REDUCED FUSION SYSTEMS OVER 2-GROUPS OF SMALL ORDER

KASPER K. S. ANDERSEN, BOB OLIVER, AND JOANA VENTURA

Abstract. We prove, when S is a 2-group of order at most 2^{9}, that each reduced fusion system over S is the fusion system of a finite simple group and is tame. It then follows that each saturated fusion system over a 2-group of order at most 2^{9} is realizable. What is most interesting about this result is the method of proof: we show that among 2-groups with order in this range, the ones which can be Sylow 2-subgroups of finite simple groups are almost completely determined by criteria based on Bender’s classification of groups with strongly 2-embedded subgroups.

A saturated fusion system over a finite p-group S is a category whose objects are the subgroups of S, whose morphisms are monomorphisms between subgroups, and which satisfy certain axioms first formulated by Puig [Pg] and motivated in part by conjugacy relations among p-subgroups of a given finite group. A saturated fusion system is realizable if it is isomorphic to the fusion system of some finite group, and is exotic otherwise. One of our main goals is to try to understand when and how exotic fusion systems can occur, especially over 2-groups.

A saturated fusion system F is reduced if O_{p}(F) = 1 and O^{p}(F) = O^{p′}(F) = F (see Definitions 1.1(c,e) and 1.10(a)). A saturated fusion system F is tame if it is realized by a group G such that the natural homomorphism from Out(G) to a certain group of outer automorphisms of F (more precisely, of an associated linking system) is split surjective (Definition 1.11). The main result in our earlier paper [AOV1] says roughly that exotic fusion systems can be detected via tameness of associated reduced fusion systems. More precisely, by [AOV1, Theorems A & B], if the “reduction” of a fusion system F is tame, then F is tame and hence realizable, while if a reduced fusion system is not tame, then it is the reduction of an exotic fusion system.

A saturated fusion system is indecomposable if it does not split as a product of fusion systems over nontrivial p-groups. We can now state our main result.

Theorem A. Let F be a reduced, indecomposable fusion system over a 2-group of order at most 2^{9}. Then F is the fusion system of a finite simple group, and is tame.

Proof. This is shown in Theorems 4.1 (for 2-groups of order at most 64), 4.3 (order 2^{7}), 5.1 (order 2^{8}), and 6.1 (order 2^{9}). □

The next theorem follows from Theorem A and the above discussion.

Theorem B. Each saturated fusion system over a 2-group of order at most 2^{9} is realizable.

2000 Mathematics Subject Classification. Primary 20D20. Secondary 20D05, 20E45, 20–04.

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

K. K. S. Andersen was partially supported by the Danish National Research Foundation (DNRF)

through the Centre for Symmetry and Deformation.

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

J. Ventura was partially supported by FCT through program POCI 2010/FEDER and project PTDC/MAT/098317/2008.

Proof. Let F be a saturated fusion system over a 2-group S of order at most 2^{9}. The reduction 𝔯𝔢𝔡(F) of F is defined in [AOV1, § 2] (see Definition 1.10 below): it is a reduced fusion system over a subquotient of S. Since 𝔯𝔢𝔡(F) is tame by Theorem A, F is realizable by [AOV1, Theorem A] (Theorem 1.12(a) below). □

Our proofs of these results are based in large part on computer computations. The starting point for doing this is the version of Alperin’s fusion theorem (re)stated in Proposition 1.4: each morphism in a saturated fusion system F is a composite of restrictions of Fautomorphisms of S and of certain “F-essential” subgroups. We refer to Definition 1.2 for the definition of F-essential.

In [OV], the second and third authors developed a procedure for determining all reduced fusion systems over a given 2-group, taking as examples two groups of order 2^{7} and two of order 2^{10}. The idea was to first determine those subgroups of a given S which could potentially be essential in some fusion system over S (the “critical” subgroups), and then study what their F-automorphism groups could be. In this paper, we first made a computer search (using Magma [Mg] and GAP [Gp]) of all 2-groups of order at most 2^{9}, to determine which of them have “enough” critical subgroups and satisfy other conditions which are necessary to support a reduced fusion system. These search criteria (listed in Proposition 2.2) are, in fact, satisfied by very few 2-groups. Reduced fusion systems over them are listed individually, using computer computations in some cases and computerfree proofs in others.

The following table shows how close these programs come to restricting attention only to groups which are Sylow 2-subgroups of simple groups.

Group order  2^{7}  2^{8}  2^{9}

Number of groups 2328 56092 ≈ 10^{7}

Nr. satisfying conditions in 2.2 9  20  34
Sylows of simple groups  6  6  10
Split as products  2  10  23
Others  1  4  1

More precisely, the number given in the third row of the table is the number of groups of the given order which satisfy the conditions in Proposition 2.2, together with the dihedral and semidihedral groups of that order, and the wreathed groups C_{2^{n}} ≀ C_{2} if there are any. (These latter were eliminated by condition (a) or (b) in Proposition 2.2, and restored afterwards.) Thus among the groups not eliminated by these formal conditions (based mostly on Bender’s theorem [Be, Satz 1]), most are either Sylow 2-subgroups of simple groups, or are products of smaller groups and cannot be Sylow 2-subgroups of simple groups nor of reduced fusion systems. Note that this dichotomy applies only in this range: the group (D_{8} ≀ C_{2}) × D_{8} of order 2^{10} is a Sylow 2-subgroup of the simple group A_{14} (and its fusion system is reduced and indecomposable).

There are many examples, especially among finite simple groups of Lie type, of different simple groups whose fusion systems (at some given prime p) are reduced and isomorphic. The following theorem gives some examples of this. We do not use this theorem, except to motivate our giving only one example (or one family of examples) of groups which realize any given fusion system. The cases most relevant to this paper are those where 𝔾 = PSL_{n} or PSp_{2n} for some n ≥ 2.

Theorem 0.1 ([BMO1, Theorem A]). Fix a prime p, a connected reductive group scheme 𝔾 over ℤ, and a pair of prime powers q and q′ both prime to p. Then the following hold, where “ G ∼_{p} H” means that the p-fusion systems of G and H are isomorphic.

(a) If ⟨q⟩ = ⟨q′⟩ as closed subgroups of ℤ_{p}^{×}, then 𝔾(q) ∼_{p} 𝔾(q′).

(b) If 𝔾 is of type A_{n}, D_{n}, or E_{6}, τ is a graph automorphism of 𝔾, and ⟨q⟩ = ⟨q′⟩ ≤ ℤ_{p}^{×}, then ^{τ}𝔾(q) ∼_{p} ^{τ}𝔾(q′).

(c) If the Weyl group of 𝔾 contains an element which acts on the maximal torus by inverting all elements, and ⟨−1, q⟩ = ⟨−1, q′⟩ ≤ ℤ_{p}^{×}, then 𝔾(q) ∼_{p} 𝔾(q′).

(d) If ⟨−q⟩ = ⟨q′⟩ ≤ ℤ_{p}^{×}, then PSU_{n}(q) ∼_{p} PSL_{n}(q′) for all n ≥ 2.

For example, by (a), if F is the fusion system of PSL_{3}(17) (for p = 2), then it is also isomorphic to the fusion system of PSL_{3}(q) for each q ≡ 17 (mod 32).

Background results on fusion systems are given in Section 1, and the precise criteria which we use in our computer searches are listed in Section 2. In Section 3, we look at the special case of reduced fusion systems over nonabelian 2-groups with abelian direct factor. Afterwards, we handle the individual cases: groups of order at most 2^{7} in Section 4, those of order 2^{8} in Section 5, and those of order 2^{9} in Sections 6–7. At the end, some standard background results about groups and representations are given in an appendix.

When G is a group and g, h ∈ G, we write [g, h] = ghg^{−1}h^{−1} for the commutator, unlike the usual notation used by group theorists (although this very rarely makes a difference in any of the formulas). Also, ^{g}x = gxg^{−1} and x^{g} = g^{−1}xg in this situation. As usual, when S is a finite p-group, Z_{2}(S) ⊴ S is the subgroup Z_{2}(S)/Z(S) = Z(S/Z(S)), and S is said to be of type G if it is isomorphic to a Sylow p-subgroup of the finite group G. When n ≥ 2 and q is a prime power, we let UT_{n}(q) ≤ SL_{n}(q) be the subgroup of strict upper triangular matrices (i.e., with ones on the diagonal). Also, we follow the usual notation for extraspecial 2-groups: 2_{+}^{1+2n} is a central product of n copies of D_{8}, while 2_{−}^{1+2n} is a central product of n−1 copies of D_{8} and one copy of Q_{8}.

All three authors would like to thank the Mathematics Institute at Copenhagen University, and especially the Centre for Symmetry and Deformation, for their hospitality while much of this work was carried out. We would also like to thank Jacob Weismann and James Avery for their support with some of the computer computations, especially those involving the groups of order 512.

1. Background results about fusion systems

In this section, we collect a few definitions and results needed in the rest of the paper. A fusion system over a finite p-group S is a category F whose objects are the subgroups of S, and whose morphism sets Hom_{F}(P, Q) ⊆ Inj(P, Q) contain all homomorphisms induced by conjugation in S. One also requires that each morphism in F factors as the composite of an isomorphism (in F) followed by an inclusion. For such F, a subgroup P ≤ S is called fully centralized (fully normalized) if |N_{S}(P)| ≥ |N_{S}(Q)| (|C_{S}(P)| ≥ |C_{S}(Q)|) for each Q in the F-isomorphism class of P. The fusion system F is saturated if it satisfies the following two axioms:

(I) (Sylow axiom) If P ≤ S is fully normalized, then P is fully centralized and Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)).

(II) (Extension axiom) If φ ∈ Iso_{F}(P, Q) where Q is fully centralized, and

N_{φ} = {g ∈ N_{S}(P) | φc_{g}φ^{−1} ∈ Aut_{S}(Q)},

then φ extends to some φ ∈ Hom_{F}(N_{φ}, S).

We refer to [BLO2, Definition 1.2], [AKO, § I.2], and [Cr, § 4.1] for more details.

For example, when G is a finite group with S ∈ Syl_{p}(G), the fusion system of G is the category F_{S}(G), where for P, Q ≤ S, Hom_{F_{S}(G)}(P, Q) = Hom_{G}(P, Q): the set of homomorphisms which are induced by conjugation in G. A fusion system F over S is realizable if F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G), and is exotic otherwise.

We say that two subgroups P, Q ≤ S are F-conjugate if they are isomorphic in the category F. Let P^{F} be the set of all subgroups of S which are F-conjugate to P.

Definition 1.1. Fix a prime p, a finite p-group S, and a saturated fusion system F over S. For each subgroup P ≤ S,

(a) P is F-centric if C_{S}(Q) = Z(Q) for each Q ∈ P^{F};

(b) P is central in F if P ≤ Z(S), and 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};

(c) P is normal in F (P ⊴ F) if P ⊴ S, and 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; and

(d) P is strongly closed in F if for each Q ≤ P and each φ ∈ Hom_{F}(Q, S), φ(Q) ≤ P.

(e) By analogy with the notation for finite groups, the maximal normal p-subgroup of a saturated fusion system F is denoted O_{p}(F), and the maximal central p-subgroup is denoted Z(F).

(f) 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}

for all P, Q ≤ S.

1.1. Essential and critical subgroups.

We next define the essential subgroups in a fusion system F, and describe how their automorphisms and those of S generate F.

Definition 1.2. (a) If p is a prime and H < G are finite groups, then H is strongly p-embedded in G if p||H|, and for each x ∈ G∖H, H ∩^{x}H has order prime to p.

(b) If F is a saturated fusion system over a finite p-group S, then a subgroup P of S is F-essential if P < S, P is F-centric and fully normalized in F, and Out_{F}(P) contains a strongly p-embedded subgroup. We let E_{F} denote the set of F-essential subgroups of S.

(c) Let S be a finite p-group. A subgroup P of S will be called critical if P < S, P is centric in S, and there are subgroups G_{0} and G of Out(P) such that

Out_{S}(P) ≤ G_{0} < G ≤ Out(P) ,

G_{0} is strongly p-embedded in G, and Out_{S}(P) ∈ Syl_{p}(G).

Note that by definition, if P is critical in S, then α(P) is critical in S for each α ∈ Aut(S).

The definition in [OV, Definition 3.1] of critical subgroups contains the additional condition that Out_{S}(P) ∩ O_{p}(Out(P)) = 1. However, this condition follows from the others, which is why we omit it here. In the situation of Definition 1.2(c), O_{p}(G) = 1 since G contains a strongly p-embedded subgroup (cf. [AKO, Proposition A.7(c)]), and so Out_{S}(P) ∩ O_{p}(Out(P)) ≤ O_{p}(G) = 1.

We refer to [Sz2, § 6.4], and also to [AKO, Proposition A.7], for some of the other properties of strongly p-embedded subgroups. We also need the following:

Lemma 1.3. Fix a prime p, let G be a finite group, and assume H < G is strongly p-embedded. Let K ⊴ G be a normal subgroup of order prime to p. Then either HK/K is strongly p-embedded in G/K, or HK = G.

Proof. Assume otherwise. By definition (and since p||HK/K|), there is g ∈ G∖HK such that p||(^{g}HK/K) ∩ (HK/K)|. Choose x ∈ ^{g}HK ∩ HK of order p. Choose x′_{1} ∈ ^{g}H and x′_{2} ∈ H such that x′_{1} ≡ x ≡ x′_{2} (mod K). Then |x′_{i}| = pm_{i} for some m_{i} prime to p, and we can choose x_{i} ∈ ⟨x′_{i}⟩ of order p such that x_{i} ≡ x (mod K).

Thus ⟨x_{1}⟩ and ⟨x_{2}⟩ are Sylow p-subgroups of K⟨x⟩, so there is k ∈ K such that ⟨x_{2}⟩ = ⟨^{k}x_{1}⟩. Then x_{2} ∈ H ∩^{kg}H, which is impossible since H is strongly p-embedded and kg ∈ G∖H. □

The next proposition, first shown by Puig, explains the importance of essential subgroups.

Proposition 1.4 ([O1, Proposition 1.10(a,b)]). For any saturated fusion system F over a finite p-group S, each morphism in F is a composite of restrictions of automorphisms in Aut_{F}(S), and of automorphisms in O^{p′}(Aut_{F}(P)) for P ∈ E_{F}.

The next lemma is an immediate consequence of the definitions: critical subgroups in S are the ones which can be essential in a saturated fusion system over S.

Proposition 1.5 ([OV, Proposition 3.2]). If F is a saturated fusion system over a finite p-group S, and P ∈ E_{F}, then P is a critical subgroup of S.

The next three propositions give some necessary conditions for a subgroup to be critical, and hence necessary conditions for it to be essential.

Proposition 1.6 ([OV, Lemma 3.4]). Fix a prime p, a finite p-group S, a subgroup P ≤ S, and a characteristic subgroup Θ ≤ P. Assume there is g ∈ N_{S}(P)∖P such that

(a) [g, P] ≤ Θ·Fr(P), and

(b) [g, Θ] ≤ Fr(P).

Then c_{g} ∈ O_{p}(Aut(P)), and hence P is not critical.

The next proposition is a consequence of Bender’s classification [Be, Satz 1] of groups with strongly 2-embedded subgroups.

Proposition 1.7 ([OV, Proposition 3.3(a,c,d)]). Fix a critical subgroup P of a finite 2-group S. Set S_{0} = N_{S}(P)/P ≅ Out_{S}(P). Then the following hold.

(a) Either S_{0} is cyclic, or Z(S_{0}) = {g ∈ S_{0} | g^{2} = 1}. If rk(Z(S_{0})) > 1, then |S_{0}| = |Z(S_{0})|^{m} for m = 1, 2, or 3.

(b) Set |S_{0}| = 2^{k}. Then rk(P/Fr(P)) ≥ 2k. If k ≥ 2, then rk([s, P/Fr(P)]) ≥ 2 for all 1 ≠ s ∈ S_{0}.

(c) Assume Z(S_{0}) ≅ C_{2}^{n} with n ≥ 2, and fix 1 ≠ s ∈ Z(S_{0}). Then rk([s, P/Fr(P)]) ≥ n.

We also need the following refinement of the last proposition.

Proposition 1.8. Fix a critical subgroup P of a finite 2-group S. Let k be such that 2^{k} = |N_{S}(P)/P | = |Out_{S}(P)|, and let Fr(P) = P_{0} ≤ ⋯ ≤ P_{r} = P be a sequence of subgroups characteristic in P. Then there is some 1 ≤ i ≤ r such that rk(P_{i}/P_{i−1}) ≥ 2k, and such that if k ≥ 2, then rk([s, P_{i}/P_{i−1}]) ≥ 2 for each 1 ≠ s ∈ Out_{S}(P).

Proof. Let Γ ≤ Out(P) be such that Out_{S}(P) ≤ Γ is a Sylow 2-subgroup and Γ has a strongly 2-embedded subgroup. Set K_{i} = C_{Γ} (P_{i}/P_{i−1}) ⊴ Γ: the kernel of the Γ -action on P_{i}/P_{i−1}. Set K = ⋂_{i=1}^{r} K_{i} (so K ⊴ Γ). Then K ≤ O_{2}(Γ) by Lemma A.1 and O_{2}(Γ) = 1 (see [AKO, Proposition A.7(c)]), so K = 1.

Fix an involution t ∈ Out_{S}(P). Choose some g ∈ Γ which does not commute with t, and choose i such that [g, t] ∉ K_{i}. Then K_{i} has odd order since all involutions in Γ are conjugate to t ∉ K_{i} (see [Sz2, 6.4.4(i)]). Also, Γ/K_{i} has at least two involutions: the images of t and gtg^{−1}. If Out_{S}(P) is cyclic or quaternion, this implies that Γ/K_{i} has a strongly 2-embedded subgroup (the centralizer of one of the involutions). Otherwise, by Bender’s theorem (see [Sz2, Theorem 6.4.2]), each strongly 2-embedded subgroup of Γ contains O_{2′}(Γ) and hence contains K_{i}, and so Γ/K_{i} still has a strongly 2-embedded subgroup by Lemma 1.3(a). So in either case, the proposition follows from [OV, Lemma 1.7], applied with V = P_{i}/P_{i−1}. □

1.2. Reduced fusion systems and tame fusion systems.

Recall that a saturated fusion system is called “exotic” if it is not isomorphic to the fusion system (over a Sylow subgroup) of any finite group. In [AOV1], we described how we could restrict attention to a smaller class of saturated fusion systems which we call reduced fusion systems, and still “detect” any exotic fusion systems (reduced or not) which might occur over p-groups of similar type. To make this more precise, we list here the main results of [AOV1]: Theorem 1.12 below. We first need some more definitions.

Let F be a saturated fusion system over a finite p-group S. The focal subgroup of F is the subgroup

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

The hyperfocal subgroup of F is the subgroup

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

Equivalently, in the definition of 𝔥𝔶𝔭(F), we can restrict to automorphisms of order prime to p. It is not hard to see that the image of the focal subgroup in S/𝔥𝔶𝔭(F) is precisely its commutator subgroup [S, S]𝔥𝔶𝔭(F)/𝔥𝔶𝔭(F) (cf. [AKO, Lemma I.7.2]). Hence 𝔥𝔶𝔭(F) is a proper subgroup of S if and only if 𝔣𝔬𝔠(F) is a proper subgroup.

As an immediate consequence of Proposition 1.4 and the definition of 𝔣𝔬𝔠(F), we have:

Proposition 1.9. For any saturated fusion system F over a finite p-group S,

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

By [BCGLO, Theorems 4.3 & 5.4], there is a unique saturated fusion subsystem O^{p}(F) ⊆ F over 𝔥𝔶𝔭(F) such that Aut_{O^{p}(F)}(P) ≥ O^{p}(Aut_{F}(P)) for all P ≤ 𝔥𝔶𝔭(F); and a unique saturated fusion subsystem O^{p′}(F) ⊆ F over S minimal with respect to the property that Aut_{O^{p′}(F)}(P) ≥ O^{p′}(Aut_{F}(P)) for all P ≤ S. This leads to the following definition.

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

(a) F is reduced if O_{p}(F) = 1, O^{p}(F) = F, and O^{p′}(F) = F;

(b) F is decomposable if F = F_{1} × F_{2} for some pair of fusion subsystems F_{1}, F_{2} ≠ 1 (i.e., over nontrivial p-groups), and is indecomposable otherwise; and

(c) F is simple if there are no proper normal subsystems E ⊴ F with 1 ≠ E ⫋ F (see [AKO, Definition I.6.1] for the definition of a normal fusion subsystem).

(d) The reduction of F is the fusion system 𝔯𝔢𝔡(F) obtained by first setting F_{0} = C_{F}(O_{p}(F))/Z(O_{p}(F)), and then applying O^{p′}(O^{p}(−)) to F_{0} repeatedly until it remains unchanged.

Here, C_{F}(Q) ⊆ F means the centralizer fusion system of Q ≤ S: the largest fusion subsystem of F in which Q is central (cf. [AKO, Definition I.5.3] or [Cr, Definition 4.26(i)]). As the names suggest, the reduction of any saturated fusion system is reduced [AOV1, Proposition 2.2].

In general, simple fusion systems are reduced and indecomposable, but the converse need not be true. For example, when p = 2, the fusion system of the wreath product A_{6} ≀ A_{5} is reduced and indecomposable but not simple. However, a reduced fusion system which has no proper nontrivial strongly closed subgroups is simple (see the proof of Proposition 1.18(d) below).

We next explain how any exotic fusion system F can be “detected” via its reduction 𝔯𝔢𝔡(F). The key to doing this is the idea of a tame fusion system. The following is one way to define this. For any finite group G, BG_{p}^{∧} denotes the p-completion of the classifying space of G, and Out(BG_{p}^{∧}) is the group of homotopy classes of homotopy equivalences from the space BG_{p}^{∧} to itself.

Definition 1.11. A saturated fusion system F over a finite p-group S is tamely realized by a finite group G if

• there is an isomorphism of fusion systems F ≅ F_{T} (G), where S ≅ T ∈ Syl_{p}(G); and

• the natural homomorphism κ_{G}: Out(G) ⟶ Out(BG_{p}^{∧}) is split surjective.

The fusion system F is tame if it is tamely realized by some finite group.

In fact, in the definition in [AOV1], we replace the group Out(BG_{p}^{∧}) by one which is defined purely algebraically, as a certain group of outer automorphisms of the centric linking system for G over S. We give the above definition here to avoid a long discussion about linking systems and their automorphisms. The two are equivalent by [BLO1, Theorem B].

The following theorem was shown in [AOV1, Theorems A, B, & C]. It is what motivated Definition 1.11.

Theorem 1.12. Let F be a saturated fusion system over a finite p-group.

(a) If 𝔯𝔢𝔡(F) is tame, then F is also tame, and in particular is realizable as the fusion system of a finite group.

(b) If F is reduced, and each indecomposable factor of F is tame, then F is tame.

(c) If F is reduced and not tame, then there is an exotic fusion system F̃ such that 𝔯𝔢𝔡(F̃) ≅ F.

Theorem 1.12 helps to explain why we only look at reduced, indecomposable fusion systems. Points (a) and (b) say that for every exotic fusion system F, the reduction 𝔯𝔢𝔡(F) is not tame, and at least one of its indecomposable factors is also not tame. In other words, each exotic fusion system is detected by some reduced, indecomposable fusion system which is not tame.

Theorem 1.12 shows the importance of determining whether a given reduced fusion system is tame. In general, rather than comparing Out(G) with Out(BG_{p}^{∧}), it is much simpler to compare Out(G) with a certain group Out(S, F) of outer automorphisms of F.

Definition 1.13. For any saturated fusion system F over a finite p-group S, let Aut(S, F) be the group of those α ∈ Aut(S) such that ^{α}F = F (the “fusion preserving automorphisms”). Set Out(S, F) = Aut(S, F)/Aut_{F}(S).

When F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G), there is a natural homomorphism from Out(G) to Out(S, F) defined by restriction (each class in Out(G) contains automorphisms of G which send S to itself). By [AOV1, §§ 1.3 & 2.2], this map factors as the composite of homomorphisms

Out(G) ⟶^{κ_{G}} Out(BG_{p}^{∧}) ⟶^{μ_{G}} Out(S, F) .

Since Out(S, F) is in general easier to describe than Out(BG_{p}^{∧}), the simplest way to prove tameness is usually by showing that μ_{G}∘κ_{G} is split surjective and that μ_{G} is injective. The following proposition, which is a special case of [AOV1, Proposition 4.2], suffices in all cases considered in this paper for showing that Ker(μ_{G}) = 1.

Proposition 1.14. Fix a finite group G and S ∈ Syl_{2}(G), and set F = F_{S}(G). Assume that at most one subgroup P ∈ E_{F} has noncyclic center. Then Ker(μ_{G}) = 1.

Proof. Let E_{F}^{0} be the set of those P ∈ E_{F} such that C_{Z(P)}(Aut_{F}(P)) < C_{Z(P)}(Aut_{S}(P)). If P ∈ E_{F} and Z(P) is cyclic, then each element of odd order in Aut_{F}(P) acts trivially on Z(P). Then C_{Z(P)}(Aut_{S}(P)) = C_{Z(P)}(Aut_{F}(P)) (recall P is fully normalized in F), and hence P ∉ E_{F}^{0}.

Thus |E_{F}^{0}| ≤ 1 under our hypotheses. By [AOV1, Proposition 4.2(d)], Ker(μ_{G}) = 1 if E_{F}^{0} = ∅, so assume E_{F}^{0} = {P }. Then by the same result, for each α ∈ Ker(μ_{G}), there is an element g_{P} ∈ C_{Z(P)}(Aut_{S}(P)) = Z(S) with the property that α = 1 if and only if g_{P} ∈ g·C_{Z(P)}(Aut_{F}(P)) for some g ∈ C_{Z(S)}(Aut_{F}(S)). Note that P ⊴ S and Aut_{F}(S) normalizes P (by the uniqueness of P), so by [AOV1, Proposition 4.2(a,c)] (with P ≤ S in the role of Q ≤ P in point (c)), g_{P} ≡ 1 modulo C_{Z(P)}(Aut_{F}(S)) = C_{Z(S)}(Aut_{F}(S)). Thus g_{P} ∈ C_{Z(S)}(Aut_{F}(S)), so α = 1 by the above remarks, applied with g = g_{P}. □

1.3. Criteria for detecting reduced fusion systems.

We now list some conditions on a finite p-group S or on a fusion system F over S which are necessary for F to be reduced (or sufficient for F to not be reduced). We begin with a simple criterion for detecting normal p-subgroups.

Proposition 1.15 ([AKO, Proposition I.4.5]). Let F be a saturated fusion system over a finite p-group S, and fix Q ⊴ S. Assume, for each P ∈ E_{F} ∪ {S}, that Q ≤ P and α(Q) = Q for each α ∈ Aut_{F}(P). Then Q ⊴ F.

The next lemma is the starting point for deciding whether or not O^{p}(F) = F.

Lemma 1.16 ([AKO, Corollary I.7.5]). For any saturated fusion system F over a finite pgroup S, F = O^{p}(F) if and only if 𝔣𝔬𝔠(F) = S. If Aut_{F}(S) is a p-group and O^{p}(F) = F, then

S = ⟨[Aut_{F}(P), P] | P ∈ E_{F}⟩.

Proof. See, e.g., [AKO, Corollary I.7.5] for a proof of the first statement. The second follows from that, Proposition 1.9, and [G, Theorems 5.1.1(ii) & 5.1.3] (Q[S, S] = S implies Q = S). □

We next look at cases where there are very few essential subgroups.

Lemma 1.17. Let S be a nontrivial finite p-group. For any saturated fusion system F over S, if |E_{F}| ≤ 1, then O_{p}(F) ≠ 1. If Out_{F}(S) = 1 and E_{F} contains exactly one S-conjugacy class, then O^{p}(F) ⫋ F. In either case, F is not reduced.

Proof. If E_{F} = ∅, then S ⊴ F by Proposition 1.15, while if E_{F} = {P }, then P ⊴ F by the same proposition. In either case, O_{p}(F) ≠ 1, so F is not reduced. If Out_{F}(S) = 1 and E_{F} = P for some S-conjugacy class P, then 𝔣𝔬𝔠(F) is contained in ⟨P, [S, S]⟩ < S, so O^{p}(F) ⫋ F, and again F is not reduced. □

In Sections 4–7, our main technique for checking that fusion systems are realizable is to list all reduced fusion systems over a given 2-group S, and then match them with simple groups having Sylow 2-subgroup S. When doing this, it is important to know that the fusion systems of the groups in question are reduced, since this is not the case for all simple groups (e.g., not for A_{5}). The following proposition, part of which is based on a theorem of Goldschmidt, gives some criteria for showing this.

Proposition 1.18. Let G be a finite nonabelian simple group. Choose S ∈ Syl_{2}(G), and set F = F_{S}(G).

