AUTOMORPHISMS OF FUSION SYSTEMS OF SPORADIC SIMPLE GROUPS

BOB OLIVER

Abstract. We prove here that with a very small number of exceptions, when G is a sporadic simple group and p is a prime such that the Sylow p-subgroups of G are nonabelian, then Out(G) is isomorphic to the outer automorphism groups of the fusion and linking systems of G. In particular, the p-fusion system of G is tame in the sense of [AOV1], and is tamely realized by G itself except when G ≅ M_{11} and p = 2. From the point of view of homotopy theory, these results also imply that Out(G) ≅ Out(BG_{p}^{∧}) in many (but not all) cases.

This paper is centered around the comparison of certain outer automorphism groups associated to a sporadic simple group: outer automorphisms of the group itself, those of its fusion at different primes, and those of its classifying space completed at different primes. In most, but not all cases (under conditions made precise in Theorem A), these automorphism groups are all isomorphic. This comparison is important when studying extensions of fusion systems, and through that plays a role in Aschbacher’s program (see, e.g., [A5]) for reproving certain parts of the classification theorem from the point of view of fusion systems.

When G is a finite group, p is a prime, and S ∈ Syl_{p}(G), the p-fusion system of G is the category F_{S}(G) whose objects are the subgroups of G, and which has morphism sets

Mor_{F_{S}(G)}(P, Q) = {φ ∈ Hom(P, Q) | φ = c_{x}, some x ∈ G with xP x^{−1} ≤ Q}.

A p-subgroup P ≤ G is called p-centric in G if Z(P) ∈ Syl_{p}(C_{G}(P)); equivalently, if C_{G}(P) = Z(P) × C′_{G}(P) for some (unique) subgroup C′_{G}(P) of order prime to p. The centric linking system of G at p is the category L_{S}^{c}(G) whose objects are the subgroups of S which are p-centric in G, and where

Mor_{L_{S}^{c}(G)}(P, Q) = T_{G}(P, Q)/C′_{G}(P) where T_{G}(P, Q) = {x ∈ G | xP x^{−1} ≤ Q}.

Note that there is a natural functor π: L_{S}^{c}(G) ⟶ F_{S}(G) which is the inclusion on objects, and which sends the class of x ∈ T_{G}(P, Q) to c_{x} ∈ Hom(P, Q). Outer automorphism groups of these systems were defined in [BLO] and later papers (see below). We say that F = F_{S}(G) is tamely realized by G if the natural homomorphism κ_{G}: Out(G) ⟶ Out(L_{S}^{c}(G)) is surjective and splits. The fusion system F is tame if it is tamely realized by some finite group.

In terms of homotopy theory, it was shown in [BLO, Theorem B] that for a finite group G and S ∈ Syl_{p}(G), there is a natural isomorphism Out(L_{S}^{c}(G)) ≅ Out(BG_{p}^{∧}). Here, BG_{p}^{∧} is the p-completion, in the sense of Bousfield-Kan, of the classifying space of G, and Out(X) means the group of homotopy classes of self equivalences of the space X. Thus F_{S}(G) is tamely realized by G if the natural map from Out(G) to Out(BG_{p}^{∧}) is split surjective.

When p = 2, our main result is easily stated: if G is a sporadic simple group, then the 2-fusion system of G is simple except when G ≅ J_{1}, and is tamely realized by G except when G ≅ M_{11}. The 2-fusion system of M_{11} is tamely realized by PSL_{3}(3).

2000 Mathematics Subject Classification. Primary 20E25, 20D08. Secondary 20D20, 20D05, 20D45.

Key words and phrases. Fusion systems, sporadic groups, Sylow subgroups, finite simple groups.

B. Oliver is partially supported by UMR 7539 of the CNRS.

For p odd, information about fusion systems of the sporadic groups at odd primes is summarized in Table 0.1. In that table, for a given group G and prime p and S ∈ Syl_{p}(G),

• a dash “—” means that S is abelian or trivial;

• “constr.” means that F_{S}(G) is constrained; and

• an almost simple group L in brackets means that F_{S}(G) is almost simple but not simple, and is shown in [A4, 16.10] to be isomorphic to the fusion system of L.

For all other pairs (G, p), F is simple by [A4, 16.10], and we indicate what is known about the nature of κ_{G}: Out(G) ⟶ Out(L_{S}^{c}(G)). In addition,

• a dagger (^{†}) marks the pairs (G, p) for which S is extraspecial of order p^{3}.

G |Out(G)|  p = 3  p = 5 p = 7  p ≥ 11
M_{12}  2  κ isom.^{†}  —  —  —
M_{24}  1  [M_{12}:2]^{†}  —  —  —
J_{2}  2  constr.^{†}  —  —  —
J_{3}  2  constr.  —  —  —
J_{4}  1  [^{2}F_{4}(2)]^{†}  —  —  11: constr.^{†}
Co^{3}  1  κ isom.  constr.^{†}  —  —
Co^{2}  1  κ isom.  constr.^{†}  —  —
Co^{1}  1  κ isom.  [SO_{5}(5)]  —  —
HS  2  —  constr.^{†}  —  —
McL  2  κ isom.  constr.^{†}  —  —
Suz  2  κ isom.  —  —  —
He  2  Out(L) = 1^{†}  —  κ isom.^{†}  —
Ly  1  κ isom.  κ isom.  —  —
Ru  1  [^{2}F_{4}(2)]^{†} [L_{3}(5):2]^{†}  —  —
O’N  2  —  —  κ isom.^{†}  —
Fi^{22}  2  κ isom.  —  —  —
Fi^{23}  1  κ isom.  —  —  —
Fi{′ \atop 24}  2  κ isom.  —  κ isom.^{†}  —
F_{5}  2  κ isom.  κ isom.  —  —
F_{3}  1  κ isom.  κ isom.^{†}  —  —
F_{2}  1  κ isom.  κ isom.  —  —
F_{1}  1  κ isom.  κ isom. κ isom. 13: κ isom.^{†}

Table 0.1. Summary of results for odd p

Here, a fusion system F = F_{S}(G) is constrained if it contains a normal p-subgroup Q ⊴ F such that C_{S}(Q) ≤ Q. The fusion system F is simple if it has no proper nontrivial normal fusion subsystems. It is almost simple if it contains a proper normal subsystem F_{0} ⊴ F which is simple, and such that C_{F}(F_{0}) = 1. We refer to [AKO, Definitions I.4.1 & I.6.1] for the definitions of normal p-subgroups and normal fusion subsystems, and to [A4, § 6] for the definition of the centralizer of a normal subsystem.

Thus when G is a sporadic simple group and p is an odd prime such that the p-fusion system F of G is simple, we show in all cases that F is tamely realized by G, and in fact that Out(G) ≅ Out(L_{S}^{c}(G)) except when G ≅ He and p = 3 (Theorem A).

Before going further, we need to define more precisely the automorphism groups which we are working with. All of the definitions given here apply to abstract fusion and linking systems (see, e.g., [AKO, § III.4.3]), but for simplicity, we always assume that F = F_{S}(G) and L = L_{S}^{c}(G) for some finite group G with S ∈ Syl_{p}(G).

Automorphisms of F = F_{S}(G) are straightforward. An automorphism α ∈ Aut(S) is fusion preserving if it induces an automorphism of the category F (i.e., a functor from F to itself which is bijective on objects and on morphisms). Set

Aut(F) = Aut(S, F) = {α ∈ Aut(S) | α is fusion preserving}
Out(F) = Out(S, F) = Aut(S, F)/Aut_{F}(S).

Here, by definition, Aut_{F}(S) = Aut_{G}(S): the automorphisms induced by conjugation in N_{G}(S). These groups have been denoted Aut(S, F) and Out(S, F) in earlier papers, to emphasize that they are groups of automorphisms of S, but it seems more appropriate here to regard them as automorphisms of the fusion system F (as opposed to the category F).

Now assume L = L_{S}^{c}(G). For each P ∈ Ob(L), set ι_{P} = [1] ∈ Mor_{L}(P, S) (the “inclusion” of P in S in the category L), and set [[P]] = {[g] | g ∈ P } ≤ Aut_{L}(P). Define

Aut(L) = Aut_{typ}^{I}(L) = {β ∈ Aut_{cat}(L) | β(ι_{P}) = ι_{β(P)}, β([[P]]) = [[β(P)]], ∀ P ∈ F^{c}}
Out(L) = Out_{typ}(L) = Aut(L)/⟨c_{x} | x ∈ N_{G}(S)⟩.

Here, Aut_{cat}(L) is the group of automorphisms of L as a category, and c_{x} ∈ Aut(L) for x ∈ N_{G}(S) sends P to ^{x}P and [g] to [^{x}g]. There are natural homomorphisms

Out(G) ⟶^{κ_{G}} Out(L) ⟶^{μ_{G}}_{≅Out(BG_{p}^{∧})} Out(F) and κ_{G} = μ_{G} ∘ κ_{G}.

Here, κ_{G} is defined by sending the class of α ∈ Aut(G), chosen so that α(S) = S, to the class of α̂ ∈ Aut(L), where α̂(P) = α(P) and α̂([g]) = [α(g)]. For β ∈ Aut(L), μ_{G} sends the class of β ∈ Aut(L) to the class of

β̂ = (S ⟶_{≅}^{g↦[g]} [[S]] ⟶_{≅}^{β|_{[[S]]}} [[S]] ⟶_{≅}^{[g]↦g} S) ∈ Aut(F) ≤ Aut(S).

Then κ_{G}: Out(G) ⟶ Out(F) is induced by restriction to S. See [AKO, § III.4.3] or [AOV1, § 1.3] for more details on these definitions.

By recent work of Chermak, Oliver, and Glauberman and Lynd, the nature of μ_{G} is now fairly well known in all cases.

Proposition 0.1 ([O2, Theorem C], [GlL, Theorem 1.1]). For each prime p, and each finite group G with S ∈ Syl_{p}(G), μ_{G}: Out(L_{S}^{c}(G)) ⟶ Out(F_{S}(G)) is surjective, and is an isomorphism if p is odd.

In fact, [O2] and [GlL] show that the conclusion of Proposition 0.1 holds for all (abstract) fusion systems and associated linking systems.

When G is a sporadic simple group and p is odd, a more direct proof that μ_{G} is an isomorphism is given in [O1, Propositions 4.1 & 4.4].

The fusion system F = F_{S}(G) is tamely realized by G if κ_{G} is split surjective, and is tame if it is tamely realized by some finite group G^{∗} with S ∈ Syl_{p}(G^{∗}) and F = F_{S}(G^{∗}). We refer to [AOV1, Theorems A & B] or [AKO, § III.6.1] for the original motivation for this definition. In practice, it is in many cases easier to study the homomorphism κ_{G}, which is why we include information about μ_{G} here. The injectivity of κ_{G}, when p = 2 and G is a sporadic simple group, follows from a theorem of Richard Lyons [Ly2, Theorem 1.1] (see the proof of Proposition 2.2).

Fusion systems of alternating groups were shown to be tame in [AOV1, Proposition 4.8], while those of finite groups of Lie type (including the Tits group) were shown to be tame in [BMO, Theorems C & D]. So the following theorem completes the study of tameness for fusion systems of the known finite nonabelian simple groups.

Theorem A. Fix a sporadic simple group G, a prime p which divides |G|, and S ∈ Syl_{p}(G). Set F = F_{S}(G) and L = L_{S}^{c}(G). Then F is tame. Furthermore, κ_{G} and μ_{G} are isomorphisms (hence F is tamely realized by G) if p = 2, or if p is odd and S is nonabelian, with the following two exceptions:

(a) G ≅ M_{11} and p = 2, in which case Out(G) = 1 and |Out(F)| = |Out(L)| = 2; and

(b) G ≅ He and p = 3, in which case |Out(G)| = 2 and Out(F) = Out(L) = 1.

Proof. By Proposition 0.1, μ_{G} is surjective in all cases, and is an isomorphism if p is odd. When p = 2, μ_{G} is injective (hence an isomorphism) by Propositions 2.1 (when |S| ≤ 2^{9}) and 5.1 (when |S| ≥ 2^{10}). Thus in all cases, κ_{G} is an isomorphism if and only if κ_{G} = μ_{G} ∘κ_{G} is an isomorphism.

When p = 2, κ_{G} is an isomorphism, with the one exception G ≅ M_{11}, by Propositions 2.1 (when |S| ≤ 2^{9}) and 2.2 (when |S| ≥ 2^{10}). When p is odd, S is nonabelian, and F is not simple, then κ_{G} is an isomorphism by Proposition 3.1. When p is odd and F is simple, κ_{G} is an isomorphism except when G ≅ He and p = 3 by Proposition 3.2. The two exceptional cases are handled in Propositions 2.1 and 3.2. □

In the first half of the paper, we compare Out(G) with Out(F): first listing some general results in Section 1, and then applying them to determine the nature of κ_{G} in Sections 2 (for p = 2) and 3 (for p odd). We then compare Out(F) with Out(L) (when p = 2) in the last half of the paper: general techniques for determining Ker(μ_{G}) are listed in Section 4, and these are applied in Section 5 to finish the proof of the main theorem.

The author plans, in a future paper with Jesper Grodal, to look more closely at the fundamental groups of geometric realizations of the categories L_{S}^{c}(G) when G is a sporadic group. This should give alternative proofs for several of the cases covered by Theorem A.

I would like to thank Michael Aschbacher for explaining to me the potential importance of these results. Kasper Andersen made some computer computations several years ago involving the Rudvalis sporadic group at p = 2; while they’re not used here, they probably gave me hints as to how to proceed in that case (one of the hardest). I also thank the referee for his many suggestions which helped simplify or clarify several arguments. I would especially like to thank Richard Lyons for the notes [Ly2] he wrote about automorphisms of sporadic groups, without which I might not have known how to begin this project.

Notation: We mostly use Atlas notation [Atl, § 5.2] for groups, extensions, extraspecial groups, etc., as well as for names (2A, 2B, 3A, …) of conjugacy classes of elements. An elementary abelian 2-group has type 2A^{n} if it is 2A-pure of rank n (similarly for an elementary abelian 3-group of type 3A^{n}); it has type 2A_{i}B_{j} … if it contains i elements of class 2A, j of class 2B, etc. Also, A_{n} and S_{n} denote the alternating and symmetric groups on n letters, E_{p^{k}} (for p prime) is an elementary abelian p-group of order p^{k}, and UT_{n}(q) (for n ≥ 2 and q a prime power) is the group of upper triangular matrices in GL_{n}(q) with 1’s on the diagonal. As usual, G^{#} = G ∖ {1} is the set of nonidentity elements of a group G, and Z_{2}(S) ≤ S (for a p-group S) is the subgroup such that Z_{2}(S)/Z(S) = Z(S/Z(S)). For groups H ≤ G and elements g, h ∈ G, ^{g}h = ghg^{−1} and ^{g}H = gHg^{−1}. For each pair of groups H ≤ G,

Aut_{G}(H) = {(x ↦ ^{g}x) | g ∈ N_{G}(H)} ≤ Aut(H) and Out_{G}(H) = Aut_{G}(H)/Inn(H).

We assume in all cases the known order of Out(G) for sporadic groups G, without giving references each time.

1. Automorphism groups of fusion systems: generalities

We give here some techniques which will be used to determine the nature of κ_{G}. We begin with the question of injectivity. Recall that |Out(G)| ≤ 2 for each sporadic simple group G.

Lemma 1.1. Fix a prime p. Let G be a finite group, fix S ∈ Syl_{p}(G), and set F = F_{S}(G).

(a) For each α ∈ Aut(G), the class [α] ∈ Out(G) lies in Ker(κ_{G}) if and only if there is α′ ∈ [α] such that |S| | |C_{G}(α′)|.

(b) Assume |Out(G)| = 2 and p is odd. If there is no α ∈ Aut(G) such that |α| = 2 and |S| | |C_{G}(α)|, then κ_{G} is injective.

(c) Assume |Out(G)| = 2. If Out_{Aut(G)}(Q) > Out_{G}(Q) for some Q ⊴ S, then κ_{G} is injective.

Proof. (a) We can assume α is chosen so that α(S) = S. If [α] ∈ Ker(κ_{G}), then α|_{S} ∈ Aut_{G}(S): conjugation by some g ∈ N_{G}(S). Set α′ = α ∘ c_{g}^{−1} ∈ Aut(G); then [α′] = [α] in Out(G), and C_{G}(α′) ≥ S.

Conversely, assume |S| | |C_{G}(α′)|. Then C_{G}(α′) ≥ ^{g}S for some g ∈ G. Set α′′ = c_{g} ∘ α′ ∘ c_{g} (composing from right to left), where c_{g} ∈ Inn(G) and c_{g}(S) = ^{g}S. Then [α′′] = [α′] = [α] in Out(G), α′′|_{S} = Id_{S}, and hence κ_{G}([α]) = κ_{G}([α′′]) = 1.

(b) If κ_{G} is not injective, then by (a), there is α ∈ Aut(G)∖Inn(G) such that |S| | |C_{G}(α)|. Since |Out(G)| = 2, |α| = 2m for some m ≥ 1. Thus |α^{m}| = 2, and |S| | |C_{G}(α^{m})|.

(c) If Q ⊴ S and Out_{Aut(G)}(Q) > Out_{G}(Q), then there is β ∈ Aut(G) ∖ Inn(G) such that β(Q) = Q and β|_{Q} ∉ Aut_{G}(Q). Since S ∈ Syl_{p}(N_{G}(Q)), we can arrange that β(S) = S by replacing β by c_{x} ∘β for some appropriate element x ∈ N_{G}(Q). We still have β|_{Q} ∉ Aut_{G}(Q), so β|_{S} ∉ Aut_{G}(S), and κ_{G}([β]) ≠ 1. Thus κ_{G} is nontrivial, and is injective if |Out(G)| = 2. □

A finite group H will be called strictly p-constrained if C_{H}(O_{p}(H)) ≤ O_{p}(H); equivalently, if F^{∗}(H) = O_{p}(H).

