A, and Â⟨t⟩ ≅ A ≀ C_{2} since c_{t} exchanges y and ay.

(d) For each x ∈ V, c_{x} commutes in Aut(Â) with σ and with c_{b} = c_{t} since [σ, V] = [t, V] = 1. Hence c_{x}(A) = A since the elements of A are the only ones in Â which are inverted by c_{b} (the two involutions in Â∖A are exchanged). Also, c_{x} sends σ-orbits to σ-orbits, and so c_{x}(a) = a^{i} and c_{x}(y) = y^{i} for some odd i. In other words, there is μ ∈ Hom(V, (ℤ/2^{n})^{×}), where 2^{n} = |A|, such that c_{x}(g) = g^{μ(x)} for each x ∈ V and each g ∈ Â.

Since Â ⊴ S, ⟨a, y^{2}⟩ = Â ∩ S_{0} ⊴ S. Also, [b, S_{0}] ≤ ⟨a^{2}⟩, and we just saw that [V, ⟨a, y^{2}⟩] ≤ ⟨a^{2}, y^{4}⟩. Thus ⟨a, y^{2}, b, V ⟩/⟨a^{2}, y^{4}⟩ ≅ C_{2}^{3} × V, and so V is cyclic since r(S) ≤ 4.

(e) Set Z = Z(Δ). Assume V ≠ 1 (otherwise there is nothing to prove). Let v ∈ V be the element of order 2. If μ(v) = 1, then Ω_{1}(Z(S)) = Z⟨v⟩, while Ω_{1}(Z(S)) ∩ [S, S] = Z. So v ∉ 𝔣𝔬𝔠(F) by Proposition 1.18(a), which contradicts Proposition 1.14(b) (F is reduced). Thus μ(v) ≠ 1, and μ is injective.

Since S = ÂV ⟨t⟩ where [t, V] = 1, [S, S] = [t, Â][V, Â] ≥ AΩ_{1}(Â). Hence C_{S}([S, S]) ≤ C_{S}(Ω_{1}(Â)) = P, so C_{S}([S, S]) ≤ C_{P} (A) = Â. Thus Â = C_{S}([S, S]) is characteristic in S. So Aut(S) is a 2-group by Lemma A.9, applied to the chain Fr(S) < ÂFr(S) < P < S (where |ÂFr(S)/Fr(S)| = 2 since a, y^{2} ∈ Fr(S)). It follows that Out_{F}(S) = 1.

Set U = Ω_{1}(Â), and let x ∈ V be a generator. If |V | ≥ 4, then for each Q < S of index 2, either Q ≥ Â and [Q, Q] ≥ [x^{2}, Â] ≥ U, or Q ≱ Â, there is x′ ∈ xÂ ∩ Q, and [Q, Q] ≥ [x′, Fr(Â)] ≥ U. Also, μ(v) = 2^{n−1} + 1, since it has order 2 and is a square in (ℤ/2^{n})^{×}. So [v, S] = [v, Â] = U, v^{2} = 1, and F is not reduced (v ∉ 𝔣𝔬𝔠(F)) by Proposition 1.18(b) applied with v in the role of g.

Thus V = ⟨v⟩ where v^{2} = 1. Since n ≥ 3, U = Ω_{1}(Â) ≤ Fr(Fr(Â)) ≤ Fr(Q) for each Q < S of index 2. If μ(v) = 2^{n−1} + 1, then [v, S] = [v, Â] = U. So by Proposition 1.18(c), v ∉ 𝔣𝔬𝔠(F) and F is not reduced, a contradiction. □

Proposition 3.14. Fix a 2-group S with r(S) ≤ 4, and a reduced, indecomposable fusion system F over S. Assume that there is some R ∈ E_{F}^{(II)} such that 𝔣𝔬𝔠(F, R) ∈ 𝒳 (S). Then S ∈ DSWG.

Proof. By Lemma 3.3 and Propositions 3.4 and 3.5, if E_{F}^{(I)} ≠ ∅, then S ≅ UT_{4}(2) or S ∈ U. Since 𝒳 (UT_{4}(2)) = ∅ by Lemma C.4(e), and 𝒳 (S) = ∅ for S ∈ U by Lemma C.9, E_{F}^{(I)} = ∅.

Since 𝒳 (S) ≠ ∅, 𝒴 (S) = ∅ by Corollary 2.5. So by Lemmas 3.7 and 3.8,

(R_{1}, R_{2}) an F-essential pair of type (II) ⟹

𝔣𝔬𝔠(F, R_{i}) ∈ 𝒳 (S), and (R_{1}, R_{2}) as in Lemma 3.8. (3.16)

Fix an F-essential pair (R_{1}, R_{2}) of type (II), and set Δ = 𝔣𝔬𝔠(F, R_{1}) ∈ 𝒳 (S). By Lemma 3.8(a,c), Δ ≅ D_{2^{n}} for n ≥ 3 or Δ ≅ Q_{2^{n}} for n ≥ 4. Let A ⊴ Δ be the cyclic subgroup of index 2, fix a generator a ∈ A, and choose b ∈ Δ∖A. Set Z = Z(Δ). Let Δ_{0} < Δ be the subgroup of order 8 which contains b. Set T = C_{S}(Δ_{0}). By Lemma 2.10(a,b), [S:T Δ] = 2.

If Δ ∈ D, then by Lemma 3.8(c,d), R_{i} = U_{i}C_{S}(U_{i}) where U_{i} ≅ C_{2}^{2} is a direct factor of R_{i} and U_{1}U_{2} ≅ D_{8}. So we can assume Δ_{0} = U_{1}U_{2}. Thus

Δ ∈ D ⟹ there is T_{0} < T such that T = T_{0} × Z.  (3.17)
Throughout the proof, when P ∈ E_{F} ∪ {S}, we write Δ_{P} = 𝔣𝔬𝔠(F, P).

Step 1: Assume E_{F}^{(III)} ≠ ∅, and fix P ∈ E_{F}^{(III)}. Then P ≥ [S, S], and hence P ≥ A by Lemma 2.10(b). We will show that either

(i) P ≥ Δ and Δ ∈ Q; or

(ii) P ≥ Δ, Z ≰ Δ_{P} ≅ C_{2}^{2} or Q_{8}, and Δ_{P} ≰ Z(S); or

(iii) P ≱ Δ, A < Δ_{P} ≅ Q_{8}, and |A| = 4; or

(iv) P ≱ Δ and S ∈ WG.

Case 1: P ≥ Δ. Since Aut_{F}(P) is generated by Aut_{S}(P) and automorphisms of odd order, it normalizes Z by Lemma B.7 (and since Δ ⊴ S).

If Δ ∈ Q, then (i) holds. So assume Δ is dihedral, and set T̂ = [Aut_{F}^{∗}(P), P]. Since Out_{S}(P) ∈ Syl_{2}(Out_{F}(P)) has order 2, O^{2}(Out_{F}(P)) has odd order by the Thompson transfer lemma [Th, Lemma 5.38(a.i)] or by Burnside’s normal pcomplement theorem [G, Theorem 7.4.3]. Hence Aut_{F}^{∗}(P) ≤ Inn(P)·H for some H < Aut_{F}^{∗}(P) of odd order, and T̂ = [H, P] by Proposition 1.14(c).

Since Z is a direct factor in T = C_{S}(Δ_{0}) by (3.17), Lemma 2.11 applies with P, U_{1}U_{2}, and H in the roles of S, Δ_{0}, and G. By that lemma, |H| = 3, T̂ ⊴ S, [T̂, Δ] = T̂ ∩ Δ = 1, and either T̂ ≅ C_{2^{m}} × C_{2^{m}} (some m ≥ 1) or T̂ ≅ Q_{8}. For g ∈ S∖P, c_{g} normalizes Aut_{F}^{∗}(P) = O^{2}(O^{2′}(Aut_{F}(P))), and hence ^{g}T̂ = T̂. Thus T̂ ⊴ S, and so Δ_{P} = T̂. If T̂ ≅ C_{2^{m}} × C_{2^{m}}, then T̂Z < T = C_{S}(Δ_{0}) since T is nonabelian, r(T_{0}) ≤ 2 since S contains a subgroup isomorphic to T_{0} × D_{8} and r(S) ≤ 4, and T_{0} ≅ C_{2^{m}} ≀ C_{2} since any extension of C_{2^{m}} × C_{2^{m}} by Out_{F}(P) ≅ Σ_{3} (with faithful action) is split (cf. [AOV2, Lemma A.8]). Hence m = 1 since r(C_{4} ≀ C_{2}) = 3, and T_{0} ≅ D_{8}. We conclude that Δ_{P} ≅ C_{2}^{2} or Q_{8} and Δ_{P} ≰ Z(S), and thus that (ii) holds.

Case 2: P ≱ Δ. Since A ≤ P, b ∉ P and A_{0} ≤ Fr(P). Also, [b, P] = [b, S] = A, so A ≰ Fr(P) by Lemma 1.8 and since P ∈ E_{F}.

Set P = P/Fr(P), and X = XFr(P)/Fr(P) for X ≤ P. Thus [b, P] = A has rank 1, and C_{P} (b) = P ∩ T Δ. By Lemma A.11, and since Out_{F}(P) acts faithfully on P (Lemma 1.7), Out_{F}(P) = Γ_{1} × Γ_{2} where Γ_{1} ≅ Σ_{3} and Γ_{2} has odd order, and where [Γ_{1}, P] has rank 2 and [Γ_{1}, P] ≰ C_{P} (b) = P ∩ T Δ. Hence Aut_{F}^{∗}(P) ≤ Inn(P)⟨σ⟩ for some σ ∈ Aut_{F}^{∗}(P) of order 3 for which [σ] ∈ Γ_{1}, and such that σ(A) ≰ T Δ. Set y = σ(a).

By Lemma 2.10(a,b), there is x ∈ S∖T Δ such that xax^{−1} = a^{1+4ℓ} (some ℓ ∈ ℤ) and xbx^{−1} = ab. Also, Δ ≅ Q̸_{8} by Lemma 3.8(a,c). We consider the following two subcases.

Case 2a: P ≱ Δ and y = σ(a) ∈ T Abx. Thus yay^{−1} = a^{4k−1} for some k ∈ ℤ, and ⟨[y, a]⟩ = ⟨a^{2}⟩ = A_{0}.

Since A, σ(A) ⊴ P, we have A_{0} = [σ(A), A] ≤ σ(A) ∩ A. So A_{0} ≤ ⟨y⟩, [y, A_{0}] = 1, and this implies that A_{0} = Z and |A| = 4. Hence Δ ≅ D_{8} since Δ ≅ Q̸_{8}. Set Q = σ(A)A ⊴ P. Then Q = A⟨y⟩ ≅ Q_{8} since yay^{−1} = a^{−1} and |y| = |a| = 4, and Q ∈ 𝒳 (S) (in particular, Q ⊴ S) since [b, y] = a^{±1}.

Now, [T, y] = [T, Q] ≤ Q ∩ T = Z since Q ⊴ S and T = C_{S}(Δ) ⊴ S. We already saw that [σ, P/Fr(P)] has rank 2 and contains the image of [b, y] = a^{±1},and hence is equal to QFr(P)/Fr(P). So [σ, P] ≤ QFr(P). Since σ^{2} ≡ c_{b}σc_{b}^{−1} (mod Inn(P)), σ^{2}(a) is P -conjugate to c_{b}(y^{−1}) ∈ y^{−1}A ⊆ Q, so σ(y) = σ^{2}(a) ∈ Q and hence σ(Q) = Q. Since σ induces the identity on P/QFr(P), it induces the identity on P/Q by Lemma A.9, so Q = [σ, P]. Hence Q = [Aut_{F}^{∗}(P), P] by Proposition 1.14(c) and since Aut_{F}^{∗}(P) ≤ Inn(P)⟨σ⟩. Since Q ⊴ S, we conclude that Q = 𝔣𝔬𝔠(F, P) = Δ_{P}, and hence that (iii) holds.

Case 2b: P ⪈ Δ and y = σ(a) ∈ T Ax. By Lemma 2.10(c), |y| = |A| implies that Z ≤ Fr(T), so Δ is quaternion (Lemma 3.8(c,d)), and ⟨y⟩ ∩ A = 1. Also, |Δ| ≥ 16 since Δ ≅ Q̸_{8}. Hence S ∈ WG by Lemma 3.13.

Step 2: From now on, we assume that S ∉ WG. We next show that there is an F-essential pair (Q_{1}, Q_{2}) of type (II) for which Δ_{Q_{1}} is dihedral. Assume otherwise; in particular, assume Δ ∈ Q. We will show that Z ⊴ F), contradicting the assumption that F is reduced.

If P = S or P ∈ E_{F}^{(III)}, then by (i)–(iv), either P ≥ Δ, or Δ_{P} = Q ≅ Q_{8} and Z(Q) = Z. In either case, each odd order element of Aut_{F}^{∗}(P) centralizes Z (by Lemma B.7 when P ≥ Δ), and hence Aut_{F}^{∗}(P) centralizes Z. If P ∈ E_{F}^{(II)}, then by (3.16) and Lemma 3.8(a,c,d), P = UC_{S}(U) for some U ≅ Q_{8} (since Δ_{P} ∈ Q by assumption), [Aut_{F}^{∗}(P), P] = U, and hence Aut_{F}^{∗}(P) acts trivially on C_{S}(U), and in particular on Z(S) ≥ Z. Since E_{F}^{(I)} = ∅, Lemma 1.15 now implies that Z ⊴ F.

Step 3: We can thus assume (R_{1}, R_{2}) was chosen so that Δ = Δ_{R_{1}} is dihedral. We next show that for each P ∈ E_{F} ∪ {S} with Δ_{P} ≠ 1, either

(v) Δ_{P} ≰ T Δ, Δ_{P} ∈ DQ, and Δ_{P} > A with index 2; or

(vi) Δ_{P} ≤ T Δ, Δ_{P} ∈ DQ, and Δ_{P} Δ/Δ ≤ Z(T Δ/Δ); or

(vii) Δ_{P} ≤ T Δ, Δ_{P} ∈ DQ or Δ_{P} ≅ C_{2}^{2}, Δ_{P} ≰ Z(S), and Z ≰ Δ_{P}; or

(viii) Δ_{P} ≅ T_{0} ≅ C_{2^{m}} × C_{2^{m}} for some m ≥ 1.

3.3. ESSENTIAL SUBGROUPS OF INDEX 2 IN S 47

By points (i)–(iv) and since Δ ∈ D and S ∉ WG, each subgroup in E_{F}^{(III)} satisfies (v) or (vii). When P = S and Out_{F}(S) ≠ 1, then by the Schur-Zassenhaus theorem (cf. [G, Theorem 6.2.1]), there is 1 ≠ G < Aut_{F}(S) of odd order such that Aut_{F}(S) = Inn(S)G, and [Aut_{F}^{∗}(S), S] = [G, S] by Proposition 1.14(c). By Lemma 2.11, [G, S] ⊴ S (so Δ_{S} = [G, S]), Δ_{S} ≤ C_{S}(Δ) ≤ T, Δ_{S} ∩ Z = 1, and [S:Δ_{S}Δ] ≤ 2 = [S:T Δ]. Hence T = Z × Δ_{S}, and Δ_{S} ≅ T_{0}. By the same lemma, Δ_{S} ≅ Q_{8} or C_{2^{m}} × C_{2^{m}}, so either (vii) or (viii) holds.

Now assume P ∈ E_{F}^{(II)}. Then Δ_{P} ∈ 𝒳 (S) by (3.16), and hence Δ_{P} ∈ DQ. If P ∈ E_{F}^{(II)} is such that Δ_{P} ≰ T Δ, and h ∈ Δ_{P} ∖T Δ, then by Lemma 2.10(a), ^{h}b = a^{i}b for some odd i. Hence ⟨[b, h]⟩ = A ≤ [S, Δ_{P}]. Also, [S, Δ_{P}] is cyclic of index 2 in Δ_{P} since Δ_{P} ∈ 𝒳 (S) (is strongly automized). Since [S, Δ_{P}] ≤ [S, S] ≤ T A and T A_{0} ⊴ T A by Lemma 2.10(a), there is no cyclic subgroup in [S, Δ_{P}] which strictly contains A, and thus A = [S, Δ_{P}] and |Δ_{P} /A| = 2. So (v) holds in this case.

If P ∈ E_{F}^{(II)} is such that neither (v) nor (vi) holds, then Δ_{P} ≤ T Δ and [T, Δ_{P}] ≰ Δ. We must show that Z ≰ Δ_{P}. Let g = ta^{i}b^{j} ∈ Δ_{P} and u ∈ T be such that t ∈ T and [u, g] ∉ Δ. Then [b, g] = a^{−2i} ∈ [S, Δ_{P}], [u, g] ≡ [u, t] (mod ⟨a^{4i}⟩ ≤ [S, Δ_{P}]), and hence [u, t] ∈ [S, Δ_{P}]∖Δ. Thus [S, Δ_{P}] is cyclic since Δ_{P} ⊴ S and Δ_{P} ∈ DQ, and 1 ≠ [u, t] ∈ [T, T] ∩ [S, Δ_{P}]. Hence Z(Δ_{P}) ≤ [T, T], and Z ≰ Δ_{P} by (3.17).

Step 4: If T is abelian, then by [AOV2, Proposition 5.1], S ∈ DSW. So assume T = T_{0} × Z is nonabelian, and set

𝒫 = {Δ_{P} = 𝔣𝔬𝔠(F, P) | P ∈ E_{F} ∪ {S} and Δ_{P} ≠ 1} .

Then S = ⟨𝒫⟩ by Proposition 1.14(b) and since F is reduced. By (v)–(viii) and since T is nonabelian, for each P ∈ E_{F} ∪ {S} with Δ_{P} ≠ 1, either Δ_{P} ∈ DQ, or Δ_{P} ≅ C_{2}^{2} and Δ_{P} ≰ Z(S).

Since T = T_{0}×Z by (3.17), T Δ/Δ ≅ T/Z ≅ T_{0} is nonabelian. Hence subgroups satisfying (v) and (vi) cannot generate S, so there is some Δ_{P} ∈ 𝒫 which satisfies (vii), and Z ≰ Δ_{P}. Then by Lemma B.6, S = S_{1} × S_{2} for S_{1}, S_{2} ∈ DSQ. So F is decomposable by [O1, Theorem B], contradicting our original assumption. □

3.3. Essential subgroups of index 2 in S

It remains to handle reduced fusion systems all of whose essential subgroups have type (III).

Proposition 3.15. Let F be a reduced, indecomposable fusion system over a 2-group S with r(S) ≤ 4, and assume all F-essential subgroups have index 2 in S. Then S is isomorphic to D_{8}, UT_{4}(2), or C_{4} ≀ C_{2}, or S ∈ U, or S has type M_{12} or Aut(M_{12}).

Proof. By Lemma 1.15, if E_{F} = ∅, then S ⊴ F, while if E_{F} = {R}, then R ⊴ F. Thus |E_{F}| ≥ 2 since F is reduced. By Lemma 3.2, for each R ∈ E_{F},

Out_{F}(R) ≅ Σ_{3}, Σ_{3}×C_{3}, or (C_{3}×C_{3}) ⋊^{−1}C_{2}, and rk(R/Fr(R)) = 4 if |Out_{F}(R)| = 18. Let Ê_{F} be the set of all pairs (R, Γ) for R ∈ E_{F} and Γ ≤ Aut_{F}(R) such that

• Γ ≥ Inn(R) and Γ =^{def} Γ/Inn(R) ≅ Σ_{3}; and

• if Out_{F}(R) ≅ (C_{3} × C_{3}) ⋊^{−1} C_{2}, then C_{R/Fr(R)}(O^{2}(Γ)) has rank 2.

Thus each R ∈ E_{F} appears in exactly one pair in Ê_{F}, except when Out_{F}(R) ≅

(C_{3} × C_{3}) ⋊^{−1} C_{2}, in which case it appears in two pairs. By the above remarks,

∀ R ∈ E_{F}, Aut_{F}^{∗}(R) = O^{2}(O^{2′}(Aut_{F}(R))) ≤ ⟨Γ | (R, Γ) ∈ Ê_{F}.⟩  (3.18)

Set Ê_{F} = {(R_{i}, Γ_{i}) | i ∈ I} for some indexing set I.

For each J ⊆ I, set R_{J} = ⋂_{j∈J} R_{j}, and let T_{J} ≤ R_{J} be the largest subgroup normalized by Γ_{j} for each j ∈ J. Also, set

Γ_{J} = ⟨γ|_{T_{J}} | γ ∈ Γ_{j} for some j ∈ J⟩ ≤ Aut_{F}(T_{J}) .

If T_{J} is centric in S, then by Theorem 1.4, there is a group G_{J} such that S ∈ Syl_{2}(G_{J}), T_{J} ⊴ G_{J}, C_{G_{J}} (T_{J}) ≤ T_{J}, and Aut_{G_{J}} (T_{J}) = Γ_{J}. Thus G_{J}/T_{J} ≅ Γ_{J}/Inn(T_{J}). When J = {i, j}, we write R_{ij} = R_{J}, T_{ij} = T_{J}, etc., and similarly with sets of one, three, or four indices. Note that if R_{i} = R_{j}, then T_{ij} = R_{ij} = R_{i} since R_{i} = R_{j} is normal in G_{i} and in G_{j}. By the maximality condition on T_{ij},

for distinct i, j ∈ I, (G_{i}/T_{ij} > S/T_{ij} < G_{j}/T_{ij}) is a primitive amalgam of the type classified by Goldschmidt in [Gd2, Theorem A].

(3.19)

Case 1: Assume T_{ij} = R_{ij} for each i, j ∈ I with i ≠ j. Then for each i, Γ_{i} normalizes R_{ij} for each j ≠ i, and hence normalizes their intersection R_{I}. Also, Aut_{F}(S) sends R_{I} to itself since it permutes the F-essential subgroups. Thus R_{I} is normalized by Aut_{F}(S) and (by (3.18)) by Aut_{F}^{∗}(R) for each R ∈ E_{F}, so R_{I} ⊴ F by Lemma 1.15. Since F is reduced, R_{I} = 1, so S is elementary abelian, which implies S ⊴ F. Case 2: Assume, for some i ≠ j, that T_{ij} is not centric in S. Set T = T_{ij} ⊴ S. By [AOV2, Theorem 4.5], there is a subgroup U ⊴ S, and a finite completion Γ of the amalgam (G_{i}/T > S/T < G_{j}/T), such that [S:T U] ≤ 2, [U, T] = 1, |U ∩ T | ≤ 2, S/T ∈ Syl_{2}(Γ), and U and Γ are one of the pairs in Table 3.1:

if T ∩ U = 1:  if |T ∩ U| = 2: if S = T U: if [S:T U] = 2:
U  U  Γ  Γ
D_{8}  Q_{16}  A_{6}  Σ_{6}
C_{4} ≀ C_{2}  does not occur  U_{3}(3)  Aut(U_{3}(3))
(C_{4} × C_{4}) ⋊^{t,−1} C_{2}^{2}  type 2M_{12}  M_{12}  Aut(M_{12})

Table 3.1

Assume U ≅ D_{8}. If S = UT, then S ≅ U × T. If [S:UT] = 2, then S/T ≅ D_{8} × C_{2} (i.e., it has type Σ_{6}) by the above table; and since U and T are both normal in S, S = U × T ⟨x⟩ for some x. In either case, by [O1, Theorem B], F is isomorphic to a product of reduced fusion systems one of which over U ≅ D_{8}. Hence S ≅ D_{8} (T = 1) since F is indecomposable.

Assume U ≅ Q_{16}, and fix a, b ∈ U such that |a| = 8 and b ∉ ⟨a⟩. Then a^{4} ∈ Z(S) since U ⊴ S. By Lemma B.7, α(a^{4}) = a^{4} for each α ∈ Aut_{F}(S) of odd order. By Lemma 1.15 and (3.18) and since O_{2}(F) = 1, there is k ∈ I and α ∈ Γ_{k} ≤ Aut_{F}(R_{k}) of order 3 such that α(a^{4}) ≠ a^{4}. By Lemma B.7 (with R_{k} in the role of S), R_{k} ∩ U must be abelian. Since [U:R_{k} ∩ U] = 2, R_{k} ∩ U = ⟨a⟩, so b ∉ R_{k}, and bα(a^{4})b^{−1} = α^{−1}(a^{4}) since c_{b}αc_{b} ≡ α^{−1} (mod Inn(R_{k})) by definition of Ê_{F}. In particular, α(a^{4}) ∉ C_{S}(U) ≥ T, so the image of α(a) in S/T ≅ D_{8} × C_{2} has order 8, which is impossible.

If U ≅ C_{4} ≀ C_{2}, then r(U) = 3, so r(T) = 1 and hence T is cyclic. If T = 1, then either S = U ≅ C_{4} ≀ C_{2}, or S is a Sylow 2-subgroup of Aut(SU_{3}(3)) and hence of type M_{12} (cf. [Gd2, Table 1]). If T ≠ 1, then Ω_{1}(Z(S)) = Ω_{1}(Z(U)Z(T)) ≅ C_{2}^{2} (recall U ⊴ S). If Ω_{1}(T) ≰ [S, S], then there are no reduced fusion systems over S by Proposition 1.18(a). If T ≠ 1 and Ω_{1}(T) ≤ [S, S], let T_{0} < T be the subgroup of index 2; then [S:T U] = 2, S/U is nonabelian since T ∩ [S, S] ≠ 1, so S = T U⟨x⟩ where x^{2} ∈ T_{0}U, and r(S/T_{0}) = r(S/T) + r(T/T_{0}) = 5 since S/T has type M_{12} (r(S/T) = 4).

Now assume U is of type M_{12} or 2M_{12}. Then T ≤ U since M_{12} has sectional 2-rank 4. So S is a Sylow 2-subgroup in M_{12} or Aut(M_{12}), or in a 2-fold central extension of one of those groups. We must eliminate this last possibility.

Assume S contains a subgroup S_{0} ≤ S of type 2M_{12}, and let Z = ⟨z⟩ ≤ Z(S) be the subgroup in the center of 2M_{12}. Let a, b, r, t ∈ S_{0} be elements whose classes a, b, r, t ∈ S_{0}/Z satisfy the presentation of Notation 4.1 (with n = 2). Set Q̂ = ⟨z, a^{2}, ab, r, t⟩ < S_{0}; thus Q̂/Z ≅ 2_{+}^{1+4}, and Z(Q̂) = ⟨z, a^{2}b^{2}⟩ ≅ C_{2}^{2} by Lemma D.3. There is α ∈ Aut_{F}(Q̂) of order 3 such that α(z) = z (hence α|_{Z(Q̂)} = Id) and C_{Q̂/Z(Q̂)}(α) = 1 (cf. [A2, Lemma 5.3(2)]). Let O_{1}, …, O_{5} be the orbits of the

α-action on (Q̂/Z(Q̂))^{#}, where (upon letting ā, b̄, r̄, t̄ ∈ S/Z(Q̂) denote the classes of a, b, r, t) O_{1} = {āb̄^{−1}, b̄^{2}t̄, āb̄t̄} and O_{2} = {āb̄, āb̄^{−1}r̄t̄, b̄^{2}r̄t̄} are the images of the two quaternion subgroups in Q̂/Z, and ā^{2} ∈ O_{3}, r̄t̄ ∈ O_{4}, and āb̄r̄t̄ ∈ O_{5} (the products of āb̄^{−1} ∈ O_{1} with the three elements in O_{2}). The elements in each O_{i} lift to elements of Q̂ with the same square x_{i} ∈ Z(Q̂) (since α|_{Z(Q̂)} = Id), where x_{1}, x_{2} ∈ a^{2}b^{2}Z, x_{3} = 1 since a^{2} is F-conjugate to a^{2}b^{2} or a^{2}b^{2}z, and x_{4} = x_{5} ∈ Z since rt is S-conjugate to abrt = ^{a}(rt). This information suffices to show that Q̂ ≅ U̸T_{3}(4) and Q̂ ≅ Q̸_{8} × Q_{8}. So either x_{4} = x_{5} = 1 and Q̂ ≅ (Q̂/Z) × Z; or [Q̂, Q̂] = Z(Q̂) and by Lemma D.2, Q̂/⟨x⟩ ≅ 2_{+}^{1+4} for exactly two of the three involutions x ∈ Z(Q̂)^{#}. In either case, Q̂/⟨a^{2}b^{2}⟩ ≅ ˆQ̸/⟨a^{2}b^{2}z⟩, so a^{2}b^{2} and a^{2}b^{2}z are not S-conjugate, and hence Z(S) = Z(Q̂) = ⟨a^{2}b^{2}, z⟩.

If S/Z has type M_{12} (i.e., S = S_{0}), then Q̂/Z(S) is the unique abelian subgroup of S/Z(S) of rank 4. If S/Z has type Aut(M_{12}) (|S/S_{0}| = 2), then an outer automorphism of M_{12} acts on Q̂/Z(Q̂) by exchanging the classes āb̄ and āb̄^{−1}, hence exchanging O_{1} and O_{2}. (See the description of the extension amalgam in [Gd2, (3.8)], or Proposition 4.3(b) below.) Thus Out_{S}(Q̂) ≅ C_{2}^{2} permutes freely a basis for Q̂/Z(S) ≅ C_{2}^{4}, and again Q̂/Z(S) is the unique abelian subgroup of S/Z(S) of rank 4 (Lemma A.4(b)). So in either case, Q̂ is characteristic in S.

If z is F-conjugate to a^{2}b^{2} or a^{2}b^{2}z, then by the extension axiom (and since Out_{F}(S) has odd order), there is β ∈ Aut_{F}(S) of odd order which permutes cyclically the involutions in Z(S) = Z(Q̂). Then β(Q̂) = Q̂, which is impossible since one of the elements in a^{2}b^{2}Z is a square in Q̂ and the other is not. So by Lemma 1.15 and since Z(F) = 1, there must be R ∈ E_{F} (of index 2) and γ ∈ Aut_{F}(R) such that γ(z) ≠ z. In particular, Z(R) > Z(S), and by Lemma A.3, R = C_{S}(V) where V = Ω_{1}(Z_{2}(S)) = ⟨z, a^{2}, b^{2}⟩ ≅ C_{2}^{3}. But each element in V ∖Z is F-conjugate to a^{2}b^{2} or a^{2}b^{2}z, so none of them can be conjugate to z. Hence this situation is impossible. Case 3: Now assume that T_{ij} is centric in S for each i, j ∈ I. By [AOV2, Lemma 4.2(e)], Out_{G_{i}}(T_{ij}) ≅ G_{i}/T_{ij} and Out_{G_{j}}(T_{ij}) ≅ G_{j}/T_{ij} both act faithfully on T_{ij}/Fr(T_{ij}) for all i, j. Since r(T_{ij}) ≤ 4, this implies that G_{i}/T_{ij} is isomorphic to a subgroup of GL_{4}(2), and hence that S/T_{ij} contains no element of order 8.

Recall that (G_{i}/T_{ij} > S/T_{ij} < G_{j}/T_{ij}) is a Goldschmidt amalgam by (3.19). If R_{i} ≠ R_{j}, then |S/T_{ij}| ≥ |S/R_{ij}| = 4, and |S/T_{ij}| ≥ 8 if T_{ij} < R_{ij}. So from the list in [Gd2, Table 1] of possible amalgams (and since C_{4} ≀ C_{2} does contain elements of order 8), we see that S/T_{ij} ≅ C_{2} (if R_{i} = R_{j}), C_{2}^{2} (type G_{1}^{3}, when T_{ij} = R_{ij}), D_{8} (type G_{2}^{2} or G_{3}), or D_{8} × C_{2} (type G_{3}^{1}). Also, when S/T_{ij} ≅ D_{8}, then R_{i}/T_{ij} = O_{2}(G_{i}/T_{ij}) ≅ C_{2}^{2}, and similarly for R_{j}/T_{ij}, regardless of whether the amalgam has type G_{2}^{2} or G_{3}.

Assume S/T_{ij} ≅ D_{8} × C_{2}. Set Γ_{ij} = Γ_{ij}/Inn(T_{ij}) for short. Then (G_{i}/T_{ij} > S/T_{ij} < G_{j}/T_{ij}) is the Σ_{6}-amalgam, and hence O^{2}(Γ_{ij}) has index 2 in Γ_{ij}. By Proposition D.1(g), Γ_{ij} contains a subgroup A_{6}, A_{7}, or GL_{3}(2) with index 2; and hence Γ_{ij} ≅ Σ_{6} since this is the only possible extension of one of these groups with Sylow 2-subgroup D_{8} × C_{2}. (Neither A_{6} × C_{2} nor GL_{3}(2) × C_{2} is contained in GL_{4}(2) ≅ A_{8}.) But this is impossible by Lemma D.6.

Thus S/T_{ij} ≅ D_{8} whenever T_{ij} < R_{ij}, and by Case 1, there is at least one such pair i, j ∈ I. Fix such i, j, and set T = T_{ij}. By Proposition D.1(g), and since O^{2}(Γ_{ij}) = Γ_{ij} by the focal subgroup theorem, Γ_{ij} ≤ Out_{F}(T) is isomorphic to A_{7}, A_{6}, or PSL_{2}(7) ≅ GL_{3}(2). Hence by [GH, Theorem II.B], T ≅ C_{2}^{3}, C_{2}^{4}, or C_{2}^{3} × C_{4}. (Recall that Out(2_{−}^{1+4}) ≅ Σ_{5}.)