(a) In all cases, O^{2}(F) = F and F is indecomposable.

(b) If S is nonabelian, and if G is not isomorphic to PSU_{3}(2^{n}) (n ≥ 2) nor to Sz(2^{2n+1}) (n ≥ 1), then O_{2}(F) = 1.

(c) If G and S satisfy the hypotheses in (b), and Aut(S) is a 2-group or O^{2′}(F) = F, then F is reduced.

(d) If G is a known simple group and F is reduced, then F is simple. In particular, this is the case whenever G and S satisfy the hypotheses in (b), and G is an alternating group, a sporadic simple group, a simple group of Lie type in defining characteristic 2, or ^{2}F_{4}(2)′.

Proof. (a) By the focal subgroup theorem for groups (cf. [G, Theorem 7.3.4]), 𝔣𝔬𝔠(F) = S ∩ [G, G]. Hence 𝔣𝔬𝔠(F) = S since G is simple, and O^{2}(F) = F by Lemma 1.16.

Assume F is decomposable: thus F = F_{1}×F_{2}, where F_{i} is over S_{i} ≠ 1 and S = S_{1}×S_{2}. In particular, S_{1} and S_{2} are both strongly closed in S with respect to G. So by [Gd2, Corollary A1], the normal closures of S_{1} and S_{2} commute with each other, which is impossible since each has normal closure G. Thus F is indecomposable.

(b) Set Q = O_{2}(F) for short. If Q ≠ 1, then 1 ≠ Z(Q) ⊴ F (see [AKO, Proposition I.4.4]), and in particular, Z(Q) is strongly closed with respect to G. But by [Gd1, Theorem A], under the given assumptions and since G is simple, no nontrivial abelian subgroup of S is strongly closed with respect to G, and thus O_{2}(F) = Q = 1.

(c) If Aut(S) is a 2-group, then Aut_{F}(S) = Inn(S) = Aut_{F}^{0}(S) in the notation of [AKO, Theorem I.7.7], and hence O^{2′}(F) = F by that theorem. Together with (a) and (b), this shows that F is reduced under the above hypotheses.

(d) Assume G is a known simple group such that F = F_{S}(G) is reduced but not simple, and let 1 ≠ E ⊴ F be a proper normal subsystem over T ⊴ S. If T = S, then O^{2′}(F) ⊆ E ⫋ F (see [AOV1, Lemma 1.26]), which is impossible since F is reduced. Thus 1 ≠ T < S where T is strongly closed. By [Ft, Theorem 1], then there is a nontrivial abelian subgroup Q ≤ T which is strongly closed in F, and Q ⊴ F by [AKO, Corollary I.4.7(a)], contradicting the assumption that F is reduced. Hence F is simple.

The last statement is shown in [As, § 16]: in 16.3 (simple groups of Lie type in characteristic 2 and ^{2}F_{4}(2)′), 16.5 (A_{n}), and 16.8 (sporadic simple groups). □

We refer to [FF] and [As, § 16] for some similar results when p is odd.

2. Computer search criteria

We now list explicitly the criteria which we use to search for 2-groups which could support reduced fusion systems, and to search for critical subgroups of a given 2-group. We will need the following application of the transfer homomorphism.

Lemma 2.1. Fix a finite p-group S. Assume there is g ∈ S which satisfies

(a) g ∉ [S, S];

(b) α(g) ≡ g (mod [S, S]) for each α ∈ Aut(S) of order prime to p; and

(c) trf_{P}^{S} ([g]) ∈ P/[P, P] is fixed by O^{p}(Aut(P)) for each critical subgroup P < S.

Alternatively, assume there is g ∈ S which satisfies

(a′) g ∉ Fr(S);

(b′) α(g) ≡ g (mod Fr(S)) for each α ∈ Aut(S) of order prime to p; and

(c′) trf_{P}^{S} ([g]) ∈ P/Fr(P) is fixed by O^{p}(Aut(P)) for each critical subgroup P < S.

In either case, every saturated fusion system over S has a normal subsystem of index p, and hence there are no reduced fusion systems over S.

Proof. We prove this for hypotheses (a)–(c). The proof for (a′)–(c′) holds by the same argument, upon replacing Q/[Q, Q] (for Q ≤ S) by Q/Fr(Q).

Let F be a saturated fusion system over S. We claim that for each isomorphism φ ∈ Iso_{F}(P, Q),

φ(trf_{P}^{S} ([g])) = trf_{Q}^{S}([g]) ∈ Q/[Q, Q] .  (1)

Here, φ: P/[P, P] ⟶^{≅} Q/[Q, Q] is the isomorphism induced by φ. Point (1) holds by (b) (and the naturality properties of the transfer) when φ = α|_{P} for some α ∈ Aut_{F}(S). If P, Q ≤ R < S where R ∈ E_{F} (hence R is critical in S by Proposition 1.5), and φ = α|_{P} for some α ∈ O^{p}(Aut_{F}(R)), then

φ(trf_{P}^{S} ([g])) = φ(trf_{P}^{R}(trf_{R}^{S}([g]))) = trf_{Q}^{R}(α(trf_{R}^{S}([g]))) = trf_{Q}^{R}(trf_{R}^{S}([g])) = trf_{Q}^{S}([g]),

where the third equality holds by (c). Point (1) now follows from Proposition 1.4: each isomorphism φ is a composite of restrictions of automorphisms of S and of F-essential (hence critical) subgroups.

By [AKO, Proposition I.8.4], there is an injective homomorphism

trf_{F}: S/𝔣𝔬𝔠(F) ⟶ S/[S, S] ,

together with proper subgroups P_{1}, …, P_{m} < S and homomorphisms φ_{i} ∈ Hom_{F}(P_{i}, S) such that for g ∈ S,

trf_{F}([g]) = ∏_{[α]∈Out_{F} (S)} α([g])· ∏_{i=1}^{m} φ_{i}(trf_{P_{i}}^{S}([g])).

Set Q_{i} = φ_{i}(P_{i}), and let φ′_{i} ∈ Iso_{F}(P_{i}, Q_{i}) be the restriction of φ_{i}. By (1), if we set k = |Out_{F}(S)|, then

trf_{F}([g]) = g^{k}· ∏_{i=1}^{m} incl_{Q_{i}}^{S}(φ′_{i}(trf_{P_{i}}^{S}([g]))) = g^{k}· ∏_{i=1}^{m} incl_{Q_{i}}^{S}(trf_{Q_{i}}^{S}([g])) = g^{k}· ∏_{i=1}^{m} g^{[S:Q_{i}]} ≠ 1

since p∤k (and p|[S:Q_{i}] for each i). Thus g ∉ 𝔣𝔬𝔠(F), so 𝔣𝔬𝔠(F) < S, and F contains a normal subgroup of index p by Lemma 1.16. □

Let S be a finite p-group. A normal subgroup P ⊴ S is called semicharacteristic in S if P is invariant under O^{p}(Aut(S)). The semicharacteristic closure of Q ≤ S is the smallest subgroup P ⊴ S containing Q which is semicharacteristic in S.

Proposition 2.2. Assume F is a reduced fusion system over a finite nonabelian 2-group S. Assume also that S is not isomorphic to D_{2^{n}} (n ≥ 3), SD_{2^{n}} (n ≥ 4), or C_{2^{n}} ≀ C_{2} (n ≥ 2). Then S satisfies the following conditions.

(a) S contains no abelian subgroup of index two.

(b) [S, S] is not cyclic.

(c) Let A denote the image of Ω_{1}(Z(S)) in S/[S, S]. Then either A = 1, or |A| > 2 and Aut(S) is not a 2-group.

(d) If Aut(S) is a 2-group, then

⋂_{M<S} Ker[S/[S, S] ⟶^{trf_{M}^{S}} M/[M, M]] = 1 and ⋂_{M<S} Ker[S/Fr(S) ⟶^{trf_{M}^{S}} M/Fr(M)] = 1,

where the intersections are taken over all maximal subgroups M < S.

(e) S has more than one critical subgroup, and S has more than one conjugacy class of critical subgroups if Aut(S) is a 2-group.

(f) For each critical subgroup P < S, let Q_{P} denote the semicharacteristic closure of [N_{S}(P), P] in P. Then ⟨[O^{2}(Aut(S)), S], Q_{P} |P critical⟩ = S.

(g) Let Q ⊴ S be any normal subgroup which is semicharacteristic in S, and which is contained in and semicharacteristic in each critical subgroup P < S. Then Q = 1.

(h) Let K ≤ S/[S, S] denote the subgroup consisting of those elements x ∈ S/[S, S] which are fixed by O^{2}(Aut(S)), and which are such that trf_{P}^{S} (x) ∈ P/[P, P] is fixed by O^{2}(Aut(P)) for each critical subgroup P < S. Then K = 1.

Similarly, let K′ ≤ S/Fr(S) denote the subgroup consisting of those elements x ∈ S/Fr(S) which are fixed by O^{2}(Aut(S)), and which are such that trf_{P}^{S} (x) ∈ P/Fr(P) is fixed by O^{2}(Aut(P)) for each critical subgroup P < S. Then K′ = 1.

Proof. By Proposition 1.5, each F-essential subgroup is a critical subgroup of S. This will be used throughout the proof.

Points (a) and (b) follow from [AOV2, Proposition 5.2(a,b)], (c) follows from [AKO, Corollary I.8.5], and (d) and (h) from Lemma 2.1. Since F-essential subgroups are critical, point (e) follows from Lemma 1.17.

(f) Let P_{1}, …, P_{m} be conjugacy class representatives for the critical subgroups of S, and let Q_{i} = Q_{P_{i}} ≤ P_{i} be the semicharacteristic closure of [N_{S}(P_{i}), P_{i}]. For each i, Q_{i} is O^{2}(Aut_{F}(P_{i}))-invariant and Aut_{S}(P_{i})-invariant, hence Aut_{F}(P_{i})-invariant. So Q_{i} ≥ [O^{2′}(Aut_{F}(P_{i})), P_{i}] since O^{2′}(Aut_{F}(P_{i})) is generated by Aut_{S}(P_{i}) ∈ Syl_{2}(Aut_{F}(P_{i})) and its conjugates in Aut_{F}(P_{i}). So if we set

H = ⟨[O^{2}(Aut_{F}(S)), S], Q_{i} | 1 ≤ i ≤ m⟩ ,

then

𝔣𝔬𝔠(F) = ⟨[Aut_{F}(S), S], [O^{2′}(Aut_{F}(P_{i})), P_{i}] | i = 1, …, m⟩
≤ [S, S]·⟨[O^{2}(Aut_{F}(S)), S], Q_{i} | i = 1, …, m⟩ = [S, S]H

where the first equality follows from Proposition 1.4. Since F is reduced, 𝔣𝔬𝔠(F) = S by Lemma 1.16. Thus [S, S]H = S, so H = S by [G, Theorem 5.1.1].

(g) Assume Q ⊴ S is normal and semicharacterisitic in S, and contained in and semicharacteristic in each critical subgroup P < S. Then if P = S or P ∈ E_{F}, Q is normalized by the action of O^{2}(Aut_{F}(P))·Aut_{S}(P) = Aut_{F}(P). Hence Q ⊴ F by, e.g., [AOV2, Proposition 1.3], and Q = 1 since F is reduced. □

We now turn to the criteria used in our computer searches to determine (possibly) critical subgroups of a finite 2-group.

Proposition 2.3. Let S be a finite 2-group, and let P ≤ S be a critical subgroup. Let S_{0} = N_{S}(P)/P. Then the following conditions hold.

(a) Let Z′ = ⟨x ∈ Z_{2}(S) | |[x, S]| ≤ 2⟩. Then either Z′ ≤ P, or there exists an involution h ∈ S such that P = C_{S}(h), [N_{S}(P):P] = 2 and [h, Z_{2}(S)] ≠ 1.

(b) P ≠ S and C_{S}(P) ≤ P.

(c) Either S_{0} is cyclic or Z(S_{0}) = Ω_{1}(S_{0}).

(d) If rk(Z(S_{0})) > 1 then |S_{0}| = |Z(S_{0})|^{m} for m = 1, 2 or 3.

(e) Let k be such that |S_{0}| = 2^{k}. Then rk(P/Fr(P)) ≥ 2k. If k ≥ 2, then for all 1 ≠ s ∈ S_{0}, rk([s, P/Fr(P)]) ≥ 2.

(f) If Z(S_{0}) ≅ C_{2}^{n} with n ≥ 2 then rk([s, P/Fr(P)]) ≥ n for all s ∈ Z(S_{0}), s ≠ 1.

(g) Let Θ = 1, Θ = Z(P) or Θ = Z_{2}(P). If g ∈ N_{S}(P) satisfies [g, P] ≤ Θ·Fr(P) and [g, Θ] ≤ Fr(P) then g ∈ P.

(h) Aut(P) is not a 2-group and Out_{S}(P) ∩ O_{2}(Out(P)) = 1.

(i) There exists a composition factor M of the 𝔽_{2}[Out(P)]-module P/Fr(P) with dim(M) ≥ 2k. If k ≥ 2, then M can be chosen such that dim([s, M]) ≥ 2 for all s ∈ S_{0}, s ≠ 1.

(j) All involutions in S_{0} are conjugate in N_{Out(P)}(S_{0}).

Proof. Points (b) and (h) hold by definition of critical subgroups. Point (a) follows from [OV, Lemma 3.6(a)], points (c) and (d) from Proposition 1.7(a), (e) from Proposition 1.7(b), (f) from Proposition 1.7(c), (g) from Proposition 1.6, and (j) from [OV, Proposition 3.3(b)]. Point (i) is shown in Proposition 1.8. □

Definition 2.4. A subgroup P of a finite 2-group S is potentially critical if it satisfies conditions (a)–(j) in Proposition 2.3.

By definition, each automorphism of S sends the set of potentially critical subgroups of S to itself.

3. Fusion systems over 2-groups with abelian direct factor

In this section, we find conditions on a 2-group S_{0} which imply that there are no reduced fusion systems over S = A × S_{0} for any abelian 2-group A ≠ 1. Very roughly, the assumption that a fusion system F over S has no normal subsystems of 2-power index implies certain extra properties, which allow us to show that A (or some other direct factor in S) is normal in F. Green correspondence plays a key role in the arguments we use, and the following elementary lemma will be useful. We refer to [Bs, §§ 3.10 & 3.12] for more details on vertices and Green correspondence.

Lemma 3.1. Let G be a finite group, let k be a field of characteristic p, and let V be a finitely generated indecomposable k[G]-module. Then for any H ≤ G, each vertex of each indecomposable summand of V |_{H} is contained in a vertex of V. In particular, if S ∈ Syl_{p}(G) and V |_{S} has a nonzero direct summand with trivial S-action, then S is a vertex of V.

Proof. Let P be a vertex of V. Then there is a k[P]-module W such that V is a direct summand of Ind_{P}^{G}(W). By the Mackey formula (cf. [Bs, Theorem 3.3.4]), (Ind_{P}^{G}(W))|_{H} is a direct sum of modules induced up from H ∩ ^{g}P for elements g ∈ G. Hence each indecomposable summand of V |_{H} has vertex contained in a subgroup of ^{g}P for some g ∈ G (and ^{g}P is also a vertex of V).

If S ∈ Syl_{p}(G) and V |_{S} has a nonzero indecomposable direct summand W with trivial S-action, then S is a vertex of W (see, e.g., [LP, Remark 4.8.11(b)]), and hence also a vertex of V. □

We will need the following lemma on Green correspondance.

Lemma 3.2. Fix a finite group G with a strongly 2-embedded subgroup H < G, and a finitely generated 𝔽_{2}[G]-module V. Fix S ∈ Syl_{2}(H) ⊆ Syl_{2}(G). Let K be the normal closure of S in H; i.e., K = O^{2′}(H). Assume 0 ≠ W ≤ V is a direct summand of V |_{H} upon which S (and hence K) acts trivially.

(a) If V is 𝔽_{2}[G]-indecomposable, then V has vertex S and Green correspondent W (over H), W is 𝔽_{2}[H/K]-irreducible, and V/W is 𝔽_{2}[S]-free and 𝔽_{2}[H]-projective.

(b) If W = ⨁_{i=1}^{n} W_{i} where each W_{i} is 𝔽_{2}[H/K]-irreducible, then V contains, as direct summand, an 𝔽_{2}[G]-submodule V^{∗} ≤ V isomorphic to the direct sum of the Green correspondents for the W_{i}.

(c) If dim(V^{∗}) > dim(W), where V^{∗} is as in (b), then either |S| = 2 and dim(V^{∗}) ≥ dim(W) + 4, or dim(V^{∗}) ≥ dim(W) + 8.

Proof. (a) Assume that V is indecomposable. By Lemma 3.1, S is a vertex of V and of every indecomposable direct summand of W. Hence by Green correspondence [Bs, Theorem 3.12.2], and since H ≥ N_{G}(S), V |_{H} has (up to isomorphism) a unique indecomposable direct summand with vertex S. Thus W is indecomposable and is the Green correspondent to V. In particular, W is irreducible as an 𝔽_{2}[H/K]-module.

Since H is strongly 2-embedded in G, ^{g}S ∩ H = 1 for all g ∈ G∖H. Green correspondence [Bs, Theorem 3.12.2(i)] then implies that V |_{H} is the direct sum of W and modules with vertex 1. Thus V |_{H} = W ⊕ F, where F is 𝔽_{2}[H]-projective. In particular V/W ≅ F is free over 𝔽_{2}[S] (by, e.g., [Se, § 14.3, Corollary 1] and since each irreducible 𝔽_{2}[S]-module has trivial action).

(b) Let V = ⨁_{i=1}^{k} V_{i} be the decomposition as a sum of indecomposable 𝔽_{2}[G]-modules, arranged so that V_{i}|_{H} has a nonzero direct summand with trivial S-action if and only if i ≤ m (some 1 ≤ m ≤ k). For i ≤ m, let U_{i} ≤ V_{i}|_{H} be the Green correspondent. The U_{i} are 𝔽_{2}[H/K]-irreducible by (a). By Green correspondence and Lemma 3.1, the U_{i} are the only indecomposable summands of V |_{H} on which S acts trivially.

By the Krull-Schmidt theorem for 𝔽_{2}[H]-modules (cf. [Bs, Theorem 1.4.6]), and since W is a direct summand of V |_{H} on which S acts trivially, there is I ⊆ {1, …, m} such that W ≅ ⨁_{i∈I} U_{i}. So V contains V^{∗} = ⨁_{i∈I} V_{i} as a direct summand. By construction, V_{i} is the Green correspondent of U_{i}.

(c) By (b), it suffices to prove this when W is 𝔽_{2}[H/K]-irreducible and V = V^{∗} is its Green correspondent. Since V > W by assumption, the action of H on W does not extend to any action of G.

Set N_{0} = C_{G}(V) ⊴ G: the kernel of the G-action on V. Since V/W ≠ 1 is a free 𝔽_{2}[S]- module by (a), N_{0} ∩ S = 1, so N_{0} has odd order. Also, HN_{0} < G since HN_{0}(W) = W while G(W) ≠ W, so HN_{0}/N_{0} is strongly 2-embedded in G/N_{0} by Lemma 1.3(a). Upon replacing G by G/N_{0} and H by HN_{0}/N_{0}, we can assume that G acts faithfully on V.

Set W = 𝔽_{2} ⊗_{𝔽_{2}} W and V = 𝔽_{2} ⊗_{𝔽_{2}} V for short. Assume W = ⨁_{i=1}^{k} W_{i}, where each W_{i} is irreducible (and nonzero) as an 𝔽_{2}[H/K]-module, and k ≥ 2. Let F ⊆ 𝔽_{2} be a finite extension of 𝔽_{2} which gives the same splitting, and set V_{F} = F ⊗_{𝔽_{2}} V and W_{F} = F ⊗_{𝔽_{2}} W. Thus W_{F} = ⨁_{i=1}^{k} W_{F i} as F [H/K]-modules, where W_{i} ≅ 𝔽_{2} ⊗_{F} W_{F i} for each i. Then V_{F} contains the direct sum of the Green correspondents V_{F i} to the W_{F i} by the same argument as that used in the proof of (b). By the Krull-Schmidt theorem, and since W_{F} |_{𝔽_{2}[H/K]} ≅ W^{ℓ} where ℓ = dim_{𝔽_{2}}(F), W_{F i}|_{𝔽_{2}[H/K]} ≅ W^{ℓ_{i}} for some 1 ≤ ℓ_{i} ≤ ℓ. If, for some i, W_{F i} is isomorphic to its own F [G]-Green correspondent, then by (b), the same must be true for each irreducible summand of W_{F i}|_{𝔽_{2}[H/K]}, which contradicts our assumption that V > W. Thus dim(V_{F i}) − dim(W_{F i}) ≥ |S| for each i, and so dim(V/W) ≥ k|S|. This proves (c) when k ≥ 2.

From now on, we assume that V is a faithful, indecomposable 𝔽_{2}[G]-module and that W is 𝔽_{2}[H/K]-irreducible. There are two cases to consider. Case (c1): O_{2′}(G) ≠ 1. Since O_{2′}(G) is solvable by the odd order theorem [FT], there is 1 ≠ N ⊴ G which is an elementary abelian p-group for some odd p. By the Frattini argument, N_{G}(SN) = N_{G}(S)·N ≤ HN, so HN/N ≥ N_{G/N}(SN/N). We claim that H ∩ N ≠ 1 and acts nontrivially on W. Assume otherwise. Regard W as an 𝔽_{2}[HN/N]- module via the isomorphism HN/N ≅ H/(H ∩ N). Since SN/N acts trivially on W, it is a vertex of W by Lemma 3.1. Let V′ be its 𝔽_{2}[G/N]-Green correspondent (note that HN/N ≥ N_{G/N}(SN/N)). Then V′ is indecomposable as an 𝔽_{2}[G]-module, and W is a direct summand of V′|_{H} with vertex S. Hence as an 𝔽_{2}[G]-module, V′ has vertex S by Lemma 3.1 and is the Green correspondent of W by [Bs, Theorem 3.12.2(i)], so V′ ≅ V by the uniqueness of Green correspondents. This is impossible, since G acts faithfully on V but not on V′, and we conclude that H ∩ N acts nontrivially. In particular, H ∩ N ≠ 1.

Let r > 1 be the multiplicative order of 2 in 𝔽_{p}^{×}. Thus all irreducible 𝔽_{2}[C_{p}]-modules with nontrivial action have dimension r. Since V is indecomposable, V |_{N} is a direct sum of irreducible 𝔽_{2}[N]-modules which are isomorphic to each other up to automorphisms in Aut_{G}(N). Hence none of those irreducible summands has trivial action, and each has dimension r. Similarly, since W is irreducible and H ∩ N ⊴ H acts nontrivially, each irreducible summand of W |_{H∩N} has dimension r. Thus r| dim(V), r| dim(W), and r| dim(V/W).

The dimension of each irreducible 𝔽_{2}[H/K]-module divides |H/K| (cf. [Se, §§ 6.5 & 15.5]). In particular, dim(W) is odd. Thus r| dim(W) is also odd, so r ≥ 3, and r|S|| dim(V/W) in this case.

Case (c2): O_{2′}(G) = 1. By Bender’s theorem [Be, Satz 1 & Lemma 2.6], O^{2′}(G) is isomorphic to PSL_{2}(2^{n}), Sz(2^{2n−1}), or PSU_{3}(2^{n}) (n ≥ 2), and hence Z(S) is elementary abelian of rank ≥ 2. If dim(V/W) < 8, then |S| < 8 since V/W is 𝔽_{2}[S]-free by (a), and so S ≅ C_{2}^{2} and G ≅ A_{5} since |Out(A_{5})| = 2 [Sz1, (3.2.17)]. But then H ≅ A_{4}, H/K ≅ C_{3}, and W has nontrivial H-action since the Green correspondent of the trivial 𝔽_{2}[A_{4}]-module is trivial. So W = 𝔽_{2} ⊗_{𝔽_{2}} W is reducible, which contradicts our assumption. □

We now prove four propositions, each showing that under certain (fairly restrictive) hypotheses on a 2-group S_{0}, there are no reduced fusion systems over A × S_{0} for any abelian 2-group A ≠ 1. Stronger results of this type will be shown in a later paper. The first proposition can be applied when S = A×S_{0} for S_{0} of type SL_{3}(2^{n}) (n ≥ 2), 2·SL_{3}(4), or Sp_{4}(2^{n}) (n ≥ 2).

Proposition 3.3. Let S be a finite 2-group containing normal abelian subgroups P_{1}, P_{2} ⊴ S such that

• S = P_{1}P_{2} and P_{1} ∩ P_{2} = Z(S);

• [S, S] ∩ Fr(Z(S)) = 1;

• P_{i} = Z(S)Ω_{1}(P_{i}) and rk(P_{i}/Z(S)) ≥ 2 for i = 1, 2; and

• for each i = 1, 2 and each g ∈ P_{i}∖Z(S), C_{S}(g) = P_{i}.

Then P_{1} and P_{2} are the only possible critical subgroups of S, and every elementary abelian subgroup of S is contained in P_{1} or P_{2}. If F is a saturated fusion system over S such that O_{2}(F) = 1, then E_{F} = {P_{1}, P_{2}}, P_{1} and P_{2} are elementary abelian, rk(P_{1}) = rk(P_{2}), and [S, S] = Z(S).

Proof. Set V_{i} = Ω_{1}(P_{i}) for short (i = 1, 2). Thus V_{i} ⊴ S and is elementary abelian, and P_{i} = V_{i}Z(S). Also, S = V_{1}V_{2}Z(S), so [S, S] ≤ V_{1} ∩ V_{2} is elementary abelian.

If g ∈ S∖(P_{1} ∪ P_{2}), then g = g_{1}g_{2}z for some g_{i} ∈ V_{i}∖Z(S) and z ∈ Z(S), so g^{2} = [g_{1}, g_{2}]z^{2} ≠ 1 since z^{2} ∈ Fr(Z(S)), Fr(Z(S)) ∩ [S, S] = 1, and C_{S}(g_{i}) = P_{i}. Thus all elements of order two in S lie in P_{1} ∪ P_{2}.

If Q ≤ S is elementary abelian, and Q ≰ Z(S) = P_{1} ∩ P_{2}, then choose g ∈ Q∖Z(S). We just showed that g ∈ P_{i} for i = 1 or 2, and so Q ≤ C_{S}(g) = P_{i}.

Let R ≤ S be a critical subgroup. If R ∉ {P_{1}, P_{2}}, then R ≰ P_{1} and R ≰ P_{2} since R is centric in S. If x ∈ Ω_{1}(Z(R)), then x ∈ P_{i} for some i as shown above. If x ∉ Z(S), then R ≤ C_{S}(x) = P_{i}, which is impossible. Thus Ω_{1}(Z(R)) ≤ Z(S), so Ω_{1}(Z(S)) = Ω_{1}(Z(R)) is characteristic in R. But for g ∈ S∖R, [g, Ω_{1}(Z(S))] = 1 and [g, R] ≤ [S, S] ≤ V_{1} ∩ V_{2} = Ω_{1}(Z(S)), which contradicts Proposition 1.6. We conclude that P_{1} and P_{2} are the only subgroups of S which could be critical.

Step 1: Let F be a saturated fusion system over S such that O_{2}(F) = 1. Then P_{1} and P_{2} must both be F-essential by Lemma 1.17. Since P_{i} = V_{i}Z(S) where V_{i} is elementary abelian, Fr(P_{1}) = Fr(Z(S)) = Fr(P_{2}). So Fr(Z(S)) ⊴ F by Proposition 1.15, and since O_{2}(F) = 1, Fr(Z(S)) = 1. Thus P_{i} = V_{i} is elementary abelian for i = 1, 2.

For x ∈ P_{1}∖Z(S),

rk(P_{1}/Z(S)) = rk(S/P_{2}) ≤ rk([x, P_{2}]) = rk(P_{2}/Z(S)) ,

where the inequality holds by Proposition 1.7(c), and the last equality since C_{P_{2}}(x) = P_{2} ∩ P_{1} = Z(S) by assumption. The opposite inequality holds by a similar argument, so rk(P_{1}/Z(S)) = rk(P_{2}/Z(S)) and rk(P_{1}) = rk(P_{2}).

Step 2: It remains to prove that Z(S) = [S, S]. Regard Z(S) as an 𝔽_{2}[Out_{F}(S)]-module with submodule [S, S]. Since |Out_{F}(S)| is odd, there is an 𝔽_{2}[Out_{F}(S)]-submodule W which is complementary to [S, S]; i.e., Z(S) = W × [S, S].

For each i = 1, 2, set G_{i} = Aut_{F}(P_{i}), T_{i} = Aut_{S}(P_{i}), and H_{i} = N_{G_{i}}(T_{i}). Since Aut_{F}(S) is generated by Inn(S) and automorphisms of odd order, each P_{i} is normalized by Aut_{F}(S). By the extension axiom, the homomorphism Aut_{F}(S) ⟶ H_{i} induced by restriction is surjective, so a subgroup of P_{i} is normalized by H_{i} if and only if it is normalized by Aut_{F}(S).