Lemma 1.2. Fix a prime p. Let G be a finite group, fix S ∈ Syl_{p}(G), and set F = F_{S}(G). Let H < G be a subgroup which contains S.

(a) If H is strictly p-constrained, then κ_{H} and μ_{H} are isomorphisms.

(b) Assume H = N_{G}(Q), where either Q is characteristic in S, or |Q| = p, Q ≤ Z(S), and Aut(F) sends each G-conjugacy class of elements of order p in Z(S) to itself. Set F_{H} = F_{S}(H) for short, and set Aut^{0}(F_{H}) = Aut(F_{H}) ∩ Aut(F) and Out^{0}(F_{H}) = Aut^{0}(F_{H})/Aut_{H}(S). Then the inclusion of Aut^{0}(F_{H}) in Aut(F) induces a surjection of Out^{0}(F_{H}) onto Out(F), and hence |Out(F)| ≤ |Out^{0}(F_{H})| ≤ |Out(F_{H})|.

If in addition, H is strictly p-constrained or κ_{H} is onto, and we set Out^{0}(H) = κ_{H}^{−1}(Out^{0}(F_{H})), then |Out(F)| ≤ |Out^{0}(H)| ≤ |Out(H)|.

Proof. (a) See, e.g., [BMO, Proposition 1.6(a)].

(b) We first claim that

Aut(F) = Aut_{G}(S)·N_{Aut(F)}(Q) ≤ Aut_{G}(S)·Aut^{0}(F_{H})  (1)

as subgroups of Aut(S). If Q is characteristic in S, then the equality is clear. If |Q| = p, Q ≤ Z(S), and each α ∈ Aut(F) sends Q to a subgroup which is G-conjugate to Q, then the equality follows from the Frattini argument (and since each subgroup of Z(S) which is G-conjugate to Q is N_{G}(S)-conjugate to Q). If α ∈ Aut(S) normalizes Q and preserves fusion in G, then it preserves fusion in H = N_{G}(Q). Thus N_{Aut(F)}(Q) ≤ Aut^{0}(F_{H}), proving the second relation in (1).

Now, Aut_{H}(S) ≤ Aut_{G}(S) ∩ Aut^{0}(F_{H}). Together with (1), this implies that the natural homomorphism

Out^{0}(F_{H}) = Aut^{0}(F_{H})/Aut_{H}(S) ⟶ Aut(F)/Aut_{G}(S) = Out(F)

is well defined and surjective. The last statement now follows from (a). □

The next lemma will be useful when determining Out(H) for the subgroups H which appear when applying Lemma 1.2(b).

Lemma 1.3. Let H be a finite group, and let Q ⊴ H be a characteristic subgroup such that C_{H}(Q) ≤ Q. Set H^{∗} = Out_{H}(Q) ≅ H/Q.

(a) There is an exact sequence

1 ⟶ H^{1}(H^{∗}; Z(Q)) ⟶ Out(H) ⟶^{R} N_{Out(Q)}(H^{∗})/H^{∗},

where R sends the class of α ∈ Aut(H) to the class of α|_{Q}.

(b) Assume R ≤ Z(Q) and R ⊴ H. Let α ∈ Aut(H) be such that α|_{R} = Id_{R} and [α, H] ≤ R. Then there is ψ ∈ Hom_{H}(Q/R, R) such that α(g) = gψ(gR) for each g ∈ Q, and hence α|_{Q} = Id_{Q} if Hom_{H}(Q/R, R) = 1. If α|_{Q} = Id_{Q}, [α, H] ≤ R, and H^{1}(H^{∗}; R) = 0, then α ∈ Aut_{R}(H).

(c) Fix a prime p, assume Q is an extraspecial or elementary abelian p-group, and set Q = Q/Fr(Q). Set H_{0}^{∗} = O^{p′}(H^{∗}), and X = N_{Out(Q)}(H^{∗})/H^{∗}. (c.i) If Q is absolutely irreducible as an 𝔽_{p}H^{∗}-module, then there is Y ⊴ X such that Y ≅ (ℤ/p)^{×}/Z(H^{∗}) and X/Y is isomorphic to a subgroup of Out(H^{∗}).

(c.ii) If Q is absolutely irreducible as an 𝔽_{p}H_{0}^{∗}-module, then there is Y ⊴ X such that Y ≅ (ℤ/p)^{×}/Z(H^{∗}) and

|X/Y | ≤ |Out(H_{0}^{∗})|/|Out_{H^{∗}}(H_{0}^{∗})|.

Here, Z(H^{∗}) acts on Q via multiplication by scalars, and we regard it as a subgroup of (ℤ/p)^{×} in that way.

Proof. (a) The exact sequence is a special case of [OV, Lemma 1.2].

(b) By assumption, there is a function ψ: Q/R ⟶ R such that α(g) = gψ(gR) for each g ∈ Q, and ψ is a homomorphism since R ≤ Z(Q). For each h ∈ H, α(h) = rh for some r ∈ R. So for g ∈ Q, since [r, Q] = 1, we get ψ(^{h}gR) = (^{h}g)^{−1}α(^{h}g) = (^{h}g)^{−1}^{rh}(α(g)) = h(g^{−1}α(g)) = ^{h}ψ(gR). Thus ψ ∈ Hom_{H}(Q/R, R).

If α|_{Q} = Id and [α, H] ≤ R, then there is χ: H^{∗} ⟶ R such that α(g) = χ(gQ)g for each g ∈ H. Then χ(ghQ) = χ(gQ)·^{g}χ(hQ) for all g, h ∈ H, so χ is a 1-cocycle. If H^{1}(H^{∗}; R) = 0, then there is r ∈ R such that χ(gQ) = r(^{g}r)^{−1} for each g ∈ H, and α is conjugation by r.

(c) If Q is absolutely irreducible as an 𝔽_{p}H^{∗}-module, then C_{Out(Q)}(H^{∗}) ≅ (ℤ/p)^{×} consists of multiplication by scalars (see [A, 25.8]), so its image Y in X = N_{Out(Q)}(H^{∗})/H^{∗} is isomorphic to (ℤ/p)^{×}/Z(H^{∗}). Also, X/Y ≅ Out_{Out(Q)}(H^{∗}): a subgroup of Out(H^{∗}). This proves (c.i).

If Q is absolutely irreducible as an 𝔽_{p}H_{0}^{∗}-module, let Y be the image of C_{Out(Q)}(H_{0}^{∗}) in X = N_{Out(Q)}(H^{∗})/H^{∗}. Then Y ≅ (ℤ/p)^{×}/Z(H^{∗}) (by [A, 25.8] again), and

|X/Y | = |N_{Out(Q)}(H^{∗})/C_{Out(Q)}(H_{0}^{∗})·H^{∗}| ≤ |N_{Out(Q)}(H_{0}^{∗})|/|C_{Out(Q)}(H_{0}^{∗})·H^{∗}|
= |Aut_{Out(Q)}(H_{0}^{∗})|/|Aut_{H^{∗}}(H_{0}^{∗})|
≤ |Aut(H_{0}^{∗})|/|Aut_{H^{∗}}(H_{0}^{∗})| = |Out(H_{0}^{∗})|/|Out_{H^{∗}}(H_{0}^{∗})|.

This proves (c.ii). □

The next lemma provides some simple tools for showing that certain representations are absolutely irreducible.

Lemma 1.4. Fix a prime p, a finite group G, and an irreducible 𝔽_{p}G-module V.

(a) The module V is absolutely irreducible if and only if End_{𝔽_{p}G}(V) ≅ 𝔽_{p}.

(b) If dim_{𝔽_{p}}(C_{V} (H)) = 1 for some H ≤ G, then V is absolutely irreducible.

(c) Assume H ≤ G is a subgroup such that V |_{H} splits as a direct sum of absolutely irreducible pairwise nonisomorphic 𝔽_{p}H-submodules. Then V is absolutely irreducible.

Proof. (a) See, e.g., [A, 25.8].

(b) Set End_{𝔽_{p}G}(V) = K: a finite extension of 𝔽_{p}. Then V can be considered as a KG-module, so [K:𝔽_{p}] divides dim_{𝔽_{p}}(C_{V} (H)) for each H ≤ G. Since there is H with dim_{𝔽_{p}}(C_{V} (H)) = 1, this implies K = 𝔽_{p}, and so V is absolutely irreducible by (a).

(c) The hypothesis implies that the ring End_{𝔽_{p}H}(V) is isomorphic to a direct product of copies of 𝔽_{p}, one for each irreducible summand of V |_{H}. Since End_{𝔽_{p}G}(V) is a subring of End_{𝔽_{p}H}(V), and is a field since V is irreducible, it must be isomorphic to 𝔽_{p}. So V is absolutely irreducible by (a). □

Lemma 1.5. Let G be a finite group, and let V be a finite 𝔽_{p}G-module.

(a) If C_{V} (O_{p′}(G)) = 0, then H^{1}(G; V) = 0.

(b) If |V | = p, and G_{0} = C_{G}(V), then H^{1}(G; V) ≅ Hom_{G/G_{0}}(G_{0}/[G_{0}, G_{0}], V ).

Proof. (a) Set H = O_{p′}(G) for short. Assume W ≥ V is an 𝔽_{p}G-module such that [G, W] ≤ V. Then [H, W] = [H, V] = V since C_{V} (H) = 0, and so W = C_{W} (H)⊕[H, W] = C_{W} (H)⊕V. Thus H^{1}(G; V) ≅ Ext_{𝔽_{p}G}^{1}(𝔽_{p}, V) = 0.

Alternatively, with the help of the obvious spectral sequence, one can show that H^{i}(G; V) = 0 for all i ≥ 0.

(b) This is clear when G acts trivially on V. It follows in the general case since for G_{0} ⊴ G of index prime to p and any 𝔽_{p}G-module V, H^{1}(G; V) is the group of elements fixed by the action of G/G_{0} on H^{1}(G_{0}; V). □

We end with a much more specialized lemma, which is needed when working with the Thompson group F_{3}.

Lemma 1.6. Set H = A_{9}. Assume V is an 8-dimensional 𝔽_{2}H-module such that for each 3cycle g ∈ H, C_{V} (g) = 0. Then V is absolutely irreducible, dim(C_{V} (T)) = 1 for T ∈ Syl_{2}(H), and N_{Aut(V)}(H)/H = 1.

Proof. Consider the following elements in A_{9}:

a_{1} = (1 2 3),  a_{2} = (4 5 6),  a_{3} = (7 8 9),
b_{1} = (1 2)(4 5),  b_{2} = (1 2)(7 8),  b_{3} = (1 2)(4 7)(5 8)(6 9).

Set A = ⟨a_{1}, a_{2}, a_{3}⟩ ≅ E_{27} and B = ⟨b_{1}, b_{2}, b_{3}⟩ ≅ D_{8}. Set V = 𝔽_{2} ⊗_{𝔽_{2}} V. As an 𝔽_{2}Amodule, V splits as a sum of 1-dimensional submodules, each of which has character A ⟶ 𝔽_{2}^{×} for which none of the a_{i} is in the kernel. There are eight such characters, they are permuted transitively by B, and so each occurs with multiplicity 1 in the decomposition of V. Thus V is AB-irreducible, and hence H-irreducible (and V is absolutely irreducible). Also, dim_{𝔽_{2}}(C_{V} (B)) = 1, so dim(C_{V} (B)) = 1, and dim(C_{V} (T)) = 1 since C_{V} (T) ≠ 0.

In particular, C_{Aut(V)}(H) ≅ 𝔽_{2}^{×} = 1, and hence N_{Aut(V)}(H)/H embeds into Out(H). So if N_{Aut(V)}(H)/H ≠ 1, then the action of H extends to one of Ĥ ≅ S_{9}. In that case, if we set x = (1 2) ∈ Ĥ, then C_{V} (x) has rank 4 since x inverts a_{1} and C_{V} (a_{1}) = 0. But the group CĤ(x)/x ≅ S_{7} acts faithfully on C_{V} (x), and this is impossible since GL_{4}(2) ≅ A_{8} contains no S_{7}-subgroup. (This argument is due to Richard Lyons [Ly2].) □

2. Automorphisms of 2-fusion systems of sporadic groups

The main result in this section is that when G is a sporadic simple group and p = 2, Out(F) ≅ Out(G) in all cases except when G ≅ M_{11}. The first proposition consists mostly of the cases where this was shown in earlier papers.

Proposition 2.1. Let G be a sporadic simple group whose Sylow 2-subgroups have order at most 2^{9}. Then the 2-fusion system of G is tame. More precisely, κ_{G} and μ_{G} are isomorphisms except when G ≅ M_{11}, in which case the 2-fusion system of G is tamely realized by PSL_{3}(3).

Proof. Fix G as above, choose S ∈ Syl_{2}(G), and set F = F_{S}(G). There are eleven cases to consider.

If G ≅ M_{11}, then Out(G) = 1. Also, F is the unique simple fusion system over SD_{16}, so by [AOV1, Proposition 4.4], |Out(F)| = 2, and κ_{G^{∗}} is an isomorphism for G^{∗} = PSU_{3}(13) (and μ_{G^{∗}} is an isomorphism by the proof of that proposition). Note that we could also take G^{∗} = PSL_{3}(3).

If G ≅ J_{1}, then Out(G) = 1. Set H = N_{G}(S). Since S ≅ E_{8} is abelian, fusion in G is controlled by H ≅ 2^{3}:7:3, and so F = F_{S}(H) and L ≅ L_{S}^{c}(H). Since H is strictly 2-constrained, Out(L) ≅ Out(F) ≅ Out(H) = 1 by Lemma 1.2(a), and so κ_{G} and μ_{G} are isomorphisms.

If G ≅ M_{22}, M_{23}, J_{2}, J_{3}, or McL, then F is tame, and κ_{G} is an isomorphism, by [AOV1, Proposition 4.5]. Also, μ_{G} was shown to be an injective in the proof of that proposition, and hence is an isomorphism by Proposition 0.1.

If G ≅ M_{12}, Ly, HS, or O’N, then F is tame, and κ_{G} and μ_{G} are isomorphisms, by [AOV3, Lemmas 4.2 & 5.2 and Proposition 6.3]. □

It remains to consider the larger cases.

Proposition 2.2. Let G be a sporadic simple group whose Sylow 2-subgroups have order at least 2^{10}. Then κ_{G} is an isomorphism.

Proof. Fix G as above, choose S ∈ Syl_{2}(G), and set F = F_{S}(G). There are fifteen groups to consider, listed in Table 2.2.

We first check that κ_{G} is injective in all cases. This follows from a theorem of Richard Lyons [Ly2, Theorem 1.1], which says that if Out(G) ≠ 1, then there is a 2-subgroup of G whose centralizer in Aut(G) = G.2 is contained in G [Ly2, Theorem 1.1]. Since that paper has not been published, we give a different argument here: one which is based on Lemma 1.1(c), together with some well known (but hard-to-find-referenced) descriptions of certain subgroups of G and of Aut(G).

The groups G under consideration for which |Out(G)| = 2 are listed in Table 2.1. In each case, N_{G}(R) has odd index in G (hence R can be assumed to be normal in S), and Out_{Aut(G)}(R) > Out_{G}(R). So κ_{G} is injective by Lemma 1.1(c).

G  Suz  He  Fi_{22}  Fi′_{24}  F_{5}
R  2_{−}^{1+6}  2^{4+4}  2^{5+8}  2_{+}^{1+12}  2_{+}^{1+8}
Out_{G}(R)  Ω_{6}^{−}(2) S_{3} × S_{3} S_{3} × A_{6}  3·U_{4}(3).2 (A_{5} × A_{5}).2
Out_{G.2}(R) SO_{6}^{−}(2)  3^{2}:D_{8}  S_{3} × S_{6} 3·U_{4}(3).2^{2} (A_{5} × A_{5}).2^{2}
Reference [GL, p.56] [W7, § 5] [A3, 37.8.2] [W8, Th.E] [NW, Th.2]

Table 2.1

It remains to prove that |Out(F)| ≤ |Out(G)|. Except when G ≅ Ru, we do this with the help of Lemma 1.2(b) applied with H as in Table 2.2. Set Q = O_{2}(H), Q = Q/Z(Q), and H^{∗} = Out_{H}(Q).

When G ≅ Co_{3}, and H = N_{G}(Z(S)) ≅ 2·Sp_{6}(2) is quasisimple, Out(H) = 1 since Out(H/Z(H)) = 1 by Steinberg’s theorem (see [GLS, Theorem 2.5.1]). Also, κ_{H/Z(H)} is surjective by [BMO, Theorem A], so κ_{H} and κ_{H} are surjective by [AOV1, Proposition 2.18]. Hence |Out(F)| ≤ |Out(H)| = 1 by Lemma 1.2(b).

If G ≅ Co_{1}, Fi_{22}, Fi_{23}, or Fi′_{24}, then Q is elementary abelian, H^{∗} ≅ M_{k} for k = 24, 22, 23, or 24, respectively, and Q is an absolutely irreducible 𝔽_{2}H^{∗}-module by [A3, 22.5]. Also, Q = J(S) (i.e., Q is the unique abelian subgroup of its rank) in each case: by [A2, Lemma 46.12.1] when G ≅ Co_{1}, and by [A3, Exercise 11.1, 32.3, or 34.5] when G is one of the Fischer groups. By [MSt, Lemma 4.1] (or by [A3, 22.7–8] when G is a Fischer group), H^{1}(H^{∗}; Q) has order 2 when G ≅ Fi′_{24} (and Q is the Todd module for H^{∗}), and has order 1 when G is one of the other Fischer groups (Q is again the Todd module) or Co_{1} (Q is the dual Todd module). So

|Out(F)| ≤ |Out(H)| ≤ |H^{1}(H^{∗}; Q)|·|Out(H^{∗})| = |Out(G)|:

the first inequality by Lemma 1.2(b), the second by Lemmas 1.3(a) and 1.3(c.i), and the equality by a case-by-case check (see Table 2.2).