If T = T_{ij} ≅ C_{2}^{3}, then Γ_{ij} = Aut_{G_{ij}}(T_{ij}) ≅ GL_{3}(2). By Lemma D.5(a), S ≅ UT_{4}(2) or S has type M_{12}. So we can assume T ≅ C̸_{2}^{3}.

If T = T_{ij} ≅ C_{4} ×C_{2}^{3}, then Γ_{ij} ≅ GL_{3}(2) ≅ Aut(T)/O_{2}(Aut(T)) (Lemma A.9). Set V = Ω_{1}(T) ≅ C_{2}^{4} and Z = Fr(T). By Lemma D.5(c), the Γ_{ij}-action on V is decomposable, that on T/Z is indecomposable, and G/[G, V] ≅ C_{4} ×_{C_{2}} SL_{2}(7). Since Z(F) = 1, there is Q ≤ S of index at most 2, together with β ∈ Aut_{F}(Q), such that β(Z) ≠ Z. We can assume that Q = C_{S}(β(Z)) (otherwise β(Z) ≤ Z(S) and we can take Q = S). Then β(Z) ≤ Z(Q), and since [T:C_{T} (g)] ≥ 4 for each involution g ∈ S∖T (rk([g, T/Z]) = 2 by Lemma D.5(b)), β(Z) ≤ V. Hence T ≤ C_{S}(β(Z)) = Q. The image in S/T ≅ D_{8} of any abelian subgroup of S/[G, V] ≅ C_{4} ×_{C_{2}} Q_{16} is cyclic, so the image of β(T) is cyclic, hence has order at most 2 since β(Z) ≤ T. Thus |T ∩β(T)| = 2^{4}, Fr(T ∩β(T)) ≤ Fr(T)∩Fr(β(T)) = 1, so T ∩ β(T) = V and hence β(V) = V. Then β(T) = T since T = C_{Q}(V), which is impossible since β(Fr(T)) ≠ Fr(T).

Thus T ≅ C̸_{4} × C_{2}^{3}. So we can now assume, for i, j ∈ I, that

T_{ij} < R_{ij} ⟹ S/T_{ij} ≅ D_{8}, R_{i}/T_{ij} ≅ R_{j}/T_{ij} ≅ C_{2}^{2}, and T_{ij} ≅ C_{2}^{4}
and Γ_{ij} ≅ A_{6}, A_{7}, or GL_{3}(2).

(3.20)

Case 3a: Assume there is a unique subgroup T ⊴ S such that S/T ≅ D_{8} and T = T_{ij} for some (possibly more than one) pair of indices i, j ∈ I. We just showed that T ≅ C_{2}^{4}. Set T_{0} = T ∩ R_{I}. Then R_{I} ≥ Fr(S) since |S/R_{i}| = 2 for each i ∈ I, so T_{0} ≥ T ∩ Fr(S) ≥ [T, S], and [T, S] ≠ 1 since T is centric in S by assumption.

Now, Aut_{F}(S) normalizes T by its uniqueness, and it normalizes T_{0} since it permutes the R_{i}. By Lemma 1.15 and (3.18), and since T_{0} ⋬ F (F is reduced), there is k ∈ I such that Γ_{k} does not normalize T_{0}. Then Γ_{k} and R_{k} have the following properties:

(a) \underline{Γ_{k} does not normalize T ∩ R_{k}:} Since Γ_{k} does not normalize the subgroup T_{0} = ⋂_{ℓ∈I}(T ∩ R_{kℓ}), there is ℓ such that Γ_{k} does not normalize T ∩ R_{kℓ}. Either R_{kℓ} > T_{kℓ} = T, in which case Γ_{k} does not normalize T = T ∩ R_{k}, or R_{kℓ} = T_{kℓ} is normalized by Γ_{k} in which case T ∩ R_{k} is not.

(b) \underline{R_{ik} = T_{ik} and R_{jk} = T_{jk}:} If R_{ik} > T_{ik}, then T_{ik} = T = T ∩ R_{k} is normalized by Γ_{k}, contradicting (a). By a similar argument, R_{jk} = T_{jk}.

(c) \underline{Γ_{k} does normalize R_{ijk}:} Since Γ_{k} normalizes T_{ik} = R_{ik} and T_{jk} = R_{jk} by (b), it also normalizes R_{ijk} = R_{ik} ∩ R_{jk}.

(d) \underline{Γ_{ij} normalizes T ∩ R_{k}:} By (b), Γ_{i} normalizes T_{ij} ∩ T_{ik} = T ∩ R_{ik} = T ∩ R_{k} and Γ_{j} normalizes T_{ij} ∩ T_{jk} = T ∩ R_{jk} = T ∩ R_{k}.

(e) \underline{R_{ijk} is nonabelian, and |R_{ijk}/(T ∩ R_{k})| = 2:} Since T ≱ Fr(S) while R_{ijk} ≥ Fr(S), R_{ijk} > T ∩ R_{k}, and so |R_{ijk}/(T ∩ R_{k})| = |R_{ij}/T | = 2. Since Γ_{ij} ≤ Aut(T) normalizes T ∩ R_{k} by (d), and since O_{2}(Γ_{ij}) = 1 by (3.20), Γ_{ij} acts faithfully on T ∩R_{k} by Lemma A.9. Hence for x ∈ R_{ijk}∖(T ∩R_{k}), [x, T ∩R_{k}] ≠ 1, and so R_{ijk} is nonabelian. Since Γ_{k} ≅ Σ_{3} acts on R_{ijk} without normalizing T ∩ R_{k} by (a) and (c), where

R_{ijk} is nonabelian by (e), it must permute T ∩ R_{k} in an orbit of three different elementary abelian subgroups of index 2 in R_{ijk}. But this is impossible: R_{ijk} would be a semidirect product of T ∩ R_{k} ≅ C_{2}^{m} (m = 3, 4) with C_{2}, |[R_{ijk}, R_{ijk}]| = 2 by Lemma A.4(a), and so R_{ijk} ≅ D_{8} × C_{2}^{m−2} contains only two elementary abelian subgroups of index 2. Case 3b: Thus there are (at least) two distinct subgroups V, W < S such that V = T_{ij} < R_{ij} and W = T_{kℓ} < R_{kℓ} for some i ≠ j, k ≠ ℓ in I. By (3.20), S/V ≅ S/W ≅ D_{8} and V ≅ W ≅ C_{2}^{4}. Set X = V ∩ W.

If [V:X] = 2, then V W/V ⊴ S/V ≅ D_{8} implies that V W = R_{ij}, and similarly, V W = R_{kℓ}. Since there are only three subgroups of index 2 in S containing V W, {R_{i}, R_{j}} ∩ {R_{k}, R_{ℓ}} ≠ ∅, and we can assume R_{i} = R_{k}. Then R_{i}/V ≅ R_{i}/W ≅ C_{2}^{2} by (3.20), so R_{i}/X ≅ C_{2}^{3}. Hence R_{i} > V W > V > X ≥ Fr(R_{i}), where [X:Fr(R_{i})] ≤ 2, and these cannot all be normalized by Γ_{i} (Lemma A.9). Hence Γ_{k} ≠ Γ_{i}, so Out_{F}(R_{i}) ≅ (C_{3} × C_{3}) ⋊ C_{2} and rk(R_{i}/Fr(R_{i})) = 4. Also, V/Fr(R_{i}) and W/Fr(R_{i}) have rank 2 and are normalized by Γ_{i}/Inn(R_{i}) and Γ_{k}/Inn(R_{i}), respectively, where C_{R_{i}/Fr(R_{i})}(O^{2}(Γ_{i})) and C_{R_{i}/Fr(R_{i})}(O^{2}(Γ_{k})) have rank 2 by definition of Ê_{F}. This implies that V/Fr(R_{i}) and W/Fr(R_{i}) are equal or complementary, and since [V:V ∩ W] = 2, they must be equal. This contradicts our assumption that V ≠ W.

Thus [V:X] ≥ 4. Since V W/V ≤ S/V ≅ D_{8}, we have V W/V ≅ V/X ≅ W/X ≅ C_{2}^{2} and so [S:V W] = 2. Since R_{i} and R_{j} are the unique subgroups of S such that R_{i}/V ≅ C_{2}^{2} ≅ R_{j}/V, V W = R_{i} or R_{j}. Thus there is an automorphism of V W of order 3 which normalizes V. Also, [V, W] = X, since otherwise r(V W) ≥ 5. So V W ≅ UT_{3}(4) by Lemma C.7(b), and hence S ∈ U. □

CHAPTER 4 Fusion systems over 2-groups of type G_{2}(q)

Throughout this chapter, we will be working with 2-groups S ∈ G, using the following notation for elements and subgroups of S.

Notation 4.1. For some n ≥ 2 and some λ = −1 or (if n ≥ 3) 2^{n−1} − 1,

S = S_{n,λ} = ⟨a, b, r, t⟩, where A =^{def} ⟨a, b⟩ ≅ C_{2^{n}} × C_{2^{n}}, ⟨r, t⟩ ≅ C_{2}^{2}, rar^{−1} = a^{λ},

rbr^{−1} = b^{λ}, tat^{−1} = b, tbt^{−1} = a. Set

Δ_{1} = ⟨ab^{−1}, a^{2^{n−1}}t⟩ ≅ Q_{2^{n+1}} U_{1} = ⟨(ab^{−1})^{2^{n−2}}, a^{2^{n−1}}t⟩ ≅ Q_{8} P_{1} = U_{1}Δ_{2}
Δ_{2} = ⟨ab^{−λ}, a^{2^{n−1}}rt⟩ ≅ Q_{2^{n+1}} U_{2} = ⟨(ab)^{2^{n−2}}, a^{2^{n−1}}rt⟩ ≅ Q_{8} P_{2} = U_{2}Δ_{1}
𝒫_{i} = {^{g}P_{i} | g ∈ S} (i = 1, 2)  Q = Δ_{1}Δ_{2}  A_{+} = A⟨r⟩ .

Define τ_{1} ∈ Aut(P_{1}), τ_{2} ∈ Aut(P_{2}), and σ ∈ Aut(A_{+}), each of order 3, by setting τ_{i}|_{C_{S}(U_{i})} = Id, σ(r) = r, and letting τ_{i} act on U_{i} and σ on A as follows:

\begin{matrix}τ_{1}|_{U_{1}} : & (ab^{−1})^{2^{n−2}} & ↦ & b^{2^{n−1}}t & ↦ & (ab)^{2^{n−2}}t & ↦ & (ab^{−1})^{2^{n−2}} \\ τ_{2}|_{U_{2}} : & (ab)^{2^{n−2}} & ↦ & b^{2^{n−1}}rt & ↦ & (ab^{−1})^{2^{n−2}}rt & ↦ & (ab)^{2^{n−2}} \\ σ|_{A}: & a & ↦ & b & ↦ & a^{−1}b^{−1} & ↦ & a \end{matrix}

Thus when n = 2, U_{i} = Δ_{i} and P_{1} = P_{2} = Q. Note that [Δ_{1}, Δ_{2}] = 1 if λ = −1, and [Δ_{1}, Δ_{2}] = ⟨(ab)^{2^{n−1}}⟩ = Z(S) if λ = −1 + 2^{n−1}. So in either case, for i = 1, 2, Aut_{Δ_{3−i}}(P_{i}) ≤ Inn(P_{i}) and hence P_{i} = U_{i}C_{S}(U_{i}).

Proposition 4.2. Assume S = S_{n,λ} is as in Notation 4.1. Then Aut(S) is a 2-group. If F is a reduced fusion system over S, then

(a) λ = −1;

(b) E_{F} = {A_{+}} ∪ 𝒫_{1} ∪ 𝒫_{2} (so E_{F} = {Q, A_{+}} if n = 2); and

(c) either n = 2, Out_{F}(Q) = ⟨[τ_{1}τ_{2}^{−1}], [c_{a}]⟩ ≅ Σ_{3}, and F is isomorphic to the fusion system of M_{12}; or

n = 2, Out_{F}(Q) = ⟨[τ_{1}], [τ_{2}], [c_{a}]⟩ ≅ (C_{3} × C_{3}) ⋊ C_{2}, and F is isomorphic to the fusion system of G_{2}(q) for each q ≡ ±3 (mod 8); or

n ≥ 3, Aut_{F}(P_{i}) = ⟨τ_{i}, Aut_{S}(P_{i})⟩ for i = 1, 2, and F is isomorphic to the fusion system of G_{2}(q) for each odd prime power q such that v_{2}(q^{2} −1) = n+1.

Proof. By Lemma A.4(b), and since |S/[S, S]| = 8, A is the unique abelian subgroup of index 4 in S. Hence Aut(S) is a 2-group by Lemma A.9, applied to the chain Fr(S) < A < A_{+} < S of characteristic subgroups. So Out_{F}(S) = 1.

We claim that

𝒳 (S) = {Δ_{1}, Δ_{2}, ⟨ab^{−1}, t⟩, ⟨ab^{−λ}, rt⟩} .  (4.1)

4. FUSION SYSTEMS OVER 2-GROUPS OF TYPE G_{2}(q) 53

Fix some Δ ∈ 𝒳 (S). By Definition 2.1, Δ = ⟨C⟩ = B ∪ {C}, where B is cyclic and C is an S-conjugacy class of elements of order 2 (if Δ ∈ D) or 4 (if Δ ∈ Q). Also, C ⊈ A since Δ is nonabelian. If C ⊆ Ar, then a^{2^{n−1}}, b^{2^{n−1}} ∈ B = A ∩ Δ, which is impossible since B is cyclic. If C ⊆ At, then since each S-conjugacy class in At has the form {(ab^{−1})^{i}gt | i ∈ ℤ} for some g ∈ A, and since (a^{i}t)^{2} = (ab)^{i}, C is the conjugacy class of t or of a^{2^{n−1}}t, and Δ = ⟨ab^{−1}, t⟩ or ⟨ab^{−1}, a^{2^{n−1}}t⟩ = Δ_{1}. By a similar argument, if C ⊆ Art, then Δ = ⟨ab^{−λ}, rt⟩ or ⟨ab^{−λ}, a^{2^{n−1}}rt⟩ = Δ_{2}.

(b) When n = 2 (hence λ = −1), E_{F} = {Q, A_{+}} by [AOV2, Proposition 3.2]. (This also follows upon making minor changes to the argument below.)

Assume n ≥ 3. If V < S is elementary abelian of rank 4, then V ∩A = Ω_{1}(A) = ⟨a^{2^{n−1}}, b^{2^{n−1}}⟩ and V A = S, so V ∩ At ≠ ∅, which is impossible since V is abelian. Hence S has rank 3, and in particular, S ∉ UV.

Thus by Theorem 3.1(a,b), E_{F}^{(I)} = ∅; and R ∈ E_{F}^{(II)} implies that 𝔣𝔬𝔠(F, R) ∈ 𝒳 (S) and R is as in Lemma 3.8(a,c,d,e). By that lemma, if R ∈ E_{F}^{(II)}, then there are V < Δ ∈ 𝒳 (S) such that V ≅ C_{2}^{2} or Q_{8}, R = V C_{S}(V), and Δ = 𝔣𝔬𝔠(F, R). Also, if V ≅ C_{2}^{2}, then V is a direct factor in R. In this last case, V is S-conjugate to ⟨(ab)^{2^{n−1}}, t⟩ or ⟨(ab)^{2^{n−1}}, rt⟩, neither of which is a direct factor in its centralizer. Thus V ≅ Q_{8} and is S-conjugate to U_{1} or U_{2}, and R is S-conjugate to P_{1} or P_{2}. Also, by definition of U_{i} in the statement of Lemma 3.8,

for i = 1, 2, P_{i} ∈ E_{F} ⟹ [Aut_{F}^{∗}(P_{i}), P_{i}] = U_{i} and 𝔣𝔬𝔠(F, P_{i}) = Δ_{i}.  (4.2)

Now assume R ∈ E_{F}^{(III)}. Thus [S:R] = 2, so ⟨ab, a^{2}⟩ = Fr(S) ≤ R, and Aut(R) is not a 2-group. Set A_{0} = Fr(S) = ⟨ab, a^{2}⟩. If R ≱ A, then R = A_{0}⟨g, h⟩ for some g ∈ Ar and h ∈ At, |[g, A_{0}]| = \frac{1}{4}|A_{0}| ≥ 8, |[h, A_{0}]| ≥ 2^{n−1} ≥ 4, and |[gh, A_{0}]| ≥ 4. So by Lemma A.4(b), A_{0} is the unique abelian subgroup of index 4 in R. Also, Aut(A_{0}) is a 2-group (Corollary A.10(a)), A_{0}⟨g⟩ is characteristic in R, so Aut(R) is a 2-group by Lemma A.9. Thus R ≥ A. If R = A⟨t⟩ or R = A⟨rt⟩, then R/[R, R] ≅ C_{2^{n}} × C_{2}, and Aut(R) is a 2-group by Corollary A.10(a) again. Thus R = A_{+} = A⟨r⟩.

This proves that E_{F} ⊆ {A_{+}}∪𝒫_{1}∪𝒫_{2}. By (4.2), if P_{i} ∈ E_{F}, then 𝔣𝔬𝔠(F, P_{i}) = Δ_{i}. If A_{+} ∈ E_{F}, then [Aut_{F}^{∗}(A_{+}), A_{+}] = 𝔣𝔬𝔠(F, A_{+}) ≤ A since A < A_{+} is characteristic of index 2. Since no two of the subgroups A, Δ_{1}, Δ_{2} generate S, E_{F} = {A_{+}} ∪ 𝒫_{1} ∪ 𝒫_{2}.

(a) Assume λ = −1 + 2^{n−1} (and hence n ≥ 3). Set U_{2}^{∗} = ^{a}U_{2}, P_{2}^{∗} = U_{2}^{∗}C_{S}(U_{2}^{∗}) = aP_{2}, and τ_{2}^{∗} = c_{a}τc_{a}^{−1} ∈ Aut_{F}(P_{2}^{∗}). Then

[U_{1}, U_{2}^{∗}] = [⟨(ab^{−1})^{2^{n−2}}, a^{2^{n−1}}t⟩, ⟨(ab)^{2^{n−2}}, ab^{−λ}a^{2^{n−1}}rt⟩] = ⟨(ab)^{2^{n−1}}⟩ = Z(S) ,

so Aut_{U_{2}^{∗}}(U_{1}) ≤ Inn(U_{1}) and Aut_{U_{1}}(U_{2}^{∗}) ≤ Inn(U_{2}^{∗}). It follows that U_{1} ≤ P_{2}^{∗} and U_{2}^{∗} ≤ P_{1}.

Set Q_{0} = U_{1}U_{2}^{∗} ≅ 2_{−}^{1+4} (Lemma C.2(a)). Since Δ_{1}Δ_{2}/Z(S) ≅ D_{2^{n}} × D_{2^{n}}, and since each g ∈ S∖Δ_{1}Δ_{2} acts on each Δ_{i}/Z(S) ≅ D_{2^{n}} by exchanging the two noncyclic subgroups of index 2, N_{S}(Q_{0}) = N_{Δ_{1}Δ_{2}}(Q_{0}) = N_{Δ_{1}}(U_{1})N_{Δ_{2}}(U_{2}) and hence |N_{S}(Q_{0})/Q_{0}| = 4. By Lemma 1.16(a), and since no essential subgroup contains N_{S}(Q_{0}) by (b), Q_{0} is fully normalized in F. Also, τ_{1}|_{Q_{0}}, τ_{2}^{∗}|_{Q_{0}} ∈ Aut_{F}(Q_{0}). For g ∈ N_{Δ_{1}}(U_{1})∖U_{1}, c_{g}|_{U_{1}} ∉ Inn(U_{1}) while c_{g}|_{U_{2}^{∗}} ∈ Inn(U_{2}^{∗}). Hence Out_{F}(Q_{0}) = Out(Q_{0}) ≅ Σ_{5} by Lemma C.2(b). Since this contradicts the Sylow axiom, we conclude that λ ≠ −1 + 2^{n−1}.

(c) Since A is the unique abelian subgroup of index 2 in A_{+}, the quotient group Out(A_{+})/O_{2}(Out(A_{+})) injects into Aut(A/Fr(A)) ≅ Σ_{3} by Lemma A.9. Thus Out_{F}(A_{+}) ≅ Σ_{3} since A_{+} is essential, and Out_{F}(A_{+}) = ⟨[γ], [c_{t}]⟩ for some γ ∈ Aut_{F}(A_{+}) of order 3. Set γ′ = [γ, c_{t}] ∈ Aut_{F}(A_{+}); then [γ′] = [γ]^{−1} in Out_{F}(A_{+}) and ⟨γ′, c_{t}⟩ ≤ Aut_{F}(A_{+}) is dihedral. So upon replacing γ by some appropriate power of γ′, we can assume that ⟨γ, c_{t}⟩ ≅ Σ_{3} as a subgroup of Aut_{F}(A_{+}). Also, C_{A}(γ) = 1 since C_{Ω_{1}(A)}(γ) = 1.

Set g = ab. Then g·γ(g)·γ^{2}(g) ∈ C_{A}(γ) = 1, [t, g] = 1, and tγ(g)t^{−1} = γ^{2}(g). Since C_{A}(γ) = 1, γ acts on the coset Ar fixing exactly one element h, and c_{t}(h) = h since c_{t} normalizes ⟨γ⟩ in Aut_{F}(A_{+}). Define φ ∈ Aut(S) by setting φ(g) = a^{−1}b^{−1}, φ(γ(g)) = a, φ(γ^{2}(g)) = b, φ(h) = r, and φ(t) = t. Upon replacing F by ^{φ}F (and γ by ^{φ}γ), we can assume that γ = σ, and hence that Out_{F}(A_{+}) = ⟨[σ], [c_{t}]⟩.

By Lemma 3.8, for i = 1, 2, Aut_{F}(P_{i}) ≅ Σ_{3} or Σ_{3} × C_{3}. In the latter case, by the extension lemma, there is an element of order 3 which extends to an element in Aut_{F}(N_{S}(P_{i})), which is impossible since Out_{F}(S) = 1 and no essential subgroup contains N_{S}(P_{i}). Thus Aut_{F}(P_{i}) = ⟨Aut_{S}(P_{i}), δ⟩ for some δ ∈ Aut_{F}^{∗}(P_{i}) of order 3. By (4.2), [δ, P_{i}] = U_{i}, so δ(U_{i}) = U_{i}, and δ|_{C_{S}(U_{i})} = Id (and Lemma A.9). So δ ∈ τ_{i}^{±1}Aut_{U_{i}}(P_{i}), and hence

i = 1, 2, n ≥ 3 ⟹ Aut_{F}(P_{i}) = ⟨Aut_{S}(P_{i}), τ_{i}⟩.  (4.3)

Case 1: Assume τ_{i} ∈ Aut_{F}(P_{i}) for i = 1, 2. Thus F is the fusion system over S generated by Inn(S), σ ∈ Aut(A_{+}), and the τ_{i} ∈ Aut(U_{i}C_{S}(U_{i})).

Let q be a prime power such that v_{2}(q^{2} − 1) = n + 1, and set G = G_{2}(q) and G = G_{2}(𝔽_{q}) > G. For any x ∈ I(G), C_{G}(x) ≅ SL_{2}(𝔽_{q}) ×_{C_{2}} SL_{2}(𝔽_{q}): this follows, for example, from the description of centralizers in [Ca2, Theorem 3.5.3]. Hence C_{G}(x) contains a subgroup SL_{2}(q)×_{C_{2}} SL_{2}(q) with index 2 (see, e.g., [Kl, Theorem A]), so the Sylow 2-subgroups of G contain Q_{2^{n+1}} ×_{C_{2}} Q_{2^{n+1}} ≅ Q with index 2, and are contained in Ŝ = Q_{2^{n+2}} ×_{C_{2}} Q_{2^{n+2}}. Fix generators c_{1}, d_{1}, c_{2}, d_{2} ∈ Ŝ, where ⟨c_{i}, d_{i}⟩ ≅ Q_{2^{n+2}}, |c_{i}| = 2^{n+1}, and z = c_{1}^{2^{n}} = c_{2}^{2^{n}} = d_{1}^{2} = d_{2}^{2} ∈ Z(Ŝ), and define χ: S ⟶ Ŝ by setting

χ(a) = c_{1}c_{2},  χ(b) = c_{1}^{−1}c_{2},  χ(r) = d_{1}d_{2},  χ(t) = c_{2}^{2^{n−1}}d_{1} .

This defines an isomorphism from S onto some T ∈ Syl_{2}(G), and χ preserves fusion in P_{1} and P_{2} since G > SL_{2}(q) ×_{C_{2}} SL_{2}(q). Since F_{T} (G) is reduced by Proposition 1.12, F ≅ F_{T} (G). Case 2: Now assume τ_{i} ∉ Aut_{F}(P_{i}) for i = 1 or 2. Thus n = 2 by (4.3), and P_{1} = P_{2} = Q ≅ 2_{+}^{1+4}. Each automorphism of Q either normalizes the Δ_{i} or exchanges them, and an automorphism of odd order must normalize them. Hence Aut_{F}^{∗}(Q) ≤ ⟨τ_{1}, τ_{2}, Inn(Q)⟩. Since S = 𝔣𝔬𝔠(F) = ⟨𝔣𝔬𝔠(F, A_{+}), 𝔣𝔬𝔠(F, Q)⟩ and 𝔣𝔬𝔠(F, A_{+}) = A, we must have 𝔣𝔬𝔠(F, Q) = [Aut_{F}^{∗}(Q), Q] = Q. So Out_{F}(Q) must be one of the groups ⟨[τ_{1}], [τ_{2}], [c_{t}]⟩, ⟨[τ_{1}τ_{2}], [c_{t}]⟩, or ⟨[τ_{1}τ_{2}^{−1}], [c_{t}]⟩, and we are assuming τ_{i} ∉ Aut_{F}(Q). If Out_{F}(Q) = ⟨[τ_{1}τ_{2}], [c_{t}]⟩, then the subgroup ⟨a^{2}, b^{2}, r⟩ is normalized by Aut_{F}(Q) and by Aut_{F}(A_{+}), and hence by Lemma 1.15 is normal in F. This is impossible since F is reduced, so Out_{F}(Q) = ⟨[τ_{1}τ_{2}^{−1}], [c_{a}]⟩.

By [A2, Lemma 5.3(2)], M_{12} contains as involution centralizer a split extension of 2_{+}^{1+4} by Σ_{3} where each of the Q_{8} factors is normal. (Note that S = Q ⋊ ⟨ar⟩.) The fusion system of M_{12} is reduced by Proposition 1.12. Hence F is the fusion system of M_{12} in this case. □

It remains to look at fusion systems over a Sylow subgroup of Aut(M_{12}).

Proposition 4.3. Set S = S_{2,−1} with the presentation in Notation 4.1. Let F be the fusion system over S generated by Inn(S), σ ∈ Aut(A_{+}), and τ_{1}τ_{2}^{−1} ∈ Aut(Q). Then F is isomorphic to the fusion system of M_{12}. Define β ∈ Aut(S) by setting β(a) = ab^{2}, β(b) = a^{2}b^{−1}, β(r) = r, and β(t) = rt.

(a) The class of β generates Out(S, F) ≅ C_{2}.

(b) Set Ŝ = S⟨u⟩, where ugu^{−1} = β(g) for g ∈ S, and u^{2} = 1. Then Ŝ is of type Aut(M_{12}).

(c) There are no reduced fusion systems over Ŝ.

Proof. Set G = M_{12}. By the proof of Proposition 4.2(c), we can identify S as a Sylow 2-subgroup of G with F = F_{S}(G).

(a) By direct computation, ^{β}σ = σ, and ^{β}τ_{i} ≡ τ_{3−i} (mod Inn(Q)) for i = 1, 2. Hence β ∈ Aut(S, F). Also, β(Δ_{1}) = Δ_{2} and β(Δ_{2}) = Δ_{1}.

Assume φ ∈ Aut(S) is fusion preserving. Then φ(A) = A since A < S is the unique abelian subgroup of index 4 (Lemma A.4(b)). By (4.1) in the proof of Proposition 4.2, either φ normalizes the subgroups Δ_{i} ≅ Q_{8}, or it exchanges them.

Assume φ(Δ_{i}) = Δ_{i} for i = 1, 2. After composing with inner automorphisms, we can assume that φ(ab) = ab and φ(ab^{−1}) = ab^{−1}. Also, φ(r) = r (the unique involution in A_{+} = A⟨r⟩ fixed by σ), and φ sends the σ-orbit of ab to itself. Since φ|_{⟨a^{2},b^{2}⟩} = Id, this proves that φ|_{A_{+}} = Id. Finally, since φ(Δ_{1}) = Δ_{1}, φ(t) = a^{i}b^{j}t for some i, j, 4|(i + j) since φ(a^{2}t) = a^{i+2}b^{j}t ∈ Δ_{1}, and 2|i since [r, φ(t)] = 1. Upon replacing φ by c_{b^{i}} ∘ φ, we can arrange that φ = Id.

Thus φ ∈ Aut(S, F) and φ(Δ_{i}) = Δ_{i} (i = 1, 2) imply that φ ∈ Inn(S). So Out(S, F) has order 2 and is generated by the class of β.

(b) Set H = N_{G}(A). Then H = AK, where K = ⟨r, t, s⟩ ≅ D_{12}, |s| = 3, [s, r] = 1, ts = s^{−1}, and ^{s}g = σ(g) for g ∈ A (see, e.g., [Gd2, (3.8)]).

By [A2, Lemma 5.9(1,3)], |Out(G)| = 2, and no α ∈ Aut(G)∖Inn(G) centralizes S. Hence by (a), Aut(G) ≅ G⟨v⟩ for some v such that c_{v}|_{S} = β and (since β^{2} = Id_{S}) v^{2} ∈ Z(S). Then ^{v}A = A, so ^{v}H = H. By Lemma A.7, applied with H, A, K, and ⟨r, t⟩ in the role of G, Q, H, and H_{0}, ^{v}⟨s⟩ = ^{g}⟨s⟩ for some g ∈ C_{A}(r, t) = ⟨a^{2}b^{2}⟩. So upon replacing v by a^{2}b^{2}v if necessary, we can assume that ^{v}⟨s⟩ = ⟨s⟩, hence that [v^{2}, H] = 1, and so v^{2} = 1. Thus S⟨v⟩ ∈ Syl_{2}(Aut(G)) is isomorphic to Ŝ.

(c) By Lemma A.4(b) and the commutator relations in Ŝ (and since |Ŝ/[Ŝ, Ŝ]| = 8), Q/Z(Ŝ) is the only abelian subgroup of rank 4 in Ŝ/Z(Ŝ), and so Q is the only extraspecial subgroup of order 2^{5} in Ŝ. Also, Z(Ŝ) = ⟨a^{2}b^{2}⟩, Z_{2}(Ŝ) = ⟨a^{2}, b^{2}⟩, and Z_{3}(Ŝ) = ⟨a^{2}, b^{2}, ab, r⟩. By Lemma A.2(b), each normal subgroup of order 8 in Ŝ contains Z_{2}(Ŝ) and is contained in Z_{3}(Ŝ), hence is abelian, and thus 𝒳 (Ŝ) = ∅. Also, 𝒴 (Ŝ) = ∅ since |Ŝ| = 2^{7} and Ŝ ≅ D̸_{8} ≀ C_{2}.

Assume there is T < Ŝ such that T ≅ UT_{3}(4). Then Z(T) ⊴ Ŝ implies Z(T) = Z_{2}(S) = ⟨a^{2}, b^{2}⟩ (Lemma A.2(b) again), so T = CŜ(⟨a^{2}, b^{2}⟩) = ⟨a, b, r, u⟩. Since I(UT_{3}(4)) = A_{1}^{#} ∪ A_{2}^{#} where A_{i} ≅ C_{2}^{4} and A_{1} ∩ A_{2} = Z(T) (Lemma C.6(a)), a subgroup of index 2 in UT_{3}(4) has at most 19 involutions, and has exactly 19 only if it contains A_{1} or A_{2}. Since |I(⟨a, b, r⟩)| = |Z(T)^{#}| + |rA| = 19, and since r(⟨a, b, r⟩) = 3, this proves that T ≅ U̸T_{3}(4), and hence that Ŝ ∉ U.

Let F be a reduced fusion system over Ŝ. Since 𝒳 (Ŝ) = ∅ = 𝒴 (Ŝ) and Ŝ ∉ U, Theorem 3.1 implies that each F-essential subgroup has index 2 in Ŝ.

If γ ∈ Aut_{F}(Ŝ) has odd order, then γ(Q) = Q, and [γ|_{Q}] ∈ Out(Q) normalizes OutŜ(Q) ≅ C_{2}^{2}. Since Out(Q) ≅ Σ_{3} ≀C_{2}, this implies that [γ|_{Q}] = 1, and hence that γ|_{Q} = Id. Since CŜ(Q) ≤ Q, γ induces the identity on Ŝ/Q, and hence γ = IdŜ by Lemma A.9. Thus Out_{F}(Ŝ) = 1.