Consider the Out_{F}(S)-module P_{i}/[S, S]. Since |Out_{F}(S)| is odd, Z(S)/[S, S] has a complement M/[S, S] in P_{i}/[S, S]. Hence P_{i} = W × M as 𝔽_{2}[Aut_{F}(S)]-modules, and in particular, W is a direct summand of P_{i} as an 𝔽_{2}[H_{i}]-module.

Since P_{i} is maximal among F-essential subgroups, each α ∈ Aut_{F}(P_{i}) which extends to Aut_{F}(Q) for any Q > P_{i} also extends to Aut_{F}(S) (Proposition 1.4), and hence lies in H_{i}. So by [AKO, Proposition I.3.3(b)] and since P_{i} ∈ E_{F}, H_{i} is strongly 2-embedded in G_{i}.

If U is an H_{i}-indecomposable summand of W, then by Lemma 3.2(a,b), the G_{i}-Green correspondent U^{∗} of U is isomorphic to a direct summand of P_{i}, and U^{∗}|_{H_{i}} ≅ U × F for some 𝔽_{2}[H_{i}]-module F such that F |_{T_{i}} is 𝔽_{2}[T_{i}]-free. For each 𝔽_{2}[T_{i}]-submodule 1 ≠ X ≤ P_{i}, T_{i} acts trivially on X ∩ Z(S) and on X/(X ∩ Z(S)), and in particular, X is not free over 𝔽_{2}[T_{i}] (recall that T_{i} ≅ C_{2}^{k} for k ≥ 2). Thus F = 1, and hence U = U^{∗}|_{H_{i}} is invariant under G_{i}.

This proves that W is an 𝔽_{2}[G_{i}]-submodule of P_{i} for i = 1, 2. Hence W ⊴ F by Proposition 1.15, so W = 1 since O_{2}(F) = 1. In particular, Z(S) = [S, S]. □

Lemma 3.4. Let S_{0} be a finite nonabelian 2-group such that Z(S_{0}) is cyclic. Let A be a finite abelian 2-group, set S = S_{0} × A, and assume F is a reduced fusion system over S. Set A^{∗} = [Aut_{F}(S), Z(S)]; thus Aut_{F}(S) normalizes A^{∗} and C_{A^{∗}}(Aut_{F}(S)) = 1. Then A^{∗} ≅ A, S = S_{0} × A^{∗}, and Ω_{1}(A^{∗}) = [Aut_{F}(S), Ω_{1}(Z(S))].

Proof. Set Z = Z(S), Z_{0} = Z(S_{0}), and Γ = Out_{F}(S) for short. Thus Γ has odd order and acts on Z, so

Z_{0} × A = Z = C_{Z}(Γ) × [Γ, Z] = C_{Z}(Γ) × A^{∗}  (1)
Ω_{1}(Z) = C_{Ω_{1}(Z)}(Γ) × [Γ, Ω_{1}(Z)]  (2)

(see [G, Theorem 5.2.3], and note in particular that C_{A^{∗}}(Γ) = 1). Since Z_{0} is cyclic, |Ω_{1}(Z_{0})| = 2; and since each nontrivial normal subgroup of S_{0} intersects nontrivially with Z_{0}, Ω_{1}(Z_{0}) ≤ [S_{0}, S_{0}] = [S, S]. By [AKO, Corollary I.8.5], and since 𝔣𝔬𝔠(F) = S by Lemma 1.16 (recall O^{2}(F) = F), Γ acts with trivial fixed subgroup on

Ω_{1}(Z)/(Ω_{1}(Z) ∩ [S, S]) = Ω_{1}(Z)/Ω_{1}(Z_{0}) .

Hence Ω_{1}(Z_{0}) = C_{Ω_{1}(Z_{0})}(Γ) = C_{Ω_{1}(Z)}(Γ) = Ω_{1}(C_{Z}(Γ)). Since [Γ, Ω_{1}(Z)] ≤ Ω_{1}(A^{∗}), and both are complements to Ω_{1}(Z_{0}) in Ω_{1}(Z) by (1) and (2), we have Ω_{1}(A^{∗}) = [Γ, Ω_{1}(Z)].

Since Ω_{1}(C_{Z}(Γ)) = Ω_{1}(Z_{0}), and C_{Z}(Γ) ∩ A^{∗} = 1 = Z_{0} ∩ A by (1), we also have C_{Z}(Γ) ∩ A = 1 = Z_{0} ∩ A^{∗} since neither intersection has any elements of order 2. Thus |A| = |A^{∗}|, and hence Z = Z_{0} × A^{∗} and A ≅ Z/Z_{0} ≅ A^{∗}. Also, S_{0} ∩ A^{∗} = 1, so S = S_{0} × A^{∗}. □

Recall that the rank of a finite p-group is the largest rank of any of its abelian subgroups.

Lemma 3.5. Let S_{0} be a finite nonabelian 2-group such that Z(S_{0}) is cyclic. Let A ≠ 1 be a nontrivial finite abelian 2-group, set S = S_{0} × A, and assume F is a reduced fusion system over S. Then for each 1 ≠ Ŵ ≤ Ω_{1}(Z(S)) which is normalized by Aut_{F}(S), there are subgroups P ∈ E_{F} and V ≤ P such that

(a) V = Ω_{1}(Z(P)), P = C_{S}(V), and V is fully normalized in F;

(b) Aut_{F}(P)(Ŵ) ≠ Ŵ; and

(c) for all Q ≤ S such that |Q| > |P |, and all β ∈ Hom_{F}(Q, S), β(Ŵ) = Ŵ.

Set W = [Aut_{F}(S), Ω_{1}(Z(S))] ≠ 1, and let P and V be any pair of subgroups satisfying (a)–(c) with Ŵ = W.

(d) If there is Γ ≤ Aut_{F}(S) such that C_{W} (Γ) = 1 and Γ (P) = P, then either (i) rk(P/A) ≥ 6, or

(ii) rk(C_{V} (N_{S}(P))/W) ≥ 3 and Γ acts nontrivially on V/W.

Proof. (a,b,c) Since F is reduced, Ŵ cannot be normal in F. Since Ŵ ≤ Z(S), it is contained in every F-essential subgroup. Hence by Proposition 1.15 (and since Aut_{F}(S) normalizes Ŵ), there is some P ∈ E_{F} and some α ∈ Aut_{F}(P) such that α(Ŵ) ≠ Ŵ. Choose P to have maximal order among all such F-essential subgroups. Then (c) holds by Proposition 1.4.

Set V = Ω_{1}(Z(P)). Thus Ŵ ≤ Ω_{1}(Z(S)) ≤ V. If V is not fully normalized, then there is φ ∈ Hom_{F}(N_{S}(V), S) such that φ(V) is fully normalized (cf. [AKO, Lemma I.2.6(c)]). Also, N_{S}(V) ≥ N_{S}(P) > P, so φ(Ŵ) = Ŵ by (c), and φ(P) is fully normalized (hence F-essential) since P is. Upon replacing P by φ(P), we can assume V is fully normalized (and hence fully centralized).

Let α ∈ Aut_{F}(P) be as above: α(Ŵ) ≠ Ŵ. Since V is fully centralized, α|_{V} ∈ Aut_{F}(V) extends to some α ∈ Aut_{F}(C_{S}(V)). If C_{S}(V) > P, then \overline{α}(Ŵ) = Ŵ by (c), contradicting our assumption on α. Hence C_{S}(V) = P, and in particular, N_{S}(V) = N_{S}(P).

(d) By Lemma 3.4, we can arrange that A = [Aut_{F}(S), Z(S)] and hence is normalized by Aut_{F}(S) and W = Ω_{1}(A) ≠ 1. Fix V ≤ P ∈ E_{F} which satisfy conditions (a)–(c) for Ŵ = W. Let Γ ≤ Aut_{F}(S) be such that C_{W} (Γ) = 1 and Γ (P) = P. Assume rk(V/W) ≤ 5 (otherwise (d.i) holds).

Set G = Out_{F}(P) and T = Out_{S}(P) ∈ Syl_{2}(G), and let H ≤ G be the subgroup generated by the classes of all β ∈ Aut_{F}(P) which extend to F-morphisms between subgroups strictly containing P. By [AKO, Proposition I.3.3(b)] and since P ∈ E_{F}, H < G and is strongly 2-embedded in G. Since H ≥ T by definition, H ≥ N_{G}(T).

Regard V = Ω_{1}(Z(P)) as an 𝔽_{2}[G]-module. By (c), W is an 𝔽_{2}[H]-submodule of V |_{H}. Set V_{0} = V ∩ S_{0}. Then V = V_{0} × W since P = (P ∩ S_{0}) × A, and V_{0} is acted upon by T = Out_{S_{0}}(P). Hence W is a direct summand of V |_{H} as 𝔽_{2}[H]-modules, since the 𝔽_{2}[T]-linear projection V ⟶ W (with kernel V_{0}) can be made 𝔽_{2}[H]-linear by averaging over the cosets in H/T.

We are thus in the situation of Lemma 3.2. By Lemma 3.2(b), V = X × V^{∗}, where X and V^{∗} are 𝔽_{2}[G]-submodules, and where V^{∗} is isomorphic to the direct sum of the Green correspondents for the simple summands of W as an 𝔽_{2}[H/O^{2′}(H)]-module. By Lemma 3.2(c) and since rk(V/W) ≤ 5, either rk(V^{∗}) ≥ rk(W)+4 and |T | = 2, or rk(V^{∗}) = rk(W). Since V^{∗}|_{H} contains a direct summand isomorphic to W, this last condition implies that V^{∗}|_{H} ≅ W (as 𝔽_{2}[H]-modules).

Assume V^{∗}|_{H} ≅ W; we show that (d.ii) holds. Since Γ (P) = P, we have [γ|_{P}] ∈ N_{H}(T) for each γ ∈ Γ, and hence the action of Γ on V factors through a subgroup Γ ≤ N_{H}(T). Then V^{∗} and W are isomorphic as 𝔽_{2}[Γ]-modules. If (d.ii) does not hold, then either Γ acts trivially on V/W or rk(C_{V} (T)/W) ≤ 2. In the latter case, Γ acts trivially on C_{V} (T)/W since it normalizes the subspace Ω_{1}(Z(S))/W ≤ C_{V} (T)/W of rank 1, and since the action of Γ factors through Γ^{∗} = Γ T/T ≤ N_{H}(T)/T of odd order. So in either case, Γ acts trivially on C_{V} (T)/W. Also, C_{V} (T) = V_{1} × V_{2}, where V_{1} = C_{C_{V} (T)}(Γ^{∗}) and V_{2} = [Γ^{∗}, C_{V} (T)] (cf. [G, Theorem 5.2.3]). Then W ≥ V_{2} since C_{V} (T)/W has trivial action, and W = V_{2} since C_{W} (Γ) = 1. Also, V^{∗} ≤ C_{V} (T) and V^{∗} ∩ V_{1} = C_{V^{∗}}(Γ) = 1 since V^{∗} ≅ W as 𝔽_{2}[Γ]-modules, so V^{∗} = V_{2} = W since they have the same rank. This is impossible, since G(W) ≠ W by (b) while G(V^{∗}) = V^{∗}.

Now assume that rk(V^{∗}) ≥ rk(W) + 4 and |T | = |N_{S}(P)/P | = 2. Recall that V_{0} = V ∩ S_{0}, so that V = V_{0}×W as 𝔽_{2}[T]-modules. Thus rk(V_{0}) = rk(V/W), and so 4 ≤ rk(V_{0}) ≤ 5. By Lemma 3.2(a), V^{∗}|_{T} ≅ (W |_{T}) × 𝔽_{2}[T]^{k} for some k, and k = 2 by the assumptions on ranks. Hence rk(V^{∗}/C_{V^{∗}}(T)) = 2, and so rk(V_{0}/C_{V_{0}}(T)) = rk(V/C_{V} (T)) ≥ 2. Since |T | = 2,

2 ≤ rk(V_{0}/C_{V_{0}}(T)) = rk([T, V_{0}]) ≤ rk(C_{V_{0}}(T)).

If P ⊴ S, i.e., if |S/P | = 2, then C_{V_{0}}(T) = Ω_{1}(Z(S_{0})) has rank at least 2, which is impossible since Z(S_{0}) is cyclic by assumption.

Thus P ⋬ S. Fix x ∈ N_{S}(P) ∖ P. Let P̃ ≠ P be another subgroup S-conjugate to P with the same normalizer P ⟨x⟩, and set Ṽ = Ω_{1}(Z(P̃)) and Ṽ_{0} = Ṽ ∩ S_{0}. Then V ∩ Ṽ = Ω_{1}(Z(P ⟨x⟩)) = C_{V} (x), so V_{0} ∩ Ṽ_{0} = C_{V_{0}}(x) = C_{V_{0}}(T), and we just saw that rk(V_{0}/C_{V_{0}}(T)) ≥ 2. If Ṽ ≤ P, then [V, Ṽ] ≤ [V, P] = 1 and rk(V_{0}Ṽ_{0}) = rk(Ṽ_{0}) + rk(V_{0}/C_{V_{0}}(T)) ≥ 4 + 2 = 6, so (d.i) holds. If Ṽ ≰ P, then we can assume x was chosen so that x ∈ Ṽ. But then [P̃, x] = 1 and hence V ∩ P̃ ≤ C_{V} (x), which is impossible since |V/(V ∩ P̃)| ≤ |P ⟨x⟩/P̃| = 2 and |V/C_{V} (x)| ≥ 4. □

Proposition 3.6. Let S_{0} be a finite 2-group such that rk(S_{0}) ≤ 5, Z(S_{0}) is cyclic, and Aut(S_{0}) is a 2-group. Then for any finite abelian 2-group A ≠ 1, there are no reduced fusion systems over S_{0} × A.

Proof. We can assume S_{0} is nonabelian; otherwise the result is clear. Set S = S_{0} × A, and assume F is a reduced fusion system over S. Using Lemma 3.4, we can arrange that A = [Aut_{F}(S), Z(S)] and hence is normalized by Aut_{F}(S). Set W = [Aut_{F}(S), Ω_{1}(Z(S))]; then W = Ω_{1}(A) and C_{W} (Aut_{F}(S)) = 1 by Lemma 3.4 again.

Fix Γ ≤ Aut_{F}(S) of odd order such that Inn(S)Γ = Aut_{F}(S) (the Schur-Zassenhaus theorem, cf. [G, Theorem 6.2.1(i)]). Then C_{W} (Γ) = C_{W} (Aut_{F}(S)) = 1 since [Inn(S), W] = 1. For each γ ∈ Γ, γ(A) = A, so γ induces an automorphism of S/A ≅ S_{0} which must be the identity since Aut(S_{0}) is a 2-group by assumption. Hence γ(P) = P for each P ≤ S which contains A, and in particular, for each P ∈ E_{F}. Also, for each such P, Γ acts trivially on Z(P)/A, and hence on Ω_{1}(Z(P))/W.

Lemma 3.5(d) now implies that rk(S_{0}) ≥ 6, which contradicts our assumptions. □

We now give two applications of Lemma 3.5 in situations where Aut(S_{0}) is not a 2group. Recall that UT_{n}(q) ≤ SL_{n}(q) denotes the subgroup of strict upper triangular matrices over 𝔽_{q}.

Proposition 3.7. Let S_{0} be a Sylow 2-subgroup of J_{2}. Then for any finite abelian 2-group A ≠ 1, there are no reduced fusion systems over S_{0} × A.

Proof. We adopt the notation used in [OV, § 4] and [O2, § 6]. Let (a ↦ a) denote the field involution of 𝔽_{4} (thus a = a^{2}). We identify S_{0} = UT_{3}(4)⟨θ⟩, where θ ( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} ) θ^{−1} =

( \begin{matrix}1 & c & b \\ 0 & 1 & a \\ 0 & 0 & 1 \end{matrix} )^{−1} for each a, b, c ∈ 𝔽_{4}, and θ^{2} = 1. For 1 ≤ i < j ≤ 3 and x ∈ 𝔽_{4}, let e_{ij}^{x} ∈ UT_{3}(4)

be the elementary matrix with unique nonzero off-diagonal entry x in position (i, j), and set E_{ij} = {e_{ij}^{x} | x ∈ 𝔽_{4}}. Set A_{1} = E_{12}E_{13} and A_{2} = E_{13}E_{23}. These are the only subgroups of S_{0} isomorphic to C_{2}^{4}, and each involution in UT_{3}(4) is in A_{1} ∪ A_{2} (see [OV, Lemma 4.1(b,c)]).

Set S = S_{0} × A, and let F be a reduced fusion system over S. Using Lemma 3.4, we can arrange that A = [Aut_{F}(S), Z(S)], and hence is normalized by Aut_{F}(S). Set W = [Aut_{F}(S), Ω_{1}(Z(S))]; then W = Ω_{1}(A) and C_{W} (Aut_{F}(S)) = 1 by Lemma 3.4 again. Let V and P be as in Lemma 3.5(a–c) for Ŵ = W. In particular, P ∈ E_{F}, V = Ω_{1}(Z(P)), and P = C_{S}(V). Set V_{0} = V ∩ S_{0} and P_{0} = P ∩ S_{0}.

If V_{0} ≰ UT_{3}(4), then there is g ∈ V_{0}∖UT_{3}(4) of order two, and C_{UT_{3}(4)}(g) ≅ Q_{8} with center ⟨e_{13}^{1}⟩ by [OV, Lemma 4.1(a)]. Hence V_{0} ≤ Ω_{1}(C_{S_{0}}(g)) = ⟨g, e_{13}^{1}⟩, with equality since V_{0} = Ω_{1}(Z(P_{0})) ≥ Z(S_{0}) = ⟨e_{13}^{1}⟩. So P = ⟨g⟩ × C_{UT_{3}(4)}(g) × A. Each class in N_{S}(P)/P is represented by some x ∈ UT_{3}(4), [x, g] ∈ Z(P) ∩ UT_{3}(4) = ⟨e_{13}^{1}⟩ since g ∈ Z(P) and x ∈ UT_{3}(4), and [x, C_{UT_{3}(4)}(g)] ≤ E_{13} ∩ P = ⟨e_{13}^{1}⟩. Hence [N_{S}(P), P] = ⟨e_{13}^{1}⟩ ≤ Fr(P), so P ∉ E_{F} by Proposition 1.6 (with Θ = 1), a contradiction.

Thus V_{0} ≤ UT_{3}(4). If V_{0} ≰ E_{13} = Z(UT_{3}(4)), then there is g ∈ (V_{0} ∩A_{i})∖E_{13} for i = 1 or 2, since all elements of order 2 in UT_{3}(4) lie in A_{1}∪A_{2}. Hence P ≤ C_{S}(g) = A_{i}×A, with equality since P is centric in S. Set Γ = {α ∈ Aut_{F}(S) | α(P) = P }: a subgroup of index 2 since A_{1}, A_{2} are the only subgroups of S_{0} isomorphic to C_{2}^{4} (and they are S-conjugate). Then Aut_{F}(S) = Inn(S)Γ, and C_{W} (Γ) = C_{W} (Aut_{F}(S)) = 1 since [Inn(S), W] = 1. So the hypothesis in Lemma 3.5(d) holds, which is impossible since rk(P/A) = 4 and rk(C_{V} (N_{S}(P))/W) = rk(C_{A_{i}}(UT_{3}(4))) = 2.

The only remaining case is that where V_{0} = E_{13} and P = UT_{3}(4)×A. This is impossible by Lemma 3.5(d) again (applied with Γ = Aut_{F}(S)), and since rk(V_{0}) = 2 and UT_{3}(4) = A_{1}A_{2} is characteristic in S_{0}. □

Proposition 3.8. For any finite abelian 2-group A ≠ 1, there are no reduced fusion systems over UT_{4}(2) × A.

Proof. Set S_{0} = UT_{4}(2) and S = S_{0} × A. We need to use the following properties of S_{0} (see, e.g., [O2, Lemma C.4(a)]): there is a unique abelian subgroup B_{0} ≤ S_{0} of order 16, B_{0} ≅ C_{2}^{4}, B_{0} ⊴ S_{0}, and S_{0}/B_{0} ≅ C_{2}^{2} acts on B_{0} by permuting a basis freely.

Assume F is a reduced fusion system over S. Using Lemma 3.4, we can arrange that A = [Aut_{F}(S), Z(S)] and hence is normalized by Aut_{F}(S).

Set B = B_{0} × A. Let P_{1}, P_{2}, P_{3} be the three subgroups of index 2 in S which contain B. Thus P_{i} = B⟨x_{i}⟩, and Z(P_{i}) = C_{B}(x_{i}), for some x_{1}, x_{2}, x_{3} ∈ S_{0} ∖ B_{0}. We first claim that

P =^{def} {P ∈ E_{F} | P = C_{S}(V) where V = Ω_{1}(Z(P))} ⊆ {B, P_{1}, P_{2}, P_{3}} .  (3)

Assume otherwise: then there is P ∈ P such that P ≱ B. Set P_{0} = P ∩ S_{0}. If P_{0} is elementary abelian, then rk(P_{0}) ≤ 3 since B_{0} is the unique abelian subgroup of order 16 in S_{0}, so rk([x, P]) = rk([x, P_{0}]) = 1 for x ∈ N_{S}(P) ∖ P such that x^{2} ∈ P. By Proposition 1.7(b), |N_{S_{0}}(P_{0})/P_{0}| = |N_{S}(P)/P | = 2. Since P is centric in S, P ≮ B. Hence either P_{0} ≤ ⟨g, B_{0}⟩ for some g ∈ P_{0} ∖ B_{0}, in which case P_{0} = ⟨g, C_{B_{0}}(g)⟩ and B ≤ N_{S}(P); or there are g, h ∈ P_{0} ∖ B_{0} with gh^{−1} ∉ B_{0}, in which case P_{0} ∩ B_{0} ≤ C_{B_{0}}(⟨g, h⟩) = Z(S_{0}) and Z_{2}(S_{0}) ≤ N_{S}(P). Thus |N_{S_{0}}(P_{0})/P_{0}| ≥ 4 in all cases, which is a contradiction.

Assume P_{0} is not elementary abelian, and set V_{0} = Ω_{1}(Z(P_{0})). Then rk(V_{0}) ≤ 2 since B_{0} is the only abelian subgroup of S_{0} of order 16, V_{0} ≰ B_{0} since P_{0} = C_{S_{0}}(V_{0}) ≱ B_{0}, so V_{0} = ⟨x, Z(S_{0})⟩ for some x ∈ S_{0} ∖ B_{0} of order 2. Then P_{0} = C_{S_{0}}(V_{0}) = ⟨x, y, C_{B_{0}}(x)⟩ for some y ∈ C_{S_{0}}(x) ∖ B_{0}⟨x⟩, ⟨x, C_{B_{0}}(x)⟩ ≅ C_{2}^{3}, and hence P_{0} ≅ C_{2} × D_{8} and [N_{S}(P), P_{0}] = Z(S_{0}) = Fr(P_{0}), which contradicts Proposition 1.6 (with Θ = 1). This proves (3).

Set

V = Ω_{1}(B)  G = Aut_{F}(B)  T = Aut_{S}(B) ∈ Syl_{2}(G)
W = Ω_{1}(A)  H = N_{G}(T)  H_{0} = C_{G}(T) .

Regard V as an 𝔽_{2}[G]-module. Since each β ∈ H = N_{G}(T) extends to some β ∈ Aut_{F}(S) by the extension axiom, W is an 𝔽_{2}[H]-submodule of V |_{H}. For each α ∈ Aut_{F}(S), α(B) = B and α|_{B} ∈ H since B is the unique abelian subgroup of index 4 in S, and hence W = [Aut_{F}(S), Ω_{1}(Z(S))] = [H, Ω_{1}(Z(S))] and C_{W} (H) = C_{W} (Aut_{F}(S)) = 1 by Lemma 3.4. Also, V = B_{0} × W as 𝔽_{2}[T]-modules, the T -linear projection V ⟶ W can be made H-linear by averaging over cosets in H/T, and hence V = B_{0}^{∗} × W for some 𝔽_{2}[H]- submodule B_{0}^{∗}. Since B_{0}^{∗}|_{T} ≅ B_{0} ≅ 𝔽_{2}[T] as 𝔽_{2}[T]-modules, B_{0}^{∗} is 𝔽_{2}[T]-indecomposable, and hence also 𝔽_{2}[H]-indecomposable.

In particular, if V is 𝔽_{2}[G]-decomposable, then all but one of its indecomposable summands has trivial action of T. Let Ŵ be an indecomposable summand with trivial T - action; then Ŵ ≤ C_{V} (T) ≤ Z(S). Also, G = O^{2′}(G)H by the Frattini argument, and O^{2′}(G) acts trivially on Ŵ since it is the normal closure of T in G. Hence Ŵ|_{H} is 𝔽_{2}[H/T]- irreducible. Also, Ŵ|_{H} ≅ Ŵ^{∗} as 𝔽_{2}[H]-modules for some Ŵ^{∗} ≤ W, so CŴ(H) = 1, and Ŵ = [H, Ŵ] ≤ [H, Ω_{1}(Z(S))] = W. Also, Ŵ is normalized by Aut_{F}(P) for each P ∈ {B, P_{1}, P_{2}, P_{3}, S} since B is characteristic in each of those subgroups. But this contradicts Lemma 3.5(a,b) and (3).

Thus V is 𝔽_{2}[G]-indecomposable. Since V |_{H} = B_{0}^{∗} × W where T acts trivially on W, T is a vertex of V (Lemma 3.1), and V |_{H} has (up to isomorphism) a unique indecomposable direct summand with vertex T by Green correspondence [Bs, Theorem 3.12.2]. Hence W is 𝔽_{2}[H/T]-irreducible. Set Γ = {α ∈ Aut_{F}(S) | α|_{B} ∈ H_{0}}; we already saw that each element of H_{0} extends to an element of Γ. If H_{0} acts nontrivially on W (equivalently, if Γ acts nontrivially on W), then C_{W} (H_{0}) = 1 (C_{W} (H_{0}) is an 𝔽_{2}[H]-submodule of W since H_{0} ⊴ H), and Γ (P) = P for P ∈ {B, P_{1}, P_{2}, P_{3}} by definition of H_{0}. This contradicts Lemma 3.5(d) and (3).

Thus H_{0} acts trivially on W, while H does not since W = [H, Ω_{1}(Z(S))]. By definition, H/H_{0} ≠ 1 acts as a group of automorphisms of T ≅ C_{2}^{2} of odd order, and hence has order 3. So rk(W) = 2 since W is irreducible. Also, 𝔽_{4} ⊗_{𝔽_{2}} W is 𝔽_{4}[H]-decomposable and the two summands are Galois conjugate, so 𝔽_{4} ⊗_{𝔽_{2}} V contains the sum of their Green correspondents (cf. Lemma 3.2(b)) which also are Galois conjugate. But this is impossible, since 𝔽_{4} ⊗_{𝔽_{2}} B_{0} ≅ 𝔽_{4}[T] as 𝔽_{4}[T]-modules and hence is indecomposable. □

4. 2-Groups of order at most 128

Reduced fusion systems over 2-groups of order at most 64 have already been handled in earlier papers. We begin by listing them. Recall that UT_{n}(q) denotes the group of upper triangular n × n matrices over 𝔽_{q} with 1’s on the diagonal.

Theorem 4.1. Let F be a reduced, indecomposable fusion system over a 2-group S of order 2^{k} ≤ 64. Then one of the following holds:

(a) S ≅ D_{2^{k}} for some 3 ≤ k ≤ 6, and F is isomorphic to the fusion system of PSL_{2}(q) where q ≡ 2^{k} ± 1 (mod 2^{k+1}).

(b) S ≅ SD_{2^{k}} for some 4 ≤ k ≤ 6, and F is isomorphic to the fusion system of PSL_{3}(q) where q ≡ 2^{k−2} − 1 (mod 2^{k−1}).

(c) S ≅ C_{4} ≀ C_{2}, and F is isomorphic to the fusion system of SL_{3}(5).

(d) S ≅ UT_{4}(2), and F is isomorphic to the fusion system of SL_{4}(2) ≅ A_{8} or of SU_{4}(2) ≅ PSp_{4}(3).

(e) S ≅ UT_{3}(4), and F is isomorphic to the fusion system of PSL_{3}(4).

(f) S is of type M_{12}, and F is isomorphic to the fusion system of M_{12} or of G_{2}(3).

In all cases, F is tame. Also, the fusion system of each of the groups listed above is simple.

Proof. If k ≤ 5, then by [AOV2, Theorem 5.3], S ≅ D_{2^{k}} (k ≥ 3), SD_{2^{k}} (k ≥ 4), or C_{4} ≀ C_{2} (k = 5). When k = 6, by [AOV2, Theorem 5.4], S is isomorphic to one of the groups D_{64}, SD_{64}, UT_{4}(2), UT_{3}(4), or is of type M_{12}.