In each of the remaining cases covered by Table 2.2, H = N_{G}(Z(S)) and is strictly 2constrained, and Q is extraspecial. We apply Lemma 1.3(a) to get an upper bound for |Out(H)|. This upper bound is listed in the fourth column of Table 2.2 in the form m = a·b, where |H^{1}(H^{∗}; Z(Q))| ≤ a and |N_{Out(Q)}(H^{∗})/H^{∗}| ≤ b. By Lemma 1.5(b), H^{1}(H^{∗}; Z(Q)) ≅ Hom(H^{∗}, C_{2}) = 1 except when G ≅ J_{4} or F_{5}, in which cases it has order 2. This explains the first factor in the fourth column. The second factor will be established case-by-case, as will be the difference between |Out(F)| and |Out(H)| when there is one (noted by an asterisk).

If G ≅ M_{24} or He, then H ≅ 2_{+}^{1+6}.L_{3}(2), and Q splits as a sum of two nonisomorphic absolutely irreducible 𝔽_{2}H^{∗}-modules which differ by an outer automorphism of H^{∗}. Hence

G |S|  H  |Out(H)| |Out(F)| |Out(G)| Reference
M_{24} 2^{10}  2_{+}^{1+6}.L_{3}(2)  2 = 1·2  1^{∗}  1  [A2, Lm. 39.1.1]
J_{4} 2^{21} 2_{+}^{1+12}.3M_{22}:2  2 = 2·1  1^{∗}  1  [KW, § 1.2]
Co^{3} 2^{10}  2·Sp_{6}(2)  1  1  1  [Fi, Lm. 4.4]
Co^{2} 2^{18}  2_{+}^{1+8}.Sp_{6}(2)  1 = 1·1  1  1  [W1, pp.113–14]
Co^{1} 2^{21}  2^{11}.M_{24}  1 = 1·1  1  1  [A2, Lm. 46.12]
Suz 2^{13}  2_{−}^{1+6}.U_{4}(2)  2 = 1·2  2  2  [W2, § 2.4]
He 2^{10}  2_{+}^{1+6}:L_{3}(2)  2 = 1·2  2  2  [He, p. 253]
Ru 2^{14}  \begin{matrix}2^{3+8}.L_{3}(2) \\ 2.2^{4+6}.S_{5} \end{matrix}  1^{∗}  1  {[A1, 12.12] \atop [AS, Th. J.1.1]}
Fi^{22} 2^{17}  2^{10}.M_{22}  2 = 1·2  2  2  [A3, 25.7]
Fi^{23} 2^{18}  2^{11}.M_{23}  1 = 1·1  1  1  [A3, 25.7]
Fi{′ \atop 24} 2^{21}  2^{11}.M_{24}  2 = 2·1  2  2  [A3, 34.8, 34.9]
F_{5} 2^{14} 2_{+}^{1+8}.(A_{5} × A_{5}).2 4 = 2·2  2^{∗}  2  [NW, § 3.1]
F_{3} 2^{15}  2_{+}^{1+8}.A_{9}  1 = 1·1  1  1  [W11, Thm. 2.2]
F_{2} 2^{41}  2_{+}^{1+22}.Co_{2}  1 = 1·1  1  1  [MS, Thm. 2]
F_{1} 2^{46}  2_{+}^{1+24}.Co_{1}  1 = 1·1  1  1  [MS, Thm. 1]

Table 2.2

N_{Out(Q)}(H^{∗}) ≅ L_{3}(2):2, and |Out(H)| ≤ |N_{Out(Q)}(H^{∗})/H^{∗}| = 2. These two irreducible submodules in Q lift to rank 4 subgroups of Q, of which exactly one is radical (with automizer SL_{4}(2)) when G ≅ M_{24} (see [A2, Lemma 40.5.2]). Since an outer automorphism of H exchanges these two subgroups, it does not preserve fusion in G when G ≅ M_{24}, hence is not in Out^{0}(H) in the notation of Lemma 1.2(b). So |Out(F)| ≤ |Out^{0}(H)| = 1 in this case.

If G ≅ J_{4}, then H ≅ 2_{+}^{1+12}.3M_{22}:2. The group 3M_{22} has a 6-dimensional absolutely irreducible representation over 𝔽_{4}, which extends to an irreducible 12-dimensional representation of 3M_{22}:2 realized over 𝔽_{2}. (See [KW, p. 487]: 3M_{22} < SU_{6}(2) < SO_{12}^{+}(2).) Hence |Out(H)| ≤ 2 by Lemmas 1.3(a,c) and 1.5(b), generated by the class of β ∈ Aut(H) of order 2 which is the identity on O^{2}(H) and on H/Z(H).

By [KW, Table 1], there is a four-group of type 2AAB in H, containing Z(Q) = Z(H) (generated by an element of class 2A), whose image in H/O_{2,3}(H) ≅ M_{22}:2 is generated by an outer involution of class 2B in Aut(M_{22}). Thus there are cosets of Z(Q) in H ∖ O^{2}(H) which contain 2A- and 2B-elements. Hence β|_{S} is not G-fusion preserving, so |Out(F)| ≤ |Out^{0}(H)| = 1 by Lemma 1.2(b).

If G ≅ Co_{2}, then H = N_{G}(z) ≅ 2_{+}^{1+8}.Sp_{6}(2). By [Sm, Lemma 2.1], the action of H/Q on Q is transitive on isotropic points and on nonisotropic points, and hence is irreducible. If Q is not absolutely irreducible, then End_{𝔽_{p}[H/Q]}(Q) ≥ 𝔽_{4} by Lemma 1.4(a), so H/Q ≅ Sp_{6}(2) embeds into SL_{4}(4), which is impossible since Sp_{6}(2) contains a subgroup of type 7:6 while SL_{4}(4) does not.

Alternatively, Q is absolutely irreducible by a theorem of Steinberg (see [GLS, Theorem 2.8.2]), which says roughly that each irreducible 𝔽_{2}Sp_{6}(𝔽_{2})-module which is “small enough” is still irreducible over the finite subgroup Sp_{6}(2).

Thus by Lemma 1.3(c.i), N_{Out(Q)}(H^{∗})/H^{∗} is isomorphic to a subgroup of Out(H^{∗}), where Out(H^{∗}) = 1 (see [GLS, Theorem 2.5.1]). This confirms the remaining entries for G in Table 2.2.

If G ≅ Suz, then H ≅ 2_{−}^{1+6}.Ω_{6}^{−}(2), H^{∗} has index 2 in Out(Q) ≅ SO_{6}^{−}(2), and |Out(H)| ≤ 2 by Lemmas 1.3(a) and 1.5(b).

If G ≅ F_{5}, then H = N_{G}(z) ≅ 2_{+}^{1+8}.(A_{5} ≀ 2) for z ∈ 2B. As described in [NW, § 3.1] and in [Ha, Lemma 2.8], O^{2}(H^{∗}) acts on Q as Ω_{4}^{+}(4) for some 𝔽_{4}-structure on Q. Also, the 2B-elements in Q ∖ Z(Q) are exactly those involutions which are isotropic under the 𝔽_{4}-quadratic form on Q ≅ 𝔽_{4}^{4}.

Now, H^{∗} has index 2 in its normalizer SO_{4}^{+}(4).2^{2} in Out(Q) ≅ SO_{8}^{+}(2), so |Out(H)| ≤ 4 by Lemmas 1.3(a) and 1.5(b). Let β ∈ Aut(H) be the nonidentity automorphism which is the identity on O^{2}(H) and on H/Z(H). To see that |Out(F)| ≤ 2, we must show that β does not preserve fusion in S.

By [NW, p. 364], if W = ⟨z, g⟩ ≅ E_{4} for z ∈ Z(H) and g ∈ H ∖ O^{2}(H), then W contains an odd number of 2A-elements, and hence g and zg are in different classes (see also [Ha, Lemma 2.9.ii]). Hence β is not fusion preserving since it doesn’t preserve G-conjugacy classes. By Lemma 1.2(b), |Out(F)| ≤ |Out^{0}(H)| ≤ 2 = |Out(G)|.

If G ≅ F_{3}, then H ≅ 2_{+}^{1+8}.A_{9}. By [Pa, § 3], the action of A_{9} on Q is not the permutation representation, but rather that representation twisted by the triality automorphism of SO_{8}^{+}(2). By [Pa, 3.7], if x ∈ H^{∗} ≅ A_{9} is a 3-cycle, then C_{Q}(x) = 1. Hence we are in the situation of Lemma 1.6, and N_{Out(Q)}(H^{∗})/H^{∗} = 1 by that lemma. So Out(H) = 1 by Lemmas 1.3(a) and 1.5(b), and Out(F) = 1.

If G ≅ F_{2} or F_{1}, then H = H_{1} ≅ 2_{+}^{1+22}.Co_{2} or 2_{+}^{1+24}.Co_{1}, respectively. Set Q = O_{2}(H) and Q = Q/Z(Q). If G ≅ F_{1}, then Q ≅ Λ̃, the mod 2 Leech lattice, and is Co_{1}-irreducible by [A2, 23.3]. If G ≅ F_{2}, then Q ≅ v_{2}^{⊥}/⟨v_{2}⟩ where v_{2} ∈ Λ̃ is the image of a 2-vector. The orbit lengths for the action of Co_{2} on Λ̃/⟨v_{2}⟩ are listed in [W1, Table I], and from this one sees that v_{2}^{⊥}/⟨v_{2}⟩ is the only proper nontrivial Co_{2}-linear subspace (the only union of orbits of order 2^{k} for 0 < k < 23), and hence that Q is Co_{2}-irreducible. The absolute irreducibility of Q (in both cases) now follows from Lemma 1.4(b), applied with H = Co_{2} or U_{6}(2):2, respectively.

Since Out(Co_{1}) ≅ Out(Co_{2}) = 1, N_{Out(Q)}(H^{∗})/H^{∗} = 1 by Lemma 1.3(c.i), and so Out(H) = 1 in both cases.

In in the remaining case, we need to work with two of the 2-local subgroups of G.

Assume G ≅ Ru. We refer to [A1, 12.12] and [AS, Theorem J.1.1] for the following properties. There are two conjugacy classes of involutions in G, of which the 2A-elements are 2-central. There are subgroups H_{1}, H_{3} < G containing S such that

H_{1} ≅ 2.2^{4+6}.S_{5} H_{3} ≅ 2^{3+8}.L_{3}(2).

Set Q_{i} = O_{2}(H_{i}) and V_{i} = Z(Q_{i}); V_{1} ≅ C_{2} and V_{3} ≅ E_{8}, and both are 2A-pure and normal in S. Also, Q_{1}/V_{1} and Q_{3} are special of types 2^{4+6} and 2^{3+8}, respectively, and Z(Q_{3}) and Q_{3}/Z(Q_{3}) are the natural module and the Steinberg module, respectively, for H_{3}/Q_{3} ≅ SL_{3}(2).

Let V_{5} < Q_{1} be such that V_{5}/V_{1} = Z(Q_{1}/V_{1}). Then V_{5} is of type 2A^{5}, and C_{Q_{1}}(V_{5}) ≅ Q_{8} × E_{16}. Also, H_{1}/Q_{1} ≅ S_{5}, and V_{5}/V_{1} and Q_{1}/C_{Q_{1}}(V_{5}) are both natural modules for O^{2}(H_{1}/Q_{1}) ≅ SL_{2}(4). Also, V_{3}/V_{1} = C_{V_{5}/V_{1}}((S/Q_{1}) ∩ O^{2}(H_{1}/Q_{1})): thus a 1-dimensional subspace of V_{5}/V_{1} as an 𝔽_{4}-vector space.

The homomorphism Q_{1}/C_{Q_{1}}(V_{5}) ⟶ Hom(V_{5}/V_{1}, V_{1}) which sends g to (x ↦ [g, x]) is injective and hence an isomorphism. So Q_{3} ∩ Q_{1} = C_{Q_{1}}(V_{3}) has index 4 in Q_{1}, and hence |Q_{3}Q_{1}/Q_{3}| = 4.

Fix β ∈ Aut(F). By Lemma 1.2(a), for i = 1, 3, κ_{H_{i}} is an isomorphism, so β extends to an automorphism β_{i} ∈ Aut(H_{i}). Since V_{3} is the natural module for H_{3}/Q_{3} ≅ SL_{3}(2), β_{3}|_{V_{3}} = c_{x} for some x ∈ H_{3}, and x ∈ N_{H_{3}}(S) since β_{3}(S) = S. Then x ∈ S since N_{H_{3}/Q_{3}}(S/Q_{3}) = S/Q_{3}, and upon replacing β by c_{x}^{−1} ∘ β and β_{i} by c_{x}^{−1} ∘ β_{i} (i = 1, 3), we can arrange that β|_{V_{3}} = Id.

Since β|_{V_{3}} = Id, β_{3} also induces the identity on H_{3}/Q_{3} ≅ L_{3}(2) (since this acts faithfully on V_{3}), and on Q_{3}/V_{3} ≅ 2^{8} (since this is the Steinberg module and hence absolutely irreducible). Since Q_{3}/V_{3} is H_{3}/Q_{3}-projective (the Steinberg module), H^{1}(H_{3}/Q_{3}; Q_{3}/V_{3}) = 0, so by Lemma 1.3(b) (applied with Q_{3}/V_{3} in the role of R = Q), the automorphism of H_{3}/V_{3} induced by β_{3} is conjugation by some yV_{3} ∈ Q_{3}/V_{3}. Upon replacing β by c_{y}^{−1} ∘ β and similarly for the β_{i}, we can arrange that [β_{3}, H_{3}] ≤ V_{3}.

Now, Q_{3}/V_{3} ≅ V̸_{3} are both irreducible 𝔽_{2}[H_{3}/Q_{3}]-modules, so Hom_{H_{3}/Q_{3}}(Q_{3}/V_{3}, V_{3}) = 0. By Lemma 1.3(b) again, applied this time with Q_{3} ≥ V_{3} in the role of Q ≥ R, β|_{Q_{3}} = Id.

Now consider β_{1} ∈ Aut(H_{1}). Since β_{1} is the identity on Q_{3} = C_{S}(V_{3}) ≥ C_{S}(V_{5}) = C_{H_{1}}(V_{5}), β_{1} ≡ Id_{H_{1}} modulo Z(C_{S}(V_{5})) = V_{5} (since c_{g} = c_{β_{1}(g)} ∈ Aut(C_{S}(V_{5})) for each g ∈ H_{1}). So by Lemma 1.3(b), there is ψ ∈ Hom_{H_{1}/Q_{1}}(Q_{1}/V_{5}, V_{5}/V_{1}) such that β(g) ∈ gψ(gV_{5}) for each g ∈ Q_{1}. Also, Im(ψ) ≤ V_{3}/V_{1} since [β, S] ≤ V_{3}, and hence ψ = 1 since V_{5}/V_{1} is irreducible. Thus [β_{1}, Q_{1}] ≤ V_{1}.

We saw that |Q_{1}Q_{3}/Q_{3}| = 4, so Aut_{Q_{1}}(V_{3}) is the group of all automorphisms which send V_{1} to itself and induce the identity on V_{3}/V_{1}. Fix a pair of generators uQ_{3}, vQ_{3} ∈ Q_{1}Q_{3}/Q_{3}. Then β(u) ∈ uV_{1} and β(v) ∈ vV_{1}, and each of the four possible automorphisms of Q_{3}Q_{1} (i.e., those which induce the identity on Q_{3} and on Q_{1}Q_{3}/V_{1}) is conjugation by some element of V_{3} (unique modulo V_{1}). So after conjugation by an appropriate element of V_{3}, we can arrange that β|_{Q_{1}Q_{3}} = Id (and still [β_{3}, H_{3}] ≤ V_{3}).

Let V_{2} < V_{3} be the unique subgroup of rank 2 which is normal in S, and set S_{0} = C_{S}(V_{2}). Thus |S/S_{0}| = 2, and S_{0}/Q_{3} ≅ E_{4}. Fix w ∈ (S_{0} ∩ Q_{1}Q_{3}) ∖ Q_{3} (thus wQ_{3} generates the center of S/Q_{3} ≅ D_{8}). Choose g ∈ N_{H_{3}}(V_{2}) of order 3; thus g acts on V_{2} with order 3 and acts trivially on V_{3}/V_{2}. So V_{3}⟨g⟩ ≅ A_{4} × C_{2}, and since |β_{3}(g)| = 3, we have β_{3}(g) = rg for some r ∈ V_{2}. Set w′ = ^{g}w ∈ S_{0}. Then S_{0} = Q_{3}⟨w, w′⟩, β(w) = w since w ∈ Q_{1}Q_{3}, and β(w′) = β(gwg^{−1}) = rgwg^{−1}r^{−1} = ^{r}w′ = w′: the last equality since w′ ∈ S_{0} = C_{S}(V_{2}). Since S = S_{0}Q_{1}, this proves that β = Id_{S}, and hence that Out(F) = 1.

This finishes the proof of Proposition 2.2. □

3. Tameness at odd primes

We now turn to fusion systems of sporadic groups at odd primes, and first look at the groups whose p-fusion systems are not simple.