Assume P < Ŝ is F-essential of index 2; thus P ≥ Fr(Ŝ) = ⟨ab, a^{2}, r⟩. If P > Q, then 𝔣𝔬𝔠(F, P) ≤ Q. If P > A and P ≱ Q, then either P = A⟨r, u⟩ ≅ A ⋊ C_{2}^{2} or P = A⟨tu⟩ ≅ A ⋊ C_{4}, and in either case, 𝔣𝔬𝔠(F, P) ≤ A. This leaves the two subgroups

Q_{1} = ⟨at, ab, a^{2}, r, u⟩ [Q_{1}, Q_{1}] = ⟨a^{2}, b^{2}, r⟩  Q_{1}/[Q_{1}, Q_{1}] ≅ C_{4} × C_{2}
Q_{2} = ⟨at, ab, a^{2}, r, tu⟩ [Q_{1}, Q_{1}] = ⟨a^{2}, b^{2}, abr⟩ Q_{2}/[Q_{2}, Q_{2}] ≅ C_{4} × C_{2}.

By Corollary A.10(a), Aut(Q_{i}) is a 2-group for i = 1, 2, so neither can be Fessential.

Thus 𝔣𝔬𝔠(F) ≤ AQ = S < Ŝ, so by Proposition 1.14(b), F is not reduced. □

CHAPTER 5 Dihedral and semidihedral wreath products

We next study reduced fusion systems over 2-groups in V. These are the groups where the set of subgroups 𝒴 (S) plays a central role. The one exception to this is the case S = UT_{4}(2) (where 𝒴 (S) = ∅) which we handle first.

Note, in the statement of the following proposition, that PSU_{4}(2) ≅ PSp_{4}(3) (cf. [Ta, Corollary 10.19]).

Proposition 5.1. Assume S = UT_{4}(2). Let F be a fusion system over S such that O_{2}(F) = 1. Then F is isomorphic either to the fusion system of GL_{4}(2), or to that of PSp_{4}(q) for each q ≡ ±3 (mod 8).

Proof. By Lemma C.4(a,b), there are unique subgroups Q, A < S such that Q ≅ 2_{+}^{1+4} and A ≅ C_{2}^{4}. Set Z = ⟨z⟩ = Z(S). Let Δ_{1}, Δ_{2} ⊴ Q be the unique subgroups isomorphic to Q_{8}; thus [Δ_{1}, Δ_{2}] = 1 and Δ_{1} ∩ Δ_{2} = Z.

Define automorphisms

γ ∈ Aut(S),  φ ∈ Z(Aut(S)),  and  τ_{1}, τ_{2} ∈ Aut(Q)

as follows. Fix some t ∈ A∖Q; then ^{t}Δ_{1} = Δ_{2} by Lemma C.4(b). Choose any τ_{1} ∈ Aut(Q) which acts with order 3 on Δ_{1} and via the identity on Δ_{2}, and set τ_{2} = c_{t}τ_{1}c_{t}^{−1} ∈ Aut(Q). Then c_{t} commutes with τ_{1}τ_{2} = τ_{2}τ_{1}, so there is γ ∈ Aut(S) of order 3 such that γ|_{Q} = τ_{1}τ_{2} and γ(t) = t. Finally, set φ(g) = g if g ∈ Q and φ(g) = zg if g ∈ S∖Q. Then φ ∈ Z(Aut(S)) since α(Q) = Q for each α ∈ Aut(S).

Let B_{1}, B_{2}, B_{3} < S be the three subgroups of index 2 which contain A. Since S ≅ C_{2} ≀ C_{2}^{2} by Lemma C.4(a), B_{i} ≅ C_{2}^{2} ≀ C_{2} for each i. Choose v_{i} ∈ I(B_{i}∖A), and set V_{i} = C_{B_{i}}(v_{i}) = ⟨v_{i}⟩ × C_{A}(v_{i}) ≅ C_{2}^{3}. Note that V_{i} = ⟨I(B_{i}∖A)⟩, so V_{i} is independent of the choice of v_{i}, and is characteristic in B_{i} since A is. By Lemma C.4(d), γ permutes the B_{i} transitively, and hence also permutes the V_{i} transitively.

We claim that

τ_{1}τ_{2}^{−1}(V_{i}) = V_{i} for i = 1, 2, 3. (5.1)

To see this, let 𝒱 be the set of all V < Q such that V ≅ C_{2}^{3}. Each involution in Q has the form g_{1}g_{2} where g_{i} ∈ Δ_{i}∖Z. Hence for each V ∈ 𝒱 , there is χ ∈ Iso(Δ_{1}, Δ_{2}) such that V = V_{χ} =^{def} ⟨z, gχ(g) | g ∈ Δ_{1}⟩. Also, V_{χ} = V_{χ′} if χ′ ∈ χ·Inn(Δ_{1}). Hence |𝒱 | = |Out(Q_{8})| = 6, each of the automorphisms τ_{1} and τ_{2} permutes 𝒱 freely, and ⟨τ_{1}, τ_{2}⟩ ≅ C_{3} × C_{3} permutes it in two orbits of length 3. Since γ permutes the set {V_{1}, V_{2}, V_{3}} ⊆ 𝒱 freely, it must be one of the orbits, is permuted freely by τ_{1}, τ_{2}, and τ_{1}τ_{2} = γ|_{Q}, and hence is pointwise fixed by τ_{1}τ_{2}^{−1}. This proves (5.1).

We next claim that there are exactly two bases B_{1}, B_{2} ⊆ A which are permuted freely by Aut_{S}(A) ≅ C_{2}^{2}, and φ(B_{1}) = B_{2}.

(5.2)

By Lemma C.4(a), there is at least one such basis B_{1}, and B_{1} ⊆ A∖Q since it is normalized by S and generates A. Thus φ(b) = bz for each b ∈ B_{1}. If φ(B_{1}) = B_{1}, then B_{1} is the union of two cosets of Z, which is impossible since ⟨B_{1}⟩ = A. Thus B_{1} and φ(B_{1}) are distinct orbits of Aut_{S}(A), so B_{1} ∪ φ(B_{1}) = A∖Q, and these are the only such bases.

Let F be a saturated fusion system over S such that O_{2}(F) = 1. By points (b.1)–(b.3) in Theorem 3.1, if R ∈ E_{F}^{(II)}, then either 𝔣𝔬𝔠(F, R) ∈ 𝒳 (S) ∪ 𝒴 (S) or S ∈ U. By Lemma C.4(e), 𝒳 (S) = ∅. Also, 𝒴 (S) = ∅ since |S| < |D_{8} ≀ C_{2}| = 2^{7}, and S ∉ U (S ≅ U̸T_{3}(4)) since A < S is the unique subgroup isomorphic to C_{2}^{4}. Thus E_{F}^{(II)} = ∅.

By Theorem 3.1(a), if R ∈ E_{F}^{(I)}, then R ≅ C_{2}^{4} or 2_{−}^{1+4}. Since [S:R] ≥ 4, this implies E_{F}^{(I)} ⊆ {A}. By Lemma C.4(c), the only subgroups of index 2 in S whose automorphism groups are not 2-groups are Q and the B_{i}. Thus E_{F} ⊆ {A, Q, B_{1}, B_{2}, B_{3}}.

For i = 1, 2, 3, Out_{F}(B_{i}) ≅ C_{Aut_{F} (A)}(c_{v_{i}})/⟨c_{v_{i}}⟩ by Lemma 1.5(a). Also, since Out_{S}(B_{i}) ∈ Syl_{2}(Out_{F}(B_{i})) and |Out_{S}(B_{i})| = 2, Out_{F}(B_{i}) has a strongly embedded subgroup if and only if O_{2}(Out_{F}(B_{i})) = 1. Hence

for i = 1, 2, 3, B_{i} ∈ E_{F} ⟺ O_{2}(C_{Aut_{F} (A)}(c_{v_{i}})) = ⟨c_{v_{i}}⟩ .  (5.3)

If Q ∉ E_{F}, then all F-essential subgroups and S contain A as a characteristic subgroup, so A ⊴ F by Lemma 1.15, and O_{2}(F) ≠ 1. Thus Q ∈ E_{F}. The images of ⟨τ_{1}τ_{2}⟩ and ⟨τ_{1}τ_{2}^{−1}⟩ in Out(Q) ≅ Σ_{3} ≀ C_{2} are the only subgroups of order 3 normalized by Out_{S}(Q). Since ⟨[τ_{1}τ_{2}], Out_{S}(Q)⟩ ≅ C_{6} while ⟨[τ_{1}τ_{2}^{−1}], Out_{S}(Q)⟩ ≅ Σ_{3}, Out_{F}(Q) must be equal to ⟨[τ_{1}τ_{2}^{−1}], Out_{S}(Q)⟩ ≅ Σ_{3} or to ⟨[τ_{1}], [τ_{2}], Out_{S}(Q)⟩ ≅ C_{3} × Σ_{3}.

If τ_{1}τ_{2} ∈ Aut_{F}(Q), then since it normalizes Aut_{S}(Q), it extends to some γ′ ∈ Aut_{F}(S) by the extension axiom. Also, γ′ ∈ {γ, φγ}, since α ∈ Aut(S) and α|_{Q} = Id_{Q} imply α ∈ ⟨φ⟩. If γ′ = φγ, then since φ ∈ Z(Aut(S)), (γ′)^{3} = φ ∈ Aut_{F}(S), which contradicts the Sylow axiom. Thus γ′ = γ. Since Q is characteristic in S, restriction induces an isomorphism Out_{F}(S) ≅ C_{Out_{F} (Q)}(Out_{S}(Q))/Out_{S}(Q) by Lemma 1.5(a), so F has either

Type (1): Out_{F}(Q) = ⟨[τ_{1}τ_{2}^{−1}], Out_{S}(Q)⟩ ≅ Σ_{3} and Out_{F}(S) = 1; or

Type (2): Out_{F}(Q) = ⟨[τ_{1}], [τ_{2}], Out_{S}(Q)⟩ ≅ C_{3} × Σ_{3} and Out_{F}(S) = ⟨[γ]⟩ ≅ C_{3}.

If F has type (1), then N_{Aut_{F} (A)}(Aut_{S}(A))/Aut_{S}(A) ≅ Out_{F}(S) = 1 by Lemma 1.5(a). Thus Aut_{F}(A) does not contain a subgroup isomorphic to A_{5}, so A ∉ E_{F} by Lemma 3.3(b,c). If E_{F} = {Q}, then Q ⊴ F. If E_{F} = {Q, B_{i}} for some i = 1, 2, 3, then since V_{i} ⊴ S is characteristic in B_{i} (as shown above), each automorphism of B_{i} sends V_{i} to itself. Also, τ_{1}τ_{2}^{−1}(V_{i}) = V_{i} by (5.1), so V_{i} ⊴ F in this case.

Since O_{2}(F) = 1, this shows that at least two of the B_{j} must be in E_{F}. Upon replacing F by ^{γ^{i}}F for appropriate i, we can arrange that B_{1}, B_{2} ∈ E_{F}. Then O_{2}(C_{Aut_{F} (A)}(c_{v_{i}})) = ⟨c_{v_{i}}⟩ for i = 1, 2 by (5.3), so by Proposition D.1(e.1), Out_{F}(A) ≅ Σ_{3} × Σ_{3}. Then c_{v_{3}} = c_{v_{1}v_{2}} ∈ Aut(A) inverts O_{3}(Aut_{F}(A)) ≅ C_{3} × C_{3}, so C_{Aut_{F} (A)}(c_{v_{3}}) = Aut_{S}(A), and B_{3} ∉ E_{F} by (5.3) again.

Now, O_{3}(Aut_{F}(A)) ≅ C_{3} ×C_{3} is determined by a choice of two complementary subgroups W_{1}, W_{2} < A of rank 2, and since O_{3}(Aut_{F}(A)) is normalized by Aut_{S}(A), each c_{v_{i}} (i = 1, 2, 3) either normalizes the W_{i} or exchanges them. Since c_{v_{3}} inverts O_{3}(Aut_{F}(A)), there is i ∈ {1, 2} such that {W_{1}, W_{2}} is the pair {⟨B′_{i}⟩, ⟨B′′_{i}⟩}, where B′_{i}, B′′_{i} ⊆ B_{i} are the two ⟨c_{g_{3}}⟩-orbits. Thus Aut_{F}(A) is determined by B_{i}, and upon replacing F by ^{φ}F, if necessary, we can assume that i = 1.

By Lemma 1.5(b), Aut_{F}(B_{i}) is determined by Aut_{F}(A) in all cases. So F is uniquely determined by the choice of Aut_{F}(A), which we just saw is determined by the choice of basis B_{1}. If F has type (2), then by Proposition D.1(e.2) (and since Out_{S}(A) permutes a basis for A), Out_{F}(A) ≅ A_{4} or A_{5}. Then B_{i} ∉ E_{F} for i = 1, 2, 3 by (5.3), so E_{F} ⊆ {A, Q}, with equality since otherwise Q ⊴ F. Hence Aut_{F}(A) ≅ A_{5} by Lemma 3.3(b), and A is the orthogonal module since rk(C_{A}(S)) = 1 (Proposition D.1(d)).

Thus the action of Aut_{F}(A) has an orbit B^{∗} of length 5, where Aut_{S}(A) ≅ C_{2}^{2} acts on B^{∗} with one orbit of length 4 and one fixed point z. Since the action is irreducible, ⟨B^{∗}⟩ = A and ∏_{g∈B^{∗}} g ∈ C_{A}(Aut_{F}(A)) = 1. So B^{∗}∖{z} is a basis for A, and hence equal to B_{1} or B_{2} by (5.2). Upon replacing F by ^{φ}F if necessary, we can assume that B^{∗} = B_{1} ∪ {z}, and hence that Aut_{F}(A) acts as the group of all even permutations of this set. Both types: We have now shown that up to isomorphism, there are at most two saturated fusion systems F_{1} and F_{2} with O_{2}(F) = 1, one of each type (1) and (2), respectively. Set G_{1} = GL_{4}(2), set G_{2} = PSp_{4}(q) for any prime power q ≡ ±3 (mod 8), and choose S_{i} ∈ Syl_{2}(G_{i}). We can take S_{1} = UT_{4}(2) ≅ S. Let Q < S_{1} be the subgroup of triangular matrices with zero in the entry (2, 3); then Q ≅ 2_{+}^{1+4} and N_{G_{2}}(Q)/Q ≅ Σ_{3}, so F_{S_{1}}(G_{1}) ≅ F_{1} has type (1).

By [CF, § 1], Sp_{4}(q) has Sylow 2-subgroups isomorphic to Q_{8} ≀ C_{2}, so S_{2} ≅

(Q_{8} ×_{C_{2}} Q_{8}) ⋊^{t} C_{2} ≅ S. By Proposition 1.12(b), O_{2}(F_{S_{2}}(G_{2})) = 1. Furthermore,

G_{2} contains (Sp_{2}(q)×_{C_{2}} Sp_{2}(q))⋊^{t}C_{2}, Sp_{2}(q) ≅ SL_{2}(q) contains a subgroup Q_{8}⋊C_{3}, and so F_{S_{2}}(G_{2}) ≅ F_{2} has type (2). □

Before continuing with the other cases, we need a lemma which helps to make more explicit how we apply the results shown in Section 3.2. Recall that for a saturated fusion system F over S and Y ⊴ S, E_{F}(Y) denotes the set of all Fessential subgroups R < S such that 𝔣𝔬𝔠(F, R) = Y.

Lemma 5.2. Let F be a saturated fusion system over a 2-group S such that r(S) ≤ 4 and 𝒴 (S) ≠ ∅. Fix Y ∈ 𝒴 (S), and assume that E_{F}(Y) ≠ ∅. Let 𝒴_{0} be the set of all Y_{0} ∈ 𝒴_{0}(S) whose normal closure is Y. Let Θ_{1}, Θ_{2} ⊴ Y and

𝒰_{F}(Y) = {U ≤ Θ_{i} | i = 1, 2, U ≅ C_{2}^{2} or Q_{8}} .

be as in Proposition 3.11(a). Then the following hold.

(a) E_{F}(Y) = E_{F}^{a}(Y) ∪ E_{F}^{c}(Y), where

• E_{F}^{a}(Y) = {Y_{0}⟨g⟩ | Y_{0} ∈ 𝒴_{0}, Aut_{F}(Y_{0}) ≅ Σ_{3} ≀ C_{2}, g ∈ N_{S}(Y_{0}),

g^{2} ∈ Y_{0}, c_{g} exchanges Y_{0} ∩ Θ_{1} and Y_{0} ∩ Θ_{2}}

• E_{F}^{c}(Y) = {UC_{S}(U) | U ∈ 𝒰_{F}(Y)}.

(b) For each R ∈ E_{F}(Y), R ≥ Y_{0} for some Y_{0} ∈ 𝒴_{0}.

(c) If P = Y_{0}⟨g⟩ ∈ E_{F}^{a}(Y), and Γ ≤ Aut(P) is such that Aut_{S}(P) ∈ Syl_{2}(Γ) and {γ|_{Y_{0}} | γ ∈ Γ} = N_{Aut_{F} (Y_{0})}(Aut_{P} (Y_{0})), then Γ = Aut_{F}(P).

Proof. We first claim that

P > Y_{0} ∈ 𝒴_{0}, |P/Y_{0}| = 2, rk([P, Y_{0}/Fr(Y_{0})]) = 2 ⟹ Y_{0} char. P . (5.4)

In all cases, Fr(Y_{0}) is characteristic in P: either Fr(Y_{0}) = 1, or Fr(Y_{0}) = Z(P); or Y_{0} ≅ Q_{8} × Q_{8}, |Z(P)| = 2, Y_{0}/Z(P) ≅ 2_{+}^{1+4}, and hence Fr(Y_{0}) = Z_{2}(P). Since Y_{0}/Fr(Y_{0}) is the unique abelian subgroup of index 2 in P/Fr(Y_{0}) by Lemma A.4(a), Y_{0} is characteristic in P.

(a) E_{F}(Y ) ⊆ E_{F}^{a}(Y ) ∪ E_{F}^{c}(Y ). By Proposition 3.9(b) and since Y ∈ 𝒴 (S), E_{F}(Y) = E_{F}^{(II)}(Y), and each F-essential pair (P_{1}, P_{2}) of type (II) in E_{F}(Y) has the form described in Lemma 3.7(a) or in Lemma 3.8(b). Set P_{12} = P_{1} ∩ P_{2}.

\underline{Case 1:} Assume (P_{1}, P_{2}) is as in Lemma 3.7(a). By that lemma, P_{12} ∈ 𝒴_{0}, Out_{F}(P_{12}) ≅ Σ_{3} ≀C_{2} or Σ_{5}, and Out_{P_{1}}(P_{12}) ≰ O^{2}(Out_{F}(P_{12})). If Out_{F}(P_{12}) ≅ Σ_{5}, then P_{12}/Fr(P_{12}) is the orthogonal module for Out_{F}(P_{12}). Since Out_{P_{1}}(P_{12}) ≰ O^{2}(Out_{F}(P_{12})), Out_{P_{1}}(P_{12}) is generated by a transposition in Σ_{5}. Thus

Out_{F}(P_{12}) ≅ Σ_{5} ⟹ rk([P_{1}, P_{12}/Fr(P_{12})]) = 1 .  (5.5)

Set U_{i} = P_{12} ∩ Θ_{i} (i = 1, 2). By Proposition 3.11(b.2), {U_{1}, U_{2}} ∈ 𝒰_{S}(P_{12}), so P_{12} = U_{1}U_{2}, U_{1} ∩ U_{2} ≤ Fr(P_{12}), and each element of Out_{S}(P_{12}) ≅ D_{8} either normalizes the U_{i} or exchanges them. So we can choose bases {b_{i1}, b_{i2}} of U_{i} (i = 1, 2) such that B = {b_{ij} | i, j = 1, 2} is a basis of P_{12}/Fr(P_{12}) permuted by Out_{S}(P_{12}). Then one of the following happens:

• The action of Out_{P_{1}}(P_{12}) exchanges U_{1} and U_{2}. In particular, a generator of this group acts on B as a product of two disjoint 2-cycles, so Out_{F}(P_{12}) ≅ Σ̸_{5} by (5.5). Thus Out_{F}(P_{12}) ≅ Σ_{3} ≀ C_{2}, and P_{1} ∈ E_{F}^{a}(Y).

• The action of Out_{P_{1}}(P_{12}) normalizes each U_{i}. Since Out_{P_{1}}(P_{12}) ≅ C_{2} is noncentral in Out_{S}(P_{12}) ≅ D_{8} (since |N_{S}(P_{1})/P_{1}| = 2), it acts on B as a 2-cycle, and there is exactly one of the groups U ∈ {U_{1}, U_{2}} for which Out_{P_{1}}(P_{12}) acts trivially on U/Fr(U) ≅ C_{2}^{2}. Then [P_{1}, U] ≤ Fr(U), so Aut_{P_{1}}(U) ≤ Inn(U), and P_{1} ≤ UC_{S}(U). If P_{1} < UC_{S}(U), then by Lemma A.1(a), there is g ∈ N_{S}(P_{1})∖P_{1} with g ∈ C_{S}(U). Since N_{S}(P_{12})/Fr(P_{12}) ≅ D_{8} ≀ C_{2} (P_{12} ∈ 𝒴_{0}) and P_{1} < N_{S}(P_{12}), there is also h ∈ N_{N_{S}(P_{12})}(P_{1})∖P_{1} which does not centralize U. Since |N_{S}(P_{1})/P_{1}| = 2, this is impossible, so P_{1} = UC_{S}(U) ∈ E_{F}^{c}(Y).

\underline{Case 2:} Now assume (P_{1}, P_{2}) has the form described in Lemma 3.8(b). By Lemma 3.8(b,d,e), P_{1} = UC_{S}(U) where U = [Aut_{F}^{∗}(P_{1}), P_{1}] ≅ C_{2}^{2} or Q_{8}. Thus Aut_{F}(U) = Aut(U), and Y = 𝔣𝔬𝔠(F, P_{1}) is the normal closure of U in S. By the same lemma, Y = ΔΔ^{∗} where {Δ, Δ^{∗}} is an S-conjugacy class, Δ, Δ^{∗} ∈ DQ, U ≤ Δ, [Δ, Δ^{∗}] ≤ Δ ∩ Δ^{∗} ≤ Z(S), and Δ ∩ Δ^{∗} = 1 if Δ, Δ^{∗} ∈ D. Also, Δ and Δ^{∗} are strongly automized in S, since Δ is the normal closure of U in a certain subgroup S_{∗} of index 2 in S.

We must show that U ∈ 𝒰_{F}(Y) (i.e., that U ≤ Θ_{i} for i = 1 or 2). Choose g ∈ S such that ^{g}Δ = Δ^{∗}, and set U^{∗} = ^{g}U and Y_{0} = UU^{∗}. If Y ∈ 𝒴_{0}, then U = Δ and U^{∗} = Δ^{∗}, so Y_{0} = UU^{∗} = Y. If Y ∉ 𝒴_{0}, then Δ ≅ D_{2^{n}} for n ≥ 3 or Q_{2^{n}} for n ≥ 4, and Y_{0} ∈ 𝒴_{0} by Lemma 2.6(a).

In either case, Y_{0} ∈ 𝒴_{0}, and {U, U^{∗}} = {Y_{0}∩Δ, Y_{0}∩Δ^{∗}} ∈ 𝒰_{S}(Y_{0}) is an N_{S}(Y_{0})- conjugacy class. Let α ∈ Aut_{F}^{∗}(P_{1}) be of odd order. Since U = [Aut_{F}^{∗}(P_{1}), P_{1}] and U ≤ Y_{0} ≤ P_{1}, α normalizes U and Y_{0}, and α|_{U} ∈ Aut_{F}(U) has order 3. So {U, U^{∗}} ∈ 𝒰_{S}(Y_{0}) is the unique element compactible with Out_{F}(Y_{0}) (unique by Lemma 2.9(b)), and hence U ∈ 𝒰_{F}(Y) by Proposition 3.11(b). E_{F}(Y ) ⊇ E_{F}^{a}(Y ) ∪ E_{F}^{c}(Y ). By Proposition 3.11(c.2), E_{F}^{c}(Y) ⊆ E_{F}(Y).

Assume P ∈ E_{F}^{a}(Y). Thus P = Y_{0}⟨g⟩ where Y_{0} ∈ 𝒴_{0}, c_{g} exchanges the two subgroups Y_{0} ∩Θ_{i}, and Out_{F}(Y_{0}) ≅ Σ_{3} ≀C_{2}. Then Y_{0} is characteristic in P by (5.4). Since Y_{0} is fully normalized (Lemma 3.10), and since Out_{P} (Y_{0}) is not Out_{F}(Y_{0})- conjugate to the center of Out_{S}(Y_{0}) ≅ D_{8} (since Out_{F}(Y_{0}) ≅ Σ_{3} ≀ C_{2}), P is fully normalized in N_{F}(Y_{0}), and hence also in F by Proposition 1.3(b). So by Lemma 1.5 and since Out_{P} (Y_{0}) is noncentral of order 2 in Out_{S}(Y_{0}) ≅ D_{8},

Out_{F}(P) ≅ N_{Out_{F} (Y_{0})}(Out_{P} (Y_{0}))/Out_{P} (Y_{0}) ≅ Σ_{3}.

Thus P ∈ E_{F}. Also, [Aut_{F}(P), P] ≤ Y_{0} by (5.4), so 𝔣𝔬𝔠(F, P) ≤ Y. Since 𝔣𝔬𝔠(F, P) ∈ 𝒴 (S) by Proposition 3.9(a), and since no element of 𝒴 (S) is strictly contained in any other (Lemma 2.4(b)), P ∈ E_{F}(Y) and thus E_{F}^{a}(Y) ⊆ E_{F}(Y).

(b) Fix R ∈ E_{F}(Y). If R ∈ E_{F}^{a}(Y), then by definition, R > Y_{0} for some Y_{0} ∈ Y_{0}. If R ∈ E_{F}^{c}(Y), then R = UC_{S}(U) for some U ∈ 𝒰_{F}(Y). Also, by definition of

𝒰_{F}(Y) (Proposition 3.11(b)), U = Y_{0} ∩ Θ_{i} for some Y_{0} ∈ 𝒴_{0}(S) and some i = 1, 2. Set U^{∗} = Y_{0}∩Θ_{3−i}. By Proposition 3.11(b.2), {U, U^{∗}} ∈ 𝒰_{S}(Y_{0}), and in particular, Y_{0} = UU^{∗} and [U, U^{∗}] ≤ Fr(U) (Definition 2.1(e)). Thus Aut_{U^{∗}}(U) ≤ Inn(U), and so Y_{0} = UU^{∗} ≤ UC_{S}(U) = R.

(c) Assume P = Y_{0}⟨g⟩ ∈ E_{F}^{a}(Y), where Y_{0} ∈ 𝒴_{0}. Recall that Y_{0} is fully normalized in F by Lemma 3.10, is F-centric by definition of 𝒴_{0}(S), and is characteristic in P by (5.4). Also, P/Y_{0} permutes freely a basis for Z(Y_{0}) if Y_{0} ≅ C_{2}^{4} or Q_{8} × Q_{8}, and |P/Y_{0}| = |Z(Y_{0})| = 2 if Y_{0} ≅ 2_{±}^{1+4}. So by Lemma 1.5(b), for any Γ ≤ Aut(P) with the given properties, Γ = Aut_{F}(P). □

We next consider wreath products Δ ≀ C_{2} for Δ ∈ DS. It is easy to see that D_{8} ≀ C_{2} is a Sylow 2-subgroup of Σ_{8} and hence of A_{10}. Since SD_{2^{n}} is a Sylow 2subgroup of GL_{2}(q) for appropriate q ≡ 3 (mod 4), SD_{2^{n}} ≀C_{2} is a Sylow 2-subgroup of the groups GL_{2}(q) ≀ C_{2} ≤ GL_{4}(q), and hence of PSL_{5}(q). We next check that D_{2^{n}} ≀ C_{2} is a Sylow 2-subgroup of PSL_{4}(q) for appropriate q.

Lemma 5.3. Fix a prime power q ≡ 3 (mod 4), and set n = 1+v_{2}(q +1). Then the Sylow 2-subgroups of PSL_{4}(q) are isomorphic to D_{2^{n}} ≀ C_{2}.

Proof. This is most easily seen via the isomorphism PSL_{4}(q) ≅ P Ω_{6}^{+}(q) (cf. [Ta, Corollary 12.21]). By [CF, Theorems 2–3] and since q ≡ (mod 4), the general orthogonal group GO_{6}^{−}(q) contains (GO_{2}^{−}(q)≀C_{2})×GO_{2}^{+}(q) with odd index, where GO_{2}^{±}(q) ≅ D_{2(q∓1)} (see [Ta, Theorem 11.4]). Thus the Sylow 2-subgroups of GO_{6}^{+}(q) are isomorphic to (D_{2^{n}} ≀ C_{2}) × C_{2}^{2}. The last factor is sent isomorphically to GO_{6}^{+}(q)/Ω_{6}^{+}(q) (in particular, −I ∈ SO_{2}^{+}(q) has nontrivial spinor norm since −1 is not a square [Ta, p. 163]). Hence Ω_{6}^{+}(q) ≅ P Ω_{6}^{+}(q) has Sylow 2-subgroup isomorphic to D_{2^{n}} ≀ C_{2}. □

The following presentation for the groups studied here will be used throughout the rest of chapter.

Notation 5.4. For some n ≥ 3, S = ⟨a_{1}, b_{1}, a_{2}, b_{2}, t⟩, where for i = 1, 2,

|a_{i}| = 2^{n−1}, |b_{i}| = 2, b_{i}a_{i}b_{i}^{−1} = a_{i}^{λ}, Δ_{i} =^{def} ⟨a_{i}, b_{i}⟩ ≅ D_{2^{n}} or SD_{2^{n}},

and λ = −1 or λ = −1 + 2^{n−2} (and λ = −1 if n = 3). Also,

[Δ_{1}, Δ_{2}] = 1, ta_{1}t^{−1} = a_{2}, tb_{1}t^{−1} = b_{2}, and t^{2} = 1.

Either Δ_{1} ∩ Δ_{2} = 1, or Δ_{1} ∩ Δ_{2} = Z(Δ_{1}) = Z(Δ_{2}). Also, set w_{i} = a_{i}^{2^{n−3}} and z_{i} = w_{i}^{2} ∈ Z(Δ_{i}), and set z = z_{1} = z_{2} if Δ_{1} ∩ Δ_{2} ≠ 1.

Proposition 5.5. Let F be a reduced fusion system over S ≅ Δ ≀ C_{2}, where Δ ∈ DS.

(a) If Δ ≅ D_{2^{n}} for n ≥ 3, then either F is isomorphic to the fusion system of PSL_{4}(q) for each q such that v_{2}(q + 1) = n − 1, or n = 3 and F is isomorphic to the fusion system of A_{10}.

(b) If Δ ≅ SD_{2^{n}} for n ≥ 4, then F is isomorphic to the fusion system of PSL_{5}(q) for each q such that v_{2}(q + 1) = n − 2.

Proof. Let S have the presentation in Notation 5.4, where Δ_{1} ∩ Δ_{2} = 1. Set Z_{∗} = ⟨a_{1}^{2^{n−3}}, a_{2}^{2^{n−3}}⟩, and set

Y_{1} = ⟨a_{1}^{2}, a_{2}^{2}, b_{1}, b_{2}⟩, Y_{2} = ⟨a_{1}^{2}, a_{2}^{2}, a_{1}b_{1}, a_{2}b_{2}⟩, Y_{3} = ⟨a_{1}a_{2}^{−1}, a_{1}a_{2}, b_{1}b_{2}, t⟩.

Thus S/Z_{∗} ≅ D_{8} ≀ C_{2} (the unique normal subgroup of index 2^{7} by Lemma 2.4(a)), Y_{1}/Z_{∗} ≅ Y_{2}/Z_{∗} ≅ C_{2}^{4}, and Y_{3}/Z_{∗} ≅ 2_{+}^{1+4}. So by Proposition 3.9, Out_{F}(S) = 1,

E_{F} = E_{F}(Y_{1}) ∪ E_{F}(Y_{2}) ∪ E_{F}(Y_{3}), E_{F}(Y_{i}) ⊆ E_{F}^{(II)} if Y_{i} ∈ 𝒴 (S),

and E_{F}(Y_{i}) ≠ ∅ for each i = 1, 2, 3.

By Lemma 2.4(b), 𝒴 (S) ⊆ {Y_{1}, Y_{2}, Y_{3}}. We claim that

𝒴 (S) = \begin{cases}\{Y_{1}, Y_{2}\} & if n = 3 \\ \{Y_{1}, Y_{2}, Y_{3}\} & if n ≥ 4. \end{cases}  (5.6)

When S ≅ D_{8} ≀ C_{2}, Y_{1}, Y_{2} ∈ 𝒴 (S) by definition, and Y_{3} ∉ 𝒴 (S) by Lemma 2.4(d) (and since [S:Y_{3}] = 4 and |S| < 2^{8}). This proves (5.6) when n = 3. For n ≥ 4, it follows from Lemma 2.6(a), except when Δ ≅ SD_{16}, in which case Y_{2} ≅ Q_{8} × Q_{8} lies in 𝒴_{0}(S) (hence in 𝒴 (S)) by definition.