The reduced fusion systems over dihedral and semidihedral 2-groups were listed in [AOV1, Propositions 4.3–4.4], and those over wreath products C_{2^{n}} ≀ C_{2} in [AOV2, Proposition 3.1]. Since all of the above groups have sectional rank at most four, the list of possible reduced fusion systems over those in (d,e,f) follows from [O2, Propositions 5.1, 6.4, & 4.2].

Tameness for all of these fusion systems is shown in [BMO2, Theorem C], except for that of M_{12}. When F = F_{S}(G) for G = M_{12}, then by [O2, Proposition 4.3(a,b)], the composite μ_{G} ∘ κ_{G} is an isomorphism from Out(G) to Out(S, F). By [AOV2, Proposition 3.2], there are at most two F-essential subgroups, of which only one (R in the notation of [AOV2] and A_{+} in [O2, § 4]) has noncyclic center. Hence Ker(μ_{G}) = 1 by Proposition 1.14, κ_{G} is an isomorphism, and F is tame.

By [O2, Theorem A], the fusion systems of all of the groups listed above are simple. □

As part of the proof of Theorem 4.1, we have also shown:

Lemma 4.2. If G is isomorphic to the sporadic simple group M_{12}, then (for p = 2) κ_{G} and μ_{G} are both isomorphisms. In particular, G tamely realizes its fusion system.

We now turn to groups of order 128, where we begin our use of a computer search to identify 2-groups which potentially could support a reduced fusion system.

Theorem 4.3. Let F be a reduced, indecomposable fusion system over a group S of order 128. Then one of the following holds:

(a) S ≅ D_{128}, and F is isomorphic to the fusion system of PSL_{2}(q) where q ≡ ±127 (mod 256).

(b) S ≅ SD_{128}, and F is isomorphic to the fusion system of PSL_{3}(q) where q ≡ 31 (mod 64).

(c) S ≅ C_{8} ≀ C_{2}, and F is isomorphic to the fusion system of SL_{3}(9).

(d) S ≅ D_{8} ≀ C_{2}, and F is isomorphic to the fusion system of A_{10} or of PSL_{4}(3).

(e) S is of type M_{22}, and F is isomorphic to the fusion system of M_{22}, M_{23}, McL, or PSL_{4}(5).

(f) S is of type J_{2}, and F is isomorphic to the fusion system of J_{2} or J_{3}.

In all cases, F is tame. Also, the fusion system of each of the groups listed above is simple.

Proof. Either S is as in (a,b,c) above, in which case F is also as in (a,b,c) by [AOV1, Propositions 4.3–4.4] or [AOV2, Proposition 3.1], or S satisfies conditions (a)–(h) in Proposition 2.2. By a computer search, this leaves the following 6 possibilities for S, where #(–) denotes its Magma/GAP identification number.

Tameness for the fusion systems in (a,b,e,f) is shown in [AOV1, Propositions 4.3–5]. Tameness for the fusion system of A_{10} was shown in [AOV1, Proposition 4.8], and for the other fusion systems in (c,d) in [BMO2, Theorem C]. All of these fusion systems are simple by [O2, Theorem A].

#928 : S ≅ D_{8} ≀ C_{2}. By [O2, Theorem A], F must be as in case (d).

#931 : S is of type M_{22}. Any reduced fusion system over S is as in (e) by [OV, Theorem 5.11]. (If F is isomorphic to the fusion system of G ≅ PΣL_{3}(4) or PΓ L_{3}(4), then 𝔣𝔬𝔠(F) = S ∩ [G, G] < S by the focal subgroup theorem [G, Theorem 7.3.4].)

#934 : S is of type J_{2}. By [OV, Theorem 4.8], F must be isomorphic to the fusion system of J_{2} or of J_{3}, and so we are in the situation of case (f). (If F is isomorphic to the fusion system of G ≅ PSL_{3}(4) ⋊ ⟨θ⟩ or PGL_{3}(4) ⋊ ⟨θ⟩, as defined in that theorem, then 𝔣𝔬𝔠(F) = S ∩ [G, G] < S by the focal subgroup theorem [G, Theorem 7.3.4].)

#1411 : This group has Magma/GAP presentation

S = ⟨s_{1}, …, s_{7} | s_{1}^{2} = s_{5}s_{6}, s_{2}^{s_{1}} = s_{2}s_{5}, s_{3}^{s_{1}} = s_{3}s_{6}, s_{3}^{s_{2}} = s_{3}s_{7},

s_{4}^{s_{1}} = s_{4}s_{5}s_{6}, s_{4}^{s_{2}} = s_{4}s_{5}⟩ .

(All other squaring and conjugation relations among the s_{i} are trivial.) It is a Sylow 2-subgroup of the 2-fold cover 2·PSL_{3}(4), although we will not use that here. It has two normal elementary abelian subgroups P_{1} = ⟨s_{3}, s_{4}, s_{5}, s_{6}, s_{7}⟩ and P_{2} = ⟨s_{1}s_{4}, s_{2}, s_{5}, s_{6}, s_{7}⟩, both isomorphic to C_{2}^{5}, and S = P_{1}P_{2} and Z(S) = P_{1} ∩ P_{2} = ⟨s_{5}, s_{6}, s_{7}⟩ ≅ C_{2}^{3}.

Let F be a saturated fusion system over S with O_{2}(F) = 1. By Proposition 3.3, E_{F} = {P_{1}, P_{2}}. For i = 1, 2, [S, P_{i}] = Z(S) = C_{P_{i}}(S) has rank 3. Also, for fixed i = 1, 2, the subgroups [x, P_{i}] for 1 ≠ x ∈ N_{S}(P_{i})/P_{i} ≅ C_{2}^{2} are the three subgroups ⟨s_{5}, s_{6}⟩, ⟨s_{5}, s_{7}⟩, and ⟨s_{6}, s_{7}⟩. By Lemma A.6, and since P_{i} ∈ E_{F} and Aut_{S}(P_{i}) ≅ C_{2}^{2}, Aut_{F}(P_{i}) = Δ_{i} × H_{i} for some Δ_{i} ≅ A_{5} and some H_{i} of odd order. By Lemma A.7, C_{P_{i}}(Δ_{i}) = C_{P_{i}}(N_{Δ_{i}}(Aut_{S}(P_{i}))) has rank one. The [x, P_{i}] must be permuted transitively by the action of N_{Δ_{i}}(Aut_{S}(P_{i})) ≅ A_{4}. Since

{s_{5}s_{6}s_{7}} = ⟨s_{5}, s_{6}, s_{7}⟩∖(⟨s_{5}, s_{6}⟩ ∪ ⟨s_{5}, s_{7}⟩ ∪ ⟨s_{6}, s_{7}⟩) ,

s_{5}s_{6}s_{7} is fixed by this subgroup. Thus C_{P_{i}}(Aut_{F}(P_{i})) = C_{P_{i}}(Δ_{i}) = ⟨s_{5}s_{6}s_{7}⟩.

Since Out_{F}(S) has odd order, Aut_{F}(S) is generated by Inn(S) and automorphisms of odd order. Hence for each α ∈ Aut_{F}(S), α(P_{i}) = P_{i} for i = 1, 2, and in particular, α fixes s_{5}s_{6}s_{7}. So ⟨s_{5}s_{6}s_{7}⟩ ⊴ F by Proposition 1.15, which contradicts the assumption that O_{2}(F) = 1. Thus there are no reduced fusion systems over F.

#2011 : S ≅ D_{8} × D_{16}. By [O1, Theorem B], F is decomposable.

#2013 : S ≅ D_{8} × SD_{16}. By [O1, Theorem B], F is decomposable. □

5. Groups of order 256

Theorem 5.1. Let F be a reduced, indecomposable fusion system over a group S of order 256. Then one of the following holds:

(a) S ≅ D_{256}, and F is isomorphic to the fusion system of PSL_{2}(q) where q ≡ ±255 (mod 512).

(b) S ≅ SD_{256}, and F is isomorphic to the fusion system of PSL_{3}(q) where q ≡ 63 (mod 128).

(c) S ≅ (Q_{16} ×_{C_{2}} Q_{16}) ⋊^{t} C_{2}, and F is isomorphic to the fusion system of PSp_{4}(7).

(d) S ≅ (C_{8} × C_{8}) ⋊^{−1,t} C_{2}^{2}, and F is isomorphic to the fusion system of G_{2}(7).

(e) S is of type Ly, and F is isomorphic to the fusion system of Lyons’ sporadic group.

(f) S is of type Sp_{4}(4), and F is isomorphic to the fusion system of Sp_{4}(4).

In all cases, F is tame. Also, the fusion system of each of the groups listed above is simple.

Proof. Either S is as in (a,b) above, in which case F is also as in (a,b) by [AOV1, Propositions 4.3–4.4], or S satisfies conditions (a)–(h) in Proposition 2.2. By a computer search, we are left with the following 18 possibilities for S, where #(–) denotes its Magma/GAP identification number. All of the fusion systems listed in the theorem are tame by [BMO2, Theorem C], except for the fusion system of Lyons’s sporadic group (#6665 below) whose tameness is shown separately.

If F is the fusion system of Sp_{4}(4), then F is simple by Proposition 1.18(d). All of the other fusion systems listed above are simple by [O2, Theorem A].

Whenever we list potentially critical subgroups of S (Definition 2.4), they were found using Magma computations based on the criteria in Proposition 2.3.

#12955, #12957, #12965, #15421, #26833, #26835, #55683 : In the first three cases, S ≅ D_{16} × D_{16}, D_{16} × SD_{16}, or SD_{16} × SD_{16}, respectively. In the last four cases, S ≅ D_{8} × S_{2} where S_{2} ≅ C_{4} ≀ C_{2}, D_{32}, SD_{32}, or C_{2}^{2} × D_{8}, respectively. All of these groups satisfy the hypotheses of [O1, Theorem B], and hence every reduced fusion system over any of them is decomposable.

#5298 : S ≅ (C_{8} ×C_{8}) ⋊^{−1,t}C_{2}^{2}. By [O2, Proposition 4.2(c)], every reduced fusion system over S is isomorphic to the fusion system of G_{2}(7).

#5352 : S ≅ (C_{8} × C_{8}) ⋊^{3,t} C_{2}^{2}. By [O2, Proposition 4.2(a)], there are no reduced fusion systems over S.

#6331 : This group has Magma/GAP presentation

S = ⟨s_{1}, …, s_{8} | s_{i}^{2} = 1 (all 1 ≤ i ≤ 8), s_{2}^{s_{1}} = s_{2}s_{4}, s_{3}^{s_{1}} = s_{3}s_{5}, s_{3}^{s_{2}} = s_{3}s_{6},

s_{4}^{s_{3}} = s_{4}s_{7}, s_{6}^{s_{1}} = s_{6}s_{7}s_{8}, s_{6}^{s_{4}} = s_{6}s_{8}, s_{7}^{s_{2}} = s_{7}s_{8}⟩ .

(All other conjugation relations among the s_{i} are trivial.) This is, in fact, a Sylow 2subgroup of 2·M_{22}, although we will not use that here.

Set

t_{1} = s_{3}, u_{1} = s_{6}, t_{2} = s_{1}s_{4}, u_{2} = s_{4}
W = ⟨s_{5}, s_{7}, s_{8}⟩ ≅ C_{2}^{3} V_{i} = W ⟨t_{i}, u_{i}⟩ ≅ C_{2}^{5} (i = 1, 2).

Note that W, V_{1}, V_{2} ⊴ S, and

W = V_{1} ∩ V_{2} = Z(V_{1}V_{2}) = [V_{1}, V_{2}] .

Also, for each i = 1, 2, S = V_{i}T_{i} where T_{i} = ⟨t_{3−i}, u_{3−i}, s_{2}⟩ ≅ D_{8} and V_{i} ∩ T_{i} = 1. Since rk(C_{V_{i}}(g)) ≤ 3 for g ∈ S∖V_{i} (i = 1, 2) and C_{V_{i}}(⟨s_{2}, u_{3−i}⟩) = ⟨s_{5}, s_{8}⟩, we have

V ≤ S, V ≅ C_{2}^{5} ⟹ V = V_{1} or V_{2}.  (1)

For i = 1, 2, define

Q_{i} = V_{i}⟨s_{2}⟩ and R_{i} = N_{S}(Q_{i}) = V_{i}⟨s_{2}, u_{3−i}⟩ .

By Magma computations, there are five S-conjugacy classes of potentially critical subgroups in S (Definition 2.4): V_{1}V_{2}, subgroups S-conjugate to the Q_{i}, and the R_{i}. Also, Aut(S) is a 2-group.

Assume there is a reduced fusion system F over S. Then the following hold. By (1), V_{i} is characteristic in Q_{i} and R_{i} for i = 1, 2.

(i) If R_{i} ∈ E_{F}, then Z(R_{i}) = Z(S) = ⟨s_{5}, s_{8}⟩ is fixed pointwise by Aut_{F}(R_{i}), and

Q_{i} ∉ E_{F}:

To see this, fix α ∈ Aut_{F}(R_{i}). Then α(Z(R_{i})) = Z(R_{i}), and α|_{Z(R_{i})} = α|_{Z(S)} extends to an automorphism in Aut_{F}(S) by the extension axiom. Since Aut_{F}(S) = Inn(S) (since Aut(S) is a 2-group), α|_{Z(R_{i})} = Id.

Since R_{i} ∈ E_{F}, there is Id ≠ α ∈ Aut_{F}(R_{i}) of odd order. If α induces the identity on R_{i}/V_{i} ≅ C_{2}^{2}, then α(Q_{i}) = Q_{i} and hence α sends each of the subgroups

Z(R_{i}) < Z(Q_{i}) < Z(R_{i})[R_{i}, R_{i}] < V_{i} < Q_{i} < R_{i}

to themselves. Since each of these subgroups is normal in R_{i} and has index two in the following one, and since α|_{Z(R_{i})} = Id, α = Id by Lemma A.1, a contradiction. So α permutes cyclically the three subgroups of index two in R_{i} which contain V_{i}, and in particular, Q_{i} is F-conjugate to V_{i}⟨u_{3−i}⟩ ⊴ S. Thus Q_{i} is not fully normalized in F, and Q_{i} ∉ E_{F}.

(ii) For i = 1, 2, V_{i} is normalized by Aut_{F}(V_{1}V_{2}):

By (1), α(V_{i}) = V_{i} for each α ∈ Aut_{F}(V_{1}V_{2}) of odd order. This also holds for α ∈ Aut_{S}(V_{1}V_{2}) since V_{i} ⊴ S. The Sylow axiom implies that Aut_{S}(V_{1}V_{2}) is a Sylow 2-subgroup of Aut_{F}(V_{1}V_{2}), so α(V_{i}) = V_{i} for all α ∈ Aut_{F}(V_{1}V_{2}).

(iii) For i = 1, 2, either R_{i} ∈ E_{F} or Q_{i} ∈ E_{F}, but not both:

If neither R_{i} ∉ E_{F} nor Q_{i} ∉ E_{F}, then V_{3−i} ⊴ F, since V_{3−i} is characteristic in Q_{3−i} and R_{3−i} by (1), and is normalized by Aut_{F}(V_{1}V_{2}) by (ii). By (i), R_{i} and Q_{i} cannot both be in E_{F}.

(iv) Either R_{1} ∈ E_{F} or R_{2} ∈ E_{F}:

By (1), V_{i} is characteristic of index two in Q_{i}, so [α, Q_{i}] ≤ V_{i} for each α ∈ Aut(Q_{i}). So if R_{1} ∉ E_{F} and R_{2} ∉ E_{F}, then 𝔣𝔬𝔠(F) ≤ V_{1}V_{2} < S.

(v) In all cases, s_{5}s_{8} is fixed by Aut_{F}(V_{1}V_{2}):

This is the only element in Z(V_{1}V_{2}) = W = ⟨s_{5}, s_{7}, s_{8}⟩ which is not of the form [x_{1}, x_{2}] for some x_{i} ∈ V_{i}.

(vi) If R_{1}, R_{2} ∈ E_{F}, then s_{5}s_{8} ∈ Z(F):

The element s_{5}s_{8} is fixed by the Aut_{F}(R_{i}) by (i), it is fixed by Aut_{F}(V_{1}V_{2}) by (v), and these are the only possible F-essential subgroups by (i). Hence ⟨s_{5}s_{8}⟩ ⊴ F by Proposition 1.15, and s_{5}s_{8} ∈ Z(F).

By (iii), (iv), and (vi), either Z(F) ≠ 1, or R_{i}, Q_{j} ∈ E_{F} for some i ≠ j.

(vii) If Q_{i} ∈ E_{F} (i = 1 or 2), then either s_{5} or s_{5}s_{8} is fixed by Aut_{F}(V_{i}):

By Lemma A.4, and since 2||Aut_{F}(V_{i})| and Aut_{F}(V_{i}) < Aut(V_{i}) by the Sylow axiom, there is a proper subgroup 1 ≠ W_{i} < V_{i} which is normalized by Aut_{F}(V_{i}). As an 𝔽_{2}[S/V_{i}]-module, V_{i}/W_{i} surjects onto 𝔽_{2} with the trivial action, and since [S, V_{i}] has index two in V_{i}, this implies that W_{i} ≤ [S, V_{i}] = W ⟨u_{i}⟩.

Let Id ≠ α ∈ Aut_{F}(Q_{i}) be of odd order. Then α induces the identity on Q_{i}/V_{i} ≅ C_{2}, and normalizes [Q_{i}, Q_{i}] = ⟨s_{8}, u_{i}⟩ ≅ C_{2}^{2} and Z(Q_{i}) = ⟨s_{5}, s_{8}, u_{i}⟩ ≅ C_{2}^{3}. By Lemma A.1, applied to the chain [Q_{i}, Q_{i}] < Z(Q_{i}) < V_{i} < Q_{i}, α acts irreducibly on V_{i}/Z(Q_{i}). Since W ⟨u_{i}⟩/Z(Q_{i}) < V_{i}/Z(Q_{i}) and W_{i} ≤ W ⟨u_{i}⟩, we get W_{i} ≤ Z(Q_{i}) = Z(S)⟨u_{i}⟩.

If W_{i} ≰ Z(S), then since W_{i} ⊴ S, W_{i} contains [t_{3−i}, u_{i}] = s_{7}, a contradiction. Thus W_{i} ≤ Z(S) = ⟨s_{5}, s_{8}⟩. The homomorphism [s_{2}, −]: V_{i} ⟶ V_{i} induces an isomorphism V_{i}/Z(Q_{i}) ⟶^{≅} [Q_{i}, Q_{i}]. Since α acts irreducibly on V_{i}/Z(Q_{i}) and α|_{V_{i}} commutes with Aut_{Q_{i}}(V_{i}) = ⟨c_{s_{2}}|_{V_{i}}⟩, α also acts irreducibly on [Q_{i}, Q_{i}]. If s_{8} ∈ W_{i}, then α normalizes ⟨s_{8}⟩ = W_{i} ∩ [Q_{i}, Q_{i}], a contradiction. Thus W_{i} = ⟨s_{5}⟩ or W_{i} = ⟨s_{5}s_{8}⟩, and one of these two elements is fixed by Aut_{F}(V_{i}).

(viii) If R_{i}, Q_{j} ∈ E_{F} for i ≠ j, then Z(F) ≠ 1:

By (vii), there is x ∈ {s_{5}, s_{5}s_{8}} ⊆ W which is fixed by Aut_{F}(V_{j}), and hence by Aut_{F}(Q_{j}). Also, x is fixed by Aut_{F}(V_{1}V_{2}) by (ii), and is fixed by Aut_{F}(R_{i}) by (i). Since each F-essential subgroup is S-conjugate to V_{1}V_{2}, R_{i}, or Q_{j} by (iii), ⟨x⟩ ⊴ F by Proposition 1.15, and hence x ∈ Z(F).

Thus in all cases, Z(F) ≠ 1, and F is not reduced.

#6661 : S ≅ (Q_{16} ×_{C_{2}} Q_{16}) ⋊^{t} C_{2}. By [O2, Proposition 5.6], every reduced fusion system over S is isomorphic to the fusion system of PSp_{4}(7).

#6662 : S ≅ (SD_{16} ×_{C_{2}} SD_{16}) ⋊^{t} C_{2}. By [O2, Proposition 5.6], there are no reduced fusion systems over S.

#6665 : S is of type Ly. Let G be Lyons’s group, assume S ∈ Syl_{2}(G), and set F = F_{S}(G). By [O2, Proposition 6.6], each reduced fusion system over S is isomorphic to F. By the same proposition, there is exactly one F-essential subgroup with noncyclic center, and Out(S, F) = 1. Hence μ_{G}: Out(BG_{2}^{∧}) ⟶ Out(S, F) is injective by Proposition 1.14, so Out(BG_{2}^{∧}) = 1, and F is tame. In fact, since Out(G) = 1 by [Ly1, Proposition 5.8], κ_{G} and μ_{G} are both isomorphisms.

#6666 : S is the nonsplit extension of UT_{3}(4) by the group ⟨ϕ, τ⟩ of field and graph automorphisms (denoted S_{ϕ,τ}^{∗} in [O2, p. 71]). By [O2, Proposition 6.6], there are no reduced fusion systems over S.

#8935 : This group has Magma/GAP presentation

S = ⟨s_{1}, …, s_{8} | s_{1}^{2} = s_{8}, s_{2}^{s_{1}} = s_{2}s_{5}, s_{3}^{s_{1}} = s_{3}s_{6}, s_{4}^{s_{1}} = s_{4}s_{8},

s_{3}^{s_{2}} = s_{3}s_{7}, s_{4}^{s_{2}} = s_{4}s_{5}⟩ .

(All other squaring and conjugation relations among the s_{i} are trivial.) Thus Z(S) = ⟨s_{5}, s_{6}, s_{7}, s_{8}⟩. If we set P_{1} = Z(S)⟨s_{1}s_{4}, s_{2}⟩ and P_{2} = Z(S)⟨s_{3}, s_{4}⟩, then P_{1} ≅ P_{2} ≅ C_{2}^{6}.

By Proposition 5.3 below, every reduced fusion system over S is isomorphic to the fusion system of Sp_{4}(4). #53366, #53380 : S ≅ C_{2}^{2} × S_{0}, where S_{0} has type M_{12} or S_{0} ≅ UT_{4}(2). By Proposition 3.6 or 3.8, respectively, there are no reduced fusion systems over S. (When S_{0} has type M_{12}, Aut(S_{0}) is a 2-group by, e.g., [AOV2, Proposition 3.2].) #55676 : S ≅ C_{2}^{2} × UT_{3}(4). By Proposition 3.3, applied with P_{i} = C_{2}^{2} × A_{i} ≅ C_{2}^{6} (where A_{1}, A_{2} < UT_{3}(4) are the two elementary abelian subgroups of rank four), there are no reduced fusion systems over S.

This finishes the proof of the theorem. □

Within the proof of Theorem 5.1, we have also shown:

Lemma 5.2. If G is isomorphic to the sporadic simple group of Lyons, then (for p = 2) κ_{G} and μ_{G} are both isomorphisms. In particular, G tamely realizes its fusion system.

It remains to describe fusion systems over the group #8935, which is a Sylow 2-subgroup of Sp_{4}(4).

Proposition 5.3. Let S be a 2-group with subgroups P_{1}, P_{2} ⊴ S such that

|S| = 2^{8}, P_{1} ≅ P_{2} ≅ C_{2}^{6}, P_{1} ∩ P_{2} = Z(S) = [S, S] ≅ C_{2}^{4} .  (2)

Then every saturated fusion system F over S such that O_{2}(F) = 1 is isomorphic to the fusion system of Sp_{4}(4).

Proof. By comparing orders, we see that S = P_{1}P_{2}. Set Z = Z(S) and P_{i} = P_{i}/Z ≅ C_{2}^{2}. Thus S/Z = P_{1} × P_{2}. Choose subgroups P_{i}^{0} < P_{i} (i = 1, 2) complementary to Z (i.e., such that P_{i} = Z ×P_{i}^{0}), and set Ŝ = P_{1}^{0} ∗ P_{2}^{0}. The commutator subgroup of Ŝ/[Ŝ, [Ŝ, Ŝ]] is central of order 2^{4} (isomorphic to the tensor product P_{1}^{0}⊗P_{2}^{0} via the commutator pairing), and hence the natural homomorphism from Ŝ to S is surjective with kernel [Ŝ, [Ŝ, Ŝ]].

Assume S^{∗} is another group of order 2^{8}, with subgroups P_{1}^{∗}, P_{2}^{∗} ≤ S^{∗} which also satisfy (2). Choose subgroups P_{i}^{∗0} < P_{i}^{∗} complementary to Z(S^{∗}), and set Ŝ^{∗} = P_{1}^{∗0} ∗ P_{2}^{∗0}. Then S^{∗} ≅ Ŝ^{∗}/[Ŝ^{∗}, [Ŝ^{∗}, Ŝ^{∗}]] by the above argument. Any pair of isomorphisms ψ_{i} ∈ Iso(P_{i}^{0}, P_{i}^{∗0}) (i = 1, 2) extends to an isomorphism ψ̂ ∈ Iso(Ŝ, Ŝ^{∗}) and hence to an isomorphism ψ ∈ Iso(S, S^{∗}). The conditions (2) thus determine S uniquely up to isomorphism.

Let Aut^{0}(S/Z) < Aut(S/Z) be the subgroup of those automorphisms which normalize the set {P_{1}, P_{2}}. The above argument, when applied with S^{∗} = S and {P_{1}^{∗0}, P_{2}^{∗0}} = {P_{1}^{0}, P_{2}^{0}}, shows that the projection S → S/Z induces a (split) surjection

Ψ: Aut(S) ⟶ Aut^{0}(S/Z) ≅ (Aut(P_{1}) × Aut(P_{2})) ⋊ C_{2} ≅ Σ_{3} ≀ C_{2} .

(Note that Ψ(Aut(S)) ≤ Aut^{0}(S/Z) since by Proposition 3.3, P_{1} and P_{2} are the unique maximal elementary abelian subgroups of S.) It follows that

for each Γ ∈ Syl_{3}(Aut(S)), Ψ(N_{Aut(S)}(Γ)) = Aut^{0}(S/Z)  (3)

by the Frattini argument applied to the normal subgroup Ker(Ψ)Γ ⊴ Aut(S).

Let F be a saturated fusion system over S such that O_{2}(F) = 1. By Proposition 3.3, E_{F} = {P_{1}, P_{2}}. Since |Out_{F}(S)| is odd and Ker(Ψ) is a 2-group by Lemma A.1,

Out_{F}(S) ≅ C_{3}^{r} for some 0 ≤ r ≤ 2.  (4)

Fix i = 1, 2. By Lemma A.6, and since Aut_{S}(P_{i}) ≅ S/P_{i} ≅ C_{2}^{2}, Aut_{F}(P_{i}) = Δ_{i} × H_{i} for some Δ_{i} ≅ A_{5} and some H_{i} of odd order. Also, rk([s, P_{i}]) = 2 for s ∈ S∖P_{i}, and [S, P_{i}] = [S, S] = Z = C_{P_{i}}(S) has rank 4. So by Lemma A.7(b), P_{i} is indecomposable as an 𝔽_{2}[Δ_{i}]-module, Z_{i} =^{def} C_{P_{i}}(Δ_{i}) has rank two, and the action of Δ_{i} ≅ A_{5} on P_{i}/Z_{i} ≅ C_{2}^{4} is isomorphic to the canonical action of SL_{2}(4) on 𝔽_{4}^{2}. Thus for g ∈ Δ_{i} of order three, C_{P_{i}/Z_{i}}(g) = 1, and hence C_{P_{i}}(g) = Z_{i}. In summary, and since Aut_{S}(P_{i}) ≤ Δ_{i},

C_{P_{i}}(g) = Z_{i} = C_{P_{i}}(Δ_{i}) ≤ C_{P_{i}}(Aut_{S}(P_{i})) = Z .

For i = 1, 2, the homomorphism

Aut_{F}(S) ⟶ N_{Aut_{F} (P_{i})}(Aut_{S}(P_{i})) ≅ A_{4} × H_{i}

induced by restriction is surjective by the extension axiom, and its kernel is a 2-group by Lemma A.1 (or Lemma A.2). Hence r ≥ 1 and H_{i} ≅ C_{3}^{r−1} (with r as in (4)).