Proposition 3.1. Let p be an odd prime, and let G be a sporadic simple group whose Sylow p-subgroups are nonabelian and whose p-fusion system is not simple. Then κ_{G} is an isomorphism.

Proof. Fix S ∈ Syl_{p}(G), and set F = F_{S}(G). By [A4, 16.10], if F is not simple, then either N_{G}(S) controls fusion in G (“G is p-Goldschmidt” in the terminology of [A4]), in which case S ⊴ F and F is constrained, or F is almost simple and is realized by an almost simple group L given explicitly in [A4, 16.10] and also in Table 0.1. We handle these two cases separately.

Case 1: Assume first that S ⊴ F and hence F is constrained. By [A4, Theorem 15.6], there are seven such cases (G, p), also listed in Table 0.1. By the tables in [GLS, Table 5.3], in each case where Out(G) ≠ 1, no involution of Aut(G) centralizes a Sylow p-subgroup. Thus κ_{G} is injective in all seven cases by Lemma 1.1(b). Set H = N_{G}(H)/O_{p′}(N_{G}(H)). Since N_{G}(S) controls p-fusion in G,

Out(F) ≅ Out(H) injects into N_{Out(S)}(Out_{G}(S))/Out_{G}(S) :  (1)

the isomorphism by Lemma 1.2(a) and the injection by 1.3(a).

In the six cases described in Table 3.1, S is extraspecial of order p^{3} and exponent p. Note

(G, p) (J_{2}, 3) (Co_{3}, 5) (Co_{2}, 5) (HS, 5) (McL, 5) (J_{4}, 11)
|Out(G)| 2 1 1 2 2 1
Out_{G}(S) C_{8} C_{24} ⋊ C_{2} 4·Σ_{4} C_{8} ⋊ C_{2} C_{3} ⋊ C_{8} 5 × 2·Σ_{4}

Table 3.1

that Out(S) ≅ GL_{2}(p). Using that PGL_{2}(3) ≅ Σ_{4}, PGL_{2}(5) ≅ Σ_{5}, and Σ_{4} is maximal in PGL_{2}(11), we see that in all cases, |Out(F)| ≤ |Out(G)| by (1). So κ_{G} and κ_{G} are isomorphisms since they are injective.

It remains to consider the case (G, p) = (J_{3}, 3), where |S| = 3^{5}. Set T = Ω_{1}(S) and Z = Z(S). By [J, Lemma 5.4], T ≅ C_{3}^{3}, T > Z ≅ C_{3}^{2}, Z ≤ [S, S], and there are two classes of elements of order 3: those in Z and those in T ∖ Z. Also, S/Z is extraspecial of order 3^{3} with center T/Z, and N_{G}(S)/S ≅ C_{8} acts faithfully on S/T and on Z.

Consider the bilinear map

Φ: S/T × T/Z ⟶^{[−,−]} Z

where Φ(gT, hZ) = [g, h]. This is nontrivial (otherwise we would have T ≤ Z), and hence is surjective since N_{G}(S)/S ≅ C_{8} acts faithfully on Z. Fix x ∈ N_{G}(S) and h ∈ T whose cosets generate the quotient groups N_{G}(S)/S and T/Z, respectively. Since x acts on S/T ≅ C_{3}^{2} with order 8, it acts via an element of GL_{2}(3)∖SL_{2}(3), and hence acts on T/Z by inverting it (recall that S/Z is extraspecial). So if we let Φ_{h}: S/T ⟶ Z be the isomorphism Φ_{h}(gT) = [g, h], then Φ_{h}(^{x}gT) = [^{x}g, h] = ^{x}[g, h^{−1}] = ^{x}Φ_{h}(gT)^{−1}. Thus if λ, λ^{3} ∈ 𝔽_{9} are the eigenvalues for the action of x on S/T (for some λ of order 8), then λ^{−1}, λ^{−3} are the eigenvalues for the action of x on Z. So there is no nontrivial homomorphism S/T ⟶ Z that commutes with the actions of x.

Let α ∈ Aut(F) be such that α|_{Z} = Id. Since α commutes with Φ, it must either induce the identity on S/T and on T/Z or invert both quotient groups, and the latter is impossible since S/Z is extraspecial. Since α is the identity on Z and on T/Z, α|_{T} is conjugation by some element of S, and we can assume (modulo Inn(S)) that α|_{T} = Id. Thus there is φ ∈ Hom(S/T, Z) such that α(g) = gφ(gT) for each g ∈ S, and φ commutes with the action of xS ∈ N_{G}(S)/S. We just showed that this is only possible when φ = 1, and conclude that α = Id_{S}.

Thus Aut(F) is isomorphic to a subgroup of Aut(Z) ≅ GL_{2}(3). Since Aut_{G}(S) ≅ C_{8} acts faithfully on Z, and the Sylow 2-subgroups of GL_{2}(3) are semidihedral of order 16, this shows that |Aut(F)| ≤ 16 and |Out(F)| ≤ 2. Since κ_{G} is injective, it is an isomorphism.

Case 2: We now show that κ_{G} is an isomorphism when F is almost simple. Let L be as in Table 0.1. If L ≅ ^{2}F_{4}(2) and p = 3, then Out(F) ≅ Out(L) = 1 = Out(G) since κ_{L} is an isomorphism by [BMO, Proposition 6.9].

Otherwise, set L_{0} = O^{p′}(L) and F_{0} = F_{S}(L_{0}). By [A4, 16.3 & 16.10], F_{0} is simple, and hence Z(F_{0}) = 1, when L_{0} ≅ M_{12} and p = 3, and when L_{0} ≅ Ω_{5}(5) or PSL_{3}(5) and p = 5. Also, κ_{L_{0}} is an isomorphism in these cases by Proposition 3.2 and [BMO, Theorem A], respectively, and L ≅ Aut(L_{0}) and |L/L_{0}| = 2 (hence Out(L) = 1) by [A4, 16.10]. If Out(F) ≠ 1, then there is α ∈ Aut(F) ∖ Aut_{F}(S) such that α|_{S_{0}} = Id, and by the pullback square in [AOV1, Lemma 2.15], this would lie in the image of a nontrivial element of Out(L) = 1. Thus Out(F) = 1, Out(G) = 1 by Table 0.1, and so κ_{G} and hence κ_{G} are isomorphisms. □

It remains to handle the cases (G, p) where the p-fusion system of G is simple.

Proposition 3.2. Let p be an odd prime, and let G be a sporadic simple group whose pfusion system is simple. Then κ_{G} is an isomorphism, except when p = 3 and G ≅ He, in which case |Out(G)| = 2 and |Out(F_{S}(G))| = 1 for S ∈ Syl_{3}(G).

Proof. Fix G and p, choose S ∈ Syl_{p}(G), set F = F_{S}(G), and assume F is simple (see Table 0.1 or [A4, 16.10]). Set L = L_{S}^{c}(G).

The centralizers of all involutions in Aut(G) are listed in, e.g., [GLS, Tables 5.3a–z]. By inspection, for each pair (G, p) in question other than (He, 3) for which Out(G) ≠ 1 (see Tables 3.2 and 3.3), there is no α ∈ Aut(G) of order 2 for which |S| divides |C_{G}(α)|. So by Lemma 1.1(b), κ_{G} is injective in all such cases.

To prove that κ_{G} is an isomorphism (with the one exception), it remains to show that |Out(F)| ≤ |Out(G)|.

Assume S is extraspecial of order p^{3}. Set H = N_{G}(S) and H^{∗} = Out_{G}(S) ≅ H/S. We list in Table 3.2 all pairs (G, p) which occur, together with a description of H^{∗} and of N_{Out(S)}(H^{∗}). To determine |N_{Out(S)}(H^{∗})/H^{∗}| in each case, just recall that GL_{2}(3) ≅ 2·S_{4} ≅ Q_{8}:S_{3}, that PGL_{2}(5) ≅ S_{5}, and that when p = 7 or 13, each subgroup of order prime to p in PGL_{2}(p) is contained in a subgroup isomorphic to D_{2(p±1)} or S_{4} (cf. [Sz1, Theorem 3.6.25]).

(G, p)  (M_{12}, 3) (He, 3) (F_{3}, 5) (He, 7) (O’N, 7) (Fi′_{24}, 7) (F_{1}, 13)
H^{∗}  2^{2}  D_{8}  4·S_{4}  3 × S_{3} 3 × D_{8} 6 × S_{3} 3 × 4·S_{4}
N_{Out(S)}(H^{∗})  D_{8}  SD_{16}  4·S_{4}  6 × S_{3} 3 × D_{16} C_{6} ≀ C_{2} 3 × 4·S_{4}
|Out(G)|  2  2  1  2  2  2  1

Ref.

[GLS, 5.3b] [He, 3.9] [W11, § 3] [He, 3.23]

[GLS, Tbl.5.3s,v]

[W10, § 11]

Table 3.2

In all cases, we have

|Out(F)| ≤ |Out(N_{G}(S))| ≤ |N_{Out(S)}(H^{∗})/H^{∗}|.

The first inequality holds by Lemma 1.2(b). The second holds by Lemma 1.3(a), applied with H = N_{G}(S), and since H^{1}(H^{∗}; Z(S)) = 0 (Lemma 1.5(b)). By Table 3.2, |N_{Out(S)}(H^{∗})/H^{∗}| = |Out(G)| in all cases. Hence |Out(F)| ≤ |Out(G)|, and so κ_{G} is an isomorphism if it is injective.

If G ≅ He and p = 3, then H^{∗} = Out_{G}(S) ≅ D_{8} permutes the four subgroups of index 3 in S ≅ 3_{+}^{1+2} in two orbits of two subgroups each. As described in [Bt, Proposition 10] (see also [GLS, Table 5.3p, note 4]), the subgroups in one of the orbits are 3A-pure while those in the other have 3A- and 3B-elements, so no fusion preserving automorphism of S exchanges them. So while |N_{Out(S)}(H^{∗})/H^{∗}| = 2, we have |Out(F)| ≤ |Out^{0}(H)| = 1 by Lemma 1.2(b). Thus κ_{G} is split surjective (and G tamely realizes F_{S}(G)), but it is not an isomorphism.

Assume |S| ≥ p^{4}. Consider the subgroups H < G described in Table 3.3. In all cases, we can assume H ≥ S.

G p Case  H  |Out(H)| |Out(G)|  K  N(−) Reference
Co^{3} 3 4  3_{+}^{1+4}.4S_{6}  1  1  3A [Fi, 5.12]
Co^{2} 3 3b  3_{+}^{1+4}.2_{−}^{1+4}.S_{5}  1  1  2_{−}^{1+4}  3A [W1, § 3]
Co^{1} 3 3a  3_{+}^{1+4}.Sp_{4}(3).2  1  1  Sp_{4}(3) 3C [Cu2, p.422]
McL 3 2  3_{+}^{1+4}.2S_{5}  2  2  2·(5:4) 3A [Fi, Lm.5.5]
Suz 3 1  3^{5}.M_{11}  2  2  J(S) [W2, Thm.]
Ly 3 1  3^{5}.(M_{11} × 2)  1  1  J(S) [Ly1, Tbl.I]
Fi^{22} 3 4  3_{+}^{1+6}.2^{3+4}.3^{2}.2  2  2  3B [W5, p.201]
Fi^{23} 3 4 3_{+}^{1+8}.2_{−}^{1+6}.3_{+}^{1+2}.2S_{4}  1  1  3B [W8, § 1.2]
Fi{′ \atop 24} 3 2  3_{+}^{1+10}.U_{5}(2):2  2  2  2·(11:10) 3B [W8, Th.B]
F_{5} 3 4  3_{+}^{1+4}.4A_{5}  2  2  3B [NW, § 3.2]
F_{3} 3 5  \begin{matrix}3^{2}.3^{3+4}.GL_{2}(3) \\ 3.[3^{8}].GL_{2}(3) \end{matrix}  1  3B^{2} 3B {[A1, 14.1–3] \atop [Pa, §§ 2,4]}
F_{2} 3 3b 3_{+}^{1+8}.2_{−}^{1+6}.SO_{6}^{−}(2)  1  1  2_{−}^{1+6}  3B [W9, § 2]
F_{1} 3 3a  3_{+}^{1+12}.2Suz.2  1  1  2·(13:6) 3B [W10, § 3]
Ly 5 4  5_{+}^{1+4}.4S_{6}  1  1  2A_{6}  5A [Ly1, Tbl.I]
F_{5} 5 3b  5_{+}^{1+4}.2_{−}^{1+4}:5:4  2  2  2_{−}^{1+4}  5B [NW, § 3.3]
F_{2} 5 3b  5_{+}^{1+4}.2_{−}^{1+4}.A_{5}.4  1  1  2_{−}^{1+4}  5B [W9, § 6]
F_{1} 5 3a  5_{+}^{1+6}.4J_{2}.2  1  1  2·(7:6) 5B [W10, § 9]
F_{1} 7 3a  7_{+}^{1+4}.3 × 2S_{7}  1  1  2·(5:4) 7B [W10, § 10]

Table 3.3

Case 1: If G ≅ Suz or Ly and p = 3, then H = N_{G}(J(S)), where J(S) ≅ E_{3^{5}} and H/J(S) ≅ M_{11} or M_{11} × C_{2}, respectively, and |Out(F)| ≤ |Out(H)| by Lemma 1.2(b). Set V = O_{3}(H) = J(S) and H^{∗} = Aut_{H}(V) ≅ H/V. Then V is the Todd module for O^{2}(H^{∗}) ≅ M_{11} (it contains 11 subgroups of type 3A permuted by H^{∗}), so H^{1}(H^{∗}; V) = 0 by [MSt, Lemma 4.1]. Also, V is absolutely 𝔽_{3}H^{∗}-irreducible since H^{∗} > 11:5. So by Lemma 1.3(c.i) and since Out(M_{11}) = 1, |N_{Aut(V)}(H^{∗})/H^{∗}| ≤ 2 if G ≅ Suz (H^{∗} ≅ M_{11}), and is trivial if G ≅ Ly. Lemma 1.3(a) now implies that |Out(H)| ≤ 2 or 1 for G ≅ Suz or Ly, respectively, and hence that |Out(F)| ≤ |Out(G)|.

For each of the remaining pairs (G, p) displayed in Table 3.3, except when G ≅ F_{3} and p = 3 (Case 5), we set Q = O_{p}(H), Q = Q/Z(Q), H^{∗} = Out_{H}(Q), H_{0} = O^{p′}(H), and H_{0}^{∗} = Out_{H_{0}}(Q). Then H is strictly p-constrained and Q is extraspecial, and hence Z(S) = Z(Q) has order p. Also, H = N_{G}(Z(Q)) = N_{G}(Z(S)) by the above references, so |Out(F)| ≤ |Out(H)| by Lemma 1.2(b), and it remains to show that |Out(H)| ≤ |Out(G)|. By Lemma 1.5(b), H^{1}(H^{∗}; Z(Q)) = 0 in each of these cases, and hence Out(H) is sent injectively into the quotient group N_{Out(Q)}(H^{∗})/H^{∗} by Lemma 1.3(a). So it remains to show that |N_{Out(Q)}(H^{∗})/H^{∗}| ≤ |Out(G)|.

Case 2: If G ≅ McL or Fi′_{24} and p = 3, then Q is an absolutely irreducible 𝔽_{p}K-module for K ≤ H^{∗} as given in Table 3.3, and hence an absolutely irreducible 𝔽_{p}H^{∗}-module. So |N_{Out(Q)}(H^{∗})/H^{∗}| ≤ 2 by Lemma 1.3(c): since |Out(2S_{5})| = 2 in the first case, and since Out(U_{5}(2).2) = 1 and Z(U_{5}(2).2) = 1 in the second case.

Case 3: If G ≅ F_{1} and p = 3, then Q splits as a sum of two absolutely irreducible 6dimensional 𝔽_{3}K-modules. Since 5^{2} | |Suz| | |H_{0}^{∗}| while 5^{2} ∤ |GL_{6}(3)|, Q is H_{0}^{∗}-irreducible, hence absolutely H_{0}^{∗}-irreducible by Lemma 1.4(c). In all other cases under consideration, Q is easily checked to be an absolutely irreducible 𝔽_{p}K-module for K ≤ H_{0}^{∗} as given in Table 3.3, and hence an absolutely irreducible 𝔽_{p}H_{0}^{∗}-module.

Thus |Out(F)| ≤ |N_{Out(Q)}(H^{∗})/H^{∗}| ≤ η·|Out(H_{0}^{∗})|/|Out_{H^{∗}}(H_{0}^{∗})| by Lemma 1.3(c.ii), where for Y as in the lemma, η = |Y | = 2 when (G, p) = (F_{5}, 5) (and H^{∗} ≱ Z(Out(Q))), and η = |Y | = 1 otherwise.

In Case 3a, we have Out(H_{0}^{∗}) = Out_{H^{∗}}(H_{0}^{∗}) in all cases, so |Out(F)| = |Out(H)| = 1. In Case 3b, we determine Out(H_{0}^{∗}) by applying Lemma 1.3(a) again, this time with

O_{2}(H_{0}^{∗}) in the role of Q. Since Out(2_{−}^{1+4}) ≅ S_{5} and Out(2_{−}^{1+6}) ≅ SO_{6}^{−}(2), the lemma implies that Out(H_{0}^{∗}) = Out_{H^{∗}}(H_{0}^{∗}) in each case, and hence that |Out(F)| ≤ η. Case 4: We show, one pair (G, p) at a time, that |N_{Out(Q)}(H^{∗})/H^{∗}| ≤ |Out(G)| in each of these five cases.