For each i = 1, 2, 3, let 𝒰_{i} = 𝒰_{F}(Y_{i}) and E_{F}(Y_{i}) = E_{F}^{a}(Y_{i}) ∪ E_{F}^{c}(Y_{i}) be as in Lemma 5.2. Let 𝒴_{0i} be the set of subgroups P ∈ 𝒴_{0}(S) whose normal closure is Y_{i}. Step 1: We first consider F-essential subgroups associated to Y_{3}. Set

Θ_{31} = ⟨a_{1}a_{2}^{−1}, z_{2}t⟩  and  Θ_{32} = \begin{cases}⟨a_{1}a_{2}, z_{2}b_{1}b_{2}t⟩ & if Δ_{i} ∈ D \\ ⟨z_{2}a_{1}a_{2}, z_{2}b_{1}b_{2}t⟩ & if Δ_{i} ∈ S. \end{cases}

Then Θ_{31} ≅ Θ_{32} ≅ Q_{2^{n}}, Θ_{31}Θ_{32} = Y_{3}, and Θ_{31}∩Θ_{32} = ⟨z_{1}z_{2}⟩. Also, [Θ_{31}, Θ_{32}] = 1 if Δ_{i} ∈ D, while [Θ_{31}, Θ_{32}] = ⟨z_{1}z_{2}⟩ = Z(S) if Δ_{i} ∈ S.

Case n = 3: If S ≅ D_{8} ≀ C_{2}, then Y_{3} ∉ 𝒴 (S) by (5.6), and E_{F}(Y_{3}) = E_{F}^{(III)} by Proposition 3.9(c) (and since Y_{3} ≅ 2_{+}^{1+4}). Since each automorphism of Y_{3} either normalizes the subgroups Θ_{3i} ≅ Q_{8} or exchanges them, Out(Y_{3}) ≅ Σ_{3} ≀ C_{2}.

Let τ_{1}, τ_{2} ∈ Aut(Y_{3}) be the automorphisms of order 3 defined by setting

τ_{1} : (a_{1}a_{2}^{−1} ↦ z_{2}t ↦  a_{1}a_{2}t  ↦ a_{1}a_{2}^{−1}) and τ_{1}|_{Θ_{32}} = Id
τ_{2} : (a_{1}a_{2} ↦ z_{2}b_{1}b_{2}t ↦ a_{1}a_{2}^{−1}b_{1}b_{2}t ↦ a_{1}a_{2}) and τ_{2}|_{Θ_{31}} = Id

Thus ⟨[τ_{1}], [τ_{2}]⟩ = O_{3}(Out(Y_{3})) = O^{2}(Out(Y_{3})). For R ∈ E_{F}(Y_{3}), [S:R] = [R:Y_{3}] = 2, Y_{3} is characteristic in R since it is the only subgroup in S of its isomorphism type (Lemma C.5(a)), and hence

Out_{F}(R) ≅ N_{Out_{F} (Y_{3})}(Out_{R}(Y_{3}))/Out_{R}(Y_{3})  (5.7)

by Lemma 1.5(a). Thus N_{Out(Y_{3})}(Out_{R}(Y_{3})) is not a 2-group, where Out(Y_{3}) ≅ Σ_{3} ≀ C_{2}, so R ≠ Y_{3}⟨a_{2}⟩, and thus R is one of the groups

R_{1} = Y_{3}⟨b_{2}⟩  or  R_{2} = Y_{3}⟨a_{2}b_{2}⟩ .

Furthermore, by (5.7), one of the following holds:

Out_{F}(Y_{3}) = ⟨[τ_{1}], [τ_{2}], Out_{S}(Y_{3})⟩ ≅ (C_{3} × C_{3}) ⋊^{−1,t} C_{2}^{2} E_{F}(Y_{3}) = {R_{1}, R_{2}}
Out_{F}(Y_{3}) = ⟨[τ_{1}τ_{2}], Out_{S}(Y_{3})⟩ ≅ C_{2} × Σ_{3}  E_{F}(Y_{3}) = {R_{1}} (5.8)
Out_{F}(Y_{3}) = ⟨[τ_{1}τ_{2}^{−1}], Out_{S}(Y_{3})⟩ ≅ C_{2} × Σ_{3}  E_{F}(Y_{3}) = {R_{2}}

Let ψ ∈ Aut(S) be the automorphism ψ(a_{i}) = a_{i}^{−1}, ψ(b_{i}) = a_{i}b_{i}, ψ(t) = t. Upon replacing F by ^{ψ}F if necessary, we can assume R_{1} ∈ E_{F}(Y_{3}). Case n ≥ 4: By Lemma 2.6(a), 𝒴_{03} is the set of subgroups S-conjugate to one of the groups

Y_{03}^{(1)} = ⟨z_{1}, w_{1}w_{2}, b_{1}b_{2}, t⟩ = ⟨w_{1}w_{2}^{−1}, z_{2}t⟩·⟨w_{1}w_{2}, z_{2}b_{1}b_{2}t⟩ ≅ 2_{+}^{1+4}
Y_{03}^{(2)} = ⟨z_{1}, w_{1}w_{2}, a_{1}a_{2}b_{1}b_{2}, t⟩

= \begin{cases}⟨w_{1}w_{2}^{−1}, z_{2}t⟩·⟨w_{1}w_{2}, z_{2}a_{1}a_{2}b_{1}b_{2}t⟩ ≅ 2_{+}^{1+4} & if Δ_{i} ∈ D \\ ⟨w_{1}w_{2}^{−1}, z_{2}t⟩·⟨w_{1}w_{2}, a_{1}a_{2}b_{1}b_{2}t⟩ ≅ 2_{−}^{1+4} & if Δ_{i} ∈ S. \end{cases}

By the uniqueness in Lemma C.3, Θ_{31} and Θ_{32} are the subgroups Θ_{i} which appear in Proposition 3.11(a). So by definition (Proposition 3.11(b)), 𝒰_{3} is the set of subgroups of Θ_{31} or of Θ_{32} isomorphic to Q_{8}, and thus the S-conjugacy class of ⟨w_{1}w_{2}^{−1}, z_{1}t⟩.

Now, Out_{F}(Y_{03}^{(i)}) ∈ 𝒜_{S}(Y_{03}^{(i)}) by Proposition 3.11(b.1). Hence Out_{F}(Y_{03}^{(i)}) = Out(Y_{03}^{(i)}) ≅ Σ_{3} ≀ C_{2} or Σ_{5}, depending on whether Y_{03}^{(i)} ≅ 2_{+}^{1+4} or 2_{−}^{1+4}. By Lemma 5.2(a), E_{F}(Y_{3}) = E_{F}^{a}(Y_{3}) ∪ E_{F}^{c}(Y_{3}), where

(i) E_{F}^{c}(Y_{3}) is the set of all P = UC_{S}(U) for U ∈ 𝒰_{3}; and

(ii) E_{F}^{a}(Y_{3}) is the union of the S-conjugacy classes of

R_{1} = Y_{03}^{(1)}⟨b_{2}⟩  and  R_{2} = Y_{03}^{(2)}⟨a_{2}b_{2}⟩.  (5.9)

By Lemma 5.2(c), Aut_{F}(R_{i}) (i = 1, 2) is uniquely determined by Aut_{F}(Y_{03}^{(i)}). For R = UC_{S}(U) ∈ E_{F}^{c}(Y_{3}) (U ∈ 𝒰_{3}), Aut_{F}^{∗}(R) is uniquely determined by Proposition 3.11(c.4): Aut_{F}^{∗}(UC_{S}(U)) = O^{2}(Inn(UC_{S}(U))⟨α⟩) for some α ∈ Aut_{F}^{∗}(UC_{S}(U)) of order 3 which normalizes U ≅ Q_{8} and acts via the identity on C_{S}(U). Step 2: We now examine subgroups in E_{F}(Y_{1})∪E_{F}(Y_{2}) and their automorphisms, by first showing how this is influenced by the subgroups in E_{F}(Y_{3}). Set

Y_{01} = ⟨z_{1}, z_{2}, b_{1}, b_{2}⟩ ≅ C_{2}^{4}
Y_{02} = \begin{cases}⟨z_{1}, z_{2}, a_{1}b_{1}, a_{2}b_{2}⟩ ≅ C_{2}^{4} & if Δ_{i} ∈ D \\ ⟨w_{1}, w_{2}, a_{1}b_{1}, a_{2}b_{2}⟩ ≅ Q_{8} × Q_{8} & if Δ_{i} ∈ S. \end{cases}

Then Y_{0i} ∈ 𝒴_{0i} in all cases, and 𝒴_{0i} is the S-conjugacy class of Y_{0i} by Lemma 2.6(a) (or by definition when Y_{0i} = Y_{i}).

Consider the subgroup R_{1} = Y_{03}^{(1)}⟨b_{2}⟩ = Y_{01}⟨w_{1}w_{2}, t⟩ (R_{1} = Y_{3}⟨b_{2}⟩ if n = 3). Since Y_{03}^{(1)} ≅ 2_{+}^{1+4} and c_{b_{2}} exchanges the two quaternion factors, R_{1} ≅ UT_{4}(2) by Lemma C.4(b), and Y_{01} < R_{1} is the unique subgroup isomorphic to C_{2}^{4} (Lemma C.4(a)). We just showed (in (5.8) and (5.9)) that R_{1} ∈ E_{F}(Y_{3}). Hence Aut_{F}(R_{1}) contains an automorphism of order 3, and it permutes cyclically the three subgroups of index 2 which contain Y_{01} (Lemma C.4(d)) and hence acts nontrivally on Aut_{R_{1}}(Y_{01}) ≅ C_{2}^{2}. So Aut_{F}(Y_{01}) ≅ Σ̸_{3} ≀ C_{2}, and hence Aut_{F}(Y_{01}) ≅ Σ_{5} by Proposition 3.11(b.1).

To identify Aut_{F}(Y_{02}), we consider three different cases: Type (1): Assume Δ_{1}, Δ_{2} ∈ D and R_{2} ∈ E_{F}(Y_{3}). Then Aut_{F}(Y_{02}) ≅ Σ_{5} by an

argument similar to the above, applied with R_{2} in place of R_{1}.

Type (2): Assume Δ_{1}, Δ_{2} ∈ D and R_{2} ∉ E_{F}(Y_{3}). By Step 1, n = 3 and S ≅ D_{8} ≀ C_{2}. Then Out_{F}(Y_{02}) ≅ Σ_{3} ≀ C_{2}, since otherwise Out_{F}(Y_{02}) ≅ Σ_{5} by Proposition 3.11(b.1), which by the extension axiom (and since R_{2} = Y_{02}⟨w_{1}w_{2}, t⟩) would imply Out_{F}(R_{2}) ≥ Σ_{3}.

Type (3): Assume Δ_{i} ∈ S. Then Y_{02} ≅ Q_{8} × Q_{8}, so Out_{F}(Y_{02}) ≅ Σ_{3} ≀ C_{2} by

Proposition 3.11(b.1).

Thus in each case, Out_{F}(P) is determined up to isomorphism for P ∈ 𝒴_{01} ∪ 𝒴_{02}.

For i = 1, 2, let {Θ_{i1}, Θ_{i2}} be as in Proposition 3.11(a). Thus Y_{i} = Θ_{i1} × Θ_{i2} where {Θ_{i1}, Θ_{i2}} is an S-conjugacy class (hence both are normal in Δ_{1}Δ_{2}), and Θ_{ij} ∈ DQ. By the Krull-Schmidt theorem (Theorem A.8(a)) (and after exchanging indices if necessary), Θ_{11} ≤ ⟨a_{m}^{2}, b_{m}⟩×⟨z_{3−m}⟩, and after reindexing if necessary, we can assume m = 1. Then b_{1}z_{2}^{j} ∈ Θ_{11} for some j = 0, 1, and ^{a_{1}}(b_{1}z_{2}^{j}) = a_{1}^{2}b_{1}z_{2}^{j} ∈ Θ_{11} since Θ_{11} ⊴ Δ_{1}Δ_{2}. Thus Θ_{11} = ⟨a_{1}^{2}, b_{1}z_{2}^{j}⟩, and hence Θ_{12} = ^{t}Θ_{11} = ⟨a_{2}^{2}, b_{2}z_{1}^{j}⟩. By a similar argument, Θ_{21} = ⟨a_{1}^{2}, a_{1}b_{1}z_{2}^{k}⟩ for some k = 0, 1, and Θ_{22} = ^{t}Θ_{21}.

Define φ_{jk} ∈ Aut(S) by setting φ_{jk}(t) = t, φ_{jk}(b_{i}) = b_{i}z_{3−i}^{j}, and φ_{jk}(a_{i}b_{i}) = a_{i}b_{i}z_{3−i}^{k}. Then upon replacing F by ^{φ_{jk}}F, we have Θ_{11} = ⟨a_{1}^{2}, b_{1}⟩ and Θ_{21} = ⟨a_{1}^{2}, a_{1}b_{1}⟩. Also, by Proposition 3.11(b), 𝒰_{1} and 𝒰_{2} are the S-conjugacy classes of

U_{1} = Θ_{11} ∩ Y_{01} = ⟨z_{1}, b_{1}⟩ and U_{2} = Θ_{21} ∩ Y_{02} = \begin{cases}⟨z_{2}, a_{2}b_{2}⟩ & if Δ_{2} ∈ D \\ ⟨w_{2}, a_{2}b_{2}⟩ & if Δ_{2} ∈ S. \end{cases}

By Lemma 3.11(b.2), for i = 1, 2, Out_{F}(Y_{0i}) is the unique subgroup of Out(Y_{0i}) of its isomorphism type (as determined above) which is compatible with the pair {Y_{0i} ∩ Θ_{i1}, Y_{0i} ∩ Θ_{i2}} ∈ 𝒰_{S}(Y_{0i}) (compatible in the sense of Definition 2.2(b)). In particular, each automorphism of order 3 of Y_{0i} ∩ Θ_{ij} (j = 1, 2) extends to an element of Aut_{F}(Y_{0i}). We refer to Lemma 2.9(c) and its proof for more details on how the above pair determines Out_{F}(Y_{0i}).

By Lemma 5.2(a), for i = 1, 2, E_{F}(Y_{i}) = E_{F}^{a}(Y_{i}) ∪ E_{F}^{c}(Y_{i}), where E_{F}^{a}(Y_{i}) = ∅ if Aut_{F}(Y_{0i}) ≅ Σ_{5}. Thus E_{F}^{a}(Y_{1}) = ∅. By Lemma 5.2(c), the F-automorphisms of P ∈ E_{F}^{a}(Y_{2}) are uniquely determined by the above information. Step 3: It remains to determine Aut_{F}(R) when R = UC_{S}(U) ∈ E_{F}^{c}(Y_{i}) for U ∈ 𝒰_{i} (i = 1, 2). By Proposition 3.11(c.3–4), [Aut_{F}^{∗}(R), R] = U in this situation, and Aut_{F}^{∗}(R) = O^{2}(Inn(R)⟨α⟩) for some α of order 3 such that α(U) = U and α induces the identity on R/U. It remains to determine α more precisely.

Consider the group P = U_{1}U_{2}. Each element of N_{S}(P) either normalizes or exchanges the two subgroups U_{i} = P ∩ Δ_{i}, and they are not S-conjugate since ⟨U_{1}^{S}⟩ = Y_{1} while ⟨U_{2}^{S}⟩ = Y_{2}. Hence N_{S}(P) = N_{Δ_{1}Δ_{2}}(P) = N_{Δ_{1}}(U_{1})N_{Δ_{2}}(U_{2}), N_{Δ_{1}}(U_{1}) ≅ D_{8}, and N_{Δ_{2}}(U_{2}) ≅ D_{8}, Q_{16}, or SD_{16}. Thus Out_{S}(P) ≅ N_{S}(P)/P ≅ C_{2}^{2}.

We claim that P is fully normalized in F. If not, then by Lemma 1.16(a), there is T ∈ E_{F} such that T ≥ N_{S}(P). Let j be such that T ∈ E_{F}(Y_{j}) (j = 1, 2, 3). If T = UC_{S}(U) for some U ∈ 𝒰_{j}, then U ≤ N_{S}(P) by Lemma A.6(c) (applied with T in the role of S), which is impossible. (Recall that if U ≅ C_{2}^{2}, then it is a direct factor in T.) If T ∈ E_{F}^{a}(Y_{j}) for j = 1, 2, then N_{S}(P) ≤ T ∩ Δ_{1}Δ_{2} ∈ 𝒴_{0j}, which is also impossible. Finally, N_{S}(P) ≰ R_{1}, R_{2} ∈ E_{F}^{a}(Y_{3}) as defined in Step 1. Hence there is no such T ∈ E_{F}, and P is fully normalized.

Let x_{1} ∈ Out_{Δ_{1}}(P) and x_{2} ∈ Out_{Δ_{2}}(P) be the generators. For each i = 1, 2, by Proposition 3.11(b.4) and since U_{i} ∈ 𝒰_{i}, there is α_{i} ∈ Aut_{F}(U_{i}C_{S}(U_{i})) of order 3 which normalizes U_{i} and induces the identity on U_{i}C_{S}(U_{i})/U_{i}. Thus α_{i}(P) = P. Set α_{i} =^{def} α_{i}|_{P} ∈ Aut_{F}(P). Since U_{i}Δ_{3−i} ≤ U_{i}C_{S}(U_{i}), [α_{i}] ∈ Out_{F}(P) normalizes (hence centralizes) Out_{Δ_{3−i}}(P) = ⟨x_{3−i}⟩. The hypotheses of Proposition D.1(e.1) thus hold, applied to the action of Out_{F}(P) on P/Fr(P) ≅ C_{2}^{4}, and hence Out_{F}(P) ≅ Σ_{3} × Σ_{3}. If Δ_{2} ∈ D, so P = U_{1}U_{2} ≅ C_{2}^{4}, then U_{1} and U_{2} are the irreducible summands of the action of ⟨α_{1}, α_{2}⟩ ≅ C_{3} × C_{3}. If Δ_{2} ∈ S, so P ≅ C_{2}^{2} × Q_{8}, then α_{1} normalizes U_{2} by a similar argument applied to P/Fr(P), and α_{2} normalizes U_{1} since α_{2}|_{Z(P)} = Id. Thus in both cases, α_{1} is the identity on U_{2} and α_{2} is the identity on U_{1}. This, together with Aut_{F}(Y_{01}) and Aut_{F}(Y_{02}), determine uniquely the automorphism groups Aut_{F}^{∗}(U_{i}C_{S}(U_{i})) for i = 1, 2, and hence determine Aut_{F}^{∗}(R) for each R ∈ E_{F}^{c}(Y_{1}) ∪ E_{F}^{c}(Y_{2}).

For example, Aut_{F}(Y_{01}) ≅ Σ_{5} is the group compatible with the pair {U_{1},^{t}U_{1}} ∈ 𝒰_{S}(Y_{01}). As shown explicitly in the proof of Lemma 2.9(c), it is the group of automorphisms of Y_{01} = ⟨z_{1}, b_{1}, z_{2}, b_{2}⟩ which permute the set

X = {b_{1}z_{2}, b_{1}z_{1}z_{2}, z_{1}z_{2}, b_{2}z_{1}, b_{2}z_{1}z_{2}} .

Any element of Aut_{F}(Y_{01}) which permutes cyclically the first three elements in X acts on U_{1} = ⟨z_{1}, b_{1}⟩ with order 3, and any element which permutes cyclically the last three acts on ^{t}U_{1} = ⟨z_{2}, b_{2}⟩ with order 3. If R = U_{1}C_{S}(U_{1}) ∈ E_{F}^{c}(Y_{1}), then Aut_{F}^{∗}(R) = O^{2}(Inn(R)⟨α⟩) where α|_{Y_{01}} permutes cyclically the first three elements in X, fixes the last two, and is the identity on U_{2}. Thus α acts on R = U_{1} × ⟨b_{2}z_{1}, a_{2}b_{2}⟩ with order 3 on the first factor and as the identity on the second factor.

We have now shown that up to isomorphism, there is at most one reduced fusion system of each of the three types listed above.

Step 4: It remains to find explicit fusion systems of each type. Assume Δ_{i} ∈ D, let q be a prime power such that v_{2}(q + 1) = n − 1, and identify S with a Sylow 2-subgroup of G_{1} = PSL_{4}(q) (Lemma 5.3). Since SL_{2}(q) contains subgroups isomorphic to Q_{8} ⋊ C_{3}, G_{1} contains subgroups Y_{3} ≅ 2_{+}^{1+4} with 9||Out_{F}(Y_{3})|. So R_{1}, R_{2} ∈ E_{F}(Y_{3}) by (5.8) (when n = 3), and F_{S}(G_{1}) has type (1). Also, F_{S}(G_{1}) is reduced by Proposition 1.12.

Next assume S ≅ D_{8} ≀ C_{2}, identify S with a Sylow 2-subgroup of Σ_{8} < A_{10}, and set G_{2} = A_{10}. There are two G_{2}-conjugacy classes of subgroups isomorphic to C_{2}^{4}, represented by

V_{1} = A_{10} ∩ ⟨(1 2), (3 4), (5 6), (7 8), (9 10)⟩
V_{2} = ⟨(1 2)(3 4), (1 3)(2 4), (5 6)(7 8), (5 7)(6 8)⟩.

Since Aut_{G_{2}}(V_{1}) ≅ Σ_{5} and Aut_{G_{2}}(V_{2}) ≅ Σ_{3} ≀ C_{2}, F_{S}(G_{2}) has type (2). Again, F_{S}(G_{2}) is reduced by Proposition 1.12.

Now assume Δ_{i} ∈ S, let q be such that v_{2}(q + 1) = n − 2, and identify S with a Sylow 2-subgroup of GL_{2}(q) ≀ C_{2} < GL_{4}(q), and hence of G_{3} = PSL_{5}(q). Thus F_{S}(G_{3}) has type (3), and F_{S}(G_{3}) is reduced by Proposition 1.12 again. □

It remains to consider the central wreath products.

Proposition 5.6. Let F be a reduced fusion system over S ≅ (Δ ×_{C_{2}} Δ) ⋊^{t} C_{2}, where Δ ∈ DS, and |Δ| = 2^{n} for n ≥ 4. Then Δ ≅ D_{2^{n}}, and F is isomorphic to the fusion system of PSp_{4}(q) for each odd prime power q such that v_{2}(q^{2} − 1) = n.

Proof. Let S have the presentation and subgroups of Notation 5.4, where n ≥ 4, z = z_{1} = z_{2}, and Δ_{1} ∩ Δ_{2} = ⟨z⟩ = Z(S). Set Z_{∗} = ⟨a_{1}^{2^{n−3}}, a_{2}^{2^{n−3}}⟩, and set

Y_{1} = ⟨a_{1}^{2}, a_{2}^{2}, b_{1}, b_{2}⟩, Y_{2} = ⟨a_{1}^{2}, a_{2}^{2}, a_{1}b_{1}, a_{2}b_{2}⟩, Y_{3} = ⟨a_{1}a_{2}^{−1}, a_{1}a_{2}, b_{1}b_{2}, t⟩.

Thus S/Z_{∗} ≅ D_{8} ≀ C_{2} (the unique normal subgroup of index 2^{7} by Lemma 2.4(a)), Y_{1}/Z_{∗} ≅ Y_{2}/Z_{∗} ≅ C_{2}^{4}, and Y_{3}/Z_{∗} ≅ 2_{+}^{1+4}. So by Lemma 2.4(b),

𝒴 (S) ⊆ {Y_{1}, Y_{2}, Y_{3}}.
For j = 1, 2, set
Θ_{1j} = ⟨a_{j}^{2}, w_{3−j}b_{j}⟩  and  Θ_{2j} = \begin{cases}⟨a_{j}^{2}, w_{3−j}a_{j}b_{j}⟩ & if Δ ∈ D \\ ⟨a_{j}^{2}, a_{j}b_{j}⟩ & if Δ ∈ S. \end{cases}

Then for i = 1, 2, Θ_{i1} ≅ Θ_{i2} ≅ Q_{2^{n−1}}, [Θ_{i1}, Θ_{i2}] = 1 since [w_{1}b_{2}, w_{2}b_{1}] = 1, Θ_{i1} ∩ Θ_{i2} = ⟨z⟩, and thus Y_{i} = Θ_{i1}Θ_{i2} ≅ Q_{2^{n−1}} ×_{C_{2}} Q_{2^{n−1}}. Also, set

Θ_{31} = ⟨a_{1}a_{2}^{−1}, t⟩ ≅ D_{2^{n−1}}  and  Θ_{32} = ⟨a_{1}a_{2}, b_{1}b_{2}t⟩ ≅ D_{2^{n−1}} ,

so that Y_{3} = Θ_{31}Θ_{32} ≅ D_{2^{n−1}} × D_{2^{n−1}}.

When n ≥ 5 or i = 3, Y_{i} ∈ 𝒴 (S) by Lemma 2.6(a). When n = 4, Y_{1}, Y_{2} ∈ 𝒴_{0}(S) by definition (and since S/⟨z⟩ ≅ D_{8} ≀ C_{2}), and hence Y_{1}, Y_{2} ∈ 𝒴 (S) since they are normal. Thus 𝒴 (S) = {Y_{1}, Y_{2}, Y_{3}}. So by Proposition 3.9(a,b), Out_{F}(S) = 1,

E_{F} = E_{F}^{(II)} = E_{F}(Y_{1}) ∪ E_{F}(Y_{2}) ∪ E_{F}(Y_{3}), and E_{F}(Y_{i}) ≠ ∅ ∀ i = 1, 2, 3. (5.10)

For each i = 1, 2, 3, set 𝒰_{i} = 𝒰_{F}(Y_{i}) as defined in Proposition 3.11(b). For i = 1, 2, if g ∈ Y_{i} = Θ_{i1}Θ_{i2} and g^{2} = z, then g ∈ Θ_{i1} ∪ Θ_{i2}. So for each U < Y_{i} with U ≅ Q_{8}, U ≤ Θ_{ij} for some j. Since all such subgroups of Y_{i} are S-conjugate to each other, this proves that

for i = 1, 2, 𝒰_{i} = {U < Y_{i} | U ≅ Q_{8}}.  (5.11)

Let E_{F}(Y_{i}) = E_{F}^{a}(Y_{i}) ∪ E_{F}^{c}(Y_{i}) be the decomposition of Proposition 5.2(a). Case 1: Assume Δ_{1}, Δ_{2} ∈ S. Set U_{1} = ⟨w_{1}, w_{2}b_{1}⟩ ∈ 𝒰_{1}, U_{2} = ⟨w_{2}, a_{2}b_{2}⟩ ∈ 𝒰_{2}, and P = U_{1}U_{2}. The given generators for U_{1} commute with those for U_{2} except that [w_{2}b_{1}, a_{2}b_{2}] = z, so P ≅ 2_{−}^{1+4} by Lemma C.2(a). For i = 1, 2, Aut_{P} (U_{i}) ≤ Inn(U_{i}) since [U_{1}, U_{2}] = Z(S), and hence P ≤ U_{i}C_{S}(U_{i}). By Proposition 3.11(c.4), there is α_{i} ∈ Aut_{F}^{∗}(U_{i}C_{S}(U_{i})) such that |α_{i}| = 3, α_{i}(U_{i}) = U_{i}, and α_{i} induces the identity on U_{i}C_{S}(U_{i})/U_{i}. Thus α_{i}(P) = P and |α_{i}|_{P} | = 3. So by Lemma C.2(b) (with α_{i} in the role of γ_{i}), Out_{F}(P) ≅ Σ_{5} or A_{5}.

Let v_{1} ∈ ⟨a_{1}⟩ be such that v_{1}^{2} = w_{1}. Then v_{1} normalizes U_{1} and centralizes U_{2}. Set η = c_{v_{1}} ∈ Aut_{S}(P). Then η|_{U_{1}} ∉ Inn(U_{1}) while η|_{U_{2}} = Id. So by Lemma C.2(b) again, Out_{F}(P) ≅ Σ_{5}.

Set Δ_{i}^{∗} = ⟨w_{3−i}a_{i}, w_{3−i}b_{i}⟩ ≅ Q_{2^{n}} (i = 1, 2). Then [Δ_{1}^{∗}, Δ_{2}^{∗}] = ⟨z⟩, so each element of S normalizes or exchanges the Δ_{i}^{∗}, and each element of N_{S}(P) either normalizes or exchanges the two subgroups U_{i} = P ∩ Δ_{i}^{∗}. Also, U_{1} and U_{2} are not S-conjugate, since ⟨U_{1}^{S}⟩ = Y_{1} while ⟨U_{2}^{S}⟩ = Y_{2}. Hence

N_{S}(P) = N_{Δ_{1}^{∗}Δ_{2}^{∗}}(P) = N_{Δ_{1}^{∗}}(U_{1})N_{Δ_{2}^{∗}}(U_{2})
= P ⟨v_{1}, v_{2}⟩ where v_{i} ∈ ⟨a_{i}⟩ and v_{i}^{2} = w_{i} for i = 1, 2 .

Thus Out_{S}(P) ≅ N_{S}(P)/P ≅ C_{2}^{2}, so Out_{S}(P) ∉ Syl_{2}(Out_{F}(P)), and P is not fully normalized in F.

By Lemma 1.16(a) and since P is not fully normalized, there is an F-essential subgroup R ≥ N_{S}(P). Let i = 1, 2, 3 be such that R ∈ E_{F}(Y_{i}). If R ∈ E_{F}^{a}(Y_{i}), then there is T < R with |R/T | = 2 and T ∈ 𝒴_{0}(S); and since |N_{S}(P)| = 2^{7}, R = N_{S}(P) and T ≅ Q_{8} × Q_{8}, which is impossible. If R = UC_{S}(U) ∈ E_{F}^{c}(Y_{i}) for some U ∈ 𝒰_{i}, then U ≤ N_{S}(P) by Lemma A.6(c) (applied with R and N_{S}(P) in the role of S and Q), so N_{S}(P) contains a direct factor C_{2}^{2} (if i = 3) or a central factor Q_{8}, which is also impossible. We thus have a contradiction, and there is no reduced fusion system over S.

Case 2: Now assume Δ_{1}, Δ_{2} ∈ D. Let 𝒴_{0i} (i = 1, 2, 3) be the set of subgroups P ∈ 𝒴_{0}(S) whose normal closure is Y_{i}. By Lemma 2.6(a) (or by definition if Y_{i} ≅ 2_{+}^{1+4}), for i = 1, 2, 𝒴_{0i} is the set of all U_{1}U_{2} ≅ 2_{+}^{1+4}, where U_{1}, U_{2} ∈ 𝒰_{i}, and U_{j} ≤ Θ_{ij} for j = 1, 2 by (5.11). By Proposition 3.11(b.1), Out_{F}(U_{1}U_{2}) = Out(U_{1}U_{2}) ≅ Σ_{3} ≀ C_{2}, and by Lemma 5.2(c), this determines Out_{F}(R) for each R ∈ E_{F}^{a}(Y_{i}). Also, E_{F}^{c}(Y_{i}) is the set of subgroups of the form R = UC_{S}(U) ≅ Q_{8} ×_{C_{2}} Q_{2^{n}} for U ∈ 𝒰_{i}. By Proposition 3.11(c.4), Aut_{F}^{∗}(R) = O^{2}(Inn(R)⟨α⟩) for some α such that |α| = 3, α(U) = U, and α|_{C_{S}(U)} = Id. Hence Aut_{F}^{∗}(R) is also determined uniquely.

It remains to examine the subgroups in E_{F}(Y_{3}). By Proposition 3.11(a), and since Y_{3} ≅ D_{2^{n−1}} × D_{2^{n−1}}, there is a product decomposition Y_{3} = Θ_{31}^{∗} × Θ_{32}^{∗} such that 𝒰_{3} is the set of subgroups of the Θ_{3i}^{∗} ≅ D_{2^{n−1}} which are isomorphic to C_{2}^{2}. By Proposition 3.11(b), the subgroups in 𝒰_{3} are all S-conjugate. By the Krull-Schmidt theorem (Theorem A.8(a)), Θ_{3i}^{∗} ≤ Θ_{3i}⟨z⟩ (after changing indices if necessary). Hence 𝒰_{3} is the S-conjugacy class of ⟨w_{1}w_{2}^{−1}, t⟩ or of ⟨w_{1}w_{2}^{−1}, tz⟩. Define φ ∈ Aut(S) by setting φ|_{Δ_{1}Δ_{2}} = Id and φ(t) = zt. Upon replacing F by ^{φ}F if necessary, we can arrange that 𝒰_{3} is the S-conjugacy class of ⟨w_{1}w_{2}^{−1}, t⟩ (and also that Θ_{3i}^{∗} = Θ_{3i}).