Fix some Γ ∈ Syl_{3}(Aut_{F}(S)). Choose γ_{1}, γ_{2} ∈ Γ of order 3 such that γ_{i}|_{P_{i}} ∈ Δ_{i}. Then Z_{i} = C_{P_{i}}(Δ_{i}) = C_{P_{i}}(γ_{i}) = C_{Z}(γ_{i}) has rank two and is normalized by Γ. If Z_{1} = Z_{2}, then Z_{1} ⊴ F by Proposition 1.15, which contradicts our assumption that O_{2}(F) = 1. Thus Z_{1} ≠ Z_{2} and ⟨γ_{1}⟩ ≠ ⟨γ_{2}⟩, so Γ = ⟨γ_{1}, γ_{2}⟩ ≅ C_{3}^{2} by (4), and r = 2. In particular, Aut_{F}(P_{i}) ≅ A_{5} × C_{3} for i = 1, 2. Also, upon regarding Z as an 𝔽_{2}[Γ]-module with Z_{i} = C_{Z}(γ_{i}), we see that Z = Z_{1} × Z_{2} where the Z_{i} are submodules. We will show that the choice of the pair (⟨γ_{1}⟩, ⟨γ_{2}⟩) completely determines F.

For i = 1, 2, choose any η_{i} ∈ Γ of order 3 such that η_{i}|_{P_{i}} ∈ H_{i}. Since η_{i}|_{P_{i}} commutes with Aut_{S}(P_{i}), η_{i} acts trivially on S/P_{i}, and hence trivially on P_{3−i}. So η_{i} must act nontrivially on P_{i} since Ker(Ψ) is a 2-group. Also, since γ_{i}|_{P_{i}} does not commute with Aut_{S}(P_{i}), γ_{i} acts nontrivially on S/P_{i} ≅ P_{3−i}, and it acts nontrivially on P_{i} = P_{i}/Z since C_{P_{i}}(γ_{i}) = Z_{i} < Z. Thus ⟨η_{1}⟩, ⟨η_{2}⟩, ⟨γ_{1}⟩, and ⟨γ_{2}⟩ are the four distinct subgroups of order 3 in Γ, and ⟨η_{1}⟩ and ⟨η_{2}⟩ are the unique subgroups of Γ of order 3 which induce the identity on P_{2} and P_{1}, respectively.

Fix ω ∈ 𝔽_{4}∖𝔽_{2}. For i = 1, 2, C_{P_{i}}(η_{i}) = 1, since C_{P_{i}}(η_{i}) = 1 and C_{Z}(η_{i}) = C_{Z}(γ_{1}^{ε_{i1}}γ_{2}^{ε_{i2}}) = 1. Hence we can make P_{i} into an 𝔽_{4}-vector space by defining multiplication by ω to be η_{i}. Thus C_{Aut(P_{i})}(η_{i}) = Aut_{𝔽_{4}}(P_{i}) ≅ GL_{3}(4). Set

G_{i} = {α ∈ Aut_{𝔽_{4}}(P_{i}) | α(Z_{i}) = Z_{i}} and Q_{i} = O_{2}(G_{i}) .

Then Q_{i} is the group of all α ∈ Aut_{𝔽_{4}}(P_{i}) which induce the identity on Z_{i} and on P_{i}/Z_{i}, Q_{i} ≅ 𝔽_{4}^{2} ≅ C_{2}^{4}, and G_{i}/Q_{i} ≅ GL_{2}(4) × C_{3} ≅ A_{5} × C_{3}^{2}. Set K_{i} = Aut_{S}(P_{i})⟨γ_{i}|_{P_{i}}⟩ ≅ A_{4}. Thus K_{i} < Δ_{i} < G_{i} for i = 1, 2.

Let Δ_{i}^{∗} < G_{i} be any subgroup such that Δ_{i}^{∗} ≅ A_{5} and K_{i} < Δ_{i}^{∗}. Then Q_{i} ∩ Δ_{i}^{∗} = 1, and Δ_{i}^{∗}Q_{i} = Δ_{i}Q_{i} = O^{3}(G_{i}) since G_{i}/Q_{i} ≅ A_{5} × C_{3}^{2}. So by Proposition A.3, applied with G = G_{i}, Q = Q_{i}, H = Δ_{i}, and H_{0} = K_{i}, we have Δ_{i}^{∗} = ^{ψ}(Δ_{i}) for some ψ ∈ C_{Q_{i}}(K_{i}). Since C_{Q_{i}}(K_{i}) ≤ C_{Q_{i}}(γ_{i}) = 1 (recall C_{P_{i}}(γ_{i}) = Z_{i}), this proves that Δ_{i}^{∗} = Δ_{i}.

Thus Aut_{F}(P_{1}) and Aut_{F}(P_{2}) are determined by the choice of Γ = ⟨γ_{1}⟩ × ⟨γ_{2}⟩. So by Proposition 1.4, F is determined by this choice. We just saw that ⟨γ_{1}⟩ and ⟨γ_{2}⟩ are the two subgroups of order 3 in Γ which act nontrivially on both P_{1} and P_{2}. If F′ is another saturated fusion system over S with O_{2}(F′) = 1, and is determined by Γ′ = ⟨γ′_{1}⟩ × ⟨γ′_{2}⟩, then there is φ ∈ Aut(S) such that ^{φ}Γ′ = Γ. Either ^{φ}⟨γ′_{i}⟩ = ⟨γ_{i}⟩ for i = 1, 2, or ^{φ}⟨γ′_{i}⟩ = ⟨γ_{3−i}⟩. By (3), there is ψ ∈ N_{Aut(S)}(Γ) which exchanges ⟨γ_{1}⟩ and ⟨γ_{2}⟩, and so either ^{φ}F′ = F or ^{ψφ}F′ = F.

Fix S^{∗} ∈ Syl_{2}(Sp_{4}(4)), and set F^{∗} = F_{S^{∗}}(Sp_{4}(4)). By the Chevalley commutator formula (see, e.g., [Ca, Theorem 5.2.2]), the two unipotent radical subgroups P_{1}^{∗}, P_{2}^{∗} < S^{∗} are isomorphic to 𝔽_{4}^{3} ≅ C_{2}^{6}, and are such that P_{1}^{∗} ∩ P_{2}^{∗} = [S^{∗}, S^{∗}] = Z(S^{∗}) ≅ C_{2}^{4}. Hence S^{∗} satisfies (2), so S^{∗} ≅ S. Also, O_{2}(F^{∗}) = 1 by Proposition 1.18(b), so F^{∗} ≅ F since there is up to isomorphism at most one such saturated fusion system over S. □

6. Groups of order 512

Theorem 6.1. Let F be a reduced, indecomposable fusion system over a 2-group S of order 512. Then one of the following holds:

(a) S ≅ D_{512}, and F is isomorphic to the fusion system of PSL_{2}(q) where q ≡ ±511 (mod 1024).

(b) S ≅ SD_{512}, and F is isomorphic to the fusion system of PSL_{3}(q) where q ≡ 127 (mod 256).

(c) S ≅ C_{16} ≀ C_{2}, and F is isomorphic to the fusion system of SL_{3}(17).

(d) S ≅ D_{16} ≀ C_{2}, and F is isomorphic to the fusion system of PSL_{4}(7).

(e) S ≅ SD_{16} ≀ C_{2}, and F is isomorphic to the fusion system of SL_{5}(3).

(f) S ≅ UT_{3}(8), and F is isomorphic to the fusion system of SL_{3}(8).

(g) S is of type A_{12}, and F is isomorphic to the fusion system of A_{12}, or of Sp_{6}(2), or of Ω_{7}(3).

(h) S ≅ ((Q_{8})^{3}/C_{2}) ⋊^{t} C_{2}, and F is isomorphic to the fusion system of PSp_{6}(3).

(i) S is of type HS, and F is isomorphic to the fusion system of the Higman-Sims sporadic group.

(j) S is of type O’N, and F is isomorphic to the fusion system of O’Nan’s sporadic group.

In all cases, F is tame. Also, the fusion system of each of the groups listed above is simple.

Proof. Either S is as in (a,b,c) above, in which case F is also as in (a,b,c) by [AOV1, Propositions 4.3–4.4] or [AOV2, Proposition 3.1], or S satisfies conditions (a)–(h) in Proposition 2.2. By a computer search, we are left with the following 31 possibilities for S, where #(−) denotes its Magma/GAP identification number. Except for the fusion systems of the sporadic groups of O’Nan and Higman-Sims (#58362 and #60329 below), all of the fusion systems listed in the theorem are tame by [BMO2, Theorem C], or (for the fusion system of A_{12}) by [AOV1, Proposition 4.8].

By [O2, Theorem A], all of the fusion systems listed above in cases (a–e) are simple. By Proposition 1.18(a,b,d), if F is any of the fusion systems in (f–j), then O^{2}(F) = F, O_{2}(F) = 1, F is indecomposable, and F is simple except possibly when F is the fusion system of Ω_{7}(3) or PSp_{6}(3). In case (g), Aut(S) is a 2-group (see the proof of Proposition 7.4), so those fusion systems are simple by Proposition 1.18(c,d). By Proposition 6.4, when S is of type PSp_{6}(3), there is a unique saturated fusion system F over S such that O_{2}(F) = 1, and since O_{2}(O^{2′}(F)) = 1 [AOV1, Lemma 1.20(e) & Proposition 1.25(b)], F = O^{2′}(F) must be simple by Proposition 1.18(c,d).

Whenever we list potentially critical subgroups of S, they were found using Magma computations based on the criteria in Proposition 2.3.

#128270, #128271, #399715, #399717, #399770, #399771 : In these six cases, S = S_{1} × S_{2} where S_{1} ≅ D_{16} or SD_{16}, and S_{2} ≅ D_{32}, SD_{32}, or C_{4} ≀ C_{2}. By [O1, Theorem B], each reduced fusion system over any of these groups is decomposable.

#420360, #420362, #6480905, #7998954, #6480855 : In these cases, S ≅ D_{8}×S_{0}, where S_{0} ≅ D_{64}, SD_{64}, UT_{4}(2), UT_{3}(4), or is of type M_{12}. By [O1, Theorem B], each reduced fusion system over S is decomposable.

#10483221, #10483222, #10493114 : In these cases, S ≅ D_{8} × S_{0} where S_{0} ≅ C_{2}^{2} × D_{16}, C_{2}^{2} × SD_{16}, or C_{2}^{3} × D_{8}, respectively. By [O1, Theorem B], there are no reduced fusion systems over S.

#7606661 : S ≅ D_{8} ×D_{8} ×D_{8}, and each reduced fusion system over S is decomposable by [O1, Theorem C].

#7530050, #7530054, #7530055, #10482003 : In these cases, S ≅ C_{2}^{2} × S_{0} where S_{0} ≅ D_{8} ≀ C_{2} or has type M_{22} or M_{12}:2; or S ≅ C_{2}^{3} × S_{0} where S_{0} has type M_{12}. In all cases, |Z(S_{0})| = 2 and rk(S_{0}) ≤ 4. Also, Aut(S_{0}) is a 2-group by [O2, Corollary A.10(c)], [OV, Lemma 5.5], the proof of [O2, Proposition 4.3(c)], or [AOV2, Proposition 3.2], respectively. So by Proposition 3.6, there are no reduced fusion systems over S.

#7530088, #10482065 : S ≅ C_{2}^{2} × S_{0} where S_{0} has type J_{2}, or S ≅ C_{2}^{3} × UT_{4}(2). By Proposition 3.7 or 3.8, respectively, there are no reduced fusion systems over S.

#10493307 : Here, S ≅ C_{2}^{3} ×UT_{3}(4). By Proposition 3.3, applied with P_{i} = C_{2}^{3} ×A_{i} ≅ C_{2}^{7} (where A_{1}, A_{2} < UT_{3}(4) are the two elementary abelian subgroups of rank four), there are no reduced fusion systems over S.

#58362 : S is of type O’N. By Proposition 6.3 below, every reduced fusion system over S is isomorphic to the fusion system of O’Nan’s group, and is tame.

#60329 : S is of type HS. By Proposition 6.3 below, every reduced fusion system over S is isomorphic to the fusion system of the Higman-Sims group, and is tame.

#60809 : S ≅ D_{16} ≀ C_{2}. By [O2, Proposition 5.5(a)], every reduced fusion system over S is isomorphic to F_{S}(PSL_{4}(7)).

#60833 : S ≅ SD_{16} ≀ C_{2}. By [O2, Proposition 5.5(b)], every reduced fusion system over S is isomorphic to F_{S}(SL_{5}(3)).

#406983 : S is of type A_{12}. By Proposition 7.4 below, each reduced fusion system over S is isomorphic to the fusion system of one of the groups A_{12}, Sp_{6}(2), or Ω_{7}(3).

#6407070 : This group has Magma/GAP presentation

S = ⟨s_{1}, …, s_{9} | s_{i}^{2} = 1 (all 1 ≤ i ≤ 9), s_{2}^{s_{1}} = s_{2}s_{6}, s_{3}^{s_{1}} = s_{3}s_{7}, s_{3}^{s_{2}} = s_{3}s_{8},

s_{4}^{s_{3}} = s_{4}s_{8}, s_{5}^{s_{2}} = s_{5}s_{8}, s_{6}^{s_{3}} = s_{6}s_{9}, s_{6}^{s_{5}} = s_{6}s_{9}, s_{7}^{s_{4}} = s_{7}s_{9}, s_{8}^{s_{1}} = s_{8}s_{9}⟩.

(All other conjugation relations among the s_{i} are trivial.) By computer computations, S has the following two potentially critical subgroups:

P_{1} = ⟨s_{1}, s_{4}, s_{5}, s_{6}, s_{7}, s_{8}, s_{9}⟩ ≅ 2_{+}^{1+6}
P_{2} = ⟨s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}, s_{8}, s_{9}⟩.

If P_{1} is critical in S, then by Lemma A.6, there is Γ ≤ Out(P_{1}) such that Γ ≅ A_{5} and Out_{S}(P_{1}) ∈ Syl_{2}(Γ). Set V = P_{1}/Fr(P_{1}) ≅ C_{2}^{6}, and let g ∈ V be the class of g ∈ P_{1}. For 1 ≠ x ∈ S/P_{1} ≅ C_{2}^{2}, [x, V] = ⟨s_{6}, s_{8}⟩, ⟨s_{7}, s_{8}⟩, or ⟨s_{6}s_{7}, s_{8}⟩, and in particular, rk([x, V]) = 2. Also, C_{V} (S) = [S, V] = ⟨s_{6}, s_{7}, s_{8}⟩. So by Lemma A.7(b), V is 𝔽_{2}[Γ]- indecomposable, C_{V} (Γ) has rank 1, and ⋂_{1≠x∈S/P_{1}}[x, V] = 1. But s_{8} lies in each of these commutators, a contradiction, and we conclude that P_{1} is not critical.

Thus E_{F} ⊆ {P_{2}} for every saturated fusion system F over S, so there are no reduced fusion systems over S by Lemma 1.17.

#7530110 : S is of type PSp_{6}(3). By Proposition 6.4 below, every reduced fusion system over S is isomorphic to the fusion system of PSp_{6}(3).

#7540630 : S ≅ C_{2}^{2} × S_{0}, where S_{0} is the group of order 128 with Magma/GAP number 1411 described in the proof of Theorem 4.3. In particular, there are two subgroups P_{1}, P_{2} < S isomorphic to C_{2}^{7}, and they satisfy the hypotheses of Proposition 3.3. Since [S, S] < Z(S), there are no reduced fusion systems over S by that proposition.

#10481201 : S ≅ UT_{3}(8). By Proposition 6.2 below, every reduced fusion system over S is isomorphic to F_{S}(SL_{3}(8)). □

It remains to handle some of the individual cases.

Proposition 6.2. Fix k ≥ 1. A saturated fusion system F over UT_{3}(2^{k}) is reduced if and only if it is isomorphic to the fusion system of PSL_{3}(2^{k}).

Proof. Since this holds for k = 1 by [AOV1, Proposition 4.3] (UT_{3}(2) ≅ D_{8}), we assume k ≥ 2 from now on. Set q = 2^{k} and S = UT_{3}(q) for short. For each 1 ≤ i < j ≤ 3, let E_{ij} ≤ S be the subgroup of strict upper triangular matrices with nonzero off-diagonal entry only in position (i, j). Thus E_{13} = Z(S) = [S, S]. Set A_{1} = E_{12}E_{13}, and A_{2} = E_{13}E_{23}. Then A_{1} ≅ A_{2} ≅ (𝔽_{q})^{2} ≅ C_{2}^{2k}, and we regard these groups as 𝔽_{q}-vector spaces.

Fix a reduced fusion system F over S. By Proposition 3.3, E_{F} = {A_{1}, A_{2}}. For each i = 1, 2, set Γ_{i} = Aut_{F}(A_{i}).

We first examine Γ_{1} = Aut_{F}(A_{1}). Fix a strongly 2-embedded subgroup H < Γ_{1} which contains Aut_{S}(A_{1}). Consider the following sets of subgroups of A_{1}:

𝒱_{0} = {C_{A_{1}}(T) | T ∈ Syl_{2}(H)} , 𝒱_{1} = {C_{A_{1}}(T) | T ∈ Syl_{2}(Γ_{1}) ∖ Syl_{2}(H)} ,
𝒱 = 𝒱_{0} ∪ 𝒱_{1} = {C_{A_{1}}(T) | T ∈ Syl_{2}(Γ_{1})} .

In particular, E_{13} = C_{A_{1}}(Aut_{S}(A_{1})) ∈ 𝒱_{0}. Since Γ_{1} permutes the elements of 𝒱 transitively (since φ(C_{A_{1}}(T)) = C_{A_{1}}(^{φ}T) for φ ∈ Γ_{1} and T ∈ Syl_{2}(Γ_{1})), rk(V) = rk(E_{13}) = k for each V ∈ 𝒱 .

For each V_{0} ∈ 𝒱_{0} and each V_{1} ∈ 𝒱_{1}, V_{0} = C_{A_{1}}(T_{0}) and V_{1} = C_{A_{1}}(T_{1}) for some T_{0} ∈ Syl_{2}(H) and T_{1} ∈ Syl_{2}(Γ_{0})∖Syl_{2}(H), and ⟨T_{0}, T_{1}⟩ contains the strongly 2-embedded subgroup ⟨T_{0}, T_{1}⟩ ∩ H. In particular, O_{2}(⟨T_{0}, T_{1}⟩) = 1, so ⟨T_{0}, T_{1}⟩ acts faithfully on A_{1}/C_{A_{1}}(⟨T_{0}, T_{1}⟩) (Lemma A.1). By [OV, Lemma 1.7(a)], rk(A_{1}/C_{A_{1}}(⟨T_{0}, T_{1}⟩)) ≥ 2k, so V_{0} ∩ V_{1} = C_{A_{1}}(⟨T_{0}, T_{1}⟩) = 1, and A_{1} = V_{0} × V_{1}.

In particular, 𝒱_{0} ∩ 𝒱_{1} = ∅. Also, when V_{0} = E_{13}, this shows that each V ∈ 𝒱_{1} contains a representative for each coset of E_{13} in A_{1}∖E_{13}. Since Aut_{S}(A_{1}) acts transitively on each such coset, this implies that ⋃_{V ∈𝒱_{1}} V ⊇ A_{1}∖E_{13}. Hence V_{0} ∩ (A_{1}∖E_{13}) = ∅ for each V_{0} ∈ 𝒱_{0}, so 𝒱_{0} = {E_{13}}. Since Γ_{1} acts transitively on 𝒱 and E_{13} ∩ V = 1 for each V ∈ 𝒱 ∖ {E_{13}}, this implies that V ∩ W = 1 for each pair V, W of distinct elements in 𝒱 . Thus A_{1}∖1 is the disjoint union of the sets V ∖1 for all V ∈ 𝒱 , so |𝒱 | = q + 1, and Aut_{S}(A_{1}) normalizes the set 𝒱_{1} = 𝒱 ∖{E_{13}}.

Fix some V ∈ 𝒱 ∖{E_{13}}. Since A_{1} = E_{13} × V, there is φ_{1} ∈ Aut(A_{1}) which induces the identity on E_{13} and on A_{1}/E_{13}, and such that φ_{1}(V) = E_{12}. Set 𝒱 ^{∗} = φ_{1}(𝒱 ). Thus E_{13} = φ_{1}(E_{13}) and E_{12} = φ_{1}(V) are both in 𝒱 ^{∗}. Also, [φ_{1}, Aut_{S}(A_{1})] = 1 since the group of automorphisms of A_{1} which induce the identity on E_{13} and on A_{1}/E_{13} is abelian, so Aut_{S}(A_{1}) also normalizes the set 𝒱 ^{∗}∖{E_{13}}. Since |𝒱 ^{∗} ∖ {E_{13}}| = q = |Aut_{S}(A_{1})| and E_{12} is stabilized only by the identity in Aut_{S}(A_{1}), Aut_{S}(A_{1}) acts transitively on this set. Hence 𝒱 ^{∗} = φ_{1}(𝒱 ) is precisely the set of all 1-dimensional 𝔽_{q}-linear subspaces of A_{1}.

Let Θ_{i}^{SL} < Θ_{i}^{GL} < Θ_{i}^{Γ L} < Aut(A_{i}) be the subgroups SL_{2}(q), GL_{2}(q), and Γ L_{2}(q), respectively, defined with respect to the canonical 𝔽_{q}-vector space structure on A_{i}. By the fundamental theorem of affine geometry [Bg, Théorème 2.6.3], N_{Aut(A_{1})}(φ_{1}(𝒱 )) = Θ_{1}^{Γ L}, and hence ^{φ_{1}}(Γ_{1}) ≤ Θ_{1}^{Γ L}. All Sylow 2-subgroups of ^{φ_{1}}(Γ_{1}) are Θ_{1}^{Γ L}-conjugate to Aut_{S}(A_{1}) and hence contained in Θ_{1}^{SL}. So O^{2′}(^{φ_{1}}(Γ_{1})) = Θ_{1}^{SL} since SL_{2}(q) is generated by any two of its Sylow 2-subgroups.

By a similar argument, there is φ_{2} ∈ Aut(A_{2}) which induces the identity on E_{13} and on A_{2}/E_{13}, and such that Θ_{2}^{SL} ≤ ^{φ_{2}}(Γ_{2}) ≤ Θ_{2}^{Γ L} and O^{2′}(^{φ_{2}}(Γ_{2})) = Θ_{2}^{SL}. Let φ ∈ Aut(S) be the unique automorphism such that φ|_{A_{i}} = φ_{i} for i = 1, 2. (Note that φ has the form φ(g) = gχ(g) for some χ ∈ Hom(S, Z(S)).) Upon replacing F by ^{φ}F, we can assume that Θ_{i}^{SL} ≤ Aut_{F}(A_{i}) ≤ Θ_{i}^{Γ L} and O^{2′}(Aut_{F}(A_{i})) = Θ_{i}^{SL} for each i = 1, 2.

Set G = PSL_{3}(q) for short, and identify S = UT_{3}(q) with its image in G. Fix a generator λ ∈ 𝔽_{q}^{×}. Let β_{1}, β_{2} ∈ Aut_{G}(S) be conjugation by the classes of the diagonal matrices diag(λ, λ^{−1}, 1) and diag(1, λ, λ^{−1}) respectively. Then Aut_{G}(S) = Inn(S)⟨β_{1}, β_{2}⟩ and

β_{1} (( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & λ^{2}a & λb \\ 0 & 1 & λ^{−1}c \\ 0 & 0 & 1 \end{matrix} ) and β_{2} (( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & λ^{−1}a & λb \\ 0 & 1 & λ^{2}c \\ 0 & 0 & 1 \end{matrix} ) .

Thus β_{1}|_{A_{2}} ∈ Θ_{2}^{SL} ≤ Aut_{F}(A_{2}). By the extension axiom, there is β′_{1} ∈ Aut_{F}(S) such that β′_{1}|_{A_{2}} = β_{1}|_{A_{2}}. Then β′_{1}|_{A_{1}} ∈ Aut_{F}(A_{1}), so β′_{1}|_{A_{1}} ∈ Θ_{1}^{Γ L} by our assumptions, and β_{1}|_{A_{1}} ∈ Θ_{1}^{Γ L} by construction. Set τ = β_{1}^{−1}β′_{1}; then τ induces the identity on A_{2} and hence on S/A_{2} (since C_{S}(A_{2}) = A_{2}) and τ|_{A_{1}} ∈ Θ_{1}^{Γ L}. It follows that τ|_{A_{1}} ∈ Aut_{A_{2}}(A_{1}), and hence that τ ∈ Inn(S). So β_{1} ∈ Aut_{F}(S).

By a similar argument, β_{2} ∈ Aut_{F}(S). Hence Aut_{G}(S) ≤ Aut_{F}(S). Also, Aut_{G}(A_{i}) = Θ_{i}^{SL}⟨β_{i}|_{A_{i}}⟩ for i = 1, 2 (the subgroup of index (3, q−1) in Θ_{i}^{GL}), so O^{2′}(Aut_{F}(A_{i})) = Θ_{i}^{SL} ≤ Aut_{G}(A_{i}) ≤ Aut_{F}(A_{i}). Since E_{F} = {A_{1}, A_{2}}, F_{S}(G) ⊆ F by Proposition 1.4 (Alperin’s fusion theorem). In addition, the A_{i} are minimal F-centric subgroups, so Aut_{G}(P) ≥ O^{2′}(Aut_{F}(P)) for each P ∈ F^{c}. (If P ∈ F^{c}∖{A_{1}, A_{2}}, then O^{2′}(Aut_{F}(P)) = Aut_{S}(P) by Proposition 1.4 and since there are no larger essential subgroups.) So by [AKO, Lemma I.7.6(a)], F contains F_{S}(G) as a subsystem of odd index in the sense of [AKO, Definition I.7.3]. Thus F = F_{S}(G) since O^{2′}(F) = F (F is reduced).

Conversely, F_{S}(G) is simple by Proposition 1.18(d). □

The 2-groups of type O’N and HS will be handled together.

Proposition 6.3. (a) If S is a group of order 512 of type O’N, then every reduced fusion system over S is isomorphic to the fusion system of O’Nan’s group, and is tame.

(b) If S is a group of order 512 of type HS, then every reduced fusion system over S is isomorphic to the fusion system of the Higman-Sims group, and is tame.

If G is either of the groups O’N or HS, then κ_{G} and μ_{G} are both isomorphisms, and hence G tamely realizes its fusion system.

Proof. We let S^{O’N} and S^{HS} be 2-groups of type O’N or HS, respectively, and set S = S^{O’N} or S^{HS} when we do not need to distinguish them. By [O’N] or [Alp2], these groups have a presentation with generators v_{1}, v_{2}, v_{3}, s, t, where

A = ⟨v_{1}, v_{2}, v_{3}⟩ ≅ C_{4}^{3} ,

and with additional relations

v_{1}^{t} = v_{3}^{−1}  v_{2}^{t} = v_{2}^{−1}  v_{3}^{t} = v_{1}^{−1}  t^{2} = 1
v_{1}^{s} = v_{2}  v_{2}^{s} = v_{3}  v_{3}^{s} = v_{1}v_{2}^{−1}v_{3}  s^{t} = s^{−1}

(in both cases), and

s^{4} = \begin{cases}v_{1}v_{3} & if S = S^{O’N} \\ 1 & if S = S^{HS}. \end{cases}

Thus S is an extension of A by D_{8}. Also, A is the unique subgroup of S isomorphic to C_{4}^{3}, and hence is characteristic in all subgroups of S which contain it.

By computer computations, there are three conjugacy classes of potentially critical subgroups in each case, with representatives

P_{1} = A⟨s^{2}, t⟩ = C_{S}(v_{1}v_{2}^{2}v_{3}^{−1})
P_{2}^{O’N} = ⟨s^{2}v_{1}, t, v_{1}v_{3}, v_{1}^{2}, v_{2}^{2}⟩ ≅ C_{4} ×_{C_{2}} 2_{+}^{1+4}
P_{2}^{HS} = ⟨sv_{1}v_{2}, t, v_{1}v_{3}, v_{1}^{2}, v_{2}^{2}⟩
P_{3} = A⟨st, s^{2}⟩ = C_{S}(v_{1}^{2}v_{2}^{2}) .

Note that P_{1}, P_{3} ⊴ S (they have index two), while N_{S}(P_{2}) = P_{1} when S = S^{O’N}, and N_{S}(P_{2}) = P_{2}⟨s⟩ when S = S^{HS}. Since P_{1} and P_{3} are the only potentially critical subgroups of index two (and are not isomorphic to each other), they are both characteristic in S.

Set w = v_{1}v_{3} and z = w^{2} = v_{1}^{2}v_{3}^{2} ∈ Z(S) ≅ C_{2}. By direct computation, Out(S) ≅ C_{2}^{4}, with explicit generators [χ_{i}] (i = 1, 2, 3, 4), where the χ_{i} ∈ Aut(S) act as follows.