If G ≅ Co_{3} and p = 3, then Q ≅ 3_{+}^{1+4} and Out(Q) ≅ Sp_{4}(3):2. Set Z = Z(Out(Q)) ≅ C_{2}. Then Out(Q)/Z ≅ PSp_{4}(3):2 ≅ SO_{5}(3) and H^{∗}/Z ≅ C_{2} × S_{6}. Under this identification, the central involution x ∈ Z(H^{∗}/Z) acts as −Id_{V} ⊕ Id_{W} for some orthogonal decomposition V ⊕ W of the natural module 𝔽_{3}^{5}; and since none of the groups Ω_{2}^{±}(3), Ω_{3}(3), or Ω_{4}^{+}(3) has order a multiple of 5, dim(V) = 4 and C_{SO_{5}(3)}(x) ≅ GO_{4}^{−}(3). Since Ω_{4}^{−}(3) ≅ PSL_{2}(9) ≅ A_{6}, this shows that C_{Out(Q)/Z}(x) = H^{∗}/Z ≅ C_{2} × S_{6}. So |N_{Out(Q)}(H^{∗})/H^{∗}| = 1.

If G ≅ F_{5} and p = 3, then Q ≅ 3_{+}^{1+4} and H^{∗} ≅ 4A_{5}. By the argument in the last case, N_{Out(Q)}(Z(H^{∗})) ≅ 4S_{6}, so |N_{Out(Q)}(H^{∗})/H^{∗}| = |N_{S_{6}}(A_{5})/A_{5}| = 2.

When G ≅ Fi_{22} and p = 3, the subgroup H ≅ 3_{+}^{1+6}.2^{3+4}.3^{2}.2 is described in [W5, p. 201]: H^{∗} can be regarded as a subgroup of GL_{2}(3) ≀ S_{3} < Sp_{6}(3).2. More precisely, 2^{3+4} < (Q_{8})^{3} (recall O_{2}(GL_{2}(3)) ≅ Q_{8}) is a subgroup of index 4, one of the factors C_{3} normalizes each Q_{8} and the other permutes them cyclically, and the C_{2} acts by inverting both factors C_{3}. Then N_{Out(Q)}(H^{∗}) ≤ GL_{2}(3) ≀ S_{3} since it must permute the three O_{2}(H^{∗})- irreducible subspaces of Q, so N_{Out(Q)}(H^{∗}) ≅ 2^{3+4}.(S_{3} × S_{3}), and |N(H^{∗})/H^{∗}| = 2.

When G ≅ Fi_{23} and p = 3, the subgroup H is described in [W8, § 1.2]. The subgroup R^{∗} = O_{2}(H^{∗}) ≅ 2_{−}^{1+6} has a unique faithful irreducible representation over 𝔽_{3}, this is 8dimensional, and N_{SL_{8}(3)}(R^{∗})/R^{∗} is sent injectively into Out(R^{∗}) ≅ SO_{6}^{−}(2) ≅ SO_{5}(3). Since H^{∗}/R^{∗} ≅ 3_{+}^{1+2}:2S_{4} is a maximal parabolic subgroup in SO_{5}(3), we get N_{Out(Q)}(H^{∗})/H^{∗} = 1.

If G ≅ Ly and p = 5, then Q is 𝔽_{5}[2A_{6}]-irreducible since 3^{2} ∤ |GL_{3}(5)|, and is absolutely irreducible since 2A_{6} is not a subgroup of SL_{2}(25) (since E_{9} is not a subgroup). Thus |N_{Out(Q)}(H^{∗})/H^{∗}| ≤ |Out(S_{6})| = 2, with equality only if the action of 2A_{6} on Q extends to 2A_{6}.2^{2}. This is impossible, since the two classes of 3-elements in 2A_{6} act differently on Q (note the action of a Sylow 3-subgroup on Q), so N_{Out(Q)}(H^{∗})/H^{∗} = 1. Case 5: When G ≅ F_{3} and p = 3, we work with two different 3-local subgroups. Set V_{1} = Z(S) and V_{2} = Z_{2}(S), and set H_{i} = N_{G}(V_{i}) and Q_{i} = O_{3}(H_{i}) for i = 1, 2. By [A1, 14.1.2 & 14.1.5] and [Pa, § 4], V_{1} ≅ C_{3}, V_{2} ≅ E_{9}, |Q_{1}| = |Q_{2}| = 3^{9}, and H_{1}/Q_{1} ≅ H_{2}/Q_{2} ≅ GL_{2}(3). Note that S ≤ H_{1} ∩ H_{2}, and |S| = 3^{10}. Also, the following hold:

(1) Set V_{5} = Z_{2}(Q_{2}). Then V_{5} = [Q_{2}, Q_{2}] ≅ E_{3^{5}}, Q_{2}/V_{5} ≅ E_{3^{4}}, V_{2} is the natural module for G_{2}/Q_{2} ≅ GL_{2}(3), and V_{5}/V_{2} is the projective absolutely irreducible PSL_{2}(3)-module of rank 3. Also, V_{5}/V_{2} = Z(Q_{2}/V_{2}), and hence Q_{2}/V_{2} is special of type 3^{3+4}. See [A1, 14.2].

(2) By [A1, 14.2.3], the quotient Q_{2}/V_{5} is G_{2}/Q_{2}-indecomposable, and is an extension of one copy of the natural SL_{2}(3)-module by another. Let R_{7} < Q_{2} be such that R_{7} > V_{5}, and R_{7}/V_{5} < Q_{2}/V_{5} is the unique H_{2}/Q_{2}-submodule of rank 2 (thus |R_{7}| = 3^{7}).

(3) We claim that C_{Q_{2}}(V_{5}) = V_{5}. Assume otherwise: then C_{Q_{2}}(V_{5}) ≥ R_{7} since it is normal in H_{2}. So V_{5} ≤ Z(R_{7}), and |[R_{7}, R_{7}]| ≤ 3 since R_{7}/V_{5} ≅ E_{9}. But [R_{7}, R_{7}] < V_{5} is normal in H_{2}, so it must be trivial, and R_{7} is abelian. This is impossible: V_{5} contains elements of all three classes of elements of order 3 [A1, 14.2.2], while the centralizer of a 3A-element is isomorphic to (3 × G_{2}(3)).2 whose Sylow 3-subgroups are nonabelian of order 3^{7}.

(4) Set V_{3} = Z_{2}(Q_{1}); then V_{3} ≅ E_{27}, and V_{3}/V_{1} is the natural module for G_{1}/Q_{1} [A1, 14.3.1]. Since V_{3} ⊴ S and V_{2} = Z_{2}(S) ≅ E_{9}, V_{3} > V_{2}. Also, V_{3}/V_{2} ≤ Z(Q_{2}/V_{2}) = V_{5}/V_{2} since |V_{3}/V_{2}| = 3. Thus V_{2} < V_{3} < V_{5}.

By [A1, 14.3.2], [Q_{1}, Q_{1}] > V_{3}, and Q_{1}/[Q_{1}, Q_{1}] ≅ E_{3^{4}} is G_{1}/Q_{1}-indecomposable and an extension of one copy of the natural SL_{2}(3)-module by another.

(5) Set W_{7} = C_{G}(V_{3}) ≥ V_{5}: a subgroup of S, hence of Q_{1} ∩ Q_{2}, of order 3^{7} [A1, 14.3.4]. We claim that W_{7}/V_{5} = Z(S/V_{5}) = C_{Q_{2}/V_{5}}(S/Q_{2}), where S/V_{5} ≅ C_{3} × (C_{3} ≀ C_{3}) by [A1, 14.2.5]. To see this, note that for each g ∈ Q_{2} such that gV_{5} ∈ C_{Q_{2}/V_{5}}(S/Q_{2}), the map x ↦ [x, g] is S/Q_{2}-linear from V_{5}/V_{2} to V_{2}, so V_{3}/V_{2} = [S, [S, V_{5}/V_{2}]] (see (1)) lies in its kernel. Thus Z(S/V_{5}) ≤ W_{7}/V_{5}, and they are equal since they both have order 9.

(6) To summarize, we have defined two sequences of subgroups

V_{2} < V_{5} < R_{7} < Q_{2} < H_{2} and V_{1} < V_{3} < W_{7} < Q_{1} < H_{1},

those in the first sequence normal in H_{2} and those in the second normal in H_{1}, where V_{m} ≅ E_{3^{m}} and |R_{7}| = |W_{7}| = 3^{7}. In addition, V_{1} < V_{2} < V_{3} < V_{5} < W_{7} < Q_{2}.

Fix β ∈ Aut(F). By Lemma 1.2(a), κ_{H_{2}} is an isomorphism, and hence β extends to an automorphism β_{2} ∈ Aut(H_{2}). Since V_{2} is the natural module for H_{2}/Q_{2} ≅ GL_{2}(3), β_{2}|_{V_{2}} = c_{x} for some x ∈ H_{2}, and x ∈ N_{H_{2}}(S) since β_{2}(S) = S. Upon replacing β by c_{x}^{−1} ∘ β and β_{2} by c_{x}^{−1} ∘ β_{2}, we can arrange that β|_{V_{2}} = Id.

Since β|_{V_{2}} = Id, β_{2} also induces the identity on H_{2}/Q_{2} ≅ GL_{2}(3) (since this acts faithfully on V_{2}), and induces ε·Id on V_{5}/V_{2} ≅ E_{27} for ε ∈ {±1} since it is absolutely irreducible. By (3), the homomorphism Q_{2}/V_{5} ⟶ Hom(V_{5}/V_{2}, V_{2}) which sends g to (x ↦ [g, x]) is injective. Since β induces the identity on V_{2} and ε·Id on V_{5}/V_{2}, it also induces ε·Id on Q_{2}/V_{5}. By (1), [Q_{2}/V_{2}, Q_{2}/V_{2}] = V_{5}/V_{2}, so β acts via the identity on V_{5}/V_{2}. Thus ε = +1, and β also induces the identity on Q_{2}/V_{5}.

Now, H^{1}(H_{2}/Q_{2}; Q_{2}/V_{5}) = 0 by Lemma 1.5(a) (and since the central involution in H_{2}/Q_{2} ≅ GL_{2}(3) inverts Q_{2}/V_{5}). So by Lemma 1.3(b), applied with H_{2}/V_{5} and Q_{2}/V_{5} in the role of H and Q = R, β_{2} ≡ c_{y} modulo V_{5} for some y ∈ Q_{2}. Upon replacing β_{2} by c_{y}^{−1} ∘ β_{2}, we can arrange that [β, H_{2}] ≤ V_{5}.

Next, note that V_{5}/V_{2} = Z(Q_{2}/V_{2}) and Hom_{H_{2}/Q_{2}}(Q_{2}/V_{5}, V_{5}/V_{2}) = 1 by (1) and (2), and H^{1}(H_{2}/Q_{2}; V_{5}/V_{2}) = 0 since V_{5}/V_{2} is H_{2}/Q_{2}-projective. So by Lemma 1.3(b), β ≡ c_{z} (mod V_{2}) for some z ∈ V_{5}. Upon replacing β_{2} by c_{z}^{−1} ∘ β_{2}, we can now arrange that [β_{2}, H_{2}] ≤ V_{2}.

By Lemma 1.3(b), β|_{Q_{2}} has the form β(u) = uχ(uV_{2}) for some χ ∈ Hom_{H_{2}/Q_{2}}(Q_{2}/V_{2}, V_{2}). Also, χ factors through Q_{2}/V_{5} since [Q_{2}, Q_{2}] = V_{5} by (1). By (2), either χ = 1, or χ is surjective with kernel R_{7}/V_{2}. In either case, β|_{R_{7}} = Id. Also, since W_{7}/V_{5} = C_{Q_{2}/V_{5}}(S/Q_{2}) by (5), χ(W_{7}/V_{5}) ≤ C_{V_{2}}(S/Q_{2}) = V_{1}. So [β, W_{7}] ≤ V_{1}.

By Lemma 1.2(a) again, κ_{H_{1}} is an isomorphism, and hence β extends to β_{1} ∈ Aut(H_{1}). Let β ∈ Aut(S/V_{1}) and β_{1} ∈ Aut(H_{1}/V_{1}) be the automorphisms induced by β and β_{1}. We have just shown that β|_{W_{7}} = Id, and that [β_{1}, S/V_{1}] ≤ V_{2}/V_{1}. By Lemma 1.3(b) again, β|_{Q_{1}/V_{1}} has the form β(g) = gψ(gW_{7}) for some ψ ∈ Hom_{H_{1}/Q_{1}}(Q_{1}/W_{7}, V_{3}/V_{1}) with Im(ψ) ≤ V_{2}/V_{1}. Since Q_{1}/W_{7} and V_{3}/V_{1} are natural modules for SL_{2}(3) by (5) and (4), ψ must be surjective or trivial. Since ψ is not surjective, β|_{Q_{1}} = Id. Also, H^{1}(H_{1}/Q_{1}; V_{3}/V_{1}) = 0 by Lemma 1.5(a), so β_{1} ∈ Aut_{V_{3}/V_{1}}(H_{1}/V_{1}) by Lemma 1.3(b).

We can thus arrange, upon replacing β_{1} by c_{w}^{−1} ∘ β_{1} for some w ∈ V_{3}, that β_{1} = Id, and hence that [β_{1}, H_{1}] ≤ V_{1}. (We can no longer claim that [β_{2}, H_{2}] ≤ V_{2}, but this will not be needed.) Set H′_{1} = [H_{1}, H_{1}]. By (4), H′_{1} ≥ Q_{1} and H′_{1}/Q_{1} ≅ SL_{2}(3). Also, V_{1} = Z(H′_{1}), so β_{1}|_{H′_{1}} has the form β_{1}(g) = gϕ(g) for some ϕ ∈ Hom(H′_{1}, V_{1}). But H′_{1} is perfect by (4) again, so ϕ = 1, and β_{1} = Id. Thus Out(F) = 1, and κ_{G} is an isomorphism.

This finishes the proof of Proposition 3.2. □

4. Tools for comparing automorphisms of fusion and linking systems

Throughout this section and the next, we assume p = 2. Many of the definitions and statements given here are well known to hold for arbitrary primes, but we restrict to this case for simplicity. In particular, a strongly embedded subgroup H < G always means a strongly 2-embedded subgroup; i.e., one such that 2||H| while 2 ∤ |H ∩^{g}H| for g ∈ G ∖ H.

Definition 4.1. Fix a finite group G, choose S ∈ Syl_{2}(G), and set F = F_{S}(G).

(a) A subgroup P ≤ S is fully normalized in F if N_{S}(P) ∈ Syl_{2}(N_{G}(P)).

(b) A 2-subgroup P ≤ G is essential if P is 2-centric in G (i.e., Z(P) ∈ Syl_{2}(C_{G}(P))), and Out_{G}(P) has a strongly embedded subgroup. Let E_{2}(G) be the set of all essential 2-subgroups of G.

(c) A subgroup P ≤ S is F-essential if P is fully normalized in F and essential in G. Let E_{F} be the set of all F-essential subgroups of G.

(d) Ẑ(F) = {W ≤ S | W elementary abelian, fully normalized in F, W = Ω_{1}(Z(C_{S}(W))), Aut_{F}(W) has a strongly embedded subgroup}.

Clearly, in the situation of Definition 4.1, E_{F} ⊆ E_{2}(G), while each member of E_{2}(G) is Gconjugate to a member of E_{F}. If W ∈ Ẑ(F) and P = C_{S}(W), then by the following lemma, restriction defines a surjection from Out_{G}(P) onto Aut_{G}(W) with kernel of odd order. Hence Out_{G}(P) also has a strongly embedded subgroup, and P ∈ E_{F}.

Lemma 4.2. Fix a finite group G and S ∈ Syl_{2}(G), and set F = F_{S}(G).

(a) If W ≤ P ≤ G are 2-subgroups such that W = Ω_{1}(Z(P)) and P ∈ Syl_{2}(C_{G}(W)), then restriction induces a surjection Out_{G}(P) ⟶ Aut_{G}(W) with kernel of odd order.

(b) If W ∈ Ẑ(F) and P = C_{S}(W), then P ∈ E_{F}.

Proof. (a) By the Frattini argument, N_{G}(W) ≤ N_{G}(P)C_{G}(W), with equality since W is characteristic in P. So the natural homomorphism

Out_{G}(P) ≅ N_{G}(P)/C_{G}(P)P ⟶ N_{G}(W)/C_{G}(W) ≅ Aut_{G}(W),

induced by restriction of automorphisms or by the inclusion N_{G}(P) ≤ N_{G}(W) is surjective with kernel (N_{G}(P) ∩ C_{G}(W))/C_{G}(P)P of odd order.

(b) If W ∈ Ẑ(F) and P = C_{S}(W), then P ∈ Syl_{2}(C_{G}(W)) and W = Ω_{1}(Z(P)) by definition. So we are in the situation of (a), and Out_{G}(P) has a strongly embedded subgroup since Aut_{G}(W) does. Also, N_{G}(P) ≤ N_{G}(W), while N_{S}(P) = N_{S}(W) ∈ Syl_{2}(N_{G}(W)) since W is fully normalized in F. Hence N_{S}(P) ∈ Syl_{2}(N_{G}(P)), so P is also fully normalized and P ∈ E_{F}. □

Our proof that Ker(μ_{G}) = 1 in all cases is based on the following proposition, which is a modified version of similar results in [AOV1] and [BMO]. In most cases handled in the next section, point (e) suffices to prove that Ker(μ_{G}) = 1.

When α ∈ Aut(L) and P is an object in L, we let α_{P} : Aut_{L}(P) ⟶ Aut_{L}(α(P)) denote the restriction of α to Aut_{L}(P).