Consider the subgroups

Y_{01} = ⟨w_{1}, b_{1}, w_{2}, b_{2}⟩ ∈ 𝒴_{01}  Y_{03}^{(1)} = ⟨w_{1}w_{2}^{−1}, t⟩ × ⟨w_{1}w_{2}, b_{1}b_{2}t⟩ ∈ 𝒴_{03}
Y_{02} = ⟨w_{1}, a_{1}b_{1}, w_{2}, a_{2}b_{2}⟩ ∈ 𝒴_{02} Y_{03}^{(2)} = ⟨w_{1}w_{2}^{−1}, t⟩ × ⟨w_{1}w_{2}, a_{1}a_{2}b_{1}b_{2}t⟩ ∈ 𝒴_{03} .

Set R_{i} = Y_{0i}⟨t⟩ ∈ E_{F}^{a}(Y_{i}) for i = 1, 2. Then

Out_{F}(R_{i}) ≅ N_{Out_{F} (Y_{0i})}(Out_{R_{i}}(Y_{0i}))/Out_{R_{i}}(Y_{0i}) ≅ Σ_{3}

by Lemma 1.5(a), and the subgroup of order 3 acts nontrivially on the group Aut_{R_{i}}(Y_{03}^{(i)}) ≅ C_{2}^{2}. Also, R_{1} = Y_{03}^{(1)} ⋊ ⟨b_{1}, w_{1}⟩ and R_{2} = Y_{03}^{(2)} ⋊ ⟨a_{1}b_{1}, w_{1}⟩, so there is α_{i} ∈ Aut_{F}(R_{i}) of order 3 which acts nontrivially on R_{i}/Y_{03}^{(i)}. Hence Aut_{F}(Y_{03}^{(i)}) ≅ Σ̸_{3}≀C_{2}. By Proposition 3.11(b.2), Aut_{F}(Y_{03}^{(i)}) ≅ Σ_{5}, and is the unique subgroup in 𝒜_{S}^{−}(Y_{03}^{(i)}) associated to {Y_{03}^{(i)} ∩Θ_{31}, Y_{03}^{(i)} ∩Θ_{32}} ∈ 𝒰_{S}(Y_{0}^{(i)}). Since each subgroup in 𝒴_{03} is S-conjugate to Y_{03}^{(1)} or Y_{03}^{(2)} (Lemma 2.6(a)), we have now determined Aut_{F}(P) for each P ∈ 𝒴_{03}. Also, E_{F}^{a}(Y_{3}) = ∅ since Aut_{F}(Y_{03}^{(i)}) ≇ Σ_{3} ≀ C_{2}.

By Lemma 5.2(a), E_{F}(Y_{3}) = E_{F}^{c}(Y_{3}): the set of all R = UC_{S}(U) for U ∈ 𝒰_{3}. By Proposition 3.11(c.4), Aut_{F}^{∗}(R) = O^{2}(Inn(R)⟨α⟩) for some α which induces the identity on U and on R/U. When U = ⟨w_{1}w_{2}^{−1}, t⟩, this shows that α normalizes Y_{03}^{(1)} and Y_{03}^{(2)}, hence is uniquely determined on those subgroups, and is uniquely determined on R = U × ⟨a_{1}a_{2}, b_{1}b_{2}t⟩ = Y_{03}^{(1)}Y_{03}^{(2)}.

This proves that F is completely determined by our choice of 𝒰_{3}. So up to isomorphism, there is at most one unique reduced fusion system over S.

Let q be any prime power such that n = v_{2}(q^{2} − 1), and set G = PSp_{4}(q). By [CF, § 1], the Sylow 2-subgroups of Sp_{4}(q) are isomorphic to Q_{2^{n}} ≀ C_{2}, and hence those of G are isomorphic to S ≅ (Q_{2^{n}} ×_{C_{2}} Q_{2^{n}}) ⋊^{t} C_{2}. The 2-fusion system of G is reduced by Proposition 1.12, and hence is isomorphic to F as just described. □

CHAPTER 6 Fusion systems over extensions of UT_{3}(4)

Recall that U is the class of all 2-groups S such that there is T ⊴ S with T ≅ UT_{3}(4) for which T/Z(T) is centric in S/Z(T). We now look at reduced fusion systems over 2-groups in U, using the following notation for their elements and subgroups. For the most part, this is the same notation as that used in [OV, § 4–5] (and also in Appendix C).

Notation 6.1. Set S_{0} = UT_{3}(4), the group of strictly upper triangular 3 × 3 matrices over 𝔽_{4}. Let e_{ij}^{a} denote the elementary matrix with nonzero entry a in position (i, j). Set

A_{1} = \{( \begin{matrix}1 & a & b \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) = e_{12}^{a}e_{13}^{b} | a, b ∈ 𝔽_{4}\} and A_{2} = \{( \begin{matrix}1 & 0 & b \\ 0 & 1 & a \\ 0 & 0 & 1 \end{matrix} ) = e_{23}^{a}e_{13}^{b} | a, b ∈ 𝔽_{4}\}.

Let a ↦ ā = a^{2} be the (nontrivial) automorphism of 𝔽_{4}, and write 𝔽_{4} = {0, 1, ω, ω̄}. Note the relation

[e_{12}^{a}, e_{23}^{b}] = e_{13}^{ab} for all a, b ∈ 𝔽_{4}. (6.1)

Let τ ∈ Aut(S_{0}) be the graph automorphism defined by transpose inverse; thus τ(e_{ij}^{a}) = e_{4−j,4−i}^{a}. Let ϕ ∈ Aut(UT_{3}(4)) be the field automorphism ϕ(e_{ij}^{a}) = eā ; ij, and set θ = ϕ ∘ τ = τ ∘ ϕ.

Set S_{ϕ,τ} = S_{0} ⋊^{ϕ,τ} ⟨ϕ, τ ⟩, the semidirect product where c_{ϕ} = ϕ, c_{τ} = τ, and ⟨ϕ, τ ⟩ ≅ C_{2}^{2}. Let S_{ϕ,τ}^{∗} = S_{0}⟨ϕ, τ ⟩ be the nonsplit extension where

S_{0} ⊴ S_{ϕ,τ}^{∗} , c_{ϕ} = ϕ, c_{τ} = τ, ϕ^{2} = e_{13}^{1}, [ϕ, τ] = 1, τ^{2} = e_{13}^{1} .

Set θ = ϕτ in both groups. Let S_{τ} , S_{θ}, S_{ϕ} < S_{ϕ,τ} and S_{τ}^{∗}, S_{θ}^{∗}, S_{ϕ}^{∗} < S_{ϕ,τ}^{∗} be the subgroups generated by S_{0}, and τ, θ, or ϕ, respectively.

Note that S_{ϕ}^{∗} is a semidirect product, since (e_{13}^{ω}ϕ)^{2} = e_{13}^{1}ϕ^{2} = 1 in S_{ϕ,τ}^{∗}. Thus the choice that θ^{2} = 1 and not ϕ^{2} = 1 was arbitrary, and was made to simplify some of the later formulas.

We first show that each S ∈ U is isomorphic to one of the groups listed above.

Lemma 6.2. Each group S ∈ U is isomorphic to one of the groups UT_{3}(4), S_{ϕ}, S_{θ}, S_{τ}, S_{τ}^{∗}, S_{ϕ,τ}, or S_{ϕ,τ}^{∗}.

Proof. Assume S ∈ U, and fix T ⊴ S such that

T ≅ UT_{3}(4) and C_{S/Z(T)}(T/Z(T)) = T/Z(T).

Since O_{2}(Out(T)) is the subgroup of all [α] ∈ Out(T) such that α induces the identity on T/Z(T) (Lemma C.8), the condition that T/Z(T) be centric in S/Z(T) implies that Out_{S}(T) ∩ O_{2}(Out(T)) = 1.

70 6. FUSION SYSTEMS OVER EXTENSIONS OF UT_{3}(4)

We identify T = S_{0} = UT_{3}(4) via some choice of isomorphism T ≅ S_{0}. Set Γ = O_{2}(Out(S_{0}))⟨[ϕ], [τ]⟩ ≤ Out(S_{0}). Then Γ ∈ Syl_{2}(Out(S_{0})) by Lemma C.8, so Out_{S}(S_{0}) is Out(S_{0})-conjugate to a subgroup of Γ. Hence after changing our choice of identification isomorphism T ≅ S_{0}, we can assume that Out_{S}(S_{0}) ≤ Γ. Since ⟨ϕ, τ⟩ permutes freely a basis for O_{2}(Out(S_{0})) ≅ C_{2}^{4} by Lemma C.8, H^{1}(⟨ϕ, τ⟩; O_{2}(Out(S_{0}))) = 0, and similarly for subgroups of ⟨ϕ, τ⟩. Hence Out_{S}(S_{0}) is Out(S_{0})-conjugate to a subgroup of ⟨[τ], [ϕ]⟩ (cf. [Br, Proposition IV.2.3]), since both are complementary to O_{2}(Out(S_{0})) in a certain subgroup of Γ. So upon changing the isomorphism T ≅ S_{0} again, we can arrange that Out_{S}(S_{0}) ≤ ⟨[τ], [ϕ]⟩.

The result now follows from the cohomology computations:

H^{2}(⟨ϕ⟩; Z(S_{0})) = H^{2}(⟨θ⟩; Z(S_{0})) = 0
H^{2}(⟨τ⟩; Z(S_{0})) ≅ Z(S_{0})
H^{2}(⟨ϕ, τ⟩; Z(S_{0})) ≅ H^{2}(⟨τ⟩; ⟨e_{13}^{1}⟩) ≅ ⟨e_{13}^{1}⟩.

These all follow from the formula H^{2}(⟨γ⟩; M) = C_{M}(γ)/⟨xγ(x)⟩ when |⟨γ⟩| = 2 (cf. [Br, pp. 58–59]), except for the first isomorphism in the third line which follows from Shapiro’s lemma (cf. [Br, Proposition III.6.2]). The three nonsplit extensions of S_{0} by τ are isomorphic via the automorphism γ_{0} ∈ Aut(S_{0}) (see Lemma C.8) which permutes transitively the set Z(T)^{#} = {e_{13}^{a} | a ∈ 𝔽_{4}^{×}}. So all of them are isomorphic to S_{τ}^{∗}. □

The following lemma about subgroups of S_{0}, S_{τ}, and S_{τ}^{∗} will also be needed.

Lemma 6.3. Assume S = S_{τ} or S_{τ}^{∗}.

(a) There are exactly three subgroups of S isomorphic to C_{4} × C_{4}: the subgroups

H_{i} = ⟨e_{12}^{1}e_{23}^{ω^{i}}, e_{12}^{ω}e_{23}^{ω^{i+1}}⟩ for i = 0, 1, 2.

(b) The only subgroups of S isomorphic to C_{2}^{4} are A_{1} and A_{2}.

Proof. Set Z_{0} = Z(S_{0}) = {e_{13}^{a} | a ∈ 𝔽_{4}} for short.

(a) If H ≤ S_{0} has order 16 and contains Z_{0}, then

H = Z_{0}⟨e_{12}^{a}e_{23}^{b}, e_{12}^{c}e_{23}^{d}⟩

for some a, b, c, d ∈ 𝔽_{4}. Since [e_{12}^{x}, e_{23}^{y}] = e_{13}^{xy} ∈ Z(S_{0}) by (6.1), H is abelian if and only if ad = bc. Thus the three subgroups H_{i} (i = 0, 1, 2) together with A_{1} ≅ C_{2}^{4} and A_{2} ≅ C_{2}^{4} are the only abelian subgroups of order 16 in S_{0}, and the H_{i} are the only ones isomorphic to C_{4} × C_{4}.

Conversely, for each i = 0, 1, 2, Ω_{1}(H_{i}) ⊆ H_{i} ∩ (A_{1} ∪ A_{2}) = Z_{0} by Lemma C.6(a), so H_{i} ≅ C_{4} × C_{4} since it is abelian of order 16.

Assume H ≤ S is such that H ≰ S_{0} and H ≅ C_{4} × C_{4}. Then Ω_{1}(H) ≤ Fr(S) < S_{0}, Ω_{1}(H) ⊆ (A_{1} ∪ A_{2}) since all elements of S_{0}∖(A_{1} ∪ A_{2}) have order 4, and Ω_{1}(H) = Z_{0} since no element of A_{i}∖Z_{0} commutes with any element of S_{0}τ. Thus H > Z_{0}.

Let g ∈ S_{0} and h ∈ S_{0}∖Z_{0} be such that H = ⟨gτ , h⟩. Then (gτ)^{2} ∈ Z_{0} implies that gτ(g) ∈ Z_{0}, and [h, gτ] = 1 implies that hτ(h)^{−1} ∈ Z_{0}. Since C_{S_{0}/Z_{0}}(τ) = H_{0}/Z_{0}, we have g, h ∈ H_{0}. Thus [h, g] = 1 since H_{0} is abelian, so [h, τ] = 1, and h = τ(h). But this is impossible: h ≡ e_{12}^{a}e_{23}^{a}h_{0} for some h_{0} ∈ Z_{0} and some 0 ≠ a ∈ 𝔽_{4}, and τ(h) = e_{23}^{a}e_{12}^{a}h_{0} = he_{13}^{a^{2}}.

(b) Assume P < S and P ≅ C_{2}^{4}. Thus |P ∩ S_{0}| ≥ 8. Since I(S_{0}) ⊆ A_{1} ∪ A_{2}, there is x ∈ (A_{i}∖Z_{0}) ∩ P for some i = 1, 2. But then P ≤ C_{S}(x) = A_{i} (recall that τ(A_{i}) = A_{3−i}), so P = A_{i}. □

We are now ready to list the reduced fusion systems over groups in the class U, beginning with S_{0} = UT_{3}(4) itself.

Proposition 6.4. Each reduced fusion system over UT_{3}(4) is isomorphic to the fusion system of PSL_{3}(4).

Proof. Set S = S_{0} = UT_{3}(4) and Z = Z(S) = {e_{13}^{a} | a ∈ 𝔽_{4}}. Let F be a saturated fusion system over S such that O_{2}(F) = 1. Each F-essential subgroup of S contains Z = Fr(S), and thus is normal in S. If P ∈ E_{F}^{(I)}, then P ≅ C_{2}^{4} by Proposition 3.4, and hence P = A_{1} or A_{2} by Lemma 6.3(b). If P ≤ S has index 2, then [P, P] = Z = Fr(S) by Lemma C.6(b), hence [g, P] ≤ Fr(P) for each g ∈ S∖P, which by Lemma 1.8 implies P is not essential. Thus E_{F} ⊆ {A_{1}, A_{2}}. If E_{F} = ∅, then S ⊴ F, while if E_{F} = {A_{i}}, then A_{i} ⊴ F. Since we are assuming O_{2}(F) = 1, E_{F} = {A_{1}, A_{2}}.

For each i = 1, 2, Aut_{S}(A_{i}) ≅ C_{2}^{2}, and C_{A_{i}}(S) = Z has rank 2. Hence by Lemma 3.3(c), Aut_{F}(A_{i}) ≅ SL_{2}(4) or GL_{2}(4), and is conjugate in Aut(A_{i}) to Aut_{G_{i}}(A_{i}), where G_{i} = PSL_{3}(4) or PGL_{3}(4).

Fix α ∈ Aut(A_{1}) such that ^{α}Aut_{F}(A_{1}) = Aut_{G_{1}}(A_{1}). Upon composing with an appropriate element of Aut_{F}(A_{1}), we can assume that α commutes with conjugation by e_{23}^{1}. Then α(Z) = Z since Z = [e_{23}^{1}, A_{1}]. Upon composing by ϕ|_{A_{1}} if necessary, we can assume that (α|_{Z})^{3} = Id, and then upon composing by an appropriate element in C_{Aut(A_{1})}(Aut_{F}(A_{1})) ≅ C_{3}, we can assume that α|_{Z} = Id (and still α commutes with conjugation by e_{23}^{1}). Since conjugation by e_{23}^{1} induces an isomorphism from A_{1}/Z to Z, α also induces the identity on A_{1}/Z.

By a similar argument, there is β ∈ Aut(A_{2}) such that β|_{Z} = Id, [β, A_{2}] ≤ Z, and ^{β}Aut_{F}(A_{2}) = Aut_{G_{2}}(A_{2}). Let φ ∈ Aut(S) be such that φ|_{A_{1}} = α and φ|_{A_{2}} = β. (Note that φ has the form φ(g) = gχ(g) for some χ ∈ Hom(S, Z(S)).) By the extension axiom (and since all automorphisms of S of odd order normalize A_{1} and A_{2}), for i = 1 or 2, Aut_{F}(S) ≅ C_{3} if Aut_{F}(A_{i}) = SL_{2}(4), and Aut_{F}(S) ≅ C_{3} × C_{3} otherwise. Thus Aut_{F}(A_{1}) ≅ Aut_{F}(A_{2}), G_{1} = G_{2}, and ^{φ}F = F_{S}(G_{1}).

We have now shown that each saturated fusion system F over S such that O_{2}(F) = 1 is isomorphic to F_{S}(PSL_{3}(4)) or F_{S}(PGL_{3}(4)). So if F is reduced, then it is the fusion system of PSL_{3}(4). □

We now look at extensions of UT_{3}(4). Since reduced fusion systems over S_{ϕ} and S_{θ} were described in [OV, § 4–5], it remains to examine fusion systems over S_{τ}, S_{τ}^{∗}, S_{ϕ,τ}, and S_{ϕ,τ}^{∗}.

Proposition 6.5. There are no reduced fusion systems over S_{τ} nor over S_{τ}^{∗}.

Proof. Assume S = S_{τ} or S_{τ}^{∗} in the notation of 6.1. Let F be any reduced fusion system over S. If P ∈ E_{F}^{(I)}, then by Propositions 3.4 and 3.5, P ≅ C_{2}^{4} or 2_{−}^{1+4}. The latter case cannot occur (P would have to be normal, and hence contain Z(S) = Z(S_{0}) ≅ C_{2}^{2} by Lemma C.9). So E_{F}^{(I)} ⊆ {A_{1}, A_{2}} by Lemma 6.3(b).

Assume P ∈ E_{F}^{(II)}. Since |S| = 2^{7} and S ≅ D̸_{8} ≀ C_{2}, 𝒴 (S) = ∅. By Lemma C.9, there are no normal dihedral or quaternion subgroups, so 𝒳 (S) = ∅. So by Theorem 3.1(b), P is in an F-essential pair of the type described in Lemma 3.7(b). This would require a subgroup T ≅ C_{2}^{4} with normalizer of order 2^{7}, which contradicts Lemma 6.3(b). Thus E_{F}^{(II)} = ∅.

Now assume P ∈ E_{F}^{(III)}; i.e., [S:P] = 2. Then by (6.1) and Lemma 6.3(a),
P ≥ Fr(S) = H_{0} = ⟨e_{12}^{1}e_{23}^{1}, e_{12}^{ω}e_{23}^{ω}⟩ ≅ C_{4} × C_{4}.

For g ∈ S∖P, [g, P] ≤ H_{0} and

[g, H_{0}] ≤ [S_{0}, S_{0}]·[ϕ, H_{0}] = Z(S_{0}) = Fr(H_{0}) ≤ Fr(P).

So by Lemma 1.8, H_{0} is not charateristic in P. Thus H_{0} is not the only subgroup of P isomorphic to C_{4} × C_{4}, H_{j} ≤ P for j ∈ {1, 2} by Lemma 6.3(a), and P ≥ H_{0}H_{j} = S_{0}. So P = S_{0} in this case.

Thus E_{F} ⊆ {A_{1}, A_{2}, S_{0}}. Also, [Aut_{F}(S), S] ≤ S_{0} since S_{0} is a characteristic subgroup of index 2 in S by Lemma C.9. Hence by Proposition 1.14(b), 𝔣𝔬𝔠(F) ≤ S_{0}, and F is not reduced. □

We now turn to fusion systems over S = S_{ϕ,τ} or S_{ϕ,τ}^{∗}. Set Z = Z(S) = ⟨e_{13}^{1}⟩. There is an epimorphism χ: S ⟶ D_{8} ≀ C_{2} with kernel Z, whose inverse is defined in Table 6.1. Here, we follow Notation 5.4 for elements a_{i}, b_{i}, z_{i}, t ∈ D_{8} ≀ C_{2}. To see

\begin{matrix}i = & χ^{−1}(a_{i}) & χ^{−1}(b_{i}) & χ^{−1}(a_{i}b_{i}) & χ^{−1}(z_{i}) & χ^{−1}(t) \\ 1 & e_{23}^{1}τ Z & e_{12}^{1}ϕZ & θZ & e_{12}^{1}e_{23}^{1}Z & e_{12}^{ω}Z \\ 2 & e_{12}^{ω}e_{23}^{ω}τ Z & ϕZ & e_{12}^{ω}e_{23}^{ω}θZ & e_{12}^{1}e_{23}^{1}e_{13}^{ω}Z \end{matrix} Table 6.1

that this is a well defined isomorphism, it suffices to check that the images under χ^{−1} of a_{1}, b_{1}, a_{1}b_{1}, and z_{1} satisfy the relations needed to lie in a dihedral group, that conjugation by e_{12}^{ω} sends each coset in the first row to the corresponding coset in the second, and that ⟨e_{23}^{1}τ , e_{12}^{1}ϕ⟩ commutes with ⟨e_{12}^{ω}e_{23}^{ω}τ , ϕ⟩ (modulo Z).

Proposition 6.6. Let F be a reduced fusion system over S, where S ∈ U and |S| = 2^{8}. Then S ≅ S_{ϕ,τ}, and F is isomorphic to the fusion system of Lyons’s group. Also, Out(S, F) = 1, and S_{τ} is the unique F-essential subgroup with noncyclic center.

Proof. By Lemma 6.2(a), S ≅ S_{ϕ,τ} or S_{ϕ,τ}^{∗}. So assume S = S_{ϕ,τ} or S_{ϕ,τ}^{∗}. We use the notation of 6.1 for elements and subgroups of S, and in particular use (6.1) (without always saying so) for commutator relations among the elements e_{ij}^{a}. Step 1: Set

Y_{1} = χ^{−1}(⟨z_{1}, b_{1}, z_{2}, b_{2}⟩) = Z(S_{0})⟨e_{12}^{1}, e_{23}^{1}, ϕ⟩
Y_{2} = χ^{−1}(⟨z_{1}, a_{1}b_{1}, z_{2}, a_{2}b_{2}⟩) = Z(S_{0})⟨e_{12}^{1}e_{23}^{1}, e_{12}^{ω}e_{23}^{ω}, θ⟩
Y_{3} = χ^{−1}(⟨a_{1}a_{2}, b_{1}b_{2}, z_{1}, τ ⟩) = S_{0} .

We first show that

Y_{1} ≅ 2_{+}^{1+4},  Y_{2} ≅ 2_{−}^{1+4},  and  Y_{1}⟨e_{12}^{ω}⟩ = A_{1}⟨e_{23}^{1}, ϕ⟩ ≅ UT_{4}(2) . (6.2)

The third isomorphism follows from Lemma C.4(a), since ⟨e_{23}^{1}, ϕ⟩ ≅ C_{2}^{2}, and the ⟨e_{23}^{1}, ϕ⟩-orbit of e_{12}^{ω} is a basis of A_{1}. Hence Y_{1} ≅ 2_{+}^{1+4} by Lemma C.4(b), and since χ(Y_{1}) ≅ Y_{1}/Z is the unique abelian subgroup of index 2 in χ(Y_{1}⟨e_{12}^{ω}⟩) = ⟨z_{1}, b_{1}, z_{2}, b_{2}, t⟩. Alternatively, Y_{1} = U_{1}U_{2} where U_{i} = χ^{−1}(⟨z_{i}, b_{i}⟩), U_{1} and U_{2} are both dihedral (if S = S_{ϕ,τ}) or quaternion (if S = S_{ϕ,τ}^{∗}), and [U_{1}, U_{2}] = 1 (so Y_{1}⟨e_{12}^{ω}⟩ ≅ UT_{4}(2) by Lemma C.4(b)). As for Y_{2}, the five elements

w_{1} = e_{13}^{ω}, w_{2} = θ, w_{3} = e_{12}^{1}e_{23}^{1}θ, w_{4} = e_{12}^{ω}e_{23}^{ω}θ, w_{5} = e_{12}^{ω}e_{23}^{ω}θ  (6.3)

all have order 2, and [w_{i}, w_{j}] = e_{13}^{1} for i ≠ j. So the associated quadratic form (gZ ↦ g^{2}) on Y_{2}/Z has exactly five isotropic points, and by Lemma A.5, it is the nondegenerate nonhyperbolic form and hence Y_{2} ≅ 2_{−}^{1+4}.

Since χ(Y_{1}) ≅ χ(Y_{2}) ≅ C_{2}^{4} and χ(Y_{3}) = χ(S_{0}) ≅ 2_{+}^{1+4}, 𝒴 (S) ⊆ {Y_{1}, Y_{2}, Y_{3}} by Lemma 2.4(b). By Definition 2.1(c,d) and since S/Z ≅ D_{8} ≀ C_{2}, Y_{1}, Y_{2} ∈ 𝒴_{0}(S), and lie in 𝒴 (S) since they are normal. If Y_{3} = S_{0} ∈ 𝒴 (S), then since S_{0} ∉ 𝒴_{0}(S) (Definition 2.1(c) again), it must be the normal closure of some Y_{0} ∈ 𝒴_{0}(S) of index 4 in S_{0} (Lemma 2.4(b)), Y_{0} ≅ C_{2}^{4}, so Y_{0} = A_{1} or A_{2} by Lemma 6.3(b), which is not possible since N_{S}(A_{1}) = S_{0}⟨ϕ⟩ ≅ D̸_{8} ≀ C_{2}. Thus

𝒴_{0}(S) = 𝒴 (S) = {Y_{1}, Y_{2}} .  (6.4)

Note also that

Aut(S) and Aut(S_{0}⟨ϕ⟩) are 2-groups:  (6.5)

the first by Corollary 2.5 and since 𝒴 (S) ≠ ∅, and the second by [OV, Lemma 5.5]. These also follow from the description of Aut(S_{0}) in Lemma C.8 and since S_{0} is characteristic in both groups. Step 2: By Proposition 3.9(a), Out_{F}(S) = 1,

E_{F} = E_{F}(Y_{1}) ∪ E_{F}(Y_{2}) ∪ E_{F}(Y_{3}), and E_{F}(Y_{i}) ≠ ∅ for each i = 1, 2, 3. (6.6)

By Proposition 3.9(c) and (6.4), E_{F}(Y_{i}) ⊆ E_{F}^{(II)} for i = 1, 2. By Proposition 3.11(b.1),

Y_{1} ≅ 2_{+}^{1+4} ⟹ Out_{F}(Y_{1}) = Out(Y_{1}) ≅ SO_{4}^{+}(2) ≅ Σ_{3} ≀ C_{2}
Y_{2} ≅ 2_{−}^{1+4} ⟹ Out_{F}(Y_{2}) = Out(Y_{2}) ≅ SO_{4}^{−}(2) ≅ Σ_{5} .

(6.7)

Next assume R ∈ E_{F}(Y_{3}). By Proposition 3.9(c) and since Y_{3} ∉ 𝒴 (S), R has type (II) or (III). If R ∈ E_{F}^{(III)}(Y_{3}), then by the same lemma, R > Y_{3} = S_{0}. Since S_{0}⟨ϕ⟩ ∉ E_{F} by (6.5), E_{F}^{(III)}(Y_{3}) ⊆ {S_{0}⟨θ⟩, S_{0}⟨τ ⟩}.

Let (R_{1}, R_{2}) be an F-essential pair of type (II) in E_{F}(Y_{3}), and set T = R_{1} ∩R_{2}. By Theorem 3.1(b), this has the type described in Lemma 3.7(b). Thus T ≅ C_{2}^{4}, |R_{1}/T | = 2, T is the L_{2}(4)-module for Aut_{F}(T) ≥ Σ_{5}, Out_{R_{1}}(T) ≰ O^{2}(Out_{F}(T)), and 𝔣𝔬𝔠(F, R_{1}) = S_{0} is the normal closure of T. Thus T = A_{1} or A_{2} by Lemma 6.3(b).

Since N_{S}(A_{1}) = S_{0}⟨ϕ⟩ and rk(C_{A_{1}}(⟨e_{23}^{1}, ϕ⟩)) = 1, and since A_{1} is the L_{2}(4)- module for O^{2}(Aut_{F}(A_{1})) ≅ A_{5}, Aut_{⟨e_{23}^{1},ϕ⟩}(A_{1}) is not contained in O^{2}(Aut_{F}(A_{1})). Hence O^{2}(Aut_{F}(A_{1})) ∩ Aut_{S}(A_{1}) = Aut_{S_{0}}(A_{1}), and similarly for A_{2}. Thus R_{1} = T ⟨x⟩ for some x ∈ S_{0}⟨ϕ⟩∖S_{0} such that x^{2} ∈ T, and this implies that R_{1} is Sconjugate to A_{1}⟨ϕ⟩.

Set H_{1} = A_{1}⟨ϕ⟩. Set N_{1} = N_{S}(H_{1}) = A_{1}⟨e_{23}^{1}, ϕ⟩ = Y_{1}⟨e_{12}^{ω}⟩. Then N_{1} ≅

UT_{4}(2) by (6.2), and UT_{4}(2) ≅ (Q_{8} ×_{C_{2}} Q_{8}) ⋊^{t} C_{2} by Lemma C.4(b). So N_{1} ∈ E_{F}^{a}(Y_{1}) ⊆ E_{F} by Lemma 5.2(a), and there is Id ≠ γ ∈ Aut_{F}(N_{1}) of odd order. By Lemma C.4(d), γ permutes transitively the three subgroups of index 2 in N_{1} which contain A_{1}. Thus H_{1} = A_{1}⟨ϕ⟩ is F-conjugate to A_{1}⟨e_{23}^{1}⟩ with normalizer S_{0}⟨ϕ⟩, so H_{1} is not fully normalized, and hence not in E_{F}. We conclude that:

E_{F}(Y_{3}) ⊆ {S_{0}⟨θ⟩, S_{0}⟨τ ⟩} .  (6.8)

Step 3: By Lemma 5.2(b), for i = 1, 2 and R_{i} ∈ E_{F}(Y_{i}), R_{i} ≥ Y_{i}, and hence Z(R_{i}) = Z(Y_{i}) = Z(S) = ⟨e_{13}^{1}⟩ by Lemma A.6(a). So by (6.6) and (6.8), the only (possible) subgroup R ∈ E_{F} with Z(R) > Z(S) is S_{0}⟨τ ⟩. Since F is reduced, Z(S) ≰ O_{2}(F) = 1, so S_{0}⟨τ ⟩ ∈ E_{F}, and there is β ∈ Aut_{F}(S_{0}⟨τ ⟩) of odd order such that β|_{Z(S_{0})} has order 3.

For each g ∈ S_{0}, (gτ)^{2} = gτ(g)τ^{2}. If gτ(g) ∈ Z(S_{0}) = {e_{13}^{x} | x ∈ 𝔽_{4}}, then g ≡ e_{12}^{a}e_{23}^{a} (mod Z(S_{0})) for some a ∈ 𝔽_{4}, which implies that gτ(g) = 1. Thus the only element of Z(S_{0}) which is the square of an element in the coset S_{0}τ is τ^{2}. Since β permutes transitively the elements of Z(S_{0})^{#}, this implies that τ^{2} = 1. In other words, S = S_{ϕ,τ}, and there are no reduced fusion systems over S_{ϕ,τ}^{∗}.

Step 4: From now on, we assume S = S_{ϕ,τ} (i.e., τ^{2} = 1). We can now write S_{θ} = S_{0}⟨θ⟩, S_{τ} = S_{0}⟨τ ⟩, and S_{ϕ} = S_{0}⟨ϕ⟩.

For i = 1, 2, let 𝒰_{i} = 𝒰_{F}(Y_{i}) be as in Proposition 3.11(b). Since Out_{F}(Y_{i}) = Out(Y_{i}) by (6.7), these sets are uniquely determined by Proposition 3.11(b.2), and they in turn determine the sets E_{F}(Y_{i}) = E_{F}^{a}(Y_{i}) ∪ E_{F}^{c}(Y_{i}) (Lemma 5.2(a)). For P ∈ E_{F}^{a}(Y_{i}), Aut_{F}(P) is uniquely determined by Lemma 5.2(c). For P ∈ E_{F}^{c}(Y_{i}), P = UC_{S}(U) for some U ∈ 𝒰_{i}, and by Proposition 3.11(c.4), Aut_{F}^{∗}(P) = O^{2}(Inn(P)⟨σ⟩) for some σ of order 3 such that σ(U) = U and σ|_{C_{S}(U)} = Id. Thus all F-automorphisms of subgroups in E_{F}(Y_{1}) ∪ E_{F}(Y_{2}) are uniquely determined by these conditions.