χ χ(v_{1}) χ(v_{2}) χ(v_{3}) χ^{O’N}(s) χ^{O’N}(t) χ^{HS}(s) χ^{HS}(t)
χ_{1} v_{1}z v_{2}z v_{3}z  s  t  s  t
χ_{2} v_{1}  v_{2}  v_{3} v_{1}^{−1}v_{2}^{2}v_{3}s  t  w^{−1}s  zt
χ_{3} v_{1}  v_{2}  v_{3} v_{1}^{−1}v_{2}^{−1}s  t  ws w^{−1}t
χ_{4} v_{1}^{−1} v_{2}^{−1} v_{3}^{−1}  v_{3}s  v_{2}t  zs  t

Note that χ_{i}(P_{j}) = P_{j} for all i, j, except for the case χ_{4}(P_{2}^{O’N}) = c_{s}(P_{2}^{O’N}) ≠ P_{2}^{O’N}. Also, Out(P_{2}^{O’N}) ≅ C_{2} × Σ_{6}, Out(P_{2}^{HS}) ≅ C_{2}^{2} × Σ_{4}, and Out(P_{1}) and Out(P_{3}) are described by the following table (where the second factor in Out(P_{3}) is centerfree):

P H^{1}(P/A; A) N_{Aut(A)}(Aut_{P} (A))/Aut_{P} (A)  Out(P)
P_{1}  ℤ/4  C_{2} × Σ_{4}  D_{8} × Σ_{4}
P_{3}  (ℤ/2)^{3}  C_{2} × Σ_{4}  C_{2}^{2} × (C_{2}^{4} ⋊ Σ_{3})

Fix a reduced fusion system F over S. Then Out_{F}(S) = 1 since Out(S) is a 2-group. Since A is characteristic in P_{1} and P_{3}, we must have P_{2} ∈ E_{F} since otherwise A ⊴ F. Also, P_{3} ∈ E_{F} since otherwise Z(S) ⊴ F by Proposition 1.15 (Z(S) ≤ Z(P_{2}) ≤ Z(P_{1}) = ⟨v_{1}v_{2}^{2}v_{3}^{−1}⟩ ≅ C_{4}, so Z(S) is characteristic in P_{1} and in P_{2}).

If S = S^{O’N}, then since P_{2} ∈ E_{F} and Out_{S}(P_{2}) ≅ C_{2}^{2}, Out_{F}(P_{2}) ≅ A_{5} × H for some H of odd order by Lemma A.6. In particular, there is ξ ∈ N_{Aut_{F} (P_{2})}(Aut_{S}(P_{2})) of order three. By the extension axiom, ξ extends to an element ξ̂ ∈ Aut_{F}(P_{1}) (recall P_{1} = N_{S}(P_{2})). Thus Aut_{F}(P_{1}) is not a 2-group, so also P_{1} ∈ E_{F} by Proposition 1.4.

If S = S^{HS}, set T = N_{S}(P_{2}) ⊴ S. Then (P_{3} ∩ T)/[P_{3}, P_{3}] = Ω_{1}(P_{3}/[P_{3}, P_{3}]), so P_{3} ∩ T is characteristic of index two in P_{3}. Thus [Aut(P_{3}), P_{3}] ≤ T and [Aut(P_{2}), P_{2}] ≤ T, so also P_{1} ∈ E_{F} by Lemma 1.16.

In both cases, set A_{0} = Ω_{1}(A) = ⟨v_{1}^{2}, v_{2}^{2}, v_{3}^{2}⟩. Since C_{S}(A_{0}) = A,

Aut_{S}(A_{0}) ≅ S/A ≅ D_{8} and Aut_{P_{i}}(A_{0}) ≅ P_{i}/A ≅ C_{2}^{2} for i = 1, 3 .

So for i = 1, 3, by Lemma A.2 (applied with G = P_{i} and H = A), restriction induces a homomorphism from Out_{F}(P_{i}) into the group N_{Aut(A_{0})}(Aut_{P_{i}}(A_{0}))/Aut_{P_{i}}(A_{0}) ≅ Σ_{3} with kernel a 2-group. This restriction homomorphism is an isomorphism since P_{i} ∈ E_{F}, so Aut_{F}(A_{0}) contains the normalizers in Aut(A_{0}) of Aut_{P_{1}}(A_{0}) and Aut_{P_{3}}(A_{0}); i.e., the two maximal parabolic subgroups in Aut(A_{0}). Thus Aut_{F}(A_{0}) = Aut(A_{0}) ≅ GL_{3}(2).

Now, O_{2}(Aut(A)) = {α ∈ Aut(A) | α|_{A_{0}} = Id} ≅ C_{2}^{9}, so Aut(A)/O_{2}(Aut(A)) ≅ Aut(A_{0}). Let G be the set of all subgroups Γ < Aut(A) such that Γ ≅ GL_{3}(2) and Aut_{S}(A) < Γ. Since Aut_{S}(A) ∩ O_{2}(Aut(A)) = 1, Aut_{F}(A) ∩ O_{2}(Aut(A)) = 1 by the Sylow axiom, and hence Aut_{F}(A) ∈ G by the extension axiom. By Proposition A.3, applied with G = Aut(A), Q = O_{2}(G), H = Aut_{F}(A), and H_{0} = Aut_{S}(A), the group C_{O_{2}(Aut(A))}(Aut_{S}(A)) = ⟨χ_{1}|_{A}, χ_{4}|_{A}⟩ ≅ C_{2}^{2} acts transitively on G via conjugation. For Γ ∈ G and χ ∈ O_{2}(Aut(A)), ^{χ}Γ = Γ if and only if [χ, Γ] ≤ Γ ∩ O_{2}(Aut(A)) = 1, in which case [χ, Aut(A)] = [χ, O_{2}(Aut(A))] = 1 since O_{2}(Aut(A)) is abelian. Thus χ_{4}|_{A} normalizes all elements in G, while χ_{1}|_{A} ∉ Z(Aut(A)) normalizes none of them. Hence |G| = 2, and its elements are exchanged by χ_{1}|_{A}.

Now fix some Γ ∈ G. We can assume, after replacing F by ^{χ_{1}}F if necessary, that Aut_{F}(A) = Γ. We claim that there is a unique possibility for Aut_{F}(P_{1}) whose elements restrict to elements of Γ, and exactly two such possibilities for Aut_{F}(P_{3}). To see this, note that for P = P_{1} or P_{3}, the image of Out_{F}(P) in N_{Aut(A)}(Aut_{P} (A))/Aut_{P} (A) is precisely N_{Γ} (Aut_{P} (A))/Aut_{P} (A) ≅ Σ_{3} by the extension axiom. When P = P_{1}, O_{3}(N_{Γ} (Aut_{P} (A))/Aut_{P} (A)) lifts to a cyclic subgroup of order 12 in Out(P_{1}), and hence there is only one choice for the subgroup of order 3 in Out_{F}(P_{1}). When P = P_{3}, it lifts to a subgroup isomorphic to C_{2} × A_{4}, so there are four subgroups of order 3, of which just two are normalized by Out_{S}(P_{3}). By direct computations, these two possibilities for Out_{F}(P_{3}) are exchanged by χ_{2}, and fixed by χ_{3} and χ_{4}.

If S = S^{O’N}, then Out(P_{2}) ≅ C_{2} ×Σ_{6}. We already saw that O^{2′}(Out_{F}(P_{2})) ≅ A_{5}, and hence Out_{F}(P_{2}) ≅ A_{5}. Also, Out_{F}(P_{2}) ≤ O^{2}(Out(P_{2})) ≅ A_{6}, A_{6} contains 12 subgroups isomorphic to A_{5} (six which act fixing a point and six which act transitively), and they are all conjugate in Aut(A_{6}) (see [Sz1, (3.2.19)]). Since each of those has five Sylow 2subgroups all lying in the same A_{6}-conjugacy class, and A_{6} has 30 subgroups isomorphic to C_{2}^{2} (15 in each of two classes), we see by counting that each C_{2}^{2} ≤ A_{6} is contained in exactly two subgroups isomorphic to A_{5}. Thus there are two possibilities for Out_{F}(P_{2}), and they are exchanged by χ_{3} and fixed by c_{s} ∘χ_{4}. (Recall that χ_{4} does not normalize P_{2}.) We now conclude that there is (up to isomorphism) at most one reduced fusion system F over S^{O’N}, and that Out(S, F) = ⟨[χ_{4}]⟩ ≅ C_{2}. By the uniqueness, F is isomorphic to the fusion system of O’Nan’s simple group, which is reduced by Proposition 1.18(d).

If S = S^{HS}, then Out(P_{2}) ≅ C_{2}^{2} × Σ_{4} contains exactly four subgroups of order three of which two are normalized by Out_{S}(P_{2}). Those two are exchanged by χ_{3} and fixed by χ_{4}. Thus there is, up to isomorphism, at most one reduced fusion system F over S^{HS}, and Out(S, F) = ⟨[χ_{4}]⟩ ≅ C_{2}. By the uniqueness, F isomorphic to the fusion system of the Higman-Sims simple group, which is reduced by Proposition 1.18(d).

In both cases, since P_{3} is the only F-essential subgroup with noncyclic center, Ker(μ_{G}) = 1 for G = O’N or HS by Proposition 1.14. When G = O’N, then by [O’N, Lemma 11.2], |Out(G)| ≤ 2. Following [O’N, p. 471], let L_{3}(7)^{∗} denote the extension of PSL_{3}(7) by its graph automorphism; G contains two conjugacy classes of subgroups isomorphic to L_{3}(7)^{∗} by [O’N, Lemma 10.6(iii)]. If μ_{G} ∘ κ_{G} is not injective, then there is α ∈ Aut(G)∖Inn(G) such that α|_{S} = Id, α exchanges the two G-conjugacy classes of subgroups isomorphic to L_{3}(7)^{∗} by [O’N, Lemma 11.1], and hence exchanges the two G-conjugacy classes of cyclic subgroups of order 16 ([O’N, Lemma 10.13]). (By [O’N, Lemma 4.3(vi)], G contains four classes of elements of order 16. Each element of order 16 in S has the form (sa)^{±1} for a ∈ A, |Aut_{S}(⟨sa⟩)| = 4 for each a, and thus there are only two classes of cyclic subgroups of order 16.) So N_{Aut(G)}(A) = N_{Inn(G)}(A)⟨α⟩ satisfies the hypotheses of [O’N, Lemma 11.3] (where V in [O’N] corresponds to A here), which is impossible by point (ii) in that lemma. Thus μ_{G} ∘ κ_{G} is injective. Since |Out(G)| = 2 (cf. [JW]), μ_{G} and κ_{G} are isomorphisms, and F = F_{S}(G) is tame.

Now assume that G = HS. Then |Out(G)| ≥ 2 by the construction of G in [HS], with equality by, e.g., [Ly2, pp. 10–11]. By [Fr, Table IIIb], there is no element α ∈ Aut(G) ∖ Inn(G) such that C_{G}(α) contains a Sylow 2-subgroup of G. Thus μ_{G} ∘ κ_{G} is injective, and hence (since μ_{G} is injective and |Out(G)| = |Out(S, F)|) μ_{G} and κ_{G} are both isomorphisms. So F = F_{S}(G) is tame. □

It remains only to consider 2-groups of type PSp_{6}(3).

Proposition 6.4. Let S be a group of order 512 of type PSp_{6}(3), and thus isomorphic

to ((Q_{8})^{3}/C_{2}) ⋊^{(12)} C_{2}. Then every saturated fusion system over S with O_{2}(F) = 1 is isomorphic to the fusion system of PSp_{6}(3).

Proof. We adopt the notation used in [OV, § 7]. We regard S as the group generated by elements a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}, τ, satisfying the following relations for i ≠ j in {1, 2, 3}:

z_{i} =^{def} [a_{i}, b_{i}] = a_{i}^{2} = b_{i}^{2},  z_{i}^{2} = 1 = z_{1}z_{2}z_{3},  [a_{i}, b_{j}] = 1 = [a_{i}, a_{j}] = [b_{i}, b_{j}];
τ^{2} = 1, τ a_{i}τ^{−1} = a_{σ(i)}, τ b_{i}τ^{−1} = b_{σ(i)} (where σ = (1 2) ∈ Σ_{3}).

Thus Q_{i} =^{def} ⟨a_{i}, b_{i}⟩ ≅ Q_{8} for i = 1, 2, 3, [Q_{i}, Q_{j}] = 1 for i ≠ j, and S contains the direct product of the subgroups Q_{i} modulo the product of their central involutions. Since

Sp_{2}(3) ≅ SL_{2}(3) ≅ Q_{8} ⋊ C_{3}, S can be identified with a subgroup of (Sp_{2}(3)^{3} ⋊^{(12)} C_{2})/C_{2} ≤ PSp_{6}(3) in the obvious way.

By computer computation, S has the following two potentially critical subgroups:

P_{1} = ⟨a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}⟩ ≅ (Q_{8})^{3}/C_{2}  (Z(P_{1}) = ⟨z_{1}, z_{2}⟩ ≅ C_{2}^{2})
P_{2} = ⟨a_{1}a_{2}, b_{1}b_{2}, a_{3}, b_{3}, z_{1}, τ ⟩ ≅ 2_{+}^{1+6}  (Z(P_{2}) = ⟨z_{3}⟩ ≅ C_{2})

both of which are characteristic in S. Thus S is the group denoted R_{2} in [OV, § 7], P_{1} is the subgroup R_{0}, and P_{2} is the subgroup Q.

The group Out(P_{1}) was described in [OV, Lemma 7.4(b,c)]. Each automorphism of P_{1} permutes the three subgroups Q_{i}Z(P_{1}). Let Aut^{0}(P_{1}) ⊴ Aut(P_{1}) be the subgroup of those elements which are the identity on Z(P_{1}); equivalently, which normalize each Q_{i}Z(P_{1}). Set Out^{0}(P_{1}) = Aut^{0}(P_{1})/Inn(P_{1}). Then

Out^{0}(P_{1}) ≅ (Σ_{4})^{3} and Out(P_{1}) ≅ Σ_{4} ≀ Σ_{3}.  (1)

The identification of Σ_{4} with the group of outer automorphisms of Q_{i}Z(P_{1}) ≅ Q_{8} × C_{2} which are the identity on Z(P_{1}) is induced by the action of this outer automorphism group on the set consisting of the four subgroups of Q_{i}Z(P_{1}) isomorphic to Q_{8}. Also, the homomorphism

Out(S) ⟶^{≅} N_{Out(P_{1})}(Out_{S}(P_{1}))/Out_{S}(P_{1}) ≅ Σ_{4} × Σ_{4}  (2)

induced by restriction is an isomorphism (by the proof of [OV, Lemma 7.4(e)]). Let η_{1}, η_{2}, η_{3} ∈ Aut^{0}(P_{1}) denote the following automorphisms of order 3:

η_{i}(a_{i}) = b_{i}, η_{i}(b_{i}) = a_{i}b_{i} (i = 1, 2, 3); and η_{i}(a_{j}) = a_{j}, η_{i}(b_{j}) = b_{j} for j ≠ i.

Also, let γ ∈ Aut(P_{1}) be the automorphism which permutes the subgroups Q_{i} cyclically:

γ(a_{i}) = a_{i+1} and γ(b_{i}) = b_{i+1} (with indices modulo 3).

Then ⟨[η_{1}], [η_{2}], [η_{3}]⟩ ≅ C_{3}^{3} is a Sylow 3-subgroup of Out^{0}(P_{1}), and ⟨[η_{1}], [η_{2}], [η_{3}], [γ]⟩ ≅ C_{3} ≀ C_{3} is a Sylow 3-subgroup of Out(P_{1}) where γη_{i}γ^{−1} = η_{i+1}. Finally, define η_{12}^{S}, η_{3}^{S} ∈ Aut(S) by setting

η_{12}^{S}|_{P_{1}} = η_{1}η_{2},  η_{3}^{S}|_{P_{1}} = η_{3},  and η_{12}^{S}(τ) = η_{3}^{S}(τ) = τ .

We next look at automorphisms of P_{2}. Set P_{2} = P_{2}/Z(P_{2}), and let x̄ ∈ P_{2} be the class of x ∈ P_{2}. Let 𝔮: P_{2} ⟶ 𝔽_{2} be the quadratic form where 𝔮(x̄) = 0 if x^{2} = 1 and 𝔮(x̄) = 1 if x^{2} = z_{3}. Then Out(P_{2}) ≅ SO(P_{2}, 𝔮). An explicit isomorphism

χ_{P_{2}} : Out(P_{2}) ⟶^{≅} Σ_{8}

was constructed in [OV, proof of Proposition 7.5], which sends the classes in Out_{S}(P_{2}) = ⟨[c_{a_{1}}], [c_{b_{1}}]⟩ ≅ C_{2}^{2} and the classes [η_{12}^{S}|_{P_{2}}] and [η_{3}^{S}|_{P_{2}}] to the permutations listed in the following table:

\begin{matrix}α & [c_{a_{1}}] & [c_{b_{1}}] & [η_{12}^{S}|_{P_{2}}] & [η_{3}^{S}|_{P_{2}}] \\ χ_{P_{2}}(α) & (1 2)(3 4) & (1 3)(2 4) & (2 3 4) & (5 7 6) \end{matrix}  (3)

Fix a saturated fusion system F over S such that O_{2}(F) = 1. By Lemma 1.17, E_{F} = {P_{1}, P_{2}}. By Lemma A.6, Out_{F}(P_{2}) = Δ × B for some Δ ≅ A_{5} and some B of odd order. By the above description of χ_{P_{2}}(Out_{S}(P_{2})), there is x ∈ {5, 6, 7, 8} such that χ_{P_{2}}(Δ) is the group of even permutations of the set {1, 2, 3, 4, x}. In particular, η_{12}^{S}|_{P_{2}} ∈ Aut_{F}(P_{2}). Also, B ≤ C_{Out(P_{2})}(Δ) ≅ Σ_{3}, and χ_{P_{2}}(B) is contained in the group of permutations of {5, 6, 7, 8}∖{x}.

Set M = ⟨η_{12}^{S}, η_{3}^{S}⟩ ≅ C_{3}^{2}. Since Out(S) ≅ Σ_{4} × Σ_{4} by (2), M ∈ Syl_{3}(Aut(S)). Hence there is φ ∈ Aut(S) such that ^{φ}Aut_{F}(S) ∩ M ∈ Syl_{3}(^{φ}Aut_{F}(S)). Upon replacing F by φF, we can assume that Aut_{F}(S) ∩ M ∈ Syl_{3}(Aut_{F}(S)). Equivalently, since Out_{F}(S) has odd order and hence is a 3-group, Aut_{F}(S) ≤ Inn(S)M.

Let M^{∗} ≤ N_{Aut(P_{2})}(Aut_{S}(P_{2})) be the image of M ≤ Aut(S) under restriction to P_{2}. By the extension axiom, this restriction sends Aut_{F}(S) onto N_{Aut_{F} (P_{2})}(Aut_{S}(P_{2})), and hence sends the Sylow 3-subgroup Aut_{F}(S) ∩ M of Aut_{F}(S) onto a Sylow 3-subgroup of the normalizer. Since this image is contained in the 3-subgroup M^{∗}, restriction sends Aut_{F}(S) ∩ M onto Aut_{F}(P_{2}) ∩ M^{∗}. So for each α ∈ M, α ∈ Aut_{F}(S) if and only if α|_{P_{2}} ∈ Aut_{F}(P_{2}) (since restriction is injective on M). Thus Out_{F}(S) = ⟨[η_{12}^{S}]⟩ or ⟨[η_{12}^{S}], [η_{3}^{S}]⟩.

Now, z_{3} ∉ Z(F) since O_{2}(F) = 1. Since ⟨z_{3}⟩ = Z(S) = Z(P_{2}) ≅ C_{2}, there is an element of Aut_{F}(P_{1}) which does not fix z_{3}. Also, c_{τ} ∈ Aut_{S}(P_{1}) exchanges z_{1} and z_{2}. Since Z(P_{1}) = ⟨z_{1}, z_{2}⟩ ≅ C_{2}^{2} and z_{3} = z_{1}z_{2}, this shows that Aut_{F}(Z(P_{1})) ≅ Σ_{3}, and that each element of Aut_{F}(Z(P_{1})) is the restriction of an element of Aut_{F}(P_{1}). Thus γ|_{Z(P_{1})} ∈ Aut_{F}(Z(P_{1})), and there is γ′ ∈ Aut_{F}(P_{1}) of odd order such that γ′|_{Z(P_{1})} = γ|_{Z(P_{1})}. In particular, γ′ ∉ Aut^{0}(P_{1}).

Let H_{0} < H_{i} < H ⊴ Aut(P_{1}) (i = 1, 2, 3) be the subgroups

H_{0} = O_{2}(Aut(P_{1})) ,  H_{i} = H_{0}⟨η_{i}⟩ ,  H = H_{1}H_{2}H_{3} ≤ Aut^{0}(P_{1}).

By (1), H/Inn(P_{1}) ≅ (A_{4})^{3}, and

H/H_{0} = O_{3}(Aut(P_{1})/H_{0}) = (H_{1}/H_{0}) × (H_{2}/H_{0}) × (H_{3}/H_{0}) ≅ (C_{3})^{3}.

Consider the homomorphism

Ψ: Aut(P_{1}) ⟶ Aut(H/H_{0}) ≅ GL_{3}(3)

induced by conjugation. Then Ker(Ψ) = H, and Im(Ψ) ≅ C_{2} ≀ Σ_{3} permutes the subgroups H_{i}/H_{0}. Also, conjugation by c_{τ} exchanges η_{1} and η_{2} and sends η_{3} to itself. Since Ψ(γ′) ≠ 1 has odd order and ⟨[c_{τ}]⟩ ∈ Syl_{2}(Out_{F}(P_{1})) by the Sylow axiom, Ψ(Aut_{F}(P_{1})) = ⟨Ψ(c_{τ}), Ψ(γ′)⟩ ≅ Σ_{3}, and Ψ(γ′) permutes cyclically the three classes η_{1}H_{0}, η_{2}H_{0}, and η_{3}^{ε}H_{0} for some ε = ±1.

Define β ∈ Aut(S) by setting

β|_{Q_{1}Q_{2}} = Id ,  β(τ) = τ ,  β(a_{3}) = a_{3}^{−1} ,  β(b_{3}) = a_{3}b_{3} .

Then conjugation by [β|_{P_{1}}] in Out(P_{1}) sends [η_{1}] and [η_{2}] to themselves and inverts [η_{3}]. So upon replacing F by ^{β}F if necessary, we can arrange that ε = 1 (i.e., Ψ(γ′) permutes cyclically the three classes η_{i}H_{0} ∈ H/H_{0}). Since β ∈ N_{Aut(S)}(Inn(S)M), we still have Out_{F}(S) = ⟨[η_{12}^{S}]⟩ or ⟨[η_{12}^{S}], [η_{3}^{S}]⟩. In particular, η_{1}η_{2} = η_{12}^{S}|_{P_{1}} ∈ Aut_{F}(P_{1}).

Set Aut_{F}^{∗}(P_{1}) = Aut_{F}(P_{1}) ∩ H for short. Since 1 ≠ η_{1}η_{2}H_{0} ∈ Aut_{F}^{∗}(P_{1})H_{0}/H_{0}, and each nontrivial subgroup of H/H_{0} normalized by Ψ(γ′) contains the class η_{1}η_{2}η_{3}H_{0} (since C_{H/H_{0}}(Ψ(γ′)) = ⟨η_{1}η_{2}η_{3}H_{0}⟩), there is η′_{3} ∈ Aut_{F}^{∗}(P_{1}) ∩ η_{3}H_{0}. Upon replacing η′_{3} by some appropriate power of η′_{3}, if necessary, we can assume |η′_{3}| = 3. Since γ′ normalizes Aut_{F}^{∗}(P_{1}), there are also elements η′_{i} ∈ Aut_{F}^{∗}(P_{1}) ∩ η_{i}H_{0} of order 3 for i = 1, 2. Thus Aut_{F}(P_{1}) ≥ Inn(P_{1})⟨η′_{1}, η′_{2}, η′_{3}, γ′, c_{τ}⟩, with equality since H_{0} ∩ Aut_{F}(P_{1}) = Inn(P_{1}).

Since c_{τ} normalizes Aut_{F}^{∗}(P_{1}), conjugation by [c_{τ}] exchanges the classes [η′_{1}] and [η′_{2}] in Out_{F}(P_{1}) and centralizes [η′_{3}]. In particular, η′_{1}η′_{2} and η′_{3} both normalize Aut_{S}(P_{1}), and hence by the extension axiom extend to elements of Aut_{F}(S). Thus Out_{F}(S) = ⟨[η_{12}^{S}], [η_{3}^{S}]⟩, and so η_{1}η_{2}, η_{3} ∈ Aut_{F}(P_{1}) by restriction. Since H_{0} ∩ Aut_{F}(P_{1}) = Inn(P_{1}), this proves that [η′_{1}η′_{2}] = [η_{1}η_{2}] and [η′_{3}] = [η_{3}] in Out^{0}(P_{1}) ≅ (Σ_{4})^{3}.

Recall that the [η_{i}] and [η′_{i}] (i = 1, 2, 3) all lie in H/Inn(P_{1}) ≅ (A_{4})^{3}. For each i, there are four subgroups of order 3 in H_{i}/Inn(P_{1}) ≅ A_{4} × C_{2}^{4}, and they generate a subgroup H_{i}^{∗} ≅ A_{4}. Thus [η_{i}], [η′_{i}] ∈ H_{i}^{∗} for i = 1, 2. Since H_{1}^{∗} ∩ H_{2}^{∗} = 1 (distinct factors isomorphic to A_{4} in H/Inn(P_{1})) and [η_{1}η_{2}] = [η′_{1}η′_{2}], we conclude that [η_{i}] = [η′_{i}] ∈ Out(P_{1}) for each i = 1, 2, 3. In particular, η_{1}, η_{2}, η_{3} ∈ Aut_{F}(P_{1}).

Set

K = ⟨[η_{1}], [η_{2}], [η_{3}]⟩ = Aut_{F}^{∗}(P_{1})/Inn(P_{1}) ≤ Out_{F}(P_{1}).

Both [γ] and [γ′] permute the elements [η_{i}] cyclically under conjugation (and γ|_{Z(P_{1})} = γ′|_{Z(P_{1})} by assumption). So [γ^{−1}γ′] ∈ C_{Out(P_{1})}(K), where C_{Out(P_{1})}(K) = K since C_{H_{i}^{∗}}([η_{i}]) = ⟨[η_{i}]⟩ for each i (H_{i}^{∗} ≅ A_{4}). Hence γ ∈ Aut_{F}(P_{1}), so

Aut_{F}(P_{1}) = Inn(P_{1})⟨η_{1}, η_{2}, η_{3}, γ, c_{τ}⟩, and Aut_{F}(S) = Inn(S)⟨η_{12}^{S}, η_{3}^{S}⟩.

Also, via the identification in (3), we see that Out_{F}(P_{2}) ≅ A_{5} ×C_{3} is the unique subgroup of this isomorphism type in Out(P_{2}) ≅ Σ_{8} which contains Out_{S}(P_{2}), [η_{12}^{S}|_{P_{2}}], and [η_{3}^{S}|_{P_{2}}].

This proves that each saturated fusion system F′ over S with O_{2}(F′) = 1 is isomorphic to F. In particular, if F′ is the fusion system of PSp_{6}(3), then O_{2}(F′) = 1 by Proposition 1.18(b), and so F ≅ F′. □

7. Fusion systems over a 2-group of type A_{12}

Throughout this section, we fix the following notation for certain elements of A_{12}:

a_{1} = (1 2)(3 4)  a_{2} = (5 6)(7 8)  a_{3} = (9 10)(11 12)
b_{1} = (1 3)(2 4)  b_{2} = (5 7)(6 8)  b_{3} = (9 11)(10 12)
μ_{12} = (1 2)(5 6)  μ_{23} = (5 6)(9 10)  τ = (1 5)(2 6)(3 7)(4 8)

and set S = ⟨a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}, μ_{12}, μ_{23}, τ⟩ ∈ Syl_{2}(A_{12}). We need to consider the following subgroups of S:

A = ⟨a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}⟩ ≅ C_{2}^{6}  Q = ⟨a_{1}, a_{2}, b_{1}b_{2}, μ_{12}, τ⟩ ≅ 2_{+}^{1+4}
N_{1} = A⟨μ_{12}, τ⟩ ≅ (C_{2}^{4} ⋊ C_{2}^{2}) × C_{2}^{2}  H_{1} = A⟨τ⟩
N_{2} = A⟨μ_{12}, μ_{23}⟩  H_{2} = A⟨μ_{23}⟩
N_{3} = Q⟨a_{3}, b_{3}, μ_{23}⟩  N_{13} = N_{1} ∩ N_{3} = Q × ⟨a_{3}, b_{3}⟩.

Note that A, Q, N_{1}, N_{2}, and N_{3} are all normal in S.

Lemma 7.1. Assume the above notation. Then A is the unique elementary abelian subgroup of rank 6 in S, and hence is characteristic in all subgroups which contain it. The subgroups N_{1}, N_{2}, N_{3}, and N_{13} are all characteristic in S, and N_{13} is characteristic in N_{1} and in N_{3}.