Proposition 4.3. Fix a finite group G, choose S ∈ Syl_{2}(G), and set F = F_{S}(G) and L = L_{S}^{c}(G). Each element in Ker(μ_{G}) is represented by some α ∈ Aut(L) such that α_{S} = Id_{Aut_{L}(S)}. For each such α, there are elements g_{P} ∈ C_{Z(P)}(Aut_{S}(P)) = Z(N_{S}(P)), defined for each fully normalized subgroup P ∈ Ob(L), for which the following hold:

(a) The automorphism α_{P} ∈ Aut(Aut_{L}(P)) is conjugation by [g_{P}] ∈ Aut_{L}(P), and g_{P} is uniquely determined by α modulo C_{Z(P)}(Aut_{F}(P)). In particular, α_{P} = Id_{Aut_{L}(P)} if and only if g_{P} ∈ C_{Z(P)}(Aut_{F}(P)).

(b) Assume P, Q ∈ Ob(L) are both fully normalized in F. If Q = ^{a}P for some a ∈ S, then we can choose g_{Q} = ^{a}g_{P}.

(c) If Q ≤ P are both fully normalized and are objects in L, then g_{P} ≡ g_{Q} modulo C_{Z(Q)}(N_{G}(P) ∩ N_{G}(Q)).

(d) Assume, for each W ∈ Ẑ(F) and P = C_{S}(W), that g_{P} ∈ C_{Z(P)}(Aut_{F}(P)) (equivalently, that α_{P} = Id_{Aut_{L}(P)}). Then α = Id.

(e) If Ẑ(F) = ∅, then Ker(μ_{G}) = 1. If |Ẑ(F)| = 1, and |Z(S)| = 2 or (more generally) Aut_{F}(Ω_{1}(Z(S))) = 1, then Ker(μ_{G}) = 1.

Proof. Points (a)–(c) are part of [AOV1, Proposition 4.2], (d) follows from [BMO, Proposition A.2(d)], and (e) combines parts (a) and (b) in [BMO, Proposition A.2]. □

The following notation will be useful in the next lemma, and in the next section.

Definition 4.4. For each finite group G and each k ≥ 0, let ℐ_{k}(G) be the set of subgroups H ≤ G such that [G:H] = 2^{k}·m for some odd m. Let ℐ_{≤k}(G) be the union of the sets ℐ_{ℓ}(G) for 0 ≤ ℓ ≤ k.

Lemma 4.5. Let H be a finite group, fix T ∈ Syl_{2}(H), and set F = F_{T} (H). Set Q = O_{2}(H), and assume C_{H}(Q) ≤ Q. Assume W ∈ Ẑ(F), and set P = C_{T} (W). Set V = Ω_{1}(Z(Q)), and set H^{∗} = Aut_{H}(V), P^{∗} = Aut_{P} (V), T^{∗} = Aut_{T} (V), and F^{∗} = F_{T^{∗}}(H^{∗}).

(a) We have W ≤ V, Aut_{H}(W) = Aut_{H^{∗}}(W) has a strongly embedded subgroup, P^{∗} is a radical 2-subgroup of H^{∗}, and N_{H^{∗}}(P^{∗})/P^{∗} has a strongly embedded subgroup.

(b) If H^{∗} is a Chevalley group (i.e., untwisted) over the field 𝔽_{2}, then P^{∗} ∈ E_{F^{∗}} ⊆ ℐ_{1}(H^{∗}). If H^{∗} ≅ SU_{2n}(2) or Ω_{2n}^{−}(2) for n ≥ 2, then P^{∗} ∈ E_{F^{∗}} ⊆ ℐ_{≤2}(H^{∗}).

(c) If H^{∗} ≅ A_{6}, A_{7}, or M_{24}, then P^{∗} ∈ E_{F^{∗}} ⊆ ℐ_{1}(H^{∗}). If H^{∗} ≅ M_{22} or M_{23}, then P^{∗} ∈ E_{F^{∗}} ⊆ ℐ_{≤2}(H^{∗}). If H^{∗} ≅ S_{5}, then P^{∗} ∈ ℐ_{≤2}(H^{∗}).

(d) If H^{∗} ≅ Aut(M_{22}), then P^{∗} ∈ E_{F^{∗}} ⊆ ℐ_{≤2}(H^{∗}), and P^{∗} ∩ O^{2}(H^{∗}) ∈ E_{2}(O^{2}(H^{∗})).

Proof. Fix W ∈ Ẑ(F), and set P = C_{T} (W) as above. Then P ∈ E_{F} by Lemma 4.2(b). Also, W = Ω_{1}(Z(P)) and P ≥ O_{2}(C_{H}(V)) = Q, and hence W ≤ Ω_{1}(Z(Q)) = V.

(a) Since V ⊴ H, each α ∈ Aut_{H}(W) extends to α ∈ Aut_{H}(V) = H^{∗}, and thus Aut_{H^{∗}}(W) = Aut_{H}(W). Hence

N_{H^{∗}}(P^{∗})/P^{∗} ≅ N_{H/Q}(P/Q)/(P/Q) ≅ N_{H}(P)/P ≅ Out_{H}(P),

so this group has a strongly embedded subgroup. In particular, P^{∗} = O_{2}(N_{H^{∗}}(P^{∗})) (see [AKO, Proposition A.7(c)]), so P^{∗} is a radical 2-subgroup of H^{∗}.

(b) Since W ≤ V, W = Ω_{1}(Z(P)) = C_{V} (P^{∗}). By (a), N_{H^{∗}}(P^{∗})/P^{∗} has a strongly embedded subgroup, and O_{2}(N_{H^{∗}}(P^{∗})) = P^{∗}.

If H^{∗} is a group of Lie type over the field 𝔽_{2}, then by the Borel-Tits theorem (see [GLS, Corollary 3.1.5]), N_{H^{∗}}(P^{∗}) is a parabolic subgroup and P^{∗} = O_{2}(N_{H^{∗}}(P^{∗})). Hence P^{∗} ∈ E_{F^{∗}} in this case. Also, O^{2′}(Out_{H^{∗}}(P)) ≅ O^{2′}(H/P) is a central product of groups of Lie type in characteristic 2 (cf. [GLS, Proposition 2.6.5(f,g)]). Since it has a strongly embedded subgroup, it must be isomorphic to SL_{2}(2) ≅ S_{3} (hence P ∈ ℐ_{1}(H^{∗})), or possibly to A_{5} ≅ SL_{2}(4) ≅ Ω_{4}^{−}(2) if H^{∗} ≅ SU_{2n}(2) or Ω_{2n}^{−}(2) for n ≥ 2 (in which case P ∈ ℐ_{2}(H^{∗})). Note that we cannot get SU_{3}(2) since we only consider even dimensional unitary groups.

(c) If H^{∗} ≅ M_{n} for n = 22, 23, 24, then by [GL, pp. 42–44], it is of characteristic 2 type, in the sense that all 2-local subgroups are strictly 2-constrained. So N_{H^{∗}}(P^{∗}) is strictly 2-constrained, P^{∗} is centric in this group, and hence P^{∗} ∈ E_{F^{∗}}. Also, E_{F^{∗}} ⊆ ℐ_{1}(H^{∗}) if H^{∗} ≅ M_{24} [OV, Proposition 6.5], while E_{F^{∗}} ⊆ ℐ_{≤2}(H^{∗}) if H^{∗} ≅ M_{22} or M_{23} [OV, Table 5.2].

The remaining cases (H^{∗} ≅ A_{6}, A_{7}, or S_{5}) are elementary.

(d) The radical 2-subgroups of H^{∗} ≅ Aut(M_{22}) are listed in [Y, Table VIII]. There are just three classes of such subgroups Q for which N(Q)/Q has a strongly embedded subgroup, of which the members of two have index 2 in a Sylow 2-subgroup and those of the third have index 4. Each of them is essential in Aut(M_{22}), and contains with index 2 an essential 2-subgroup of M_{22}. □

We will need to identify the elements of Ẑ(F), when F = F_{S}(G) for a sporadic group G and S ∈ Syl_{2}(G). In most cases, it will turn out that Ẑ(F) = {Z_{2}(S)}, which is why we need some tools for identifying this subgroup.

Lemma 4.6. Let S be a 2-group, and assume W ≤ S is elementary abelian. If [S:C_{S}(W)] = 2, then W ≤ Z_{2}(S) and rk(W) ≤ 2·rk(Z(S)).

Proof. Set Q = C_{S}(W) for short; Q ⊴ S since it has index 2. Then W ≤ Ω_{1}(Z(Q)), and upon replacing W by Ω_{1}(Z(Q)), we can arrange that W ⊴ S.

Fix x ∈ S ∖ Q. Since x^{2} ∈ Q = C_{S}(W), we have [W, S] = [W, x] ≤ C_{W} (x) ≤ Z(S). So W ≤ Z_{2}(S), and rk(W) ≤ 2·rk(Z(S)). □

Lemma 4.7. Fix a finite group G and a Sylow 2-subgroup S ∈ Syl_{2}(G).

(a) If G is one of the sporadic groups J_{4}, Co_{2}, Co_{1}, Suz, Ru, Fi′_{24}, F_{5}, F_{3}, F_{2}, or F_{1}, then |Z(S)| = 2 and Z_{2}(S) ≅ E_{4}. If G ≅ Co_{2}, then Z_{2}(S) has type 2ABB, while in all other cases, the three involutions in Z_{2}(S) lie in the same G-conjugacy class.

(b) If G ≅ Fi_{22}, then Z_{2}(S) ≅ E_{8} is of type 2A_{2}B_{3}C_{2} and contains a subgroup of type 2B^{2}. If G ≅ Fi_{23}, then Z_{2}(S) ≅ E_{16}.

(c) If G ≅ HS, O’N, or Co_{3}, then |Z(S)| = 2 and Z_{2}(S) ≅ C_{4} × C_{2}.

Proof. (a) In each of these cases, we choose Q ⊴ S and H = N_{G}(Q) as follows, where H^{∗} = H/Q ≅ Aut_{H}(Q):

G Co_{1}  Suz  Ru  F_{5}  F_{3} F_{2} F_{1} J_{4} Co_{2} Fi′_{24}
Q 2_{+}^{1+8} 2_{−}^{1+6} 2.2^{4+6} 2_{+}^{1+8} 2_{+}^{1+8} 2_{+}^{1+22} 2_{+}^{1+24} E_{2^{11}} E_{2^{10}} E_{2^{11}}
H^{∗} Ω_{8}^{+}(2) Ω_{6}^{−}(2) S_{5} Ω_{4}^{+}(4):2 A_{9} Co_{2} Co_{1} M_{24} M_{22}:2 M_{24}

References for all of these subgroups are given in the next section.

Assume that |Z(Q)| = 2; i.e., that we are in one of the first seven cases. Then |Z(S)| = 2, and Z_{2}(S) ≤ Q since H^{∗} acts faithfully on Q/Z(Q). Set Q = Q/Z(Q), so that Z_{2}(S)/Z(S) = C_{Z(Q)}(S/Q). If Q is extraspecial, then rk(Z_{2}(S)) ≥ 2: since Q has an odd number of isotropic points (cf. [Ta, Theorem 11.5]), at least one is fixed by S.

When G ≅ Co_{1} or Suz, Q is the natural (orthogonal) module for H^{∗}, so |C_{Q}(S)| = 2 (see [Cu, Theorem 6.15] or [GLS, Theorem 2.8.9]), and hence Z_{2}(S) ≅ E_{4}.

When G ≅ Ru, Q is special of type 2^{4+6}, Z_{2}(Q) ≅ E_{32}, and H/Q acts on Z_{2}(Q)/Z(Q) via the natural action of ΣL_{2}(4) [W4, § 1.4]. So |C_{Z(Q)}(S/Q)| = 2 in this case, and Z_{2}(S) ≅ E_{4}.

When G ≅ F_{5}, a Sylow 2-subgroup of O^{2}(H/Q) ≅ Ω_{4}^{+}(4) acts on Q ≅ (𝔽_{4})^{4} with 1dimensional fixed subgroup. This subgroup lifts to V_{3} < Q, where V_{3} ≅ E_{8} and Aut_{G}(V_{3}) ≅ GL_{3}(2) (see [NW, p. 365]). Thus [V_{3}, S] > Z(S), so Z_{2}(S) < V_{3}, and Z_{2}(S) ≅ E_{4}.

When G ≅ F_{3}, Q as an 𝔽_{2}A_{9}-module satisfies the hypotheses of Lemma 1.6 by [Pa, 3.7], and hence |C_{Q}(S/Q)| = 2 by that lemma.

Assume G ≅ F_{1} or F_{2}. Thus H^{∗} ≅ Co_{1} or Co_{2}, respectively. Set T = S/Q ∈ Syl_{2}(H^{∗}), and let V ⊴ T and K = N_{H^{∗}}(V) be such that K ≅ 2^{11}.M_{24} or 2^{10}.M_{22}:2 and V = O_{2}(K). By [MStr, Lemmas 3.7.b & 3.8.b], |C_{Q}(V)| = 2, and hence |C_{Q}(S/Q)| = 2. So Z_{2}(S) ≅ E_{4} in both cases.

In the remaining three cases, Q is elementary abelian. When G ≅ Co_{2}, Q ≅ E_{2^{10}} is the Golay module (dual Todd module) for H^{∗} ≅ M_{22}:2. Let K < H^{∗} be the hexad subgroup K ≅ 2^{4}:S_{6}, chosen so that K > S^{∗} = S/Q, and set R = O_{2}(K) ≅ E_{16}. Set Q_{1} = C_{Q}(R) and Q_{5} = [R, Q]. By [MStr, Lemma 3.3.b], rk(Q_{1}) = 1, rk(Q_{5}) = 5, and Q_{5}/Q_{1} is the natural module for S_{6} ≅ Sp_{4}(2). Hence Z(S) = C_{Q}(S^{∗}) = Q_{1}, and Z_{2}(S)/Z(S) = C_{Q/Q_{1}}(S^{∗}) = C_{Q_{5}/Q_{1}}(S^{∗}) also has rank 1. So Z_{2}(S) ≅ E_{4}. The two elements in Z_{2}(S) ∖ Z(S) are Sconjugate, and do not lie in 2C since C_{G}(x) ∈ ℐ_{3}(G) for x ∈ 2C (see [W1, Table II]). By [W1, Table II] again, each 2A-element acts on the Leech lattice with character −8, so a subgroup of type 2A^{2} would act fixing only the zero vector, hence cannot be in Co_{2}. Thus Z_{2}(S) has type 2ABB.

Assume G ≅ Fi′_{24} or J_{4}. In both cases, Q ≅ E_{2^{11}} is the Todd module for H^{∗} ≅ M_{24} (see [A3, 34.9] and [J, Theorem A.4]). Let K < H/Q be the sextet subgroup K ≅ 2^{6}:3S_{6}, chosen so that K > S^{∗} = S/Q ∈ Syl_{2}(H^{∗}), and set R = O_{2}(K) ≅ E_{64}. By [MStr, Lemma 3.5.b], there are 𝔽_{2}K-submodules Q_{1} < Q_{7} < Q of rank 1 and 7, respectively, where Q_{1} = C_{Q}(R) and Q_{2} = [R, Q], and where K/R ≅ 3S_{6} acts on Q_{7}/Q_{1} as the dual module to R. Thus Z(S) = Q_{1} and Z_{2}(S)/Z(S) = C_{Q_{7}/Q_{1}}(S^{∗}) ≅ R/[S^{∗}, R]. Since S^{∗} ≅ UT_{5}(2) contains only two subgroups of rank 6, one easily sees that |R/[S^{∗}, R]| = 2, and hence Z_{2}(S) ≅ E_{4}.

In all of the above cases except Co_{2}, S contains a normal elementary abelian subgroup V of rank at least 2 all of whose involutions lie in the same G-conjugacy class. We refer to the lists of maximal 2-local subgroups in the next section, where we can take V = V_{i} = Z(O_{2}(H_{i})), for i = 2 (when G ≅ Co_{1}, Suz, F_{2}, or F_{1}), i = 3 (when G ≅ J_{4}, Ru, or F_{5}), or i = 5 (for G ≅ Fi′_{24} or F_{3}). Since each normal subgroup of order at least 4 contains Z_{2}(S), the involutions in Z_{2}(S) also lie in the same class.

(b) When G ≅ Fi_{22} and S ∈ Syl_{2}(G), Z(S) = ⟨z⟩ has order 2, and H = C_{G}(z) ≅ (2 × 2_{+}^{1+8}):U_{4}(2):2. Set Q = O_{2}(H). Then O^{2}(H/Q) acts faithfully on Q = Q/Z(Q) as a 4-dimensional unitary space over 𝔽_{4}, so dim_{𝔽_{4}}(C_{Q}(S ∩ O^{2}(H))) = 1 [Cu, Theorem 6.15]. An

involution hQ with h ∈ H ∖ O^{2}(H) acts as a field automorphism on the unitary space Q, so dim_{𝔽_{2}}(C_{Q}(S)) = 1. Since |Z(Q)| = 4, this proves that |Z_{2}(S)| ≤ 8.

To see that Z_{2}(S) does contain a subgroup of rank 3, consider a hexad group V ≅ E_{32} normal in S, generated by six transpositions {a_{1}, …, a_{6}} (where a_{1} ⋯ a_{6} = 1), ordered so that Aut_{S}(V) = ⟨(1 2)(3 4), (1 2)(5 6), (1 3)(2 4)⟩. Then

Z(S) = C_{V} (S) = ⟨a_{5}a_{6}⟩ and Z_{2}(S) = ⟨a_{1}a_{2}, a_{3}a_{4}, a_{5}, a_{6}⟩ is of type 2A_{2}B_{3}C_{2},

and ⟨a_{1}a_{2}, a_{3}a_{4}⟩ < Z_{2}(S) has type 2B^{2}.