The elements w_{1}, …, w_{5} ∈ Y_{2} of (6.3) are permuted under the action of Out_{F}(Y_{2}) ≅ Σ_{5}, and Out_{S_{θ}}(Y_{2}) is generated by the elements c_{e_{12}^{1}} and c_{e_{12}^{ω}}, corresponding to the permutations (2 3)(4 5) and (2 4)(3 5), respectively. Let ρ ∈ Out_{F}(Y_{2}) be an automorphism of order 3 which induces the 3-cycle (3 4 5) (i.e., ρ permutes cyclically the elements w_{3}, w_{4}, w_{5}). Then ρ normalizes Aut_{S_{θ}}(Y_{2}), and hence by the extension axiom extends to some ρ̂ ∈ Aut_{F}(S_{θ}). We showed in Step 3 that there is β ∈ Aut_{F}(S_{τ}) of order 3 (and thus E_{F}(Y_{3}) = {S_{θ}, S_{τ} } by (6.8)). In particular, [ρ̂|_{S_{0}}, θ], [β|_{S_{0}}, τ] ∈ Inn(S_{0}).

Recall the description of Aut(S_{0}) before Lemma C.8:

Aut(S_{0}) = O_{2}(Aut(S_{0}))·(Γ_{0} × Γ_{1}), where Γ_{0} = ⟨γ_{0}, θ⟩ ≅ Σ_{3}, Γ_{1} = ⟨γ_{1}, τ⟩ ≅ Σ_{3}.

Hence ρ̂|_{S_{0}} ≡ γ_{1}^{±1} and β|_{S_{0}} ≡ γ_{0}^{±1} (mod O_{2}(Aut(S_{0}))),

Out_{F}(S_{0}) = ⟨Out_{S}(S_{0}), [ρ̂|_{S_{0}}], [β|_{S_{0}}]⟩ ≅ Σ_{3} × Σ_{3}

(it cannot be larger by the Sylow axiom), and O_{2}(Out(S_{0}))·Out_{F}(S_{0}) = Out(S_{0}). So by Lemma A.7, applied with G = Out(S_{0}), Q = O_{2}(G), H_{0} = Out_{S}(S_{0}), and H = ⟨Out_{S}(S_{0}), [γ_{0}], [γ_{1}]⟩, there is φ_{0} ∈ O_{2}(Aut(S_{0})) such that

[φ_{0}] ∈ C_{O_{2}(Out(S_{0}))}(Out_{S}(S_{0})) and ^{φ_{0}}Aut_{F}(S_{0}) = ⟨Aut_{S}(S_{0}), γ_{0}, γ_{1}⟩.

By Lemma C.8, Out_{S}(S_{0}) ≅ C_{2}^{2} permutes freely a basis for O_{2}(Out(S_{0})) ≅ C_{2}^{4}, so rk(C_{O_{2}(Out(S_{0}))}(Out_{S}(S_{0}))) = 1. Define ψ ∈ Aut(S) by setting ψ(g) = g for g ∈ Y_{1}Y_{2} and ψ(g) = e_{13}^{1}g for g ∈ S∖Y_{1}Y_{2}. Since |S/Y_{1}Y_{2}| = 2 and e_{13}^{1} ∈ Z(S), this does define an automorphism of S. Also, [ψ|_{S_{0}}] centralizes Out_{S}(S_{0}) since it extends to S, ψ|_{S_{0}} ∈ O_{2}(Aut(S_{0})) since it is the identity modulo Z(S_{0}), and ψ|_{S_{0}} ∉ Inn(S_{0}). (Recall that [g, S_{0}] = Z(S_{0}) for g ∈ S_{0}∖Z(S_{0}).) We have now shown

C_{O_{2}(Out(S_{0}))}(Out_{S}(S_{0})) = ⟨[ψ|_{S_{0}}]⟩ ≠ 1 where ψ ∈ Aut(S), ψ|_{Y_{1}Y_{2}} = Id. (6.9)

Thus φ_{0} ∈ Inn(S_{0})⟨ψ|_{S_{0}}⟩, and hence φ_{0} extends to some φ ∈ Aut(S). Upon replacing F by ^{φ}F, we can assume that Aut_{F}(S_{0}) = ⟨Aut_{S}(S_{0}), γ_{0}, γ_{1}⟩.

Define γ̇_{0} ∈ Aut(S_{τ}) and γ̇_{1} ∈ Aut(S_{θ}) by setting γ̇_{i}|_{S_{0}} = γ_{i}, γ̇_{0}(τ) = τ, and γ̇_{1}(θ) = θ. Since H^{1}(⟨θ⟩; Z(S_{0})) = 0 (since θ exchanges the elements in the basis {e_{13}^{ω}, eω̄ ; 13} ⊆ Z(S_{0})), there is a unique extension of γ_{1} to γ̇_{1} ∈ Out(S_{θ}) (unique modulo c_{e_{13}^{1}}), and Aut_{F}(S_{θ}) = ⟨Aut_{S}(S_{θ}), γ̇_{1}⟩.

The choice of extension of γ_{0} to S_{τ} is not unique. Set

G = {α ∈ Aut(S_{τ}) | α|_{S_{0}} ∈ ⟨γ_{0}, θ⟩} and V = {α ∈ G | α|_{S_{0}} = Id}.

Then V ≅ Hom(S_{τ} /S_{0}, Z(S_{0})) ≅ C_{2}^{2}, and hence G ≅ Σ_{4}. Also, V ∩ Aut_{F}(S_{τ}) = 1 by the Sylow axiom. For any γ^{∗} ∈ Aut_{F}(S_{τ}) such that γ^{∗}|_{S_{0}} = γ_{0}, we have G∩Aut_{F}(S_{τ}) = ⟨γ^{∗}, c_{θ}⟩. By Lemma A.7 (or by a direct check since G ≅ Σ_{4}), there is α ∈ C_{V} (c_{θ}) such that ^{α}(γ̇_{0}) = γ^{∗}. Then either α = Id, or α(τ) = e_{13}^{1}τ. In either case, α extends to an automorphism α̂ of S, and upon replacing F by α̂F, we can arrange that Aut_{F}(S_{τ}) = ⟨Aut_{S}(S_{τ}), γ̇_{0}⟩. Also, Out_{F}(S) = 1 by (6.5). Step 5: We have now shown that up to isomorphism, there is at most one reduced fusion system F over S = S_{ϕ,τ}. By [Ly, Proposition 2.5], Lyons’ group Ly contains a subgroup isomorphic to 3McL:2 with odd index. Also, S_{ϕ} is isomorphic to a Sylow 2-subgroup of McL, and this group has an outer automorphism whose restriction to S_{0} is τ (see, e.g., [AOV1, Table 4.1] and the proof of Proposition 4.5 there). So by Lemma 6.2(a), Aut(McL) and hence Ly have Sylow 2-subgroups isomorphic to S_{ϕ,τ} or S_{ϕ,τ}^{∗}. Since Aut(S_{ϕ,τ}) and Aut(S_{ϕ,τ}^{∗}) are both 2-groups by Corollary 2.5, the fusion system of Ly is reduced by Proposition 1.12. Hence the Sylow 2-subgroups of Ly are isomorphic to S_{ϕ,τ}, and its fusion system is isomorphic to F.

It remains to prove that Out(S, F) = 1. Fix φ ∈ Aut(S, F), and set φ_{0} = φ|_{S_{0}}. Upon replacing φ by some other element of φ ∘ Inn(S), we can assume that φ_{0} ∈ O_{2}(Aut(S_{0}))⟨γ_{0}, γ_{1}⟩. Since Aut(S) is a 2-group by (6.5), this implies that φ_{0} ∈ O_{2}(Aut(S_{0})). Also, [φ_{0}] ∈ Out(S_{0}) centralizes Out_{S}(S_{0}) = ⟨τ, ϕ⟩, since φ_{0} = φ|_{S_{0}} where φ normalizes each of the subgroups S_{θ}, S_{τ}, and S_{ϕ} (since they are pairwise nonisomorphic). So φ_{0} ∈ Inn(S_{0})⟨ψ|_{S_{0}}⟩ by (6.9). Since φ is fusion preserving, φ_{0} normalizes Aut_{F}(S_{0}) = ⟨Aut_{S}(S_{0}), γ_{0}, γ_{1}⟩, and hence [φ_{0}, Aut_{F}(S_{0})] ∈ Inn(S_{0}) (recall that φ_{0} ∈ O_{2}(Aut(S_{0}))). Since [ψ|_{S_{0}}, γ_{0}] ∉ Inn(S_{0}) (since γ_{0} does not normalize Y_{1}Y_{2} ∩ S_{0}), this proves that φ_{0} ∈ Inn(S_{0}). So without changing [φ] ∈ Out(S), we can assume that φ_{0} = Id.

Let g, h ∈ Z(S_{0}) be such that φ(τ) = gτ and φ(ϕ) = hϕ. The relations (hϕ)^{2} = 1 = [gτ , hϕ] imply that g, h ∈ ⟨e_{13}^{1}⟩. If g = e_{13}^{1}, then [γ̇_{0}, φ|_{S_{τ}}] sends τ to e_{13}^{ω}τ, so φ|_{S_{τ}} does not normalize Aut_{F}(S_{τ}). Thus g = 1, so φ ∈ Aut_{Z(S_{0})}(S), and hence Out(S, F) = 1. □

APPENDIX A Background results about groups

We collect here several general results about finite groups, especially p-groups, and their automorphisms.

Lemma A.1. (a) If P < S are p-groups for some prime p, then P < N_{S}(P).

(b) If P < S are p-groups, and P is characteristic in N_{S}(P), then P ⊴ S.

Proof. Part (a) is shown, for example, in [Sz1, Theorem 2.1.6]. To prove (b), assume P is characteristic in N_{S}(P). Then each g ∈ N_{S}(N_{S}(P)) normalizes P, so N_{S}(N_{S}(P)) = N_{S}(P), and hence S = N_{S}(P) by (a). □

Recall that Z_{i}(G) denotes the i-th term in the upper central series for G: Z_{0}(G) = G, and Z_{i}(G)/Z_{i−1}(G) = Z(G/Z_{i−1}(G)) for i ≥ 1.

Lemma A.2. Let S be a p-group, and let Q ⊴ S be a normal subgroup.

(a) If Q ≠ 1, then Q ∩ Z(S) ≠ 1.

(b) Assume |Z_{k}(S)| = p^{k} for some k ≥ 1. Then either Q = Z_{i}(S) for some i ≤ k, or Q > Z_{k}(S). If |Q| = p^{k+1}, then Z_{k}(S) < Q ≤ Z_{k+1}(S).

Proof. Point (a) holds since Q ∩ Z(S) = C_{Q}(S), and |C_{Q}(S)| ≡ |Q| ≡ 0 (mod p). Hence Z(S) ≤ Q if |Z(S)| = p. If |Z(S)| = p and |Q| = p^{2}, then Q/Z(S) ≤ Z(S/Z(S)) by (a) again, so Q ≤ Z_{2}(S). This proves (b) when k = 1, and the general case follows by induction on k. □

Lemma A.3. Fix a p-group S. Assume Q < S is such that [S:Q] = p and Z(Q) > Z(S). Then Q = C_{S}(x) for some x ∈ Z_{2}(S)∖Z(S). If Ω_{1}(Z(Q)) ≰ Z(S), then x can be chosen with order p.

Proof. Set S = S/Z(S), and set P = P Z(S)/Z(S) for each P ≤ S. For each x ∈ Z(Q)∖Z(S), Q ≤ C_{S}(x) < S, and Q = C_{S}(x) since [S:Q] = p. Also, Z(Q) ⊴ S since Q ⊴ S, so Z(Q) ∩ Z(S) ≠ 1 by Lemma A.2(a), and x can be chosen so that x ∈ Z_{2}(S). If Ω_{1}(Z(Q)) ≰ Z(S), then a similar argument, applied to Ω_{1}(Z(Q)), shows that x can be chosen in Ω_{1}(Z(Q)) ∩ Z_{2}(S). □

The next lemma involves abelian subgroups of index 2 or 4 in a 2-group.

Lemma A.4. Let S be a nonabelian 2-group, and let A ⊴ S be a normal abelian subgroup.

(a) If [S:A] = 2, and |[g, A]| ≥ 4 for g ∈ S∖A, then A is the unique abelian subgroup of index 2 in S.

(b) If [S:A] = 4, and |[g, A]| ≥ 4 for each g ∈ S∖A, then either A is characteristic in S, or S/[S, S] surjects onto (S/A) × (S/A). If in addition, |[g, A]| ≥ 8 for some g ∈ S∖A, or |S/[S, S]| ≤ 8, then A is the unique abelian subgroup of index 4 in S.

Proof. Note, for each g ∈ G, that |[g, A]| = |A/C_{A}(g)|, since C_{A}(g) is the

kernel of the homomorphism A ⟶^{a↦[b,a]} A, while [g, A] is its image.

(a) If [S:A] = 2, and B < S is another abelian subgroup of index 2, then for g ∈ B∖A, |[g, A]| = |A/C_{A}(g)| ≤ |A/(B ∩ A)| = 2.

(b) Assume [S:A] = 4, and |[g, A]| ≥ 4 for each g ∈ S∖A. If B ⊴ S is another abelian subgroup of index 4, then AB = S, since otherwise A and B are both abelian of index 2 in AB, contradicting (a). Also, for each g ∈ S, where g = ab for a ∈ A and b ∈ B, |[g, A]| = |[b, A]| = |A/C_{A}(b)| ≤ |A/(A ∩ B)| = 4. If B is normal, then S/[S, S] has as quotient S/(A ∩ B) ≅ (S/A) × (S/B), so |S/[S, S]| ≥ 16, and S/B ≅ S/A if B = φ(A) for some φ ∈ Aut(S).

If B ⋬ S, set S_{0} = N_{S}(B). Then [S:S_{0}] = [S_{0}:B] = 2 since B < S_{0} < S by Lemma A.1, B ≠ ^{x}B < S_{0} for x ∈ S∖S_{0}, and thus S_{0} = B·^{x}B. Set B_{0} = B ∩^{x}B. Then B_{0} ≤ Z(S_{0}), so S_{0} ∩ B_{0}A is abelian of index 2 in B_{0}A, |[g, A]| = 2 for g ∈ B_{0}∖A by (a), and this contradicts the original assumption. So B must be normal. □

The next lemma, on quadratic forms over 𝔽_{2}, is very elementary and presumably well known, but we have been unable to find references.

Lemma A.5. Let V be an 𝔽_{2}-vector space of dimension 2n (some n ≥ 1), and let 𝔮: V ⟶ 𝔽_{2} be a quadratic form. Then

\{ \begin{matrix}|𝔮^{−1}(0)| = 2^{2n−1} ± 2^{n−1} & if 𝔮 is nondegenerate \\ |𝔮^{−1}(0)| ≡ 0 (mod 2^{n}) & if 𝔮 is degenerate. \end{matrix}  (A.1)

If dim(V) = 4 and 𝔮 is nondegenerate, then either

• 𝔮 is hyperbolic, 𝔮^{−1}(1) = V_{1}^{#} ∪ V_{2}^{#} for some complementary pair of 2-dimen-sional subspaces V_{1}, V_{2} < V, and Aut(V, 𝔮) ≅ SO_{4}^{+}(2) ≅ Σ_{3} ≀ C_{2}; or

• 𝔮^{−1}(0)∖{0} is a set of 5 points permuted transitively by Aut(V, 𝔮) ≅ SO_{4}^{−}(2) ≅ Σ_{5}.

Proof. Point (A.1) is easily checked when n = 1. So fix n ≥ 2, and assume (A.1) holds for n − 1. If 𝔮 is degenerate (V^{⊥} ≠ 0), then either there is v ∈ V^{⊥} such that 𝔮(v) = 1, in which case |𝔮^{−1}(0)| = 2^{2n−1} since 𝔮(x + v) = 𝔮(x) + 1 for each x ∈ V; or V = V_{1} ⊥ V_{2} where rk(V_{1}) = 2 and 𝔮|_{V_{1}} = 0. In this last case, set 𝔮_{2} = 𝔮|_{V_{2}}; then |𝔮^{−1}(0)| = 4|𝔮_{2}^{−1}(0)| where |𝔮_{2}^{−1}(0)| ≡ 0 (mod 2^{n−2}) by the induction hypothesis. (See [Ta, Theorem 11.5] for a formula which applies to nondegenerate forms over arbitrary finite fields.)

Now assume 𝔮 is degenerate. Then V = V_{1} ⊥ V_{2}, where rk(V_{1}) = 2, and 𝔮_{i} = 𝔮|_{V_{i}} is nondegenerate for i = 1, 2. Hence 𝔮_{1}^{−1}(0) = 2+η and 𝔮_{2}^{−1}(0) = 2^{2n−3} +ε2^{n−2} for some η, ε ∈ {±1}, and so

|𝔮^{−1}(0)| = |𝔮_{1}^{−1}(0)|·|𝔮_{2}^{−1}(0)| + |𝔮_{1}^{−1}(1)|·|𝔮_{2}^{−1}(1)|
= (2 + η)(2^{2n−3} + ε2^{n−2}) + (2 − η)(2^{2n−3} − ε2^{n−2})
= 2^{2n−1} + εη2^{n−1}.

When dim(V) = 4, each nondegenerate form is equivalent to 𝔮_{1} (the hyperbolic form) or 𝔮_{2}, where

𝔮_{1}(x_{1}, x_{2}, x_{3}, x_{4}) = x_{1}x_{2} + x_{3}x_{4}
𝔮_{2}(x_{1}, x_{2}, x_{3}, x_{4}) = x_{1}x_{2} + x_{3}^{2} + x_{3}x_{4} + x_{4}^{2}

(see, e.g., [A1, § 21] or [Sz1, Proposition 3.5.10]). The properties listed above are easily checked. □

The next two lemmas are is more specialized.

Lemma A.6. Let S be a 2-group, and let Q ≤ S be such that r(Q/Z(Q)) = r(S).

(a) In all cases, C_{S}(Q) ≤ Q. In particular, Z(S) ≤ Z(N_{S}(Q)) ≤ Z(Q), and Z(S) = Z(Q) if |Z(Q)| = 2.

(b) Assume that Q is special of type 2^{2+4} (i.e., Z(Q) = [Q, Q] ≅ C_{2}^{2} and Q/Z(Q) ≅ C_{2}^{4}), and that Z(N_{S}(Q)) < Z(Q). Assume also that all involutions in Q are central, or more generally, that the number of classes gZ(Q) ∈ (Q/Z(Q))^{#} such that g^{2} = 1 is even. Then Z(Q) = Z_{2}(S).

(c) If U ⊴ S is such that [S, U] ≤ Fr(U) ≤ Z(S), then U ≤ Q.

Proof. (a) Since r(S) ≥ r(QC_{S}(Q)/Z(Q)) = r(Q/Z(Q)) + r(C_{S}(Q)/Z(Q)) for any pair Q ≤ S, the assumption r(S) = r(Q/Z(Q)) implies that C_{S}(Q) = Z(Q) ≤ Q.

(b) Assume Q is special of type 2^{2+4}, and Z(N_{S}(Q)) < Z(Q). Thus Z(S) = Z(N_{S}(Q)) = ⟨z_{0}⟩ for some z_{0} ∈ Z(Q)^{#}. Let z_{1}, z_{2} ∈ Z(Q) be the other two involutions. Set V = Q/Z(Q), let 𝔮: V ⟶ Z(Q) be the quadratic map 𝔮(gZ(Q)) = g^{2}, and let 𝔮_{i}: V ⟶ Z(Q)/⟨z_{i}⟩ ≅ 𝔽_{2} (i = 0, 1, 2) be the quadratic form induced by 𝔮. Set m = |𝔮^{−1}(1)|, and for i = 0, 1, 2, set n_{i} = |𝔮^{−1}(z_{i})|. Note that n_{1} = n_{2}, since there is g ∈ N_{S}(Q) such that ^{g}z_{1} = z_{2}.

By assumption, the number k of classes in (Q/Z(Q))^{#} which lift to involutions in Q is even, so m = k+1 is odd. For each i, by point (A.1) in Lemma A.5, m+n_{i} = 𝔮_{i}^{−1}(0) is even, and thus n_{i} is odd. Hence |𝔮_{0}^{−1}(0)| = m + n_{0} = 16 − 2n_{1} ≡ 2 (mod 4), so 𝔮_{0} is nondegenerate by (A.1) again. In particular, Z(Q/⟨z_{0}⟩) = Z(Q)/⟨z_{0}⟩, and thus Z_{2}(S)/⟨z_{0}⟩ = Z(S/⟨z_{0}⟩) = Z(Q)/⟨z_{0}⟩ by (a).

(c) Set U_{0} = Fr(U) ⊴ S for short. Then QU/U_{0} = (U/U_{0})(QU_{0}/U_{0}), where U/U_{0} is elementary abelian and [U/U_{0}, QU_{0}/U_{0}] = 1, so either U ≤ QU_{0} or r(QU/U_{0}) > r(QU_{0}/U_{0}). If r(QU/U_{0}) > r(QU_{0}/U_{0}), then r(S) > r(QU_{0}/U_{0}) = r(Q/(Q ∩ U_{0})) ≥ r(Q/Z(Q)) since Q ∩ U_{0} ≤ Q ∩ Z(S) ≤ Z(Q). Since this contradicts our hypothesis, U ≤ QU_{0}, so (U ∩ Q)Fr(U) = U, and hence U ≤ Q (cf. [G, § 5.1]). □

Lemma A.7 ([OV, Proposition 1.8]). Fix a prime p, a finite group G, and a normal abelian p-subgroup Q ⊴ G. Let H ≤ G be such that Q ∩ H = 1, and let H_{0} ≤ H be of index prime to p. Consider the set

H = {H′ ≤ G | H′ ∩ Q = 1, QH′ = QH, H_{0} ≤ H′}.

Then for each H′ ∈ H, there is g ∈ C_{Q}(H_{0}) such that H′ = ^{g}H.

Throughout the rest of the chapter, we recall some general results about automorphisms.

Theorem A.8 (Krull-Schmidt theorem). Let G be a finite group, and assume G = G_{1} × ⋯ × G_{k} for some sequence of indecomposable subgroups 1 ≠ G_{i} ⊴ G.

(a) If G = G_{1}^{∗} × ⋯ × G_{ℓ}^{∗} is a second factorization into nontrivial indecomposables, then k = ℓ, and there are σ ∈ Σ_{k} and β ∈ Aut(G) such that β ≡ Id_{G} (mod Z(G)) and β(G_{i}) = G_{σ(i)}^{∗}.

(b) For any α ∈ Aut(G), there is σ ∈ Σ_{k} such that α(G_{i}Z(G)) = G_{σ(i)}Z(G) for each i.

Proof. Point (a) is a special case of the Krull-Schmidt theorem in the form shown in [Sz1, Theorem 2.4.8]. Note that by [Sz1, 1.6.18], a “normal automorphism” of G is one which is the identity modulo Z(S). Point (b) follows from (a), applied with G_{i}^{∗} = α(G_{i}). □

Lemma A.9. Fix a prime p, a p-group S, a subgroup P_{0} ≤ Fr(S), and a sequence of subgroups

P_{0} ⊴ P_{1} ⊴ ⋯ ⊴ P_{k} = S

all normal in S. Set

A = {α ∈ Aut(S) | ∀ 0, ≤ i ≤ k − 1, α(P_{i}) = P_{i} and [α, P_{i+1}] ≤ P_{i}} ≤ Aut(S) :

the group of automorphisms which induce the identity on each of the quotient groups P_{i}/P_{i−1}. Then A is a p-group. If the P_{i} are all characteristic in S, then A ⊴ Aut(S), and hence A ≤ O_{p}(Aut(S)).

Proof. See, e.g., [G, Theorems 5.1.4 & 5.3.2]. □

As an easy exercise, Lemma A.9 implies the following list of 2-groups whose automorphism groups are 2-groups.

Corollary A.10. For a 2-group S, Aut(S) is a 2-group if any of the following hold: either

(a) S is cyclic, or S/[S, S] ≅ C_{2^{m}} × C_{2^{n}} for m ≥ n ≥ 2; or

(b) S ≅ D_{2^{k}} with k ≥ 3, or S ≅ Q_{2^{k}} or SD_{2^{k}} with k ≥ 4; or

(c) S ≅ D_{8} × D_{8}, D_{8} ≀ C_{2}, or D_{8} × C_{2}.

Proof. Each of these follows upon applying Lemma A.9 to an appropriate chain of characteristic subgroups of S. When S = S_{1} × S_{2} for S_{i} ≅ D_{8}, there is a simpler argument using the Krull-Schmidt theorem (Theorem A.8): each automorphism of S normalizes or exchanges the subgroups S_{i}Z(S) ≅ D_{8} × C_{2}, where Aut(D_{8} × C_{2}) is a 2-group. □

Lemma A.11 ([O1, Proposition 2.3]). Fix an abelian 2-group A, and a subgroup G ≤ Aut(A) with |G| = 2m for some odd m. Assume, for each x ∈ I(G), that x ∉ Z(G) and [x, A] ≅ C_{2^{n}} (some n ≥ 1). Set G_{1} = O^{2′}(G), G_{2} = C_{G}(G_{1}), and A_{1} = [G_{1}, A]. Then G_{1} ≅ Σ_{3}, G_{2} has odd order, G = G_{1}×G_{2}, and A_{1} ≅ C_{2^{n}} ×C_{2^{n}}.

APPENDIX B Subgroups of 2-groups of sectional rank 4

We list here some properties of 2-groups S with r(S) ≤ 4, starting with the case r(S) = 2.

Since r(P) ≤ r(P/Q)+r(Q) when Q ⊴ P are p-groups, all noncyclic metacyclic p-groups have sectional rank 2. The converse to this also holds when p = 2, as shown in the following lemma. It is not true for odd p: the nonabelian groups of order p^{3} and exponent p have sectional rank 2 and are not metacyclic.

Lemma B.1. The following hold for any 2-group S with r(S) = 2.

(a) S is metacyclic.

(b) If S contains a subgroup isomorphic to D_{8} or Q_{8}, then S ∈ DSQ.

(c) If Aut(S) is not a 2-group, then S ≅ C_{2^{k}} × C_{2^{k}} for some k, or S ≅ Q_{8}.

Proof. (a) Assume otherwise, and let S be a counterexample of minimal order. Thus r(S) = 2, and S is not metacyclic. Also, S is nonabelian, so there is a central involution z ∈ Z(S) ∩ [S, S]. Set Z = ⟨z⟩.

By the minimality assumption, S/Z is metacyclic. Hence we can choose a, b ∈ S such that S = ⟨z, a, b⟩ where A =^{def} ⟨a, z⟩ is normal; and A is noncyclic since otherwise S would be metacyclic. Also, [S, S] = [b, A] = ⟨[b, a]⟩ (since [b, z] = 1), so [S, S] is cyclic. Since z ∈ [S, S] ≤ A, and z is not a square in A, this implies that [a, b] = z and [S, S] = ⟨z⟩.

Thus S/Z is abelian. So we can assume that a and b were chosen so that S/Z = ⟨aZ⟩ × ⟨bZ⟩. Also, z ∉ ⟨b⟩, since otherwise S would be metacyclic. Set |a| = 2^{j} and |b| = 2^{k}. Then either j = k = 1 and S ≅ D_{8}; or one of the subgroups ⟨z, a^{2}, b⟩ or ⟨z, a, b^{2}⟩ is abelian of rank 3. In either case, this contradicts our original assumption on S.

(b) Since S is metacyclic by (a), there are elements a, b ∈ S such that S = ⟨a, b⟩, ⟨a⟩ ⊴ S, |a| = 2^{k}, |S/⟨a⟩| = 2^{ℓ}, and bab^{−1} = a^{j} (j odd). If P ≤ S is isomorphic to D_{8} or Q_{8}, then the image of P in S/⟨a⟩ must have order 2, and thus P ∩ ⟨a⟩ ≅ C_{4}. Since the elements in P ∖⟨a⟩ invert P ∩ ⟨a⟩, b^{2^{ℓ−1}} inverts P ∩ ⟨a⟩, so ℓ = 1 (b^{2^{ℓ−1}} cannot be a square), and j = −1 or 2^{k−1}−1 since j^{2} ≡ 1 (mod 2^{k}). Thus S ∈ DSQ.

(c) See [Cr, Proposition 6.7]. □

The following well known result is needed frequently.

Proposition B.2. Let S be a 2-group such that S/[S, S] ≅ C_{2}^{2}. Then either S ≅ C_{2}^{2}, or S ∈ DSQ.

Proof. See, e.g., [G, Theorem 5.4.5]. □

Lemma B.3. Let S be a 2-group such that r(S) ≤ 4, and let P, Q ⊴ S be normal nonabelian subgroups such that |Z(P)| = |Z(Q)| = 2 and Z(P) ≠ Z(Q). Then [P, Q] = P ∩ Q = 1, and C_{S}(P Q) ≤ P Q (i.e., P Q is centric).

Proof. Since P, Q ⊴ S, Z(P), Z(Q) ⊴ S, and hence Z(P)Z(Q) ≤ Z(S). If P ∩ Q ≠ 1, then P ∩ Q ≥ Z(P) by Lemma A.2(a) (and since P ∩ Q ⊴ P), so Z(P) ≤ Q ∩ Z(S) ≤ Z(Q), a contradiction. Thus [P, Q] ≤ P ∩ Q = 1.

In particular, r(P Q/Z(P Q)) = 4, since r(X/Z(X)) ≥ 2 for any nonabelian 2-group X. So C_{S}(P Q) ≤ P Q by Lemma A.6(a). □

The next lemma is a corollary of Lemma B.3.

Lemma B.4. Let S be a 2-group such that r(S) ≤ 4, and let P < S be a nonabelian subgroup such that |Z(P)| = 2. Set S_{0} = C_{S}(Z(P)), and assume P ⊴ S_{0}. Then either S_{0} = S, or r(P) = 2 and [S:S_{0}] = 2.

Proof. Assume S_{0} < S. Choose g ∈ N_{S}(S_{0})∖S_{0} such that g^{2} ∈ S_{0}, and set Q = ^{g}P and S_{1} = S_{0}⟨g⟩. Then Z(P) ≠ Z(Q) since g ∉ S_{0} = C_{S}(Z(P)), so by Lemma B.3, [P, Q] ≤ P ∩ Q = 1 and C_{S}(P Q) = Z(P Q). Hence 4 ≥ r(P Q) ≥ 2r(P) ≥ 2r(P/Z(P)) ≥ 4 implies that r(P) = r(P/Z(P)) = 2 and r(P Q/Z(P Q)) = 4.

Set Ẑ = Z(P)Z(Q) for short. Since Z(S) ≤ C_{S}(P Q) = Ẑ, Z(S) < Ẑ ≅ C_{2}^{2} is the subgroup of order 2 distinct from Z(P) and Z(Q). Hence Z(P Q/Z(S)) = Ẑ/Z(S). By Lemma A.6(a), and since r(P Q/Ẑ) = 4, C_{S/Z(S)}(P Q/Z(S)) = Ẑ/Z(S), so Ẑ = Z_{2}(S) ⊴ S. Thus [S:S_{0}] = [S:C_{S}(Ẑ)] = 2 since |Aut_{S}(Ẑ)| = 2. □

The next two lemmas involve 2-groups of sectional rank 2 or 4 which contain several normal dihedral or quaternion subgroups.

Lemma B.5. Fix a 2-group S such that r(S/Z(S)) = 2, and a subgroup Z ≤ Z(S) of order 2. Let 𝒫 be a set of subgroups of S such that S = Z(S)⟨𝒫⟩, and such that for each P ∈ 𝒫, P Z(S) ⊴ S, and either P ∈ DQ and Z(P) = Z, or P ≅ C_{2}^{2} and P ∩ Z(S) = Z. Assume also that at least one subgroup in 𝒫 is nonabelian. Then there is a subgroup P̂ ⊴ S such that P̂ ∈ DSQ, P̂Z(S) = S, and P̂ ∩ Z(S) = Z.

Proof. For each X ≤ S, let X = XZ(S)/Z(S) be the image of X in S = S/Z(S), and set 𝒫 = {P | P ∈ 𝒫}. Thus P ⊴ S for each P ∈ 𝒫, and S = ⟨𝒫 ⟩. Since r(S) = 2 and S is generated by involutions (since each P is generated by involutions), S ∉ SQ and S/[S, S] ≅ C_{2}^{2}, so S ∈ D by Proposition B.2.

If S ≅ C_{2}^{2}, then S = P for any nonabelian subgroup P ∈ 𝒫, and we set P̂ = P.

Now assume S is nonabelian. We must find P̂ ⊴ S such that P̂ ∈ DSQ, P̂ = S, and P̂ ∩ Z(S) = Z. Let H_{1}, H_{2} < S be the two noncyclic subgroups of index 2 (recall S ∈ D). Choose x, y ∈ S whose images x, y ∈ S have order 2, such that each of x and y lies in some P ∈ 𝒫, x ∈ S∖H_{1}, and y ∈ S∖H_{2}. (This is possible since each P ∈ 𝒫 is generated by elements of order 2.) Thus ⟨x, y⟩ = S. Set P̂ = ⟨x, y⟩. Then

• [P̂, P̂] = [S, S] ≥ Z since P̂Z(S) = S; and

• x^{2}, y^{2} ∈ Z, since each lies in some P ∈ 𝒫 and P ∩ Z(S) = Z.