Proof. We first check the first statement. If P < S, P ≠ A, and P ≅ C_{2}^{6}, then P ∩ A ≤ C_{A}(P A/A), so rk(C_{A}(P A/A)) ≥ 6−rk(P A/A) (i.e., P A/A is a nontrivial offender in S/A on A). It is straightforward to check that S/A ≅ D_{8}, that rk(C_{A}(g)) = 4 for g ∈ S/A of order 2, and that rk(C_{A}(V)) = 3 if V < S/A and V ≅ C_{2}^{2}. So there is no such subgroup P, and A is the unique elementary abelian subgroup of rank 6.

If N < S is such that N ≥ A and N/A ≅ C_{2}^{2}, then N = N_{1} or N_{2}. Since N_{1} ≅ N̸_{2} ([N_{2}, N_{2}] = Z(N_{2}) while [N_{1}, N_{1}] ≠ Z(N_{1})), both subgroups are characteristic in S.

Set Z = ⟨a_{1}a_{2}⟩ = [N_{1}, N_{1}] ∩ Z(N_{1}): a subgroup characteristic in N_{1} and in S. Then N_{13}/Z ≅ C_{2}^{6}, S/N_{13} ≅ C_{2}^{2}, rk(C_{N_{13}/Z}(g)) ≤ 4 for each g ∈ S ∖ N_{13}, and rk(C_{N_{13}/Z}(S)) = 2. So N_{13}/Z is the unique elementary abelian subgroup of S/Z of rank 6, and N_{13} is characteristic in N_{1} and in S. Also, the three subgroups of S containing N_{13} with index 2 (N_{1}, N_{3}, and N_{13}⟨b_{1}μ_{23}⟩) are pairwise nonisomorphic — they have commutator subgroups ⟨a_{1}, a_{2}, b_{1}b_{2}⟩, ⟨a_{1}, a_{2}, a_{3}, μ_{12}⟩, and ⟨a_{1}, a_{2}, a_{3}, b_{1}b_{2}μ_{12}⟩, respectively — and hence all three are characteristic in S.

Since N_{13}/Z(N_{3}) ≅ C_{2}^{5} is the unique abelian subgroup of index two in N_{3}/Z(N_{3}), N_{13} is characteristic in N_{3}. □

Lemma 7.2. Assume the above notation.

(a) There is an automorphism ν_{3} ∈ Aut(N_{3}) of order 3 which takes values as follows:

\begin{matrix}g & a_{1} & a_{2} & a_{3} & b_{3} & b_{1}b_{2} & μ_{12} & μ_{23} & τ \\ ν_{3}(g) & a_{2}μ_{12} & a_{1}μ_{12} & a_{3} & b_{3} & a_{1}a_{2}b_{1}b_{2}τ & a_{2} & μ_{23} & b_{1}b_{2} \end{matrix}

The action of ν_{3} on

Q = ⟨a_{1}b_{1}b_{2}μ_{12}, a_{1}τ⟩ ×_{⟨a_{1}a_{2}⟩} ⟨a_{1}b_{1}b_{2}τ, b_{1}b_{2}μ_{12}⟩ ≅ Q_{8} ×_{C_{2}} Q_{8}

has order three on each factor. Also, ν_{3} commutes in Aut(Q) with c_{μ_{23}}, where c_{μ_{23}} exchanges the two quaternion factors.

(b) Aut(N_{3})/O_{2}(Aut(N_{3})) ≅ Σ_{3}, generated by the classes of ν_{3} and c_{b_{1}}.

(c) If α ∈ Aut(N_{3}) has order 3, then α|_{Z(N_{13})} = Id, and α acts on N_{13}/Z(N_{13}) ≅ Q/Z(Q) with C_{N_{13}/Z(N_{13})}(α) = 1. If in addition, α(Q) = Q, then ⟨^{φ}α⟩ = ⟨ν_{3}⟩ for some φ ∈ Aut_{Q}(N_{3}).

(d) Let Δ ≤ Aut(N_{3}) be such that Δ > Aut_{S}(N_{3}) and Δ/Inn(N_{3}) ≅ Σ_{3}. Then there is φ ∈ Aut(S) such that ^{φ}Δ = ⟨Aut_{S}(N_{3}), ν_{3}⟩.

Proof. (a) This is easily checked. Note that [ν_{3}|_{Q}, c_{μ_{23}}] = Id in Aut(Q) since ν_{3} is a homomorphism and ν_{3}(μ_{23}) = μ_{23}.

(b) Consider the chain of subgroups P_{0} < P_{1} < P_{2} < N_{3}, where

P_{0} = Fr(N_{3}) = ⟨a_{1}, a_{2}, a_{3}, μ_{12}⟩, P_{1} = P_{0}Z(N_{13}) = P_{0}⟨b_{3}⟩, P_{2} = N_{13} = P_{1}⟨b_{1}b_{2}, τ⟩.

Since N_{13} is characteristic in N_{3} by Lemma 7.1, each of the P_{i} is characteristic in N_{3}. So by Lemma A.1, the kernel of the induced homomorphism Aut(N_{3}) ⟶ Aut(P_{2}/P_{1}) ≅ Σ_{3} is contained in O_{2}(Aut(N_{3})). Also, the images of ν_{3} and c_{b_{1}} in Aut(P_{2}/P_{1}) generate this group, so Aut(N_{3})/O_{2}(Aut(N_{3})) ≅ Σ_{3}.

(c) Assume α ∈ Aut(N_{3}) has order 3. Since α normalizes the chain

[N_{13}, N_{13}]_{=⟨a_{1}a_{2}⟩} < Z(N_{3})_{=⟨a_{1}a_{2},a_{3}⟩} < Z(N_{13})_{=⟨a_{1}a_{2},a_{3},b_{3}⟩}

of characteristic subgroups, α|_{Z(N_{13})} = Id by Lemma A.1.

Consider the subgroups P_{1} = Z(N_{13})⟨a_{2}, μ_{12}⟩ and N_{13} = P_{1}⟨b_{1}b_{2}, τ⟩. Since

[μ_{23}, b_{1}b_{2}] = a_{2} and [μ_{23}, τ] = μ_{12} ,

[μ_{23}, −] sends N_{13}/P_{1} isomorphically to P_{1}/Z(N_{13}), and this isomorphism commutes with α since α(μ_{23}) ∈ μ_{23}N_{13}. So α induces an automorphism of order 3 on P_{1}/Z(N_{13}) since it induces an automorphism of order 3 on N_{13}/P_{1} by the proof of (b). Thus α acts on N_{13}/Z(N_{13}) ≅ Q/Z(Q) with C_{N_{13}/Z(N_{13})}(α) = 1.

Now assume in addition that α(Q) = Q. Then C_{Q}(α) = Z(Q), so α and ν_{3} both act nontrivially on each of the quaternion factors of Q. They also commute with c_{μ_{23}}|_{Q} modulo Aut_{N_{13}}(Q) = Inn(Q). Since c_{μ_{23}} exchanges the two quaternion factors of Q, this implies that α|_{Q} is congruent to ν_{3}|_{Q} or ν_{3}^{−1}|_{Q} modulo Inn(Q). Thus ⟨α|_{Q}⟩ and ⟨ν_{3}|_{Q}⟩ are Sylow 3-subgroups of Inn(Q)⟨α|_{Q}⟩, so upon replacing α by α^{−1} if necessary, we can arrange that ^{φ}α|_{Q} = ν_{3}|_{Q} for some φ ∈ Aut_{Q}(N_{3}). We already showed that α|_{Z(N_{13})} = Id, so ^{φ}α|_{N_{13}} = ν_{3}|_{N_{13}}.

Thus ^{φ}α = β ∘ ν_{3} for some β ∈ Aut(N_{3}) such that β|_{N_{13}} = Id. Then β(μ_{23}) = gμ_{23} for some g ∈ Z(N_{13}), and in particular, β has order at most 2 and commutes with ν_{3}. Thus β = Id since β ∘ ν_{3} has order 3, and hence ^{φ}α = ν_{3}.

(d) Set T_{3} = ⟨a_{3}, b_{3}⟩ ⊴ S; thus N_{13} = Q × T_{3}. For each χ ∈ Hom(Q, T_{3}), set Q_{χ} = {gχ(g) | g ∈ Q}. Each subgroup of N_{13} = Q × T_{3} isomorphic to Q is sent isomorphically to Q by projection to the first factor, and hence is equal to Q_{χ} for some unique χ.

Let Q ⊇ Q_{0} be the sets of subgroups of N_{13} isomorphic to Q which are normal in N_{3} or in S, respectively. For example, Q ∈ Q_{0}. Then

Q = {Q_{χ} | χ ∈ Hom_{Aut_{N_{3}}(N_{13})}(Q, T_{3})} and Q_{0} = {Q_{χ} | χ ∈ Hom_{Aut_{S}(N_{13})}(Q, T_{3})},

where Hom_{X}(Q, T_{3}) is the set of group homomorphisms which commute with the action of X.

If χ ∈ Hom(Q, T_{3}) is such that ν_{3}(Q_{χ}) = Q_{χ}, then χ(ν_{3}(g)) = χ(g) for each g ∈ Q (recall ν_{3}|_{T_{3}} = Id), and χ = 1 since Q = [ν_{3}, Q]. Thus Q is the only member of Q normalized by ν_{3}.

Fix Δ ≤ Aut(N_{3}) such that Δ > Aut_{S}(N_{3}) and Δ/Inn(N_{3}) ≅ Σ_{3}. Choose α ∈ Δ of order three. Since Aut(N_{3})/O_{2}(Aut(N_{3})) ≅ Σ_{3} by (b), there is ψ ∈ O_{2}(Aut(N_{3})) such that ⟨^{ψ}α⟩ = ⟨ν_{3}⟩. Let χ be such that Q_{χ} = ψ^{−1}(Q) ∈ Q. Since Q is the only member of Q normalized by ν_{3}, Q_{χ} is the only member normalized by α. Since c_{b_{1}} normalizes Inn(N_{3})⟨α⟩, ^{b_{1}}Q_{χ} = Q_{χ} by the uniqueness of Q_{χ}, so Q_{χ} ⊴ S, and Q_{χ} ∈ Q_{0}. In particular, χ is Aut_{S}(N_{13})-linear.

We claim that there is φ ∈ Aut(S) such that φ|_{A} = Id, φ(μ_{23}) = μ_{23}, φ(μ_{12}) = μ_{12}χ(μ_{12}), and φ(τ) = τχ(τ). Since χ is Aut_{S}(N_{13})-linear, Q∩A = [b_{1}, Q] ≤ Ker(χ). Thus φ(g) = gχ(g) for all g ∈ Q. Since T_{3} ≤ Z(N_{1}) (where N_{1} = AQ), φ|_{N_{1}} is a well defined automorphism. To see that φ is well defined, it remains to check that χ(^{μ_{23}}g) = ^{μ_{23}}χ(g) for all g ∈ Q, and this holds since χ is Aut_{S}(N_{13})-linear.

By construction, φ(Q_{χ}) = Q. So upon replacing Δ by ^{φ}Δ and α by ^{φ}α, we can arrange that α(Q) = Q. By (c), ⟨^{η}α⟩ = ⟨ν_{3}⟩ for some η ∈ Aut_{Q}(N_{3}). Thus α ≡ ν_{3}^{±1} (mod Inn(N_{3})), so Δ = ⟨Aut_{S}(N_{3}), α⟩ = ⟨Aut_{S}(N_{3}), ν_{3}⟩. □

Lemma 7.3. Assume the above notation. Set A_{0} = Z(N_{2}) = ⟨a_{1}, a_{2}, a_{3}⟩, and let

R_{0} : Aut(N_{2}) ⟶ Aut(A_{0})

be the homomorphism induced by restriction.

(a) There is an automorphism ν_{2} ∈ Aut(N_{2}) of order 3 which takes values as follows:

\begin{matrix}g & a_{1} & b_{1} & a_{2} & b_{2} & a_{3} & b_{3} & μ_{12} & μ_{23} \\ ν_{2}(g) & a_{2} & b_{2} & a_{3} & b_{3} & a_{1} & b_{1} & μ_{23} & μ_{12}μ_{23} \end{matrix}

(b) For each β ∈ Ker(R_{0}), β induces the identity on N_{2}/A_{0}. Furthermore, Ker(R_{0}) = O_{2}(Aut(N_{2})) is an elementary abelian 2-group, and Im(R_{0}) = ⟨ν_{2}|_{A_{0}}, c_{τ}|_{A_{0}}⟩ ≅ Σ_{3}.

(c) Let Δ ≤ Aut(N_{2}) be such that Δ > Aut_{S}(N_{2}) and Δ/Inn(N_{2}) ≅ Σ_{3}. Then there is ξ ∈ Aut(S) such that ^{ξ}Δ = ⟨Aut_{S}(N_{2}), ν_{2}⟩ and ξ(τ) = τ.

Proof. (a) This is easily checked.

(b) Set K = Ker(R_{0}) for short, and fix β ∈ K. Then β(A) = A since A is characteristic in N_{2} by Lemma 7.1. The commutators

[μ_{12}, A] = ⟨a_{1}, a_{2}⟩,  [μ_{23}, A] = ⟨a_{2}, a_{3}⟩,  and [μ_{12}μ_{23}, A] = ⟨a_{1}, a_{3}⟩

are all distinct, so for x, y ∈ N_{2} such that [x, A] = [y, A], we have x ≡ y (mod A). Since [β(x), A] = β([x, A]) = [x, A] for x ∈ N_{2}, this shows that β induces the identity on N_{2}/A. Thus β sends C_{A}(μ_{12}) = A_{0}⟨b_{3}⟩, C_{A}(μ_{23}) = A_{0}⟨b_{1}⟩, and C_{A}(μ_{12}μ_{23}) = A_{0}⟨b_{2}⟩ to themselves, and hence induces the identity on A/A_{0}.

Let h_{12}, h_{23} ∈ A be such that β(μ_{ij}) = μ_{ij}h_{ij}. Then h_{12} ∈ C_{A}(μ_{12}) = A_{0}⟨b_{3}⟩ since (μ_{12}h_{12})^{2} = 1, and h_{23} ∈ A_{0}⟨b_{1}⟩ and h_{12}h_{23} ∈ A_{0}⟨b_{2}⟩ by similar arguments. Hence h_{12}, h_{23} ∈ A_{0}, so β induces the identity on N_{2}/A_{0}, and β ∈ O_{2}(Aut(N_{2})) by Lemma A.1.

Thus K ≤ O_{2}(Aut(N_{2})). Each α ∈ Aut(N_{2}) permutes the subgroups [x, A] for x ∈ {μ_{12}, μ_{23}, μ_{12}μ_{23}}; and thus permutes their pairwise intersections ⟨a_{i}⟩ for i = 1, 2, 3. So Im(R_{0}) ≅ Σ_{3}, generated by ν_{2}|_{A_{0}} and c_{τ}|_{A_{0}}, and K = O_{2}(Aut(N_{2})). Also, K is elementary abelian, since each β ∈ K has the form β(g) = gχ(g) for some χ ∈ Hom(N_{2}, A_{0}) ≅ C_{2}^{15} (and the resulting bijection K ≅ Hom(N_{2}, A_{0}) is an isomorphism).

(c) Again set K = Ker(R_{0}) = O_{2}(Aut(N_{2})). Let Δ ≤ Aut(N_{2}) be a subgroup such that Δ > Aut_{S}(N_{2}) and Δ/Inn(N_{2}) ≅ Σ_{3}. Thus Δ ∩ K = Inn(N_{2}). By Proposition A.3 (applied with Out(N_{2}), K/Inn(N_{2}), ⟨Out_{S}(N_{2}), [ν_{2}]⟩, and Out_{S}(N_{2}) in the roles of G, Q, H, and H_{0}), there is ξ_{0} ∈ K such that [ξ_{0}, c_{τ}] ∈ Inn(N_{2}) and ^{ξ_{0}}Δ = ⟨ν_{2}, Aut_{S}(N_{2})⟩. Since K = C_{K}(ν_{2}) × [ν_{2}, K] (see [G, Theorem 5.2.3]) and both factors are normalized by c_{τ}, we can choose ξ_{0} ∈ [ν_{2}, K].

Now, [ν_{2}, Inn(N_{2})] has an 𝔽_{2}-basis {c_{b_{1}b_{3}}, c_{b_{2}b_{3}}, c_{μ_{12}μ_{23}}, c_{μ_{23}}} permuted freely by c_{τ}. Hence each element of [ν_{2}, K]/[ν_{2}, Inn(N_{2})] which is centralized by c_{τ} lifts to an element of [ν_{2}, K] which commutes with c_{τ} in Aut(N_{2}). In particular, there is ξ_{1} ≡ ξ_{0} (mod [ν_{2}, Inn(N_{2})]) such that ξ_{1} ∈ C_{K}(c_{τ}). Since ξ_{1} commutes with c_{τ} in Aut(N_{2}), it extends to ξ ∈ Aut(S) such that ξ(τ) = τ, and we still have ^{ξ}Δ = ⟨ν_{2}, Aut_{S}(N_{2})⟩. □

The following proposition is essentially a special case of the main theorem in Ron Solomon’s paper [So1], where he lists the finite simple groups (more generally, the “fusionsimple” groups) with Sylow 2-subgroups isomorphic to S. (See also [So2, Theorem 1.1].) But our method of proof, based on analysis of the possible essential subgroups, is somewhat different.

Proposition 7.4. Let S be a 2-group of type A_{12}. Then every reduced fusion system over S is isomorphic to the fusion system of one of the groups A_{12}, Sp_{6}(2), or Ω_{7}(3). The subgroups N_{1}, N_{2}, and N_{3} are essential in all three fusion systems, H_{1} is essential in the fusion system of A_{12}, and these are the only essential subgroups up to S-conjugacy.

Proof. By a computer search, the only potentially critical subgroups of S (up to conjugacy) are N_{1}, N_{2}, N_{3}, H_{1}, and H_{2}. Of these, the N_{i} are characteristic of index two in S (Lemma 7.1), while N_{S}(H_{i}) = N_{i} for i = 1, 2. Also, Aut(S) is a 2-group by Lemma A.1, applied to the sequence Fr(S) < N_{1} ∩ N_{2} ∩ N_{3} < N_{1} ∩ N_{2} < N_{1} < S of characteristic subgroups.

Assume F is a reduced fusion system over S. Since Out(S) is a 2-group, Out_{F}(S) = 1. By Lemma 1.16,

S = ⟨[Aut_{F}(P), P] | P ∈ E_{F}⟩.  (1)

Step 1: E_{F} ⫌ {N_{2}, N_{3}}. By Lemma 7.1, A is a characteristic subgroup of each Fessential subgroup except N_{3} (if it is essential). Hence N_{3} ∈ E_{F}, since otherwise A ⊴ F by Proposition 1.15.

Since N_{13} is characteristic of index 2 in N_{3} by Lemma 7.1, [Aut_{F}(N_{3}), N_{3}] ≤ N_{13} ≤ N_{1}. Since A is characteristic of index two in H_{2} by Lemma 7.1, [Aut_{F}(H_{2}), H_{2}] ≤ A ≤ N_{1}. So N_{2} ∈ E_{F}, since otherwise [Aut_{F}(P), P] ≤ N_{1} for all P ∈ E_{F}, contradicting (1).

Since Aut(N_{3})/O_{2}(Aut(N_{3})) ≅ Σ_{3} by Lemma 7.2(b), Out_{F}(N_{3}) ≅ Σ_{3}. By Lemma 7.2(d), there is φ ∈ Aut(S) such that ^{φ}Aut_{F}(N_{3}) = ⟨Aut_{S}(N_{3}), ν_{3}⟩. So upon replacing F by ^{φ}F, we can assume that Aut_{F}(N_{3}) = ⟨Aut_{S}(N_{3}), ν_{3}⟩.

Since Aut(N_{2})/O_{2}(Aut(N_{2})) ≅ Σ_{3} by Lemma 7.3(b), Out_{F}(N_{2}) ≅ Σ_{3}. By Lemma 7.3(c), there is ξ ∈ Aut(S) such that ξ(τ) = τ and ^{ξ}Aut_{F}(N_{2}) = ⟨ν_{2}, Aut_{S}(N_{2})⟩. Upon replacing F by ^{ξ}F, we can arrange that Aut_{F}(N_{2}) = ⟨ν_{2}, Aut_{S}(N_{2})⟩. Also, ξ sends Q = ⟨τ, [τ, S]⟩ to itself, so (^{ξ}ν_{3})(Q) = Q, and ^{ξ}ν_{3} ∈ Inn(N_{3})⟨ν_{3}⟩ by Lemma 7.2(c). So we still have Aut_{F}(N_{3}) = ⟨Aut_{S}(N_{3}), ν_{3}⟩.

By direct computations, [ν_{3}, N_{3}] = Q, and hence

[Aut_{F}(N_{2}), N_{2}][Aut_{F}(N_{3}), N_{3}] = ⟨a_{1}, a_{2}, a_{3}, b_{1}b_{2}, b_{2}b_{3}, μ_{12}, μ_{23}, τ⟩ < S .  (2)

So by (1), E_{F} ⫌ {N_{2}, N_{3}}.

Step 2: H_{2} ∉ E_{F}, and at least one of N_{1}, H_{1} lies in E_{F}. Since ν_{2} ∈ Aut_{F}(N_{2}), H_{2} is F-conjugate to ν_{2}^{−1}(H_{2}) = A⟨μ_{12}⟩. Since A⟨μ_{12}⟩ is normal in S, H_{2} is not fully normalized in F, and hence is not F-essential. Thus at least one of the subgroups N_{1} or H_{1} is F-essential.

Step 3: Aut_{F}(N_{1}) normalizes Q. Set T_{3} = ⟨a_{3}, b_{3}⟩, so that N_{13} = N_{1} ∩N_{3} = Q×T_{3}. Recall that N_{13} is characteristic in N_{1} by Lemma 7.1.

By [O1, Proposition 3.2(b,c)], Out(N_{13})/O_{2}(Out(N_{13})) ≅ Out(Q) × Aut(T_{3}) ≅ (Σ_{3} ≀ C_{2}) × Σ_{3}. By direct computation, Out_{S}(N_{13}) ≅ C_{2}^{2} embeds into this quotient group: the generators [c_{b_{1}}] and [c_{μ_{23}}] are both nontrivial in Aut(Q/Z(Q)), and only the second acts nontrivially on T_{3}. Since N_{13} ⊴ S, Out_{S}(N_{13}) ∈ Syl_{2}(Out_{F}(N_{13})), so Out_{F}(N_{13}) ∩ O_{2}(Out(N_{13})) = 1. Hence Out_{F}(N_{13}) also embeds into (Σ_{3} ≀ C_{2}) × Σ_{3}.

Thus any two elements of odd order in Out_{F}(N_{13}) commute. So if α ∈ Aut_{F}(N_{1}) has odd order, then [α|_{N_{13}}, ν_{3}|_{N_{13}}] ∈ Inn(N_{13}). Since [N_{13}, N_{13}] ≤ Q, this shows that α sends Q = [ν_{3}, N_{13}] to itself. So O^{2}(Aut_{F}(N_{1})) normalizes Q, Aut_{S}(N_{1}) ∈ Syl_{2}(Aut_{F}(N_{1})) normalizes Q since Q ⊴ S, and hence Aut_{F}(N_{1}) normalizes Q.

We claim that in fact,

Aut_{F}(N_{1}) = {α ∈ Aut(N_{1}) | α|_{A} ∈ Aut_{F}(A), α(Q) = Q}.  (3)

Since A is characteristic in N_{1} by Lemma 7.1, Aut_{F}(N_{1}) normalizes A as well as Q, and hence Aut_{F}(N_{1}) is contained in the right hand side of (3).

If α ∈ Aut(N_{1}) is such that α|_{A} ∈ Aut_{F}(A) and α(Q) = Q, then α|_{A} normalizes Aut_{N_{1}}(A). So by the extension axiom (and since C_{S}(A) = A ≤ N_{1}), there is β ∈ Aut_{F}(N_{1}) such that β|_{A} = α|_{A}. Set P = ⟨a_{1}, b_{1}, a_{2}, b_{2}, τ, μ_{12}⟩ = Q⟨b_{1}⟩. Then α^{−1}β induces the identity on A, hence on P ∩ A, and it sends P to P since it sends Q to Q. So by Lemma A.2 (and since P ∩ A is centric in P and Aut_{P} (P ∩ A) = ⟨c_{τ}, c_{μ_{12}}⟩ permutes freely the basis {b_{1}, a_{1}b_{1}, b_{2}, a_{2}b_{2}} of P ∩ A), we have (α^{−1}β)|_{P} ∈ Inn(P). Let x ∈ P be such that (α^{−1}β)|_{P} = c_{x}|_{P}; then x ∈ C_{P} (P ∩ A) = P ∩ A since (α^{−1}β)|_{A} = Id. Hence (α^{−1}β)|_{A} = Id = c_{x}|_{A}, and since N_{1} = AP, α^{−1}β = c_{x} ∈ Inn(N_{1}). Thus α ∈ Aut_{F}(N_{1}), and this finishes the proof of (3).

Step 4: The group Aut_{F}(A). Set

Γ = Aut_{F}(A) and Γ_{0} = ⟨Aut_{S}(A), ν_{2}|_{A}⟩ ≅ Σ_{4} .  (4)

Thus O_{2}(Γ_{0}) = ⟨c_{μ_{12}}, c_{μ_{23}}⟩, and Aut_{S}(A) ≅ D_{8} is a Sylow 2-subgroup of Γ. We will show that

Γ ≅ A_{7}, GL_{3}(2), or C_{3}^{3} ⋊ Σ_{4} .  (5)

By the Gorenstein-Walter theorem on groups with dihedral Sylow 2-subgroups [GW, Theorem 1], Γ/O_{2′}(Γ) is isomorphic to a subgroup of PΓ L_{2}(q) which contains PSL_{2}(q) (q odd), to A_{7}, or to D_{8}. In the first case, |PSL_{2}(q)|||Γ |||GL_{6}(2)| and 16 ∤ |PSL_{2}(q)| imply that q ≤ 9. The third case is impossible, since Γ_{0}/O_{2′}(Γ_{0}) ≅ Σ_{4} is not a subquotient of D_{8}. Hence Γ/O_{2′}(Γ) is isomorphic to one of the groups PGL_{2}(3) ≅ Σ_{4}, PGL_{2}(5) ≅ Σ_{5}, PSL_{2}(7) ≅ GL_{3}(2), PSL_{2}(9) ≅ A_{6}, or A_{7}.

Assume O_{2′}(Γ) ≠ 1. By Lemma A.5, for some odd prime p, there is a normal elementary abelian p-subgroup 1 ≠ K ⊴ Γ, and N_{Aut(A)}(K) ≥ Γ ≥ Γ_{0} ≅ Σ_{4}. Also, O_{2}(Γ_{0}) cannot centralize K, since with respect to the basis {a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}} of A,

C_{Aut(A)}(⟨c_{μ_{12}}, c_{μ_{23}}⟩) = \{( \begin{matrix}I & X \\ 0 & I \end{matrix}) | X ∈ M_{3}(𝔽_{2})\} ≅ C_{2}^{9}.  (6)

Thus O^{2}(N_{Aut(A)}(K)) ≥ O^{2}(Γ_{0}) ≥ O_{2}(Γ_{0}) contains involutions which do not centralize K, and from the list of normalizers in Lemma A.5, this is possible only if K ≅ C_{3}^{3} and Γ ≤ N_{Aut(A)}(K) ≅ Σ_{3} ≀ Σ_{3}. Since [Γ:Γ_{0}] is odd, Γ = KΓ_{0} ≅ C_{3}^{3} ⋊ Σ_{4}.

Now assume O_{2′}(Γ) = 1. We already showed that Γ ≅ Σ_{4}, Σ_{5}, A_{6}, A_{7}, or GL_{3}(2), and it remains to eliminate the first three possibilities. Since A is characteristic in all subgroups of S which contain it (Lemma 7.1), C_{P} (Aut_{F}(P)) ≥ C_{A}(Γ) for P = S and for P ∈ E_{F} ∖ {N_{3}} ⊆ {N_{1}, N_{2}, H_{1},^{μ_{23}}H_{1}}. Since C_{N_{3}}(Aut_{F}(N_{3})) ≥ ⟨a_{1}a_{2}a_{3}⟩ = C_{A}(Γ_{0}) ≥ C_{A}(Γ) by a direct check, Proposition 1.15 implies that C_{A}(Γ) ⊴ F, and hence that

C_{A}(Γ) ≤ O_{2}(F) = 1 .  (7)

In particular, Γ > Γ_{0}, and Γ ≅ Σ̸_{4}.