When G ≅ Fi_{23} and S ∈ Syl_{2}(G), Z(S) ≅ E_{4} contains involutions x, y, z in each of the three classes 2A, 2B, and 2C, respectively. Also, C_{G}(x) ≅ 2Fi_{22}, so we can identify S/⟨x⟩ as a Sylow 2-subgroup of Fi_{22}, whose center lifts to a pair of elements of class 2B and 2C in G. Thus S/Z(S) ≅ T/Z(T) when T ∈ Syl_{2}(Fi_{22}), we already saw that |Z(T/Z(T))| = 4, and so |Z_{2}(S)| = 16. All involutions in Fi_{22} lift to involutions in 2·Fi_{22} < G, so Z_{2}(S) is elementary abelian.

(c) When G ≅ HS or O’N, this follows from the descriptions by Alperin [Alp, Corollary 1] and O’Nan [O’N, § 1] of S as being contained in an extension of the form 4^{3}.L_{3}(2). (In terms of their presentations, Z(S) = ⟨v_{1}^{2}v_{3}^{2}⟩, while Z_{2}(S) = ⟨v_{1}v_{3}, v_{1}^{2}v_{2}^{2}⟩.) When G ≅ Co_{3}, it follows from a similar presentation of S ≤ 4^{3}.(2 × L_{3}(2)) (see, e.g., [OV, § 7]). □

5. Injectivity of μ_{G}

We are now ready to prove, when p = 2, that Ker(μ_{G}) = 1 for each of the sporadic groups G not handled in Proposition 2.1. This will be done in each case by determining the set Ẑ(F) and then applying Proposition 4.3. One can determine Ẑ(F) using the lists of radical 2-subgroups found in [Y] and other papers. However, we decided to do this instead using lists of maximal 2-local subgroups, to emphasize that the details needed to prove this result are only a small part of what is needed to determine the radical subgroups.

Proposition 5.1. Assume p = 2, and let G be a sporadic simple group whose Sylow 2subgroups have order at least 2^{10}. Then Ker(μ_{G}) = 1.

Proof. There are fifteen groups to consider, and we go through the list one or two at a time. In each case, we fix S ∈ Syl_{2}(G) and set F = F_{S}(G), L = L_{S}^{c}(G), and Ẑ = Ẑ(F). When we list representatives for the conjugacy classes of maximal 2-local subgroups of G, we always choose them so that each such H satisfies S ∩ H ∈ Syl_{2}(H). In particular, if H has odd index in G, then H ≥ S and hence O_{2}(H) ⊴ S and Z(O_{2}(H)) ⊴ S (making the choice of H unique in most cases).

In four of the cases, when G ≅ M_{24}, He, Co_{2}, or Fi_{23}, Ẑ has two members, and we use Proposition 4.3(b,c,d) to prove that μ_{G} is injective. In all of the other cases, |Ẑ| = 1 and |Z(S)| = 2, and we can apply Proposition 4.3(e). Recall that by Proposition 4.3, each class in Ker(μ_{G}) contains an element α ∈ Aut(L) which acts as the identity on Aut_{L}(S).

Note that whenever |Z(S)| = 2 and W ≅ E_{4} is normal in S, [S:C_{S}(W)] = 2, and hence W ≤ Z_{2}(S) by Lemma 4.6.

For convenience, we sometimes write A ∼_{H} B to mean that A is H-conjugate to B, and A ≤_{H} B to mean that A is H-conjugate to a subgroup of B.

G ≅ M_{24}, He : We identify S with UT_{5}(2), the group of (5×5) upper triangular matrices over F_{2}. Let e_{ij} ∈ S (for i < j) be the matrix with 1’s on the diagonal, and with unique nonzero off-diagonal entry 1 in position (i, j). Set W_{1} = ⟨e_{15}, e_{25}⟩ and W_{4} = ⟨e_{14}, e_{15}⟩, Q_{i} = C_{S}(W_{i}) for i = 1, 4, and Q_{14} = Q_{1} ∩ Q_{4}. By [OV, Propositions 6.2 & 6.9], Q_{1} and Q_{4} are essential in G, and are the only essential subgroups with noncyclic center. Hence by Lemma 4.2, Ẑ = {W_{1}, W_{4}}. Also, Q_{14} = A_{1}A_{2}, where A_{1} and A_{2} are the unique subgroups of S of type E_{64}, and hence Q_{14} = J(S) is characteristic in S, Q_{1}, and Q_{4}.

Fix α ∈ Aut(L) which is the identity on Aut_{L}(S). By Proposition 4.3(a), there are elements g_{P} ∈ C_{Z(P)}(Aut_{S}(P)), chosen for each P ≤ S which is fully normalized in F and 2-centric in G, such that α|_{Aut_{L}(P)} is conjugation by [g_{P}]. Then g_{Q_{1}} = g_{Q_{14}} = g_{Q_{4}} ∈ Z(S) by point (c) in the proposition, since for i = 1, 4, C_{Z(Q_{14})}(N_{G}(Q_{i})) = 1. Set g = g_{Q_{1}}; upon replacing α by c_{g}^{−1} ∘ α, we can arrange that α|_{Aut_{L}(Q_{i})} = Id for i = 1, 4 without changing α|_{Aut_{L}(S)}. Hence Ker(μ_{G}) = 1 by Proposition 4.3(d).

G ≅ J_{4} : By [KW, § 2], there are four conjugacy classes of maximal 2-local subgroups, represented by:

H_{1} ≅ 2_{+}^{1+12}.3M_{22}:2, H_{3} ≅ 2^{3+12}.(Σ_{5} × L_{3}(2)), H_{10} ≅ 2^{10}:L_{5}(2), H_{11} ≅ 2^{11}:M_{24}.

Set Q_{i} = O_{2}(H_{i}) and V_{i} = Z(Q_{i}) ≅ E_{2^{i}}. Note that H_{10} ∈ ℐ_{1}(G), while H_{i} ≥ S for i ≠ 10.

Fix W ∈ Ẑ and set P = C_{S}(W). Then N_{G}(W) ≤_{G} H_{i} for some i, in which case P ≥_{G} Q_{i} and W ≤_{G} V_{i} by Lemma 4.5(a). Thus i > 1 since rk(W) ≥ 2. By Lemma 4.5(b,c), Aut_{P} (V_{i}) ∈ E_{2}(Aut_{H_{i}}(V_{i})) ⊆ ℐ_{1}(Aut_{H_{i}}(V_{i})), and hence P ∈ ℐ_{1}(H_{i}).

Thus either [S:P] = 2, in which case W = Z_{2}(S) ≅ E_{4} by Lemmas 4.6 and 4.7(a); or i = 10 and [S:P] = 4. In the latter case, since H_{10}/V_{10} ≅ L_{5}(2) acts on V_{10} as Λ^{2}(𝔽_{2}^{5}), we have rk(W) = rk(C_{V_{10}}(P/V_{10})) ≤ rk(C_{V_{10}}([S^{∗}, S^{∗}])) = 2 for S^{∗} ∈ Syl_{2}(H_{10}/V_{10}). So W ≅ E_{4} in all cases.

By [KW, Table 1], there are two classes of four-groups in G whose centralizer has order a multiple of 2^{19}, denoted AAA^{(1)} and ABB^{(1)}, with centralizers of order 2^{20}·3·5 and 2^{19}·3·5, respectively. Thus AAA^{(1)} ∼_{G} Z_{2}(S) (Lemma 4.6), and W lies in one of the two classes. Since Aut_{G}(ABB^{(1)}) is a 2-group, W ≁_{G} ABB^{(1)}. Hence Ẑ = {Z_{2}(S)}, and μ_{G} is injective by Proposition 4.3(e).

G ≅ Co_{3} : By [OV, Proposition 7.3], there is at most one essential subgroup with noncyclic center (denoted R_{1}); and R_{1} ∈ E_{F} since otherwise N_{G}(Z(S)) would control fusion in G. Also, Out_{G}(R_{1}) ≅ S_{3} and Z(R_{1}) ∈ Ẑ by [OV, Propsition 7.5]. So |Ẑ| = 1, and Ker(μ_{G}) = 1 by Proposition 4.3(e). (In fact, it is not hard to see that Ẑ = {Ω_{1}(Z_{2}(S))}.)

G ≅ Co_{2} : By [W1, pp. 113–114], each 2-local subgroup of G is contained up to conjugacy in one of the following subgroups:

H_{1} ≅ 2_{+}^{1+8}.Sp_{6}(2), H_{4} ≅ 2^{4+10}.(S_{3} × S_{5}), H_{5} ≅ (2^{4} × 2_{+}^{1+6}).A_{8}, H_{10} ≅ 2^{10}:M_{22}:2

K_{1} ≅ U_{6}(2):2, K_{2} ≅ McL, K_{3} ≅ M_{23}.

For i = 1, 4, 5, 10, set Q_{i} = O_{2}(H_{i}) and V_{i} = Z(Q_{i}) ≅ E_{2^{i}}.

Recall (Lemma 4.7(a)) that Z_{2}(S) has type 2ABB. Set Z_{2}(S) = {1, x, y_{1}, y_{2}}, where x ∈ 2A and y_{1}, y_{2} ∈ 2B. Thus Z(S) = ⟨x⟩, H_{1} = C_{G}(x), and we can assume H_{5} = C_{G}(y_{1}).

Fix W ∈ Ẑ, and set P = C_{S}(W). Then W ≥ Z(S), so W ∩ 2A ≠ ∅. If rk(W) = 2, then W must have type 2A^{2}. Since each 2A-element acts on the Leech lattice with character −8 [W1, Table II], W would fix only the zero element, and hence cannot be contained in Co_{2}. Thus rk(W) ≥ 3. If [S:P] = 2, then W ≤ Z_{2}(S) by Lemma 4.6, which is impossible since rk(Z_{2}(S)) = 2. So [S:P] ≥ 4.

If N_{G}(W) ≤_{G} K_{2} ≅ McL or N_{G}(W) ≤_{G} K_{3} ≅ M_{23}, then by the list of essential subgroups in these groups in [OV, Table 5.2], rk(W) = rk(Z(P)) ≤ 2. So these cases are impossible.

The subgroup K_{1} ≅ U_{6}(2):2 in Co_{2} is the stabilizer of a triple of 2-vectors in the Leech lattice [Cu1, pp. 561–2], which we can choose to have the form (4, 4, 0, …), (0, −4, 4, …), and (−4, 0, −4, …). Using this, we see that the maximal parabolic subgroups 2_{+}^{1+8}:U_{4}(2):2, 2^{9}:L_{3}(4):2, and 2^{4+8}:(S_{3} × S_{5}) in K_{1} can be chosen to be contained in H_{1}, H_{10}, and H_{4}, respectively. If N_{G}(W) ≤_{G} K_{1}, then it is contained in one of the maximal parabolics by the Borel-Tits theorem, and so N_{G}(W) is also conjugate to a subgroup of one of the H_{i}.

Thus in all cases, we can assume that N_{G}(W) ≤ H_{i} for some i = 1, 4, 5, 10. Then P ≥ Q_{i} and W = Z(P) ≤ V_{i}, so i ≠ 1.

Assume i = 5, and recall that Fr(Q_{5}) = ⟨y_{1}⟩. The image of W in V_{5}/⟨y_{1}⟩ ≅ E_{16} has rank at least 2 since rk(W) ≥ 3, so Aut_{N_{G}(W)}(V_{5}/⟨y_{1}⟩) is the stabilizer subgroup of a projective line and plane in A_{8} ≅ SL_{4}(2) (a line and plane determined by S). So there is at most one member of Ẑ whose normalizer is in H_{5} = C_{G}(y_{1}), and it has rank 3 if it exists.

Now, V_{4} ≥ Z_{2}(S) since it is normal in S. Since Z_{2}(S) has type 2ABB, S_{5} must act on V_{4}^{#} with orbits of order 5 and 10, and has type 2A_{5}B_{10}. So if i = 4, then W is a rank 3 subgroup of the form 2A_{3}B_{4} (the centralizer of a 2-cycle in S_{5}). There is exactly one 2B-element in W whose product with each of the other 2B-elements is in class 2A, so N_{G}(W) ≤_{G} H_{5} = N_{G}(2B): a case which we have already handled.

Assume i = 10, and set H^{∗} = H_{10}/V_{10} ≅ Aut_{H_{10}}(V_{10}) ≅ Aut(M_{22}) and P^{∗} = P/V_{10}. By Lemma 4.5(d), P^{∗}∩O^{2}(H^{∗}) is an essential 2-subgroup of O^{2}(H^{∗}) ≅ M_{22}. Since P ∉ ℐ_{1}(H_{10}), P^{∗} ∩ O^{2}(H^{∗}) has the form 2^{4}:2 < 2^{4}:S_{5} (the duad subgroup) by [OV, Table 5.2], and this extends to P^{∗} ≅ 2^{5}:2 < 2^{5}:S_{5} < Aut(M_{22}). But V_{10}.2^{5} has center V_{4} (see [MStr, Lemma 3.3]), and so we are back in the case i = 4.

Thus Ẑ = {W_{1}, W_{2}}, where rk(W_{i}) = 3 and N_{G}(W_{i}) ≤ C_{G}(y_{i}) ∼_{G} H_{5} for i = 1, 2. (These also correspond to the two 2-cycles in Aut_{S}(V_{4}) < S_{5}.) Set P_{i} = C_{S}(W_{i}). Fix α ∈ Aut(L) which is the identity on Aut_{L}(S), and let g_{i} = g_{P_{i}} ∈ C_{W_{i}}(Aut_{S}(P_{i})) = Z_{2}(S) (i = 1, 2) be as in Proposition 4.3. Thus α|_{Aut_{L}(P_{i})} is conjugation by g_{i}. Since y_{i} ∈ Z(N_{G}(P_{i})), we can replace g_{i} by g_{i}y_{i} if necessary and arrange that g_{i} ∈ Z(S). Then g_{1} = g_{2} by Proposition 4.3(b) and since P_{1} and P_{2} are S-conjugate. Upon replacing α by c_{g_{1}}^{−1} ∘α, we can arrange that α|_{Aut_{L}(Q_{i})} = Id for i = 1, 4 without changing α|_{Aut_{L}(S)}. Hence Ker(μ_{G}) = 1 by Proposition 4.3(d).

G ≅ Co_{1} : There are three conjugacy classes of involutions in G, of which those in 2A are 2-central. By [Cu2, Theorem 2.1], each 2-local subgroup of G is contained up to conjugacy in one of the subgroups

H_{1} ≅ 2_{+}^{1+8}.Ω_{8}^{+}(2), H_{2} ≅ 2^{2+12}.(A_{8} × S_{3}), H_{4} ≅ 2^{4+12}.(S_{3} × 3S_{6}), H_{11} ≅ 2^{11}M_{24};

K_{1} ≅ (A_{4} × G_{2}(4)):2, K_{2} ≅ (A_{6} × U_{3}(3)):2.

Curtis also included Co_{2} in his list, but it is not needed, as explained in [W1, p. 112]. Set Q_{i} = O_{2}(H_{i}) and V_{i} = Z(Q_{i}) ≅ E_{2^{i}}.

Assume W ∈ Ẑ. Then W ≥ Z(S), so W ∩2A ≠ ∅. If W ∩2C ≠ ∅, then N_{G}(W) ≤ H_{i} for some i = 1, 2, 4, 11 by [Cu2, Lemma 2.2] (where the involution centralizer in the statement is for an involution of type 2A or 2C). If W contains no 2C-elements, then by the argument given in [Cu2, p. 417], based on the action of the elements on the Leech lattice, a product of distinct 2A-elements in W must be of type 2A. So in this case, ⟨W ∩ 2A⟩ is 2A-pure, and its normalizer is contained in some H_{i} by [Cu2, Lemma 2.5] (together with Wilson’s remark [W1, p. 112]).

Set P = C_{S}(W); then P ≥ Q_{i} and hence W ≤ V_{i}. Also, i ≠ 1 since rk(W) > 1. By Lemmas 4.5(b,c), Aut_{P} (V_{i}) ∈ E_{2}(Aut_{H_{i}}(V_{i})) ⊆ ℐ_{1}(Aut_{H_{i}}(V_{i})). Since H_{i} ≥ S, we have [S:P] = 2, and W = Ω_{1}(Z(P)) ≤ Z_{2}(S) by Lemma 4.6, with equality since |Z_{2}(S)| = 4 by Lemma 4.7. It follows that Ẑ = {Z_{2}(S)}, and Ker(μ_{G}) = 1 by Proposition 4.3(e).

G ≅ Suz : By [W2, § 2.4], there are three classes of maximal 2-local subgroups which are normalizers of 2A-pure subgroups, represented by

H_{1} ≅ 2_{−}^{1+6}.Ω_{6}^{−}(2),  H_{2} ≅ 2^{2+8}.(A_{5} × S_{3}),  H_{4} ≅ 2^{4+6}.3A_{6}.

Fix W ∈ Ẑ, and set P = C_{S}(W). Since W ≥ Z(S), it contains 2A-elements, and since ⟨W ∩2A⟩ is 2A-pure by [W2, p. 165], N_{G}(W) ≤ H_{i} for some i ∈ {1, 2, 4}. Then P ≥ O_{2}(H_{i}) and W ≤ V_{i} =^{def} Z(O_{2}(H_{i})) by Lemma 4.5(a), so i ≠ 1 since rk(W) ≥ 2. Hence i = 2 or 4, so Aut_{G}(V_{i}) ≅ S_{3} or A_{6}, and Aut_{P} (V_{i}) ∈ E_{2}(Aut_{G}(V_{i})) ⊆ ℐ_{1}(Aut_{G}(V_{i})) by Lemma 4.5(b,c). So [S:P] = 2, and W ≤ Z_{2}(S) by Lemma 4.6, with equality since |Z_{2}(S)| = 4 by Lemma 4.7. Thus Ẑ = {Z_{2}(S)}, and Ker(μ_{G}) = 1 by Proposition 4.3(e).