Hence P̂/[P̂, P̂] ≅ C_{2}^{2}. So P̂ ∈ DSQ by Proposition B.2, and P̂∩Z(S) = Z(P̂) ≥ Z with equality since |Z(P̂)| = 2. □

Lemma B.6. Fix a 2-group S with r(S) ≤ 4. Assume S = ⟨𝒫⟩, where 𝒫 is a set of normal subgroups of S such that for each P ∈ 𝒫, either P ∈ DQ, or P ≅ C_{2}^{2} and |P ∩ Z(S)| = 2. Assume also that not all of the P ∈ 𝒫 have the same center (or intersection with Z(S) when P ≅ C_{2}^{2}). Then there are subgroups S_{1}, S_{2} ⊴ S such that S_{1}, S_{2} ∈ DSQ, S = S_{1}S_{2}, S_{1} ∩ S_{2} = 1, and for some partition 𝒫 = 𝒫_{1} ⨿ 𝒫_{2} of 𝒫, S_{i} ≤ ⟨𝒫_{i} ⟩ ≤ S_{i}Z(S) for i = 1, 2.

Proof. Set Z = {P ∩ Z(S) | P ∈ 𝒫}, and let Z_{1}, ⋯, Z_{m} be the distinct subgroups in Z. By assumption, m ≥ 2, and each Z_{i} has order 2. For each i, set 𝒫_{i} = {P ∈ 𝒫 | P ≥ Z_{i}}. Set Δ_{i} = ⟨𝒫_{i} ⟩.

Assume i ≠ j, P ∈ 𝒫_{i}, and Q ∈ 𝒫_{j}. If P and Q are both nonabelian, then P ∩ Q = 1 by Lemma B.3. If P ≅ C_{2}^{2} and Q is nonabelian, then P ∩ Q ⊴ Q and P ∩ Z(S) = Z_{i} ≠ Z_{j} = Z(Q) again imply P ∩ Q = 1. If P ≅ Q ≅ C_{2}^{2}, then either P ∩ Q = 1, or |P ∩ Q| = 2, P = Z_{i}(P ∩ Q), and Q = Z_{j}(P ∩ Q). Thus [P, Q] = 1 in all cases, so [Δ_{i}, Δ_{j}] = 1 for i ≠ j.

If Δ_{i} is abelian for some i, then Δ_{i} ≤ Z(S) since it commutes with the other Δ_{j}, which is impossible since no P ∈ 𝒫 is contained in Z(S). Thus each Δ_{i} is nonabelian. If, for some i, all subgroups in 𝒫_{i} are abelian, then there are P, Q ∈ 𝒫_{i} such that P ≅ Q ≅ C_{2}^{2} and [P, Q] ≠ 1; and P Q ≅ D_{8} since they are both normal in S. So after replacing P and Q by P Q in this situation, we can assume each 𝒫_{i} contains a nonabelian subgroup.

If m ≥ 3, then for P_{i} ∈ 𝒫_{i} nonabelian (i = 1, 2, 3), P_{3} ≤ C_{S}(P_{1}P_{2}), and C_{S}(P_{1}P_{2}) = Z(P_{1}P_{2}) by Lemma B.3. Since this is impossible, m = 2, S = Δ_{1}Δ_{2}, [Δ_{1}, Δ_{2}] = 1, and hence Δ_{1} ∩ Δ_{2} ≤ Z(S) and Z(Δ_{i}) ≤ Z(S) (i = 1, 2).

For each X ≤ S, let X = XZ(S)/Z(S) be the image of X in S = S/Z(S). Then S = Δ_{1} × Δ_{2}, so r(Δ_{i}) = 2 (i = 1, 2) since r(S) ≤ 4 and Δ_{i} ≅ Δ_{i}/Z(Δ_{i}) is noncyclic. So by Lemma B.5, applied with Δ_{i}, 𝒫_{i}, and Z_{i} in the role of S, 𝒫, and Z, there is S_{i} ⊴ Δ_{i} such that S_{i} ∈ DSQ, S_{i} = Δ_{i}, and Z(S_{i}) = S_{i} ∩ Δ_{i} = Z_{i}. By Lemma B.3, Z(S) ≤ S_{1}S_{2}, so S = S_{1}S_{2} ≅ S_{1} × S_{2}. □

Lemma B.7. Let S be a 2-group such that r(S) ≤ 4, and let Q ⊴ S be a normal nonabelian subgroup such that |Z(Q)| = 2. Then for every α ∈ Aut(S) of odd order, α(Z(Q)) = Z(Q).

Proof. If α(Z(Q)) ≠ Z(Q), then Q, α(Q), and α^{2}(Q) are three normal nonabelian subgroups with distinct centers of order 2. So by Lemma B.3, α^{2}(Q) ≤ C_{S}(Qα(Q)) = Z(Qα(Q)), which is impossible. □

APPENDIX C Some explicit 2-groups of sectional rank 4

In this chapter, we collect some (mostly) technical results about subgroups and automorphisms of certain 2-groups, such as 2_{−}^{1+4}, UT_{4}(2), and UT_{3}(4). We begin with products of dihedral groups.

Lemma C.1. Assume S ∈ D × D: a product of two nonabelian dihedral groups. Then there is a unique abelian subgroup A < S of index 4 and rank 2. Of the three elements in Z(S)^{#}, exactly two are squares of elements in S∖A.

Proof. Fix dihedral subgroups D_{i} = ⟨a_{i}, b_{i}⟩ (i = 1, 2) such that S = D_{1} × D_{2} and [D_{i}:⟨a_{i}⟩] = 2. If A < S is abelian of index 4, then for each i, the image A_{i} ≤ D_{i} of A under the projection is abelian, so [D_{i}:A_{i}] = 2 and A = A_{1}A_{2}. Each A_{i} is cyclic since A has rank 2, so A = ⟨a_{1}, a_{2}⟩ is the unique such subgroup.

Let z_{i} be the generator of Z(D_{i}). If g ∈ S∖A, then either g ∈ b_{1}A and g^{2} ∈ ⟨a_{2}^{2}⟩, or g ∈ b_{2}A and g^{2} ∈ ⟨a_{1}^{2}⟩, or g ∈ b_{1}b_{2}A and g^{2} = 1. Thus z_{1} and z_{2} can occur as squares of such elements, while z_{1}z_{2} cannot. □

We now look at certain 2-groups, beginning with 2_{−}^{1+4}.

Lemma C.2. Assume S = Δ_{1}Δ_{2}, where Δ_{1} ≅ Δ_{2} ≅ Q_{8}, [Δ_{1}, Δ_{2}] = Δ_{1} ∩ Δ_{2} = Z(Δ_{1}) = Z(Δ_{2}), and |C_{Δ_{1}}(Δ_{2})| = 4. Then the following hold.

(a) S ≅ 2_{−}^{1+4}. There are exactly five involutions in S/Z(S) ≅ C_{2}^{4} which lift to involutions in S, and they are permuted transitively by Out(S) ≅ Σ_{5}.

(b) Let Γ ≤ Aut(S) be a subgroup which contains Inn(S). Assume, for each i = 1, 2, that there is γ_{i} ∈ Γ of order 3 such that γ_{i}(Δ_{i}) = Δ_{i} and γ_{i}|_{Δ_{i}} ≠ Id. Then [Aut(S):Γ] ≤ 2. If in addition, there is η ∈ Γ such that η(Δ_{i}) = Δ_{i} (i = 1, 2), η|_{Δ_{1}} ∉ Inn(Δ_{1}), and η|_{Δ_{2}} ∈ Inn(Δ_{2}), then Γ = Aut(S).

Proof. Set Z = Z(Δ_{1}) = Z(Δ_{2}), V = S = S/Z, and X = XZ/Z for X ≤ S. Let 𝔮: V ⟶ Z be the quadratic form 𝔮(gZ) = g^{2}, and let 𝔟 be its associated bilinear form (𝔟(gZ, hZ) = [g, h]).

(a) If 𝔟 is degenerate, then dim(V^{⊥}) ≥ 2, so V^{⊥} = (Δ_{1})^{⊥} = (Δ_{2})^{⊥}, which is impossible since |C_{Δ_{1}}(Δ_{2})| = 4 (dim(Δ_{1} ∩ Δ_{2}^{⊥}) = 1). Thus 𝔟 and 𝔮 are nondegenerate, S ≅ 2̸_{+}^{1+4} since that group contains exactly two quaternion subgroups (and they commute), and hence S ≅ 2_{−}^{1+4} ≅ Q_{8} ×_{C_{2}} D_{8}.

By Lemma A.5, there are exactly five isotropic points in V (which lift to involutions in S), and Out(S) ≅ SO(V, 𝔮) ≅ SO_{4}^{−}(2) ≅ Σ_{5} is the group of all permutations of this set.

(b) By assumption, for i = 1, 2, |γ_{i}| = 3, γ_{i}(Δ_{i}) = Δ_{i}, and γ_{i}|_{Δ_{i}} ≠ Id. Thus C_{S}(γ_{i}) ≤ C_{S}(Δ_{i}), so C_{S}(⟨γ_{1}, γ_{2}⟩) = Z(S) = Z. Each subgroup of Σ_{5} generated by two elements of order 3 is isomorphic to A_{3}, A_{4}, or A_{5}, and so ⟨[γ_{1}], [γ_{2}]⟩ ≅ A_{5} (as a subgroup of Out(S) ≅ Σ_{5}) since it doesn’t fix any of the five isotropic points in V = S.

If η(Δ_{i}) = Δ_{i} for i = 1, 2 and η|_{Δ_{2}} ∈ Inn(Δ_{2}), then the induced automorphism η ∈ Aut(V, 𝔮) is the identity on Δ_{2} and sends Δ_{1} to itself. Hence rk(C_{V} (η)) = 3 (since C_{Δ_{1}}(η) ≠ 1), so η permutes the isotropic points in V as a 2-cycle. Thus [η] ∉ O^{2}(Out(S)) ≅ A_{5}, and Γ = Aut(S). □

The next lemma involves other “near central products” of dihedral and quaternion groups.

Lemma C.3. Assume S = Δ_{1}Δ_{2}, where Δ_{1}, Δ_{2} ∈ DSQ, |Δ_{1}|, |Δ_{2}| ≥ 16, and [Δ_{1}, Δ_{2}] ≤ Δ_{1} ∩ Δ_{2} = Z(S). Then there is a unique pair of subgroups Θ_{1}, Θ_{2} ∈ Q such that S = Θ_{1}Θ_{2}, |Θ_{i}| = |Δ_{i}| (i = 1, 2), and [Θ_{1}, Θ_{2}] ≤ Θ_{1} ∩ Θ_{2} = Z(S).

Proof. Set B_{i} = Z_{2}(Δ_{i}) ≅ C_{4} for short. Then Z_{2}(S) = B_{1}B_{2}, since
S/Z(S) = (Δ_{1}/Z(S)) × (Δ_{2}/Z(S))  (C.1)

by assumption. Also, B_{i} ≤ Fr(Δ_{i}) for i = 1, 2 since |Δ_{i}| ≥ 16, so [Δ_{i}, B_{3−i}] = 1, and Δ_{i}Z_{2}(S) = Δ_{i}B_{3−i} ≅ Δ_{i} ×_{C_{2}} C_{4}.

Again fix i = 1, 2. We claim that

there is a unique Θ_{i}^{∗} ≤ Δ_{i}Z_{2}(S) such that Θ_{i}^{∗} ∈ Q and |Θ_{i}^{∗}| = |Δ_{i}|.  (C.2)

To see this, fix x, y ∈ Δ_{i} such that Δ_{i} = ⟨x, y⟩, x^{2}, y^{2} ∈ Z(S), and ⟨xy⟩ < Δ_{i} has index 2. There are x′ ∈ xB_{i} and y′ ∈ yB_{i} (unique modulo Z(S)) such that |x′| = 4 = |y′|. Set Θ_{i}^{∗} = ⟨x′, y′⟩. Then Θ_{i}^{∗}/Z(S) ∈ D since it is generated by two involutions, and hence Θ_{i}^{∗} ∈ Q. Also, Θ_{i}^{∗}B_{i} = Δ_{i}B_{i}, Θ_{i}^{∗} ∩B_{i} = Z(Θ_{i}^{∗}) = Z(S), and so |Θ_{i}^{∗}| = |Δ_{i}|. Any other quaternion subgroup of index 2 in Δ_{i}B_{i} must contain elements of order 4 in xB_{i} and yB_{i}, hence contains x′ and y′, and is equal to Θ_{i}^{∗}.

By the Krull-Schmidt theorem (Theorem A.8(a)), applied to the factorization of S/Z(S) in (C.1), for any Θ_{1}, Θ_{2} ∈ Q such that S = Θ_{1}Θ_{2} and [Θ_{1}, Θ_{2}] ≤ Θ_{1} ∩ Θ_{2} = Z(S), Θ_{i} ≤ Δ_{i}Z_{2}(S) (possibly after an exchange of indices). Hence Θ_{i} = Θ_{i}^{∗} exists and is uniquely determined by (C.2). □

We next look at the group UT_{4}(2).

Lemma C.4. Set S = UT_{4}(2).

(a) There is a unique abelian subgroup A ≤ S of order 16, A ≅ C_{2}^{4}, S splits over A, and Out_{S}(A) ≅ C_{2}^{2} permutes freely a basis for A (thus S ≅ C_{2} ≀ C_{2}^{2}).

(b) There is a unique extraspecial subgroup Q ≤ S of index 2, Q ≅ 2_{+}^{1+4}, and

S ≅ (Q_{8} ×_{C_{2}} Q_{8}) ⋊^{t} C_{2} ≅ (D_{8} ×_{C_{2}} D_{8}) ⋊^{t} C_{2}.

(c) If P < S has index 2, then either P = Q, or P ≥ A, or P^{ab} ≅ C_{4} × C_{2} and Aut(P) is a 2-group. All involutions in S are in A ∪ Q.

(d) If Id ≠ α ∈ Aut(S) has odd order, then |α| = 3, and α permutes transitively the three subgroups of index 2 which contain A.

(e) There is no normal subgroup P ⊴ S with P ∈ DQ.

Proof. Fix V = (𝔽_{2})^{4} with basis {b_{1}, b_{2}, b_{3}, b_{4}}. For each 0 ≤ i ≤ 4, set V_{i} = ⟨b_{j} | 1 ≤ j ≤ i⟩. Set G = Aut(V) ≅ GL_{4}(2). We identify S with the group of all α ∈ G which normalize the chain 0 < V_{1} < V_{2} < V_{3} < V. Set Z = Z(S).

For σ ∈ Σ_{4}, define ψ_{σ} ∈ G by setting ψ_{σ}(b_{i}) = b_{σ(i)} for each i.

(a) Set H = {α ∈ G | α(V_{2}) = V_{2}} and A = O_{2}(H). The map (α ↦ α − Id) defines an isomorphism A ⟶^{≅} Hom(V/V_{2}, V_{2}) ≅ C_{2}^{4}, and H = A ⋊ (G_{12} × G_{34}). Hence S is H-conjugate to A⟨ψ_{(1 2)}, ψ_{(3 4)}⟩, and ⟨ψ_{(1 2)}, ψ_{(3 4)}⟩ ≅ C_{2}^{2} permutes freely the canonical basis for A. Thus S ≅ C_{2} ≀ C_{2}^{2}. So by Lemma A.4(b), and since |S/[S, S]| = 8, A is the unique abelian subgroup of index 4 in S.

(e) Assume P ⊴ S and P ∈ DQ. Then P ≰ A since it is nonabelian. If g ∈ P ∖A, then P ≥ [g, A] = C_{A}(g) ≅ C_{2}^{2} since P is normal, so P ≥ ⟨g, C_{A}(g)⟩ which is abelian of order 8. This is impossible.

(b) Set K = C_{G}(Z) = {α ∈ G | α(V_{1}) = V_{1}, α(V_{3}) = V_{3}} and Q = O_{2}(K). Set

W_{1} = {α ∈ G | [α, V] ≤ V_{1}}  and  W_{2} ={α ∈ G | α|_{V_{3}} = Id} .

Then W_{1} ≅ W_{2} ≅ C_{2}^{3}, W_{1}W_{2} = Q, and W_{1} ∩ W_{2} = Z. Also, W_{1}, W_{2} ⊴ S, so [W_{1}, W_{2}] ≤ W_{1} ∩ W_{2} = Z, with equality since Q is nonabelian by (a). Since Q contains no abelian subgroups of order 2^{4} by (a) again, it must be extraspecial, and so Q ≅ 2_{+}^{1+4} ≅ Q_{8} ×_{C_{2}} Q_{8} since it contains elementary abelian subgroups of rank 3. Since neither of the two quaternion subgroups of Q is normal in S by (e), we get

S ≅ (Q_{8} ×_{C_{2}} Q_{8}) ⋊^{t} C_{2}. Finally, the explicit isomorphisms D_{8} ×_{C_{2}} D_{8} ≅ Q_{8} ×_{C_{2}} Q_{8} constructed in [G, p. 205] and in [Sz1, pp. 139–140] extend to isomorphisms

(D_{8} ×_{C_{2}} D_{8}) ⋊^{t} C_{2} ≅ (Q_{8} ×_{C_{2}} Q_{8}) ⋊^{t} C_{2} between the semidirect products.

Since [S, Q] = [S, S] has order 2^{3}, [S, Q/Z] has rank 2. So by Lemma A.4(a), applied with Q/Z < S/Z in the role of A < S, Q/Z is the unique abelian subgroup of index 2 in S/Z, and Q is the unique extraspecial subgroup of index 2 in S.

(c) Since S/Q and S/A are elementary abelian, [S, S] ≤ Fr(S) ≤ Q ∩ A. Since Aut_{S}(A) permutes freely a basis of A by (a), |[S, S]| = |[S, A]| = 8, and thus [S, S] = Fr(S) = Q ∩ A. So there are 7 subgroups of S of index 2, including Q and the three which contain A.

Let T_{1}, T_{2}, T_{3} < S be the three subgroups of index 2 in [S, S] which contain Z = Z(Q). By (b), S/Z ≅ C_{2}^{2} ≀ C_{2}, where [S, Q/Z] = [S, S]/Z ≅ C_{2}^{2}. Hence S/T_{i} ≅ D_{8} × C_{2} for each i = 1, 2, 3. Let P_{i} < S be the subgroup such that P_{i} > T_{i} and P_{i}/T_{i} ≅ C_{4} × C_{2}. Then T_{i}/Z is nonabelian since Q/Z is the unique abelian subgroup of S/Z of index 2, so P_{i}^{ab} = P_{i}/T_{i} ≅ C_{4} × C_{2}, and Aut(P_{i}) is a 2-group by Corollary A.10(a). Also, since Q/T_{i} ≅ C_{2}^{3} (since Q/Z is elementary abelian), Ω_{1}(P_{i}/T_{i}) = Z(S/T_{i}) ≤ Q/T_{i}, so I(P_{i}) ⊆ Q, and hence Q ≱ A. Thus P_{1}, P_{2}, P_{3} are the three subgroups of index 2 which contain neither Q nor A.

In particular, if g ∈ I(S) and g ∉ Q, then g ∉ P_{1} ∪ P_{2} ∪ P_{3}. Since each element of S is contained in at least three subgroups of index 2, g is contained in all three of the subgroups which contain A, and thus g ∈ A.

(d) If Id ≠ α ∈ Aut(S) has odd order, then by Lemma A.9 and since |A/Fr(S)| = 2, α has order 3 and acts nontrivially on S/A, and hence permutes transitively the three subgroups of index 2 containing A. □

Lemma C.5. Set S = D_{8} ≀ C_{2}.

(a) There are exactly two normal subgroups V_{1}, V_{2} ⊴ S isomorphic to C_{2}^{4}, one (normal) subgroup Q ⊴ S isomorphic to 2_{+}^{1+4}, and no subgroups isomorphic to 2_{−}^{1+4}. The images of V_{1}, V_{2}, and Q in S/[S, S] ≅ C_{2}^{3} have order 2 and are linearly independent. Also, Q ∩ V_{i} ≅ C_{2}^{3} for i = 1, 2.

(b) If P < D_{8} ≀ C_{2} has index 2, and Aut(P) is not a 2-group, then P ≅ UT_{4}(2) ≅

(Q_{8} ×_{C_{2}} Q_{8}) ⋊^{t} C_{2}. There are two such subgroups in D_{8} ≀ C_{2}.

Proof. Set S = ⟨a_{1}, b_{1}, a_{2}, b_{2}, t⟩ ≅ D_{8} ≀ C_{2}, where |a_{i}| = 4, |b_{i}| = 2, Δ_{i} =^{def}

⟨a_{i}, b_{i}⟩ ≅ D_{8}, [Δ_{1}, Δ_{2}] = 1, t^{2} = 1, and ta_{i}t^{−1} = a_{3−i}, tb_{i}t^{−1} = b_{3−i}. Set Q = ⟨a_{1}a_{2}, b_{1}b_{2}, a_{1}^{2}, t⟩ ≅ 2_{+}^{1+4}. Set Z = Z(S) = ⟨a_{1}^{2}a_{2}^{2}⟩ and Z_{2} = Z_{2}(S) = ⟨a_{1}^{2}, a_{2}^{2}⟩.

(a) If R ⊴ S and |R| ≥ 4, then R ≥ Z_{2} by Lemma A.2(b). If R ≅ C_{2}^{4}, then R ≤ C_{S}(Z_{2}) ≅ D_{8} × D_{8}, and of the four subgroups in D_{8} × D_{8} isomorphic to C_{2}^{4}, only V_{1} = Z_{2}⟨b_{1}, b_{2}⟩ and V_{2} = Z_{2}⟨a_{1}b_{1}, a_{2}b_{2}⟩ are normal in S.

If R ≤ S and R ≅ 2_{+}^{1+4} or 2_{−}^{1+4}, then Z(R) = Z, R/Z and Q/Z are both abelian of index 4 in S/Z, and so R = Q ≅ 2_{+}^{1+4} by Lemma A.4(b). The images of V_{1}, V_{2}, and Q in S^{ab} are generated by the classes of b_{1}, a_{1}b_{1}, and t, respectively, and thus are independent. Also, Q ∩ V_{1} = Z_{2}⟨b_{1}b_{2}⟩ ≅ C_{2}^{3} and Q ∩ V_{2} = Z_{2}⟨a_{1}b_{1}a_{2}b_{2}⟩ ≅ C_{2}^{3}.

(b) Assume P < S has index 2. If Z(P) > Z, then by Lemma A.3, P = C_{S}(Z_{2}) ≅ D_{8} × D_{8}, and Aut(P) is a 2-group by Corollary A.10(c). If Z(P) = Z and Aut(P) is not a 2-group, then Aut(P/Z) is not a 2-group by Lemma A.9. By

Lemma C.4(b,c), and since S/Z ≅ (Q_{8} ×_{C_{2}} Q_{8}) ⋊^{t} C_{2}, either P/Z is extraspecial (hence P ≅ D_{8} × D_{8}), or P > Q.

Of the three subgroups of index 2 which contain Q,

P_{1} = ⟨a_{1}a_{2}, b_{1}, b_{2}, a_{1}^{2}, t⟩ ≅ UT_{4}(2) and P_{2} = ⟨a_{1}a_{2}, a_{1}b_{1}, a_{2}b_{2}, a_{1}^{2}, t⟩ ≅ UT_{4}(2),

while P_{3} = ⟨a_{1}, a_{2}, b_{1}b_{2}, t⟩ contains the sequence Fr(P) < C_{Q}(Z_{2}) < C_{P_{3}}(Z_{2}) < P_{3} of characteristic subgroups. (Note that Z_{2} = Ω_{1}(Z_{2}(P_{3})).) So Aut(P_{3}) is a 2-group by Lemma A.9. □

Throughout the rest of the chapter, we work with the group UT_{3}(4). We use the following notation, taken from [OV, §§ 4–5] and from Notation 6.1, for certain subgroups of UT_{3}(4). Set

A_{1} = \{( \begin{matrix}1 & a & b \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) = e_{12}^{a}e_{13}^{b} | a, b ∈ 𝔽_{4}\} and A_{2} = \{( \begin{matrix}1 & 0 & b \\ 0 & 1 & a \\ 0 & 0 & 1 \end{matrix} ) = e_{23}^{a}e_{13}^{b} | a, b ∈ 𝔽_{4}\}.

Thus A_{1} = O_{2}(𝔓_{1}) and A_{2} = O_{2}(𝔓_{2}), where 𝔓_{1}, 𝔓_{2} are the two maximal parabolic subgroups in SL_{3}(4) containing UT_{3}(4). Also,

Z(UT_{3}(4)) = A_{1} ∩ A_{2} = {e_{13}^{a} | a ∈ 𝔽_{4}}.

Lemma C.6. Set S = UT_{3}(4), and Z = Z(S) = A_{1} ∩ A_{2}.

(a) All involutions in S lie in A_{1} ∪ A_{2}, and each elementary abelian subgroup of S is contained in A_{1} or in A_{2}. For each g ∈ A_{i}∖Z (i = 1, 2), C_{A_{3−i}}(g) = [g, A_{3−i}] = Z.

(b) For each S_{0} < S of index 2, [S_{0}, S_{0}] = [S, S] = Z.

Proof. (a) For each g ∈ A_{1}∖Z and h ∈ A_{2}∖Z, g = e_{12}^{a}e_{13}^{b} and h = e_{23}^{c}e_{13}^{d} for some a, b, c, d ∈ 𝔽_{4} with a, c ≠ 0, and [g, h] = [e_{12}^{a}, e_{23}^{c}] = e_{13}^{ac} ≠ 1. Thus C_{A_{2}}(g) = [g, A_{2}] = Z, and similarly for h.

If g ∈ S∖(A_{1} ∪ A_{2}), then g = e_{12}^{a}e_{23}^{b}e_{13}^{c} for some a, b, c ∈ 𝔽_{4} where a, b ≠ 0, and g^{2} = e_{13}^{ab} ≠ 1. Thus I(S) = A_{1}^{#} ∪ A_{2}^{#}. Since no element of A_{1}∖Z commutes with any element of A_{2}∖Z, each elementary abelian subgroup of S is contained in A_{1} or in A_{2}.

(b) Assume S_{0} < S has index 2. If S_{0} ∩ A_{1} < A_{1}, then S = A_{1}S_{0}. So for any g ∈ (S_{0} ∩ A_{1})∖Z, [S_{0}, S_{0}] ≥ [g, S_{0}] = [g, S] = Z by (a). If S_{0} ∩ A_{1} = A_{1}, then S_{0} ∩ A_{2} < A_{2}, and a similar argument applies. □

The next lemma gives some criteria for characterizing UT_{3}(4).

Lemma C.7. Fix a 2-group S of order 2^{6}, and set Z = Z(S). Assume that S is special of type 2^{2+4}; i.e., Z = [S, S] ≅ C_{2}^{2} and S/Z ≅ C_{2}^{4}. Assume also that there are subgroups B_{1}, B_{2} < S such that B_{1}B_{2} = S and B_{1} ≅ B_{2} ≅ C_{2}^{4}. If either

(a) [g, B_{1}] = Z for each g ∈ B_{2}∖Z, or

(b) there is Id ≠ α ∈ Aut(S) of odd order such that α(B_{1}) = B_{1},

then S ≅ UT_{3}(4).

Proof. Fix V_{i} < B_{i} which is complementary to Z; thus V_{i} ≅ C_{2}^{2}. Let χ: V_{1} × V_{2} ⟶ Z be the biadditive commutator map χ(v, w) = [v, w].

(a) For each v_{2} ∈ V_{2}^{#}, [v_{2}, B_{1}] = Z by assumption, so χ(−, v_{2}) ∈ Hom(V_{1}, Z) is surjective, and hence an isomorphism. Thus χ(v_{1}, v_{2}) ≠ 1 for each pair (v_{1}, v_{2}) ∈ V_{1}^{#} × V_{2}^{#}, and hence χ(v_{1}, −) is an isomorphism for each v_{1} ∈ V_{1}^{#}.

Fix any e_{i} ∈ V_{i}^{#}, and set e = [e_{1}, e_{2}] ∈ Z^{#}. Choose any isomorphism ρ: Z ⟶^{≅} (𝔽_{4}, +) such that φ(e) = 1, and set

ρ_{1} : V_{1} ⟶_{≅}^{χ(−,e_{2})} Z ⟶_{≅}^{ρ} 𝔽_{4}  and  ρ_{2} : V_{2} ⟶_{≅}^{χ(e_{1},−)} Z ⟶_{≅}^{ρ} 𝔽_{4}.

Set μ = ρ ∘ χ ∘ (ρ_{1}^{−1} × ρ_{2}^{−1}): 𝔽_{4} × 𝔽_{4} ⟶ 𝔽_{4}. By construction, μ is biadditive, μ(1, a) = a = μ(a, 1) for each a ∈ 𝔽_{4}, and μ(a, −) and μ(−, a) are isomorphisms for each a ≠ 0. So if a ≠ 0, 1, then μ(a, a) ∉ {0, a}, μ(1 + a, 1 + a) = 1 + μ(a, a) ≠ 0, and hence μ(a, a) = 1 + a = a^{2}. So μ(x, y) = xy for all x, y ∈ 𝔽_{4}.

Now define α: S ⟶ UT_{3}(4) by setting α(v_{1}) = e_{12}^{ρ_{1}(v_{1})}, α(v_{2}) = e_{23}^{ρ_{2}(v_{2})}, and α(z) = e_{13}^{ρ(z)} for all v_{i} ∈ V_{i} and z ∈ Z. Then by the relation [e_{12}^{a}, e_{23}^{b}] = e_{13}^{ab} in UT_{3}(4), α is an isomorphism.

(b) Let α ∈ Aut(S) be of odd order k > 1, and such that α(B_{1}) = B_{1}. Since the induced action of ⟨α⟩ on (S/B_{1})×(B_{1}/Z)×Z is faithful by Lemma A.9, k = |α| = 3. Since Z = [S, S] = [S, B_{1}], α induces a nontrivial action on S/B_{1} or on B_{1}/Z (or both).

Assume S ≅ U̸T_{3}(4). By the proof of (a), there are elements v_{i} ∈ V_{i}^{#} such that χ(v_{1}, v_{2}) = 1. If α(v_{1}) ∈ v_{1}Z, then α acts trivially on B_{1}/Z and hence nontrivially on S/B_{1}, so S = B_{1}⟨v_{2}, α(v_{2})⟩, v_{1} ∈ Z(S), contradicting the assumption that Z(S) = Z = [S, S]. Thus α(v_{1}) ∉ v_{1}Z, and α(v_{2}) ∉ v_{2}B_{1} by a similar argument.

Let v′_{i} ∈ V_{i} be the unique elements such that v′_{1} ∈ α(v_{1})Z and v′_{2} ∈ α(v_{2})B_{1}. Then V_{i} = ⟨v_{i}, v′_{i}⟩, α^{2}(v_{1}) ∈ v_{1}v′_{1}Z, α^{2}(v_{2}) ∈ v_{2}v′_{2}B_{1}, and χ(v_{1}, v_{2}) = 1 = χ(v′_{1}, v′_{2}) = χ(v_{1}v′_{1}, v_{2}v′_{2}). Thus [S, S] = ⟨χ(V_{1}, V_{2})⟩ = ⟨χ(v_{1}, v′_{2})⟩ has rank 1, which contradicts the assumption that [S, S] = Z. We conclude that S ≅ UT_{3}(4). □

Fix ω ∈ 𝔽_{4}∖{0, 1}, and let (a ↦ ā) ∈ Aut(𝔽_{4}) be the field automorphism. Thus 𝔽_{4} = {0, 1, ω, ω̄}. Define γ_{0}, γ_{1}, ϕ, τ ∈ Aut(UT_{3}(4)) by setting

γ_{0} (( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & ωa & ω̄b \\ 0 & 1 & ωc \\ 0 & 0 & 1 \end{matrix} )  γ_{1} (( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & ωa & b \\ 0 & 1 & ω̄c \\ 0 & 0 & 1 \end{matrix} )

ϕ (( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & ā & b̄ \\ 0 & 1 & c̄ \\ 0 & 0 & 1 \end{matrix})  τ (( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & c & b \\ 0 & 1 & a \\ 0 & 0 & 1 \end{matrix} )^{−1} .

Thus γ_{0} and γ_{1} are conjugation by diag(ω, 1, ω̄) and diag(1, ω̄, 1), respectively, while ϕ and τ are restrictions of field and graph automorphisms of SL_{3}(4). Set

Γ_{0} = ⟨γ_{0}, τϕ⟩  and  Γ_{1} = ⟨γ_{1}, τ⟩ .

As subgroups of Aut(UT_{3}(4)), Γ_{0} ≅ Γ_{1} ≅ Σ_{3} and [Γ_{0}, Γ_{1}] = 1.

Lemma C.8. Let R be the group of automorphisms of UT_{3}(4) which induce the identity on UT_{3}(4)/Z(UT_{3}(4)). There are isomorphisms

Aut(UT_{3}(4)) = R·(Γ_{0} × Γ_{1}) ≅ C_{2}^{8} ⋊ (Σ_{3} × Σ_{3}),
Out(UT_{3}(4)) = (R/Inn(UT_{3}(4)))·(Γ_{0} × Γ_{1}) ≅ C_{2}^{4} ⋊ (Σ_{3} × Σ_{3}) .