We claim that

Γ_{0} < Γ_{1} ≤ Γ, Γ_{1} ≅ Σ_{5} or A_{6} ⟹ C_{A}(Γ_{1}) = ⟨a_{1}a_{2}a_{3}⟩ .  (8)

Fix such Γ_{1}, and assume there is H < Γ_{1} such that H ≅ A_{5}, Γ_{1} = ⟨Γ_{0}, H⟩, and Γ_{0} ∩ H ≅ A_{4}. Under this assumption, since C_{A}(⟨μ_{12}, μ_{23}⟩) = ⟨a_{1}, a_{2}, a_{3}⟩ = [⟨μ_{12}, μ_{23}⟩, A] (and since rk([x, A]) = rk([μ_{12}, A]) = 2 for each involution x ∈ H), Lemma A.7(b) implies that C_{A}(H) = C_{A}(H ∩ Γ_{0}) = C_{A}(Γ_{0}) = ⟨a_{1}a_{2}a_{3}⟩, and hence that (8) holds.

We now check that such an H exists under the hypotheses of (8). If Γ_{1} ≅ Σ_{5}, then H = O^{2}(Γ_{1}) satisfies the above conditions. If Γ_{1} ≅ A_{6}, then there is at least one pair of subgroups Γ_{0}^{∗}, H^{∗} < Γ such that Γ_{0}^{∗} ≅ Σ_{4}, H^{∗} ≅ A_{5}, Γ_{1} = ⟨Γ_{0}^{∗}, H^{∗}⟩, and Γ_{0}^{∗} ∩ H^{∗} ≅ A_{4}; and we can take Γ_{0}^{∗} = Γ_{0} since the subgroups of A_{6} isomorphic to Σ_{4} are permuted transitively by Aut(A_{6}). (This last claim holds by [Sz1, (3.2.20)], and since the two A_{6}- conjugacy classes of subgroups isomorphic to Σ_{4} contain elements of order 3 in different classes.) This finishes the proof of (8). By (7) and (8), Γ ≅ Σ̸_{5} and Γ ≅ A̸_{6}, so (5) holds.

Step 5: Three distinct fusion systems over S. Set â = a_{1}a_{2}a_{3} for short. We will need to refer to automorphisms σ_{1}, σ_{2} ∈ Aut(S), defined by setting (for i = 1, 2, 3):

σ_{1}(a_{i}) = a_{i}  σ_{1}(b_{i}) = a_{i}b_{i}  σ_{1}(μ_{12}) = μ_{12}  σ_{1}(μ_{23}) = μ_{23}  σ_{1}(τ) = τ
σ_{2}(a_{i}) = a_{i}  σ_{2}(b_{i}) = âb_{i}  σ_{2}(μ_{12}) = μ_{12}  σ_{2}(μ_{23}) = μ_{23}  σ_{2}(τ) = τ.

For each j = 1, 2, σ_{j}|_{N_{2}} commutes with ν_{2}, and σ_{j}|_{N_{3}} commutes with ν_{3} modulo Inn(N_{3}). Commutativity with ν_{2} is easily checked. Commutativity with ν_{3} follows from Lemma 7.2(c), applied with ^{σ_{j}}ν_{3} in the role of α. Hence we can replace F by ^{σ_{1}}F or ^{σ_{2}}F without changing Aut_{F}(N_{2}) or Aut_{F}(N_{3}) (or Γ_{0}).

Consider the basis B = {b_{1}, a_{1}b_{1}, b_{2}, a_{2}b_{2}, b_{3}, a_{3}b_{3}} for A, and set B̂ = B ∪ {â}. Each element of Γ_{0} < Aut(A) permutes the elements of B and sends â to itself.

We claim that

H_{1} fully normalized in F ⟹ Out_{F}(H_{1}) ≅ C_{Γ} (c_{τ})/⟨c_{τ}⟩.  (9)

By Lemma A.2, restriction induces a homomorphism R: Out_{F}(H_{1}) ⟶ C_{Γ} (c_{τ})/⟨c_{τ}⟩, and Ker(R) is a 2-group. Also, R is surjective by the extension axiom. If H_{1} is fully normalized, then Out_{S}(H_{1}) = ⟨[c_{μ_{12}}]⟩ ∈ Syl_{2}(Out_{F}(H_{1})), so R is injective.

Assume Γ ≅ A_{7}. Then Γ contains two conjugacy classes of subgroups isomorphic to C_{2}^{2}, represented by the two subgroups of this type in any Sylow 2-subgroup. Of these, each subgroup in one of the classes contains a subgroup of order three in its centralizer, while those in the other class have normalizer isomorphic to Σ_{4}. By (6), O_{2}(Γ_{0}) = ⟨c_{μ_{12}}, c_{μ_{23}}⟩ must be of the latter type, so Γ_{0} = N_{Γ} (⟨c_{μ_{12}}, c_{μ_{23}}⟩), and there is a unique subgroup Γ_{1} < Γ such that Γ_{0} < Γ_{1} ≅ A_{6}. By (8), â ∈ C_{A}(Γ_{1}), and hence lies in a Γ -orbit X of length |Γ/Γ_{1}| = 7. By the above remarks about subgroups of A_{7}, Γ_{0} acts on X with two orbits: of lengths 1 and 6. By a direct check, the action of Γ_{0} on A has exactly two orbits of length 6: B and σ_{2}(B) (the orbits of b_{1} and of âb_{1}). So after replacing F by ^{σ_{2}}F if necessary, we can assume that X = B̂, and hence that Γ is the group of even permutations of B̂. Since Γ determines Aut_{F}(N_{1}) by (3), and since H_{1} is not fully normalized (hence not F-essential) by (9) (C_{Γ} (c_{τ})/⟨c_{τ}⟩ ≅ C_{2} × Σ_{3}), Γ determines F. More precisely, there is at most one reduced fusion system F over S for which Γ = Aut_{F}(A), Aut_{F}(N_{2}), and Aut_{F}(N_{3}) are as just described; and any other reduced fusion system over S with Aut_{F}(A) ≅ A_{7} is isomorphic to it. Also, by Step 2, N_{1} is F-essential in this case since H_{1} is not.

Assume Γ ≅ GL_{3}(2). Since â = a_{1}a_{2}a_{3} is fixed by Γ_{0} but not by Γ, it must lie in an orbit X of length 7 = |Γ/Γ_{0}|. We claim that any transitive action of GL_{3}(2) on a set X of 7 elements (with isotropy subgroups isomorphic to Σ_{4}) is the group of all automorphisms of (X ∪{0}, +) under some group structure such that (X ∪{0}, +) ≅ 𝔽_{2}^{3}. To see this, note that the isotropy subgroup must be the normalizer of some P ≅ C_{2}^{2} (since it is maximal), and hence the stabilizer subgroup of a 1- or 2-dimensional subspace in 𝔽_{2}^{3} (since P is conjugate to one of two subgroups in any given Sylow 2-subgroup). Hence the elements of X are in natural correspondence with the elements of 𝔽_{2}^{3} ∖ {0} or (𝔽_{2}^{3})^{∗} ∖ {0}.

Since the six elements of X ∖{â} are permuted transitively by Γ_{0}, we can assume (after replacing F by ^{σ_{2}}F if necessary) that they form the basis B, and hence that Γ permutes the set B̂. Thus Γ is the group of all automorphisms of (B̂ ∪ {0}, +) for some group structure such that (B̂ ∪ {0}, +) ≅ (𝔽_{2})^{3}. Since the fixed set of an automorphism is a subgroup, we have â + b_{i} = a_{i}b_{i} for each i = 1, 2, 3 (corresponding to automorphisms in ⟨c_{μ_{12}}|_{A}, c_{μ_{23}}|_{A}⟩). Hence there are exactly two such structures on which Γ_{0} acts via automorphisms: one in which b_{1} + b_{2} = b_{3}, and the other in which b_{1} + b_{2} = a_{3}b_{3}. So we can assume, after replacing F by ^{σ_{1}}F if necessary, that Γ is the group of automorphisms of (B̂ ∪ {0}, +) where b_{1} + b_{2} = b_{3}. Again in this case, Γ determines Aut_{F}(N_{1}) by (3), H_{1} is not fully normalized (hence not F-essential) by (9) (C_{Γ} (c_{τ})/⟨c_{τ}⟩ ≅ C_{2}^{2}), and thus Γ determines F. Also, by Step 2, N_{1} is F-essential since H_{1} is not.

If Γ ≅ C_{3}^{3} ⋊ Σ_{4}, then by Lemma A.5 (applied with K ≅ C_{3}^{3}), there is a decomposition A = A_{1} × A_{2} × A_{3} (rk(A_{i}) = 2 for each i) such that each element of Γ permutes the subgroups A_{1}, A_{2}, A_{3}. The subgroup O_{2}(Γ_{0}) = ⟨c_{μ_{12}}, c_{μ_{23}}⟩ < Aut_{S}(A) normalizes each A_{i} and acts on it nontrivially, so there is a unique basis B_{i} of A_{i} which is permuted by O_{2}(Γ_{0}). Then B_{1}∪B_{2}∪B_{3} is a basis of A which is permuted transitively by Γ_{0}. After replacing F by σ_{2}F if necessary, we can assume that B_{1}∪B_{2}∪B_{3} = B, and hence (possibly after permuting the indices) that B_{i} = {b_{i}, a_{i}b_{i}} for each i. Thus Γ = ⟨Aut_{S}(A), ν_{2}|_{A}, α_{1}, α_{2}, α_{3}⟩, where the α_{i} are defined by setting

α_{i}(a_{i}) = b_{i}, α_{i}(b_{i}) = a_{i}b_{i},  α_{i}(a_{j}) = a_{j}, α_{i}(b_{j}) = b_{j}  (for j ≠ i.)

By (3), Aut_{F}(N_{1}) is uniquely determined by this choice of Γ = Aut_{F}(A).

Since C_{O_{3}(Γ)}(c_{τ}) = ⟨α_{1}α_{2}, α_{3}⟩ is not isomorphic to C_{O_{3}(Γ)}(c_{μ_{12}}) = ⟨α_{3}⟩, c_{τ} and c_{μ_{12}} are not conjugate in Γ = Aut_{F}(A), and hence H_{1} = A⟨τ⟩ is not F-conjugate to A⟨μ_{12}⟩. So H_{1} is fully normalized in F. By (9), Out_{F}(H_{1}) ≅ C_{Γ} (c_{τ})/⟨c_{τ}⟩ ≅ C_{3} × Σ_{3}, where the quotient is generated by the classes of α_{3}, α_{12} =^{def} α_{1}α_{2}, and c_{μ_{12}}. Thus H_{1} ∈ E_{F} and

Aut_{F}(H_{1}) = ⟨Aut_{S}(H_{1}), ν_{1}|_{H_{1}}, η⟩

for some ν_{1} ∈ Aut_{F}(N_{1}) and η ∈ Aut_{F}(H_{1}) of order three such that ν_{1}|_{A} = α_{3} and η|_{A} = α_{12}. (Note that for P = N_{1} or H_{1}, Ker[Aut_{F}(P) ⟶ Aut_{F}(A)] is a 2-group, by Lemma A.2 and since A is centric in P.) Also, [ν_{1}, c_{μ_{23}}] ≠ 1 in Out_{F}(N_{1}) (since they do not commute after restriction to Z(N_{1}) = ⟨a_{1}a_{2}, a_{3}, b_{3}⟩), so [c_{μ_{23}}] ∉ Z(Out_{F}(N_{1})), and N_{1} ∈ E_{F}.

By Step 3, ν_{1}(Q) = Q, so ν_{1} induces the identity on A∩Q and (since C_{Q}(A∩Q) = A∩Q) on Q/(A∩Q) ≅ C_{2}^{2}. Hence ν_{1}|_{Q} = Id_{Q} by Lemma A.1 and since |ν_{1}| = 3. Also, η(τ) = τg for some g ∈ A, g·α_{12}(g)·α_{12}^{2}(g) = 1 since |η| = 3, g ∈ C_{A}(τ) since |η(τ)| = 2, and thus g ∈ ⟨a_{1}a_{2}, b_{1}b_{2}⟩. So after replacing η by c_{a_{1}^{i}b_{1}^{j}} ∘ η for appropriate i, j, we can assume that η(τ) = τ. Thus Aut_{F}(H_{1}), and hence F, is uniquely determined in this case.

Step 6: Identifying the fusion system F. We have now shown that F is isomorphic to one of three fusion systems, F_{1}, F_{2}, or F_{3}, where Aut_{F_{1}}(A) ≅ A_{7}, Aut_{F_{2}}(A) ≅ GL_{3}(2), and Aut_{F_{3}}(A) ≅ C_{3}^{3} ⋊ Σ_{4}.

When G = Ω_{7}(3), the group of all elements of G which act up to sign on an orthonormal basis of 𝔽_{3}^{7} is isomorphic to C_{2}^{6}⋊A_{7} and has odd index in G. This follows from the formula for |G| (cf. [Ta, p. 166]). Since S splits over A and the two A_{7}-actions are isomorphic (see Step 5), S is isomorphic to any S_{1} ∈ Syl_{2}(G). Since Out(S_{1}) is a 2-group, F_{S_{1}}(G) is reduced by Proposition 1.18(c). Hence Aut_{G}(A) ≅ A_{7} (it can’t be any bigger by Step 4), and so F_{S_{1}}(G) ≅ F_{1} in this case.

The group G = Sp_{6}(2) has a maximal parabolic subgroup H ≅ M_{3}^{s}(2) ⋊ GL_{3}(2) (the stabilizer of a maximal isotropic subspace), where M_{3}^{s}(2) ≅ C_{2}^{6} is the group of symmetric 3 × 3 matrices over 𝔽_{2}, and A ∈ GL_{3}(2) acts on it via X ↦ AXA^{t}. By the order formulas for symplectic groups [Ta, p. 70], H has odd index in G. Also, the GL_{3}(2)-orbit of ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) has order 7 and stabilizer subgroup isomorphic to Σ_{4}, and the other six elements form a basis for M_{3}^{s}(2). So this is the same action as that described in Step 5, S_{2} ≅ S for S_{2} ∈ Syl_{2}(G), and F_{S_{2}}(G) ≅ F_{2} by Step 5 (and since F_{S_{2}}(G) is reduced by Proposition 1.18(c)).

Finally, when G = A_{12} and A < S are as defined here, then S ∈ Syl_{2}(G) by definition, F_{S}(G) is reduced by Proposition 1.18(c), Aut_{Σ_{12}}(A) ≅ Σ_{3} ≀ Σ_{3} and hence Aut_{G}(A) ≅ C_{3}^{3} ⋊ Σ_{4}, and so F_{S}(G) ≅ F_{3}.

This finishes the proof that every reduced fusion system over S is isomorphic to the 2-fusion system of Ω_{7}(3), Sp_{6}(2), or A_{12}. □

Appendix A. Background results on groups and representations

We collect here some background results on groups and their representations which are needed elsewhere in the paper. The first ones involve automorphisms and extensions of p-groups.

Lemma A.1. Fix a prime p, a finite p-group P, a subgroup P_{0} ≤ Fr(P), and a sequence of subgroups

P_{0} < P_{1} < ⋯ < P_{k} = P

all normal in P. Set

A = {α ∈ Aut(P) | x^{−1}α(x) ∈ P_{i−1}, all x ∈ P_{i}, all i = 1, …, k} ≤ Aut(P) :

the group of automorphisms which leave each P_{i} invariant, and which induce the identity on each quotient group P_{i}/P_{i−1}. Then A is a p-group. If the P_{i} are all characteristic in P, then A ⊴ Aut(P), and hence A ≤ O_{p}(Aut(P)).

Proof. See, for example, [G, Theorems 5.1.4 & 5.3.2]. □

When H ⊴ G, we let Aut(G, H) be the group of automorphisms of G which normalize H, and set Out(G, H) = Aut(G, H)/Inn(G).

Lemma A.2. Fix a group G and a normal subgroup H ⊴ G such that C_{G}(H) ≤ H (i.e., H is centric in G). Then there is an exact sequence

1 ⟶ H^{1}(G/H; Z(H)) ⟶^{η} Out(G, H) ⟶^{R}

N_{Out(H)}(Out_{G}(H))/Out_{G}(H) ⟶^{χ} H^{2}(G/H; Z(H)), (1)

where R is induced by restriction, and where all maps except (possibly) χ are homomorphisms. If H is abelian and the extension of H by G/H is split, then R is onto. If Z(H) has exponent p for some prime p, and has a basis which is permuted freely by G/H under conjugation, then R is an isomorphism.

Proof. See [OV, Lemma 1.2 & Corollary 1.3]. □

Proposition A.3 ([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′ = gHg^{−1}.

The next lemma involves subgroups of GL_{5}(2).

Lemma A.4. If G acts linearly, faithfully, and irreducibly on 𝔽_{2}^{5}, then either G has odd order, or G ≅ GL_{5}(2).

Proof. See [Wg, Theorem 1.1(ii)]. Wagner’s theorem deals more generally with irreducible subgroups of PSL_{5}(2^{a}) for arbitrary a ≥ 1, but only case (ii) in the theorem applies when a = 1. □

The next three lemmas involve representations over 𝔽_{2}.

Lemma A.5. Set G = GL_{6}(2), and assume H ≤ G is a subgroup such that O_{2′}(H) ≠ 1. Then for some odd prime p, H contains a normal elementary abelian p-subgroup 1 ≠ K ⊴ H, and hence H is contained up to conjugation in one of the following normalizers:

N_{G}(C_{3}) :  Σ_{3} × GL_{4}(2), (GL_{2}(4) ⋊ C_{2}) × Σ_{3}, GL_{3}(4) ⋊ C_{2}
N_{G}(C_{3}^{2}) :  (Σ_{3} ≀ C_{2}) × Σ_{3}, (GL_{2}(4) ⋊ C_{2}) × Σ_{3}, C_{3}^{3} ⋊ (C_{2} × Σ_{3})
N_{G}(C_{3}^{3}) :  Σ_{3} ≀ Σ_{3}
N_{G}(C_{5}) :  (C_{15} ⋊ C_{4}) × Σ_{3}
N_{G}(C_{7}) :  (C_{7} ⋊ C_{3}) × GL_{3}(2), C_{7}^{2} ⋊ C_{6}, GL_{2}(8) ⋊ C_{3}
N_{G}(C_{7}^{2}) :  (C_{7} ⋊ C_{3}) ≀ C_{2}
N_{G}(C_{31}) :  C_{31} ⋊ C_{5} .

When K ≅ C_{3}^{3} acts on V ≅ 𝔽_{2}^{6}, there is a decomposition V = V_{1} × V_{2} × V_{3} such that each element of N_{G}(K) ≅ Σ_{3} ≀ Σ_{3} permutes the subgroups V_{i} ≅ 𝔽_{2}^{2}.

Proof. Since O_{2′}(H) ≠ 1 is solvable by the odd order theorem [FT], it contains an elementary abelian p-subgroup K which is characteristic (for some odd p). Since |G| = 2^{15}·3^{4}·5·7^{2}·31, K must be isomorphic to one of the groups listed above.

The list of normalizers now follows from elementary representation theoretic considerations. When K ∈ Syl_{p}(G), then there is only one possibility for K up to conjugacy, and its normalizer is as described above. For example, when p = 5, 31, the irreducible 𝔽_{2}[C_{p}]- modules have dimensions 4 and 5, respectively (the smallest n such that p|(2^{n} − 1)), and N_{G}(C_{p}) ≅ ((𝔽_{2^{n}})^{×} ⋊ C_{n}) × GL_{6−n}(2).

In all other cases, N_{G}(K) depends on the description of 𝔽_{2}^{6} as an 𝔽_{2}[K]-module. For example, when K ≅ C_{3}^{2}, the module could be a sum of two irreducible modules of dimension two (and a 2-dimensional module with trivial action), or three irreducibles of dimension two of which two are isomorphic, or three pairwise nonisomorphic irreducible modules of dimension 2. The three possible normalizers listed above correspond to these three choices. □

Lemma A.6. Let P be a finite 2-group such that rk(P/Fr(P)) ≤ 6. Assume G ≤ Out(P) has a strongly 2-embedded subgroup, where S ∈ Syl_{2}(G) and S ≅ C_{2}^{2}. Then G ≅ A_{5} × C_{3}^{s} for some s ≤ 2.

Proof. Set H = O_{2′}(G). Since G has a strongly 2-embedded subgroup, O^{2′}(G/H) ≅ A_{5} by Bender’s theorem [Be, Satz 1]. Since |Out(A_{5})| = 2 (see [Sz1, (3.2.17)]), G/H ≅ A_{5}. In particular, O_{2}(G) = 1, and by Lemma A.1, G acts faithfully on P/Fr(P) ≅ C_{2}^{r} (r ≤ 6).

If H ≠ 1, then since G is isomorphic to a subgroup of GL_{6}(2), Lemma A.5 implies that there is K ⊴ G such that K ≅ C_{3}, C_{3}^{2}, or C_{7}. Also, K ≅ C̸_{7} since 5 ∤ |GL_{3}(2)| and 5 ∤ |GL_{2}(8)|. From the list of possibilities for N_{GL_{6}(2)}(C_{3}^{s}) for s = 1, 2, we see that G ≅ H × (G/H) where H = K ≅ C_{3} or C_{3}^{2}. (Note that GL_{2}(4) ≅ C_{3} × A_{5}.) □

There are four distinct irreducible 𝔽_{2}[A_{5}]-modules: the trivial module, the natural 2dimensional 𝔽_{2}[SL_{2}(4)]-module and its Galois conjugate, and the natural 4-dimensional module for A_{5} (cf. [Se, § 18.6]). The first and last are realizable over 𝔽_{2}, while the 2dimensional modules are realizable over 𝔽_{4} and hence induce a 4-dimensional irreducible 𝔽_{2}[A_{5}]-module by restriction of scalars. Thus there are three irreducible 𝔽_{2}[A_{5}]-modules: the trivial module 𝔽_{2}, and two 4-dimensional modules W_{1} and W_{2} described as follows:

• W_{1} is generated by a G-orbit of five elements whose sum is zero; and

• W_{2} ≅ 𝔽_{4}^{2} with the canonical action of G ≅ SL_{2}(4).

We keep this notation for the irreducible modules in the statement of the following lemma.

Lemma A.7. Fix G ≅ A_{5}. Let V be a finitely generated 𝔽_{2}[G]-module such that for each involution x ∈ G, dim([x, V]) = 2. Then the composition factors of V include exactly one irreducible 4-dimensional module. Also, the following hold for any S ∈ Syl_{2}(G).

(a) If [S, V] ≰ C_{V} (S), then V ≅ 𝔽_{2}^{k} ⊕ W_{1} (k = dim(V) − 4).

(b) If [S, V] ≤ C_{V} (S), then there is a composition factor of V isomorphic to W_{2}, and V is indecomposable if and only if [S, V] = C_{V} (S). Also, C_{V} (G) = C_{V} (N_{G}(S)) and dim(C_{V} (G)) = dim(C_{V} (S)) − 2 in this case. If V is indecomposable and C_{V} (G) ≠ 0, then ⋂_{1≠x∈S}[x, V] = 0.

Proof. As noted above, each irreducible 𝔽_{2}[G]-module is isomorphic to 𝔽_{2}, W_{1}, or W_{2}. If each composition factor of V is 1-dimensional, then for an appropriate choice of basis, G would act on V via upper triangular matrices, and thus via the identity (Lemma A.1).

By a direct check, dim([x, W_{j}]) = 2 for 1 ≠ x ∈ S and j = 1, 2. If the composition factors of V include more than one 4-dimensional module, then this would mean dim([x, V]) ≥ 4. Since dim([x, V]) = 2 for 1 ≠ x ∈ S, there is exactly one such composition factor.

Let V_{1} ≤ V_{2} ≤ V be submodules such that V_{2}/V_{1} is 4-dimensional and irreducible. Then G acts trivially on V_{1} and on V/V_{2} by the above remarks. In particular, we can assume they were chosen so that V_{1} = C_{V} (G). Also, dim([x, V_{2}]) = 2 for 1 ≠ x ∈ S, and thus [x, V_{2}] = [x, V] and [S, V_{2}] = [S, V]. Likewise, dim([x, V_{2}]) = dim([x, V_{2}/V_{1}]) for 1 ≠ x ∈ S implies V_{1} ∩ [x, V_{2}] = 0, and hence C_{V_{2}}(x) surjects onto C_{V_{2}/V_{1}}(x). Since G acts trivially on V_{1}, this shows that C_{V_{2}/V_{1}}(S) = C_{V_{2}}(S)/V_{1}.

Upon combining these observations, we see that

[S, V] ≤ C_{V} (S) ⟺ [S, V_{2}] = [S, V] ≤ C_{V} (S) ∩ V_{2} = C_{V_{2}}(S)
⟺ [S, V_{2}] + V_{1} ≤ C_{V_{2}}(S)
⟺ [S, V_{2}/V_{1}] ≤ C_{V_{2}}(S)/V_{1} = C_{V_{2}/V_{1}}(S) .

Since [S, W_{2}] = C_{W_{2}}(S) while [S, W_{1}] ≰ C_{W_{1}}(S), we conclude that [S, V] ≤ C_{V} (S) if and only if V_{2}/V_{1} ≅ W_{2}.

Since W_{1} is free (hence projective) as an 𝔽_{2}[S]-module, it is also projective as an 𝔽_{2}[G]- module, as is seen by averaging over cosets in G/S. Hence W_{1} is also injective by, e.g., [Bs, Proposition 3.1.2]. So if V_{2}/V_{1} ≅ W_{1}, then V ≅ W_{1} ⊕ 𝔽_{2}^{k} where k = dim(V) − 4.

Now assume V_{2}/V_{1} ≅ W_{2}, and thus [S, V] ≤ C_{V} (S). If V is decomposable, then it contains a nonzero summand with trivial action, in which case [S, V] < C_{V} (S). In other words, V is indecomposable if [S, V] = C_{V} (S).

Conversely, assume [S, V] < C_{V} (S). Fix a set X ⊆ G of representatives for the left cosets gS ⊆ G. For each v ∈ C_{V} (S), set v^{∗} = ∑_{g∈X} gv. Then v^{∗} ∈ C_{V} (G) = V_{1} and v^{∗} ≡ |G/S|·v (mod [G, V]), so v ∈ V_{1} + [G, V] ≤ V_{2}. Hence C_{V} (S) ≤ V_{2}, and C_{V} (S)/V_{1} = C_{V_{2}/V_{1}}(S) = [S, V_{2}/V_{1}] ≤ ([S, V] + V_{1})/V_{1} by the above remarks. Thus [S, V] ≱ V_{1} since [S, V] < C_{V} (S). So there are v ∈ V_{1} and φ ∈ Hom_{S}(V, 𝔽_{2}) such that φ(v) ≠ 0. Define ψ ∈ Hom_{G}(V, 𝔽_{2}) by ψ(x) = ∑_{g∈X} φ(g^{−1}x); then ψ(v) = |G/S|φ(v) ≠ 0. Thus the inclusion ⟨v⟩ < V is split by the map (x ↦ ψ(x)v), and V is decomposable.

It remains to prove the claims in the last two sentences in (b). Since V is the direct sum of a trivial module and an indecomposable module, it suffices to show them when V is indecomposable, and thus when [S, V] = C_{V} (S). Recall that V_{1} = C_{V} (G). Also, C_{V} (S) = [S, V] ≤ [G, V] ≤ V_{2}, and thus C_{V} (S)/V_{1} = C_{V_{2}}(S)/V_{1} = C_{V_{2}/V_{1}}(S) ≅ C_{W_{2}}(S) is 2-dimensional. So dim(C_{V} (G)) = dim(C_{V} (S)) − 2, and C_{V} (N_{G}(S)) = C_{V} (G) since C_{W_{2}}(N_{G}(S)) = 0.

Recall that dim([x, V]) = 2 and [x, V] ∩ C_{V} (G) = 0 for 1 ≠ x ∈ S. Set V_{0} = ⋂_{1≠x∈S}[x, V]. Either V_{0} = 0; or dim(V_{0}) = 1 and V_{0} ≤ C_{V} (N_{G}(S)) = C_{V} (G), which is impossible; or V_{0} is 2-dimensional, hence is equal to [S, V] = C_{V} (S), and so C_{V} (G) = 0. □

In fact, up to isomorphism, there are two distinct indecomposable 𝔽_{2}[A_{5}]-modules of dimension five and three of dimension six which satisfy the hypotheses of Lemma A.7(b). The three of dimension six are the permutation module for the A_{5}-action on A_{5}/D_{10}, the group S_{2}(𝔽_{4}) of symmetric (2 × 2) matrices over 𝔽_{4} with the canonical action of A_{5} ≅ SL_{2}(4), and the dual of this last module.

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Center for Mathematical Sciences, LTH, Box 118, 22100 Lund, Sweden

E-mail address: kksa@maths.lth.se

Université Paris 13, Sorbonne Paris Cité, LAGA, UMR 7539 du CNRS, 99, Av. J.-B. Clément, 93430 Villetaneuse, France.

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

Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049–001 Lisboa, Portugal

E-mail address: jventura@math.ist.utl.pt