G ≅ Ru : There are two conjugacy classes of involutions, of which the 2A-elements are 2-central. By [W4, § 2.5], the normalizer of each 2A-pure subgroup is contained up to conjugacy in one of the following subgroups:

H_{1} ≅ 2.2^{4+6}.S_{5} H_{3} ≅ 2^{3+8}.L_{3}(2) H_{6} ≅ 2^{6}.G_{2}(2).

Set Q_{i} = O_{2}(H_{i}) and V_{i} = Z(Q_{i}). For each i = 1, 3, 6, V_{i} is elementary abelian of rank i and 2A-pure.

Fix W ∈ Ẑ, and set P = C_{S}(W) ∈ E_{F}. Then W ≥ Z(S), so W contains 2A-elements. Since the subgroup W_{0} = ⟨W ∩ 2A⟩ is 2A-pure [W4, p. 550], N_{G}(W) ≤ N_{G}(W_{0}) ≤ H_{i} for i ∈ {1, 3, 6}. Since H_{i} is 2-constrained, P ≥ Q_{i} = O_{2}(H_{i}) and W ≤ V_{i} by Lemma 4.5(a). Hence i ≠ 1, since rk(W) ≥ 2.

For i = 3, 6, Aut_{G}(V_{i}) is a Chevalley group over 𝔽_{2}, so by Lemma 4.5(b), Aut_{P} (V_{i}) ∈ ℐ_{1}(Aut_{G}(V_{i})), and hence P ∈ ℐ_{1}(H_{i}). So |P | = 2^{13} (if i = 3) or 2^{11} (if i = 6). Also, W is 2A-pure since V_{i} is. By [W4, § 2.4], there are four classes of subgroups of type 2A^{2}, of which only one has centralizer of order a multiple of 2^{11}, and that one must be the class of Z_{2}(S) (Lemma 4.7). So W = Z_{2}(S) if i = 3, or if i = 6 and rk(W) = 2.

As explained in [W4, § 2.5], if W ≤ V_{6} and rk(W) ≥ 3, then either N_{G}(W) ≤_{G} H_{1}, or N_{G}(W) is in the normalizer of a group of the form 2A^{2} which must be conjugate to Z_{2}(S) by the above remarks, or C_{G}(W) = V_{6}. The first case was already handled. If N_{G}(W) ≤_{G} N_{G}(Z_{2}(S)), then N_{G}(W) ≤_{G} H_{3} by [W4, p. 550], and this case was already handled. If C_{G}(W) = V_{6}, then W = P = V_{6}, which is impossible since G_{2}(2) does not have a strongly embedded subgroup. Thus Ẑ = {Z_{2}(S)}, and μ_{G} is injective by Proposition 4.3(e).

G ≅ Fi_{22}, Fi_{23}, or Fi′_{24} : It will be simplest to handle these three groups together. Their maximal 2-local subgroups were determined in [W5, Proposition 4.4], [Fl], and [W8, Theorem D], and are listed in Table 5.1. To make it clearer how 2-local subgroups of one Fischer group lift to larger ones, we include the maximal 2-local subgroups in Fi_{21} ≅ PSU_{6}(2) (the maximal parabolic subgroups by the Borel-Tits theorem), and give the normalizers in Fi_{24} of the maximal 2-local subgroups of Fi′_{24}. Also, we include one subgroup which is not

PSU_{6}(2) = Fi_{21}  Fi_{22}  Fi_{23}  Fi_{24}
K_{1}  2·Fi_{22}  (2 × 2·Fi_{22}).2
K_{2}  2·Fi_{21}  2^{2}·Fi_{21}.2  (2 × 2^{2}·Fi_{21}).S_{3}
K_{3}  S_{4} × Sp_{6}(2)  S_{4} × Ω_{8}^{+}(2):S_{3}
H_{1}  2_{+}^{1+8}:U_{4}(2) (2×2_{+}^{1+8}:U_{4}(2)).2 (2^{2}×2_{+}^{1+8}).(3×U_{4}(2)).2 (2_{+}^{1+12}).3U_{4}(3).2^{2}
H_{2} 2^{4+8}:(A_{5} × S_{3}) 2^{5+8}:(A_{6} × S_{3})  2^{6+8}:(A_{7} × S_{3})  2^{7+8}:(A_{8} × S_{3})
H_{3}  2^{9}:M_{21}  2^{10}.M_{22}  2^{11}.M_{23}  2^{12}.M_{24}
H_{4}  2^{6}.Sp_{6}(2)  [2^{7}.Sp_{6}(2)]  2^{8}:SO_{8}^{−}(2)
H_{5}  2^{3+12}(SL_{3}(2)×S_{6})

Table 5.1

maximal: H_{4} ≤ Fi_{23} is contained in K_{1}.

As usual, set Q_{i} = O_{2}(H_{i}) and V_{i} = Z(Q_{i}). For each of the four groups Fi_{n}, H_{i} ≥ S for i = 1, 2, 3, 5. We write K_{i}^{(n)}, H_{i}^{(n)}, Q_{i}^{(n)}, or V_{i}^{(n)} when we need to distinguish K_{i}, H_{i}, Q_{i}, or V_{i} as a subgroup of Fi_{n}.

Each of the groups Fi_{n} for 21 ≤ n ≤ 24 is generated by a conjugacy class of 3-transpositions. By [A3, 37.4], for 22 ≤ n ≤ 24, Fi_{n} has classes of involutions J_{m}, for m = 1, 2, 3 when n = 22, 23 and for 1 ≤ m ≤ 4 when n = 24. Each member of J_{m} is a product of m commuting transpositions (its factors): a unique such product except when n = 22 and m = 3 (in which case each x ∈ J_{3} has exactly two sets of factors) and when n = 24 and m = 4. Note that J_{1} = 2A, J_{2} = 2B, and J_{3} = 2C in Fi_{22} and Fi_{23}, while J_{2} = 2A and J_{4} = 2B in Fi′_{24} (and the other two classes are outer automorphisms).

In all cases, K_{1}, K_{2}, and H_{1} are normalizers of sets of (n−22), (n−21), and (n−20) pairwise commuting transpositions. Also, H_{3} is the normalizer of the set of all n transpositions in S; these generate Q_{3} = V_{3} of rank n − 12, and form a Steiner system of type (n − 19, n − 16, n). Then H_{2} is the normalizer of a pentad, hexad, heptad, or octad of transpositions: one of the members in that Steiner system. From these descriptions, one sees, for example, that a subgroup of type K_{i} (i = 1, 2) or H_{i} (i = 1, 2, 3) in Fi_{22} lifts to a subgroup of type K_{i} or H_{i}, respectively, in 2·Fi_{22} < Fi_{23} and in 2Fi_{22}.2 < Fi′_{24}.

By [W5, Lemma 4.2], each 2B-pure elementary abelian subgroup of Fi_{22} (2B = J_{2}) supports a symplectic form for which (x, y) = 1 exactly when conjugation by y exchanges the two factors of x. Then V_{4}^{(22)} is characterized as a subgroup of type 2B^{6} with nonsingular symplectic form. Since each 2B-element in Fi_{22} lifts to a 2B- and a 2C-element in 2·Fi_{22} < Fi_{23}, H_{4}^{(22)} lifts to H_{4}^{(23)} of the form 2^{7}.Sp_{6}(2).

By [W8, Corollary 3.2.3], each elementary abelian subgroup of G ≅ Fi′_{24} supports a symplectic form where (x, y) = 1 if and only if y is in the “outer half” of C_{G}(x) ≅ 2·Fi_{22}.2 or 2_{+}^{1+12}.3U_{4}(3):2. By [W8, Proposition 3.3.3], the form on V_{4}^{(24)} ≅ E_{2^{8}} is nonsingular, and V_{4}^{(24)} contains elements in both classes 2A = J_{2} and 2B = J_{4}. If x ∈ V_{4} ∩ 2A, then V_{4} ∩ O^{2}(C_{G}(x))/⟨x⟩ has rank 6 with nonsingular symplectic form in Fi_{22}, and hence (C_{H_{4}}(x) ∩ O^{2}(C_{G}(x)))/⟨x⟩ is conjugate to H_{4}^{(22)}. Thus H_{4}^{(24)} contains a lifting of H_{4}^{(22)} via the inclusion 2·Fi_{22} < Fi′_{24}.

Fix W ∈ Ẑ, and set P = C_{S}(W). If N_{G}(W) ≤ K_{i} for i = 1 or 2, then since W ≥ Z(S), and O_{2}(K_{i}) does not contain involutions of all classes represented in Z(S) (note that O_{2}(K_{i}^{(24)}) ∩ Fi′_{24} is 2A-pure for i = 1, 2), we have W = (W ∩ F^{∗}(K_{i}))/O_{2}(K_{i}) ≠ 1. Thus N_{G}(W) is a 2-local subgroup of F^{∗}(K_{i})/O_{2}(K_{i}) ≅ Fi_{22} or Fi_{21}, and hence is contained up to conjugacy in one of its maximal 2-local subgroups. So (after applying this reduction twice if i = 1), N_{G}(W) ≤ H_{i} for some 1 ≤ i ≤ 4. We will see below that we can also avoid the case N_{G}(W) ≤ K_{3} (when G ≅ Fi_{23} or Fi′_{24}), and hence that in all cases, N_{G}(W) ≤_{G} H_{i} for some 1 ≤ i ≤ 5.

When G ≅ Fi_{22}, we just showed that (up to conjugacy) we can assume N_{G}(W) ≤ H_{i} for some i = 1, 2, 3, 4. If i = 4, then by Lemma 4.5(b), W = C_{V_{4}}(P/V_{4}) where P/V_{4} ∈ E_{2}(H_{4}/V_{4}) and H_{4}/V_{4} ≅ Sp_{6}(2), so W must be totally isotropic with respect to the symplectic form on V_{4} described above. But in that case, by [W5, Lemma 3.1], the subgroup W^{∗} > W generated by all factors of involutions in W is again elementary abelian, and N_{G}(W) ≤ N_{G}(W^{∗}) ≤ H_{j} for some j = 1, 2, 3.

Thus N_{G}(W) ≤ H_{i} where i ∈ {1, 2, 3}, H_{i} is 2-constrained, and so P = C_{S}(W) ≥ O_{2}(H_{i}) and W = Ω_{1}(Z(P)) ≤ V_{i}. Also, i ≠ 1 since V_{1} has type 2AAB (so Aut_{G}(V_{1}) is a 2-group). Hence i = 2, 3, and H_{i} ∈ ℐ_{0}(G). By Lemma 4.5(c), Aut_{P} (V_{i}) ∈ E_{2}(Aut_{H_{i}}(V_{i})), and either [S:P] = 2, or i = 3 and [S:P] = 4. In this last case, P/V_{3} ≅ 2^{4}:2 is contained in a duad subgroup D ≅ 2^{4}:S_{5} in M_{22}. Also, O_{2}(D) ≅ E_{16} permutes V_{3} ∩ 2A in five orbits of length 4, each of which forms a hexad together with the remaining two transpositions. Hence C_{V_{3}}(O_{2}(D)) has type 2AAB, and cannot contain W.

Thus [S:P] = 2, and hence rk(W) = 2 and W ≤ Z_{2}(S) by Lemma 4.6. By Lemma 4.7(b), Z_{2}(S) has rank 3 and type 2A_{2}B_{3}C_{2}. Since Aut_{G}(W) is not a 2-group, W must be the 2B-pure subgroup of rank 2 in Z_{2}(S). (Note that the factors of the involutions in W form a hexad.) Thus |Ẑ| = 1, and Ker(μ_{G}) = 1 by Proposition 4.3(e).

When G ≅ Fi_{23}, W = Ω_{1}(Z(P)) strictly contains Z(S). Hence rk(W) ≥ 3, and W contains involutions of each type 2A, 2B, and 2C. If |W ∩ 2A| = 1 2, or 3, then N_{G}(W) ≤ K_{1}, K_{2}, or H_{1}, respectively, while if |W ∩ 2A| ≥ 4, then N_{G}(W) ≤ H_{2} or H_{3}, depending on whether or not the transpositions in W are contained in a heptad. So by the above remarks, we can assume in all cases that N_{G}(W) ≤ H_{i} for some i = 1, 2, 3, 4. Since H_{i} is strictly 2-constrained, P ≥ Q_{i} and W ≤ V_{i}. If i = 1, then W = V_{1} since it has rank at least 3, and thus W has type 2A_{3}B_{3}C. The case i = 4 can be eliminated in the same way as it was when G ≅ Fi_{22}.

Assume N_{G}(W) ≤ H_{2} and W ≤ V_{2}, where Aut_{G}(V_{2}) ≅ A_{7}. Write V_{2} ∩ 2A = {a_{1}, …, a_{7}}, permuted by Aut_{G}(V_{2}) ≅ A_{7} in the canonical way. Then (up to choice of indexing), Aut_{P} (V_{2}) is one of the two essential subgroups P_{1}^{∗} = ⟨(1 2)(3 4), (1 2)(5 6)⟩ and P_{2}^{∗} = ⟨(1 2)(3 4), (1 3)(2 4)⟩. Set W_{j} = C_{V_{2}}(P_{j}^{∗}) and P_{j} = C_{S}(W_{j}); thus P_{j}^{∗} = Aut_{P_{j}}(V_{2}) and hence [S:P_{j}] = 2. Also, W_{1} = ⟨a_{1}a_{2}, a_{3}a_{4}, a_{5}a_{6}⟩ has type 2AB_{3}C_{3}, and W_{2} = ⟨a_{5}, a_{6}, a_{7}⟩ has type 2A_{3}B_{3}C (thus W_{2} ∼_{G} V_{1}).

If N_{G}(W) ≤ H_{3} and W ≤ V_{3}, then Aut_{G}(V_{3}) ≅ M_{23} has three essential subgroups, of which two are contained in the heptad group 2^{4}:A_{7} and one in the triad group 2^{4}:(3 × A_{5}):2. In the first case, the subgroup 2^{4} acts on V_{3} ∩2A fixing a heptad, and we are back in the case N_{G}(W) ≤ H_{2}. In the second case, the subgroup 2^{4} fixes a rank 3 subgroup in V_{3} generated by three tranpositions, and so the essential subgroup 2^{4}:2 fixes only Z(S).

Thus Ẑ = {W_{1}, W_{2}}, where W_{1}, W_{2} ≤ Z_{2}(S) by Lemma 4.6, and W_{1}, W_{2} < V_{2}. Also, σ_{2} = (5 6 7) normalizes P_{2} and Q_{2} and permutes the three 2B-elements in W_{1} cyclically, while σ_{1} = (1 3 5)(2 4 6) normalizes P_{1} and Q_{2} and permutes the three 2A-elements in W_{2} cyclically.

Fix α ∈ Aut(L) which is the identity on Aut_{L}(S). Let g_{P} ∈ C_{Z(P)}(Aut_{S}(P)), for all P ∈ Ob(L) fully normalized in F, be as in Proposition 4.3. Thus α|_{Aut_{L}(P)} is conjugation by g_{P}. Set g = g_{Q_{2}} ∈ C_{Z(Q_{2})}(Aut_{S}(Q_{2})) = Z(S). Upon replacing α by c_{g}^{−1} ∘ α, we can arrange that g_{Q_{2}} = 1, and hence that α is the identity on Aut_{L}(Q_{2}). Since Z(S) = Z(N_{G}(S)) (recall Z(S) has type 2ABC), α is still the identity on Aut_{L}(S).

Set P_{j} = C_{S}(W_{j}) (j = 1, 2). By Proposition 4.3(c) and since σ_{j} normalizes P_{j} and Q_{2}, g_{P_{1}} ≡ g_{Q_{2}} = 1 modulo ⟨W_{1} ∩ 2A⟩, and g_{P_{2}} ≡ g_{Q_{2}} = 1 modulo ⟨W_{2} ∩ 2C⟩. Also, ⟨W_{1} ∩ 2A⟩ ≤ Z(N_{G}(P_{1})) and ⟨W_{2} ∩ 2C⟩ ≤ Z(N_{G}(P_{2})) (since N_{G}(P_{i}) ≤ N_{G}(W_{i})). Thus α|_{Aut_{L}(P_{j})} = Id for j = 1, 2, so α = Id by Proposition 4.3(d). This proves that Ker(μ_{G}) = 1.

When G ≅ Fi′_{24}, since W ≥ Z(S), it contains at least one 2B-element (recall 2A = J_{2} and 2B = J_{4}). By Propositions 3.3.1, 3.3.3, 3.4.1, and 3.4.2 in [W8] (corrected in [LW, § 2]), the normalizer of every elementary abelian 2-subgroup of G is contained up to conjugacy in K_{1}, K_{2}, or one of the H_{i} for i ≤ 5, except when it is 2A-pure and the symplectic form described above is nonsingular. So we can assume that N_{G}(W) is contained in one of these groups. Together with earlier remarks, this means that we can eliminate all of the K_{i}, and assume that N_{G}(W) ≤ H_{i} for some 1 ≤ i ≤ 5. So P ≥ Q_{i} and W ≤ V_{i}, and i ≠ 1 since rk(V_{1}) = 1.

By Lemma 4.5(b,c), W = C_{V_{i}}(P^{∗}), where P^{∗} = Aut_{P} (V_{i}) is an essential 2-subgroup of H_{i}^{∗} = Aut_{H_{i}}(V_{i}). If i = 2, 3