Also, the group ⟨ϕ, τ⟩ ≅ C_{2}^{2} permutes freely a basis for O_{2}(Out(UT_{3}(4))) ≅ C_{2}^{4}.

Proof. Set S = UT_{3}(4) for short. The above descriptions of Aut(S) and of Out(S) are proven in [OV, Lemma 4.5(a)].

To see that ⟨ϕ, τ⟩ permutes freely a basis for O_{2}(Out(S)) ≅ C_{2}^{4}, let R_{i} ≤ Aut(S) (i = 1, 2) be the group of automorphisms which induce the identity on A_{3−i} and on S/Z(S). Thus R/Inn(S) = (R_{1}/Aut_{A_{2}}(S))×(R_{2}/Aut_{A_{1}}(S)), and τ exchanges these two factors. So it suffices to prove that ϕ acts nontrivially on R_{1}/Aut_{A_{2}}(S) ≅ C_{2}^{2}; which is easily checked. For example, let ρ ∈ R_{1} be the automorphism ρ(e_{12}^{1}) = e_{12}^{1}e_{13}^{ω} and ρ(e_{12}^{ω}) = e_{12}^{ω} (and ρ|_{A_{2}} = Id). Then ϕρϕ^{−1}ρ^{−1} sends e_{12}^{1} to e_{12}^{1}e_{13}^{1} and e_{12}^{ω} to e_{12}^{ω}e_{13}^{ω}, and is not in Inn(UT_{3}(4)). □

Recall that we define U to be the family of 2-groups S such that there is T ⊴ S where T ≅ UT_{3}(4) and C_{S/Z(T)}(T/Z(T)) = T/Z(T).

Lemma C.9. If S ∈ U, then there is a unique normal subgroup T ⊴ S such that T ≅ UT_{3}(4). For each Q ⊴ S with |Q| ≥ 4, Q ≥ Z(T) and Q ∉ DQ.

Proof. Fix T ⊴ S such that T ≅ UT_{3}(4) and C_{S/Z(T)}(T/Z(T)) = T/Z(T). Set Z = Z(T) for short. For each Z_{0} < Z of order 2, T/Z_{0} ≅ 2_{+}^{1+4}, so Z(T/Z_{0}) = Z/Z_{0}, and Z(S/Z_{0}) = Z/Z_{0} since C_{S/Z}(T/Z) = T/Z. If Q ⊴ S and |Q| ≥ 4, then Q ∩ Z ≥ Q ∩ Z(S) ≠ 1 (Lemma A.2(a)), so either Q ≥ Z or |Q ∩ Z| = 2. In the latter case, set Z_{0} = Q ∩ Z; then (Q/Z_{0}) ∩ Z(S/Z_{0}) = (Q/Z_{0}) ∩ (Z/Z_{0}) ≠ 1 by Lemma A.2(a) again, and hence Q ≥ Z.

If Q ∈ DQ, then Q ≅ D_{8} since it contains a normal subgroup isomorphic to C_{2}^{2}. So Q/Z ≤ Z(S/Z) ≤ T/Z by Lemma A.2(a), and Q is abelian, a contradiction.

In particular, if U ⊴ S and U ≅ UT_{3}(4), then Z(U) = Z by the first paragraph, applied with Z(U) in the role of Q. By Lemma C.8, and since the automorphisms in R induce the identity on T/Z(T), S/T ≅ Aut_{S}(T/Z) ≅ C_{2}^{k} for k ≤ 2, and this group acts on T/Z by permuting freely a basis. Hence T/Z is the unique abelian subgroup of order 2^{4} in S/Z by Lemma A.4(a,b), so U = T. □

APPENDIX D Actions on 2-groups of sectional rank at most 4

When studying automorphisms of 2-groups of sectional rank 4, it is natural to begin by looking at subgroups of GL_{4}(2).

Proposition D.1. Assume V = 𝔽_{2}^{4}, H < G = Aut(V), and S ∈ Syl_{2}(H).

(a) If O_{2′}(H) ≠ 1, then H is contained in one of the following subgroups:

\begin{matrix}N_{G}(C_{3}) ≅ Σ_{3} × Σ_{3} & N_{G}(C_{3}) ≅ GL_{2}(4) ⋊ C_{2} ≅ (C_{3} × A_{5}) ⋊ C_{2} \\ N_{G}(C_{3} × C_{3}) ≅ Σ_{3} ≀ C_{2} & N_{G}(C_{5}) ≅ C_{15} ⋊^{2} C_{4} N_{G}(C_{7}) ≅ C_{7} ⋊^{2} C_{3}. \end{matrix} (D.1)

(b) If O_{2′}(H) = 1 and O_{2}(H) ≠ 1, then O_{2}(H) is centric in H.

(c) If O_{2′}(H) = 1 and O_{2}(H) = 1, then H is isomorphic to one of the groups A_{n} for 5 ≤ n ≤ 8, Σ_{5}, Σ_{6}, or GL_{3}(2). There are two G-conjugacy classes of subgroups isomorphic to A_{5} or Σ_{5}, three classes of subgroups isomorphic to GL_{3}(2), and a unique class in each of the other cases.

(d) If H ≅ A_{5} or Σ_{5}, then either V is the L_{2}(4)-module for H (the natural module for SL_{2}(4) ≅ A_{5}), or V is the orthogonal module (the natural module for SO_{4}^{−}(2) ≅ Σ_{5}, and the reduced permutation module). In the former case, H acts transitively on V^{#}, and C_{S}(V) = [S, V] has rank 2. In the latter case, H acts on V^{#} with orbits of length 5 and 10, rk(C_{S}(V)) = 1, and rk([S, V]) = 3.

(e) Assume that S ≅ C_{2}^{2}, and that S permutes a basis of V.

(e.1) If there are distinct involutions x_{1}, x_{2} ∈ S such that S ≰ O_{2}(C_{H}(x_{i})), then H ≅ Σ_{3} × Σ_{3}.

(e.2) If the three involutions in S are all H-conjugate, then H ≅ A_{4} or A_{5}.

(f) Assume S ≅ D_{8}. Assume there is a noncentral involution x ∈ S such that x ∉ O^{2}(H), and such that O_{2}(C_{H}(x)) = ⟨x⟩. Then H ≅ Σ_{3} ≀ C_{2}, Σ_{5}, or (A_{5} × C_{3}) ⋊ C_{2} ≅ ΓL_{2}(4).

(g) Assume S ≅ D_{8} and O^{2}(H) = H. Then H ≅ A_{6}, A_{7}, or GL_{3}(2). The first two are unique up to conjugacy, while Aut(V) contains three conjugacy classes of subgroups isomorphic to GL_{3}(2).

Proof. Since G ≅ A_{8} (cf. [Ta, Corollary 6.7]), this is equivalent to looking at subgroups of A_{8}.

(a) Recall that each minimal normal subgroup of H is isomorphic to a product of simple groups isomorphic to each other (cf. [G, Theorem 2.1.5]). So if O_{2′}(H) ≠ 1, then for some odd prime p, H ≤ N_{G}(A) for some elementary abelian p-subgroup A ⊴ H. Since |G| = 2^{6}·3^{2}·5·7, either A ∈ Syl_{p}(G) for p = 3, 5, 7, or A is in one of the two G-conjugacy classes of subgroups of order 3.

(b) If O_{2′}(H) = 1, then the generalized Fitting subgroup F^{∗}(H) is a central product E(H)O_{2}(H), where E(H) is generated by nonabelian quasisimple subgroups (cf. [A1, 31.12] or [AKO, Theorem A.13(b)]). If O_{2}(H) ≠ 1, then since the centralizer of each involution in A_{8} (hence in H) is solvable, E(H) = 1, and F^{∗}(H) = O_{2}(H) is centric in H (cf. [A1, 31.13] or [AKO, Theorem A.13(c)]).

(c) If O_{2′}(H) = 1 = O_{2}(H), then F^{∗}(H) ≠ 1 is a product of nonabelian simple groups. The nonabelian simple subgroups of G ≅ A_{8} are well known. For example, this follows from Burnside’s list [Bu, § 146] of primitive permutation groups of degree at most 8. Each simple subgroup of G is isomorphic to A_{n} for 5 ≤ n ≤ 8 or to GL_{3}(2). Also, there are two G-conjugacy classes of subgroups isomorphic to A_{5} (one each of degree 5 and 6), and three classes of subgroups isomorphic to GL_{3}(2) (two of degree 7 and one of degree 8).

Set H_{0} = F^{∗}(H). Since no product of two nonabelian simple subgroups is contained in G (the centralizer of each nonabelian simple subgroup is solvable), H_{0} is simple, and H ≤ N_{G}(H_{0}). Since O_{3}(H) = 1, H ≅ (C_{3} × A_{5}) ⋊ C_{2}.

(d) When H ≅ A_{5} or Σ_{5}, we already saw that there are only two possibilities (up to isomorphism) for V as an 𝔽_{2}[H]-module. Hence V must be the L_{2}(4)-module or the orthogonal module, as defined above. The other properties are immediate.

(e) Assume S ≅ C_{2}^{2} permutes a basis for V: either transitively or in two orbits of length 2. If O_{2′}(H) ≠ 1, then H is contained in one of the normalizers listed in (D.1). By inspection, H must be isomorphic to C_{2} × Σ_{3} or Σ_{3} × Σ_{3}. (If H ≤ N_{G}(A) ≅ (C_{3} × A_{5}) ⋊ 2, where |A| = 3, then H ≰ C_{G}(A) since a Sylow 2-subgroup of C_{G}(A) permutes no basis, so H ≅ Σ_{3} × C_{2} or Σ_{3} × Σ_{3}.) Hence the situation of (e.2) cannot occur, and H ≅ Σ_{3} × Σ_{3} in the situation of (e.1).

Now assume O_{2′}(H) = 1. If O_{2}(H) ≠ 1, then it is centric in H by (b). Since H ≠ S (that would satisfy neither of the conditions (e.1) nor (e.2)), H ≅ A_{4}. If O_{2}(H) = 1, then H ≅ A_{5} by (c) and since S ≅ C_{2}^{2}.

(f) Assume that S ≅ D_{8}, and that there is a noncentral involution x ∈ S such that x ∉ O^{2}(H) and O_{2}(C_{H}(x)) = ⟨x⟩.

If O_{2′}(H) ≠ 1, then by (D.1) (and since S ≅ D_{8}), H ≅ (C_{3} × A_{4}) ⋊ C_{2}, (C_{3} × A_{5}) ⋊ C_{2}, or Σ_{3} ≀ C_{2}. The first of these cannot occur by the assumption that there be x ∈ S with O_{2}(C_{H}(x)) = ⟨x⟩.

If O_{2′}(H) = 1 and O_{2}(H) ≠ 1, then O_{2}(H) is centric in H by (b), so H ≅ D_{8} or H ≅ Σ_{4}. In both cases, the centralizer of each involution in H is a 2-group, which contradicts our assumption. If O_{2′}(H) = O_{2}(H) = 1, then by (c) (and since O^{2}(H) < H), H ≅ Σ_{5}.

(g) Since S ≅ D_{8} and O^{2}(H) = H, all involutions in S are H-conjugate by the focal subgroup theorem, so O_{2}(H) = 1. By inspection of the different cases in (D.1), O_{2′}(H) = 1. So by (c), H ≅ A_{6}, A_{7}, or GL_{3}(2). □

Lemma D.2. Let S be a special 2-group of type 2^{2+4}; i.e., [S, S] = Z(S) ≅ C_{2}^{2} and S/Z(S) ≅ C_{2}^{4}. Assume α ∈ Aut(S) is an automorphism of order 3 such that C_{S/Z(S)}(α) = 1 and [α, Z(S)] = 1. Then either S ≅ Q_{8} × Q_{8} or UT_{3}(4); or else S/⟨z⟩ ≅ 2_{+}^{1+4} for two of the three elements z ∈ Z(S)^{#}, S/⟨z⟩ ≅ Q_{8} × C_{2}^{2} for the third, and there is a unique subgroup T < S with T ≅ C_{2}^{4}.

Proof. Set Z = Z(S) and V = S/Z. Let z_{1}, z_{2}, z_{3} ∈ Z^{#} be the three distinct elements. Then α induces an automorphism α_{i} of S/⟨z_{i}⟩ for each i = 1, 2, 3, and hence S/⟨z_{i}⟩ is isomorphic to 2_{+}^{1+4} or Q_{8} × C_{2}^{2}. (It is nonabelian since [S, S] = Z, and none of the other nonabelian central extensions, 2_{−}^{1+4}, C_{2} × (C_{4} ×_{C_{2}} Q_{8}), nor C_{2}^{2} × D_{8}, has an automorphism of order 3 of the required type.) There are five α-invariant subspaces in V of rank 2, of which two have nontrivial squares in 2_{+}^{1+4}, and four have nontrivial squares in Q_{8} × C_{2}^{2}. For each i = 1, 2, 3, let θ(z_{i}) be the number of those subgroups the squares of whose elements are z_{i}. The θ(z_{i}) all have the same parity since the sum of any two of them is 2 or 4, and hence (θ(z_{1}), θ(z_{2}), θ(z_{3})) is (up to permutation) one of the triples (1, 1, 1), (1, 1, 3), or (0, 2, 2). In the first case, S ≅ UT_{3}(4) by Lemma C.7(a), while in the second, S ≅ Q_{8} × Q_{8}. In the third case, S is a pullback of two copies of 2_{+}^{1+4} as described above, and the “exceptional” subspace lifts to a characteristic subgroup isomorphic to C_{2}^{4}. □

Lemma D.3. Fix a 2-group S, and an elementary abelian subgroup Z ≤ Z(S)∩ Fr(S) such that S/Z ≅ 2_{+}^{1+4}, Q_{8} × Q_{8}, or UT_{3}(4). Then Fr(S) is elementary abelian, and Z(S)/Z = Z(S/Z).

Proof. Let π: S ⟶ S/Z be the projection, and set Ẑ = π^{−1}(Z(S/Z)) ≥ Z(S). When S/Z ≅ 2_{+}^{1+4} or Q_{8} × Q_{8}, the relations [g^{2}, g] = 1 for g ∈ S suffice to show that [S, Ẑ] = 1, and hence that Ẑ = Z(S).

Assume S/Z ≅ UT_{3}(4). We identify these two groups, and use the notation of 6.1 (also used in Appendix C) for elements of UT_{3}(4). For g ∈ S/Z, let ĝ ∈ S denote some element in π^{−1}(g). If g, h ∈ S/Z are such that ⟨g, h⟩ ≅ C_{4} × C_{4}, then [ĝ, ĥ] ∈ Z, so [ĝ, ĥ^{2}] = 1 and [ĝ, g^{2}] = 1. Since ⟨g^{2}, h^{2}⟩ = Z(S/Z), this shows that [ĝ, Ẑ] = 1 for such g. For each u, v ∈ 𝔽_{4}^{×}, ⟨e_{12}^{u}e_{23}^{v}, e_{12}^{uω}e_{23}^{vω}⟩ ≅ C_{4} × C_{4} (by Lemma 6.3(a) or by the relation [e_{12}^{x}, e_{23}^{y}] = e_{13}^{xy}), so [e_{12}^{u}e_{23}^{v}, Ẑ] = 1. Sine S/Z = UT_{3}(4) is generated by such elements, [S, Ẑ] = 1, and Ẑ = Z(S).

Thus Ẑ = Z(S) = Fr(S) (so Z(S/Z) = Z(S)/Z) in all cases. Hence [S, S] is elementary abelian, since [g, h]^{2} = [g, h^{2}] = 1 for g, h ∈ S. Also, Fr(S) = [S, S]Z since Fr(S/Z) = [S/Z, S/Z], and hence Fr(S) is also elementary abelian. □

We say that a finite group G is strictly 2-constrained if O_{2}(G) is centric in G; equivalently, F^{∗}(G) = O_{2}(G). Let 2Cons_{4} denote the class of all finite groups which are strictly 2-constrained with sectional 2-rank at most 4. Throughout the rest of the chapter, we list a few results about the structure of such groups. Some of these are taken entirely or in part from [GH].

Lemma D.4 ([GH, II.4.1]). Assume G ∈ 2Cons_{4} is such that G/O_{2}(G) ≅ A_{5}. Then O_{2}(G) is isomorphic to C_{2}^{4} or D_{8} ×_{C_{2}} Q_{8}.

Lemma D.5. Assume G ∈ 2Cons_{4} is such that G/O_{2}(G) ≅ GL_{3}(2). Choose S ∈ Syl_{2}(G), and set Q = O_{2}(G) ⊴ S. Then Q ≅ C_{2}^{3}, C_{2}^{4}, or C_{4} × C_{2}^{3}.

(a) If Q ≅ C_{2}^{3}, then S ≅ UT_{4}(2) or S has type M_{12}, and in either case, r(S) = 4.

(b) If Q ≅ C_{2}^{4}, then either Q is decomposable as an Aut_{G}(Q)-module, or Q is indecomposable with an invariant subgroup of rank 1 or 3. If Q is decomposable, then [G, Q] ≅ C_{2}^{3} and G/[G, Q] ≅ SL_{2}(7). If Q is indecomposable, then for each involution α ∈ Aut_{G}(Q), rk([α, Q]) = 2.

(c) Assume Q ≅ C_{4} × C_{2}^{3}, and set V = Ω_{1}(Q). Then V is decomposable as an Aut_{G}(Q)-module with invariant submodule [G, V] of rank 3, Q/Fr(Q) is indecomposable, and G/[G, V] ≅ C_{4} ×_{C_{2}} SL_{2}(7).

Proof. Set Γ = Out_{G}(Q) ≅ GL_{3}(2) for short. The possibilities for Q are listed in [GH, Proposition II.3.1]. (a,b) Point (a) is shown in [GH, Lemma II.3.4], and the first statement in (b) in [GH, Lemma II.3.7]. If Q ≅ C_{2}^{4} and is Γ-decomposable, then since G does not have a direct factor C_{2} (that would imply r(S) = 5 by (a)), G/[G, Q] ≅ SL_{2}(7) by [GH, Lemma II.3.8].

If Q ≅ C_{2}^{4} and the conclusion of the last statement in (b) is not true, then rk([α, Q]) = 1 (and hence rk(C_{Q}(α)) = 3) for each involution α ∈ Γ since the involutions are all Γ-conjugate. Also, Γ ≅ GL_{3}(2) is generated by three involutions (e.g., the three elementary matrices e_{12}, e_{23}, and e_{31}). So rk(C_{Q}(Γ)) ≥ 1 and rk([Γ, Q]) ≤ 3, with equality in each case since Γ acts faithfully. Thus Q is decomposable.

(c) Assume Q ≅ C_{4} × C_{2}^{3}, and set V = Ω_{1}(Q) and Z = Fr(Q). By [GH, Lemma II.3.12], G/V ≅ C_{2} × GL_{3}(2), and hence Q/Z is Γ-indecomposable by (b).

Fix an involution α ∈ Γ, and choose g ∈ Q∖V. Then [α, g] ∈ V, and [α, g] ∈ C_{V} (α) since α^{2} = Id. Since Q/Z is indecomposable, rk(C_{Q/Z}(α)) = 2 = rk(C_{V/Z}(α)) by (b), so C_{Q}(α) ≤ V, [α, gv] ≠ 1 for each v ∈ V, and hence [α, g] ∉ [α, V]. Thus C_{V} (α) > [α, V], which by (b) implies that V is decomposable.

Thus V = Z × W, where G/Q ≅ GL_{3}(2) acts faithfully on W. In particular, W = [G, V], and Q/[G, V] ≅ C_{4}. So G/[G, V] is a (central) extension of Q/[G, V] ≅ C_{4} by G/Q ≅ GL_{3}(2), O^{2}(G)/[G, V] ≅ SL_{2}(7) by (b) (and since G/V ≅ C_{2} × GL_{3}(2)), and thus G/[G, V] ≅ C_{4} ×_{C_{2}} SL_{2}(7). □

Lemma D.6. If G ∈ 2Cons_{4}, then G/O_{2}(G) ≅ Σ̸_{6}.

Proof. This is shown in [GH, Theorem II.B], but since the proof there is somewhat long and indirect, we give a different argument here. Assume G is strictly 2-constrained with G/O_{2}(G) ≅ Σ_{6}; we will show that G has sectional 2-rank at least 5. Set Q = O_{2}(G). If rk(Q/Fr(Q)) > 4, then we are done, so upon replacing G by G/Fr(Q), we can assume that Q ≅ C_{2}^{4}.

Fix a surjection ψ: G −↠ Σ_{6} with kernel Q. Since there is a unique conjugacy class of subgroup Σ_{6} in GL_{4}(2) (Proposition D.1(c)), we can identify Q with the group of subsets of even order in {1, 2, 3, 4, 5, 6}, modulo the relation of identifying each subset with its complement (and where ψ induces the obvious action of Σ_{6}). Consider the subgroups

T̂ = ⟨(1 2), (3 4), (5 6)⟩ ≤ Σ_{6}  and  T = T̂ ∩ A_{6} .

Then T̂ ≅ C_{2}^{3}, T ≅ C_{2}^{2}, and T̂ acts via the identity on Q_{0} = ⟨12, 34⟩ and on Q/Q_{0}. There is a subgroup Γ ≤ Σ_{6} such that Γ ≅ A_{5} and T ∈ Syl_{2}(Γ) (defined via the permutation action of A_{5} on its six Sylow 5-subgroups, or on the six pairs of opposite vertices in an icosahedron).

Set G_{0} = ψ^{−1}(Γ), Ĥ = ψ^{−1}(T̂) and H = ψ^{−1}(T). Then G_{0} ≅ Q ⋊ Γ, since by [GH, Lemma II.2.6], any extension of C_{2}^{4} by A_{5} splits. Since C_{Q}(T) = Q_{0} has rank 2, Q is the L_{2}(4)-module for Γ (Proposition D.1(d)), so G_{0} is isomorphic to a parabolic subgroup in SL_{3}(4), and H ≅ UT_{3}(4). Let R be the unique subgroup R < H such that R ≠ Q and R ≅ C_{2}^{4} (see Lemma C.6(a)).

Let x ∈ Ĥ be such that ψ(x) = (1 2)(3 4)(5 6). Then c_{x} ∈ Aut(H) induces the identity on Q_{0}, Q/Q_{0}, and H/Q. Also, c_{x}(R) = R since Q, R < H are the only subgroups isomorphic to C_{2}^{4}. Since [x, H] ≤ Q, [x, R] ≤ R ∩ Q = Q_{0}. Thus c_{x} induces the identity on R/Q_{0} and on Q/Q_{0}, and hence on H/Z(H) = QR/Q_{0}.

Let y ∈ G be such that ψ(y) = (1 3 5)(2 4 6). Since [x, y] ∈ Ker(ψ) = Q, [x^{2}, y] ≡ [x, y]^{2} = 1 (mod [x, Q] ≤ Q_{0}). Since C_{Q/Q_{0}}(y) = 1, this implies that x^{2} ∈ Q_{0}, and hence that Ĥ/Q_{0} ≅ (Q/Q_{0}) × T̂ ≅ C_{2}^{5}. □

Two more lemmas of this type are needed. By (C_{3} × C_{3}) ⋊ C_{4}, we mean the semidirect product where C_{4} acts faithfully on C_{3} × C_{3}.

Lemma D.7. Assume G ∈ 2Cons_{4} is such that G/O_{2}(G) is isomorphic to Σ_{3} ≀ C_{2} or (C_{3} × C_{3}) ⋊ C_{4}. Then O_{2}(G) is isomorphic to one of the groups C_{2}^{4}, Q_{8} ×_{C_{2}} Q_{8}, or Q_{8} × Q_{8}.

Proof. Set Q = O_{2}(G). Since Σ_{3} ≀ C_{2} ≅ (C_{3} × C_{3}) ⋊ D_{8} contains a subgroup isomorphic to (C_{3} × C_{3}) ⋊ C_{4}, it suffices to prove this when G/Q ≅ (C_{3} × C_{3}) ⋊ C_{4}. Note that G/Q contains no normal subgroup of order 3. Set H = O^{2}(G),so that H/Q ≅ C_{3} × C_{3}.

Set Z = Fr(Q) and V = Q/Z. By Lemma A.9, H/Q acts faithfully on V, so rk(V) = 4. Let V_{1}, V_{2} < V be the two subgroups of rank 2 in which are normal in H/Z. Thus V = V_{1} × V_{2}. Let

𝔮: V = V_{1} × V_{2} ⟶ Z  and  𝔟: V × V ⟶ Z

be the quadratic and bilinear maps where 𝔮(xZ) = x^{2} and 𝔟(xZ, yZ) = [x, y]. Case 1: Assume Z = Fr(Q) is elementary abelian and [H, Z] ≠ 1. Thus H/Q acts nontrivially on Z, and since G/Q contains no normal subgroup of order 3, H/Q must act faithfully on Z. Hence Z/C_{Z}(H) has rank at least 4. Since r(Q) ≤ 4, we have rk(Z) = 4 and C_{Z}(H) = 1. Let Z_{1}, Z_{2} < Z be the two subgroups of rank 2 normalized by H. For i = 1, 2, let 𝔮_{i}: V → Z_{i} and 𝔟_{i}: V × V → Z_{i} be the composites of 𝔮 and 𝔟 with projection to Z_{i}.

For each i, j = 1, 2, there is g ∈ H of order 3 which acts nontrivially on both V_{i} and Z_{j}. Since 𝔮_{j}|_{V_{i}}: V_{i} → Z_{j} commutes with the action of g, it is either a bijection or zero, and in particular is linear. Hence 𝔮|_{V_{1}} and 𝔮|_{V_{2}} are both linear. They are both nonzero, since otherwise the preimage of V_{1} or of V_{2} in Q would have rank 6. Thus they are injective. Also, Im(𝔮|_{V_{1}}) + Im(𝔮|_{V_{2}}) is normalized by the action of G/Q, hence must be equal to Z, and thus 𝔮|_{V_{1}} ⊕ 𝔮|_{V_{2}} is an isomorphism. Since V_{1}, V_{2}, Z_{1}, and Z_{2} are the only subgroups of rank 2 in V or Z normalized by H, we can assume (after reindexing if necessary) that 𝔮(V_{1}) = Z_{1} and 𝔮(V_{2}) = Z_{2}.

Thus, for i = 1, 2, there is g ∈ G of order 3 which acts trivially on V_{i} and Z_{i} and nontrivially on V_{3−i} and Z_{3−i}. Hence for each v ∈ V_{i}, 𝔟_{i}(v, w) ∈ Z_{i} is independent of w ∈ V_{3−i}^{#}, and since ∏_{w∈V_{3−i}^{#}} 𝔟_{i}(v, w) = 1 (𝔟_{i} is bilinear), 𝔟_{i}(v, V_{3−i}) = 1. Thus 𝔟_{i}(V_{1}, V_{2}) = 1 for i = 1, 2, so 𝔟(V_{1}, V_{2}) = 1, and 𝔮 is linear.

This proves that Q ≅ C_{4}^{4}. Hence G contains a subgroup isomorphic to T ≅ (C_{4} × C_{4}) ≀ C_{2}, which is impossible since r(T) = 5. So this case is impossible. Case 2: Next assume Z = Fr(Q) is elementary abelian and [H, Z] = 1. If Z = 1, then Q ≅ C_{2}^{4}, so assume Z ≠ 1. Since 𝔮 commutes with the actions of H/Q on V = Q/Z and on Z, it sends all elements of V_{1}^{#} to the same element z_{1} ∈ Z, and all elements of V_{2}^{#} to the same element z_{2} ∈ Z.

Let g ∈ H be an element of order 3 which acts nontrivially on V_{1} and trivially on V_{2}, and fix w ∈ V_{2}. Then 𝔟(v, w) ∈ Z is constant for v ∈ V_{1}^{#} (since the three elements are permuted transitively by g), ∏_{v∈V_{1}^{#}} 𝔟(v, w) = 1 since 𝔟 is bilinear, and so 𝔟(V_{1}, w) = 1.

Thus 𝔟(V_{1}, V_{2}) = 1, and Z = Fr(Q) = ⟨Im(𝔮))⟩ = ⟨z_{1}, z_{2}⟩. So either z_{1} = z_{2} and Q ≅ 2_{+}^{1+4}, or z_{1} ≠ z_{2} and Q ≅ Q_{8} × Q_{8}. Case 3: Now assume Z is not elementary abelian, and assume G is a minimal example of this type. Set Z_{0} = Fr(Z). By minimality, Z_{0} is elementary abelian and central, and Q/Z_{0} ≅ 2_{+}^{1+4} or Q_{8} × Q_{8} by Steps 1 and 2. But then Z = Fr(Q) is elementary abelian by Lemma D.3, a contradiction. □

Lemma D.8. Assume G ∈ 2Cons_{4} is such that G/O_{2}(G) ≅ D_{10}. Then O_{2}(G) is isomorphic to C_{2}^{4} or 2_{−}^{1+4}, or is of type U_{3}(4).

Proof. Set Q = O_{2}(G) for short. Fix σ, τ ∈ Aut(Q) such that |σ| = 5, τ^{2} ∈ Inn(Q), and ⟨[σ], [τ]⟩ = Out_{G}(Q) ≅ D_{10}. Then Out_{G}(Q) acts faithfully on Q/Fr(Q) by Lemma A.9, C_{Q/Fr(Q)}(σ) = 1, and rk(C_{Q/Fr(Q)}(τ)) = 2. So σ acts on Q/Fr(Q) with three free orbits of involutions each of which contains an element of C_{Q/Fr(Q)}(τ) and hence is τ-invariant. Fix a_{1}, a_{2}, a_{3}, a_{4}, a_{5} ∈ Q whose classes a_{i} generate Q/Fr(Q) (hence generate Q), whose product is trivial in Q/Fr(Q), and with indices (modulo 5) chosen such that σ(a_{i}) = a_{i+1} and τ(a_{i}) = a_{−i}.

Assume C_{Q}(σ) ≠ 1. Then by [GH, Lemma I.3.9], Q is special of type 2^{k+4} where 1 ≤ k ≤ 4. Set Z = Fr(Q) = Z(Q) ≅ C_{2}^{k}. Then C_{Z}(σ) ≠ 1, and hence [σ, Z] = 1 since rk(Z) ≤ 4. Thus there are x, y, z ∈ Z such that x = a_{i}^{2}, y = (a_{i}a_{i+1})^{2} = [a_{i}, a_{i+1}], and z = (a_{i}a_{i+2})^{2} = [a_{i}, a_{i+2}] for all i (indices again taken modulo 5). Then 1 = (a_{1}a_{2}a_{3}a_{4}a_{5})^{2} = x^{5} ∏_{1≤i<j≤5}[a_{i}, a_{j}] = xyz, so Z = Fr(Q) = ⟨x, y, z⟩ has rank at most 2. By [GH, Lemma I.3.9] again, Q is isomorphic to 2_{−}^{1+4} or is of type U_{3}(4).

Now assume C_{Q}(σ) = 1. Assume G and Q are minimal such that Q ≅ C̸_{2}^{4}, and set Z = Fr(Q). By minimality, Z ≤ Z(Q) and is elementary abelian. Also, rk(Z) ≥ 4 since σ acts nontrivially, and rk(Z) = 4 since r(Q) = 4. If Q is abelian, then Q ≅ C_{4}^{4}, G has Sylow subgroups isomorphic to (C_{4} × C_{4}) ≀ C_{2} of sectional rank 5, so G ∉ 2Cons_{4}.

Thus Q is nonabelian, and so [Q, Q] = Z. For each i, set x_{i} = [a_{i}, a_{i+1}a_{i−1}]. Then σ(x_{i}) = x_{i+1}, and x_{1}x_{2}x_{3}x_{4}x_{5} ∈ C_{Z}(σ) = 1. Also,

x_{i} = [a_{i}, a_{i+1}a_{i−1}] = [a_{i}, a_{i−1}]·σ([a_{i}, a_{i−1}])
= [a_{i}, a_{i+2}a_{i−2}] = [a_{i}, a_{i−2}]·σ^{2}([a_{i}, a_{i−2}]).

Since C_{Z}(σ) = 1, this implies that [a_{i}, a_{i−1}] = x_{i+1}x_{i−2} and [a_{i}, a_{i−2}] = x_{i+1}x_{i+2} for each i. In particular, Z = [Q, Q] = ⟨x_{1}, …, x_{5}⟩. Also, (a_{i})^{2} ∈ C_{Z}(σ^{2i}τ), so either (a_{i})^{2} = 1, or one of the following holds:

(a_{i})^{2} = x_{i} ⟹ (a_{i+1}a_{i−1})^{2} = (a_{i+2}a_{i−2})^{2} = x_{i}
(a_{i})^{2} = x_{i+1}x_{i−1} ⟹ (a_{i+1}a_{i−1})^{2} = 1
(a_{i})^{2} = x_{i+2}x_{i−2} ⟹ (a_{i+2}a_{i−2})^{2} = 1.

If any element of Q∖Z has order 2, then Q contains an abelian subgroup of rank 5. If (a_{i})^{2} = x_{i}, then Z⟨a_{i}, a_{i+1}a_{i−1}⟩ ≅ Q_{8} × C_{2}^{3}. In either case, r(Q) ≥ 5, and so G ∉ 2Cons_{4}. □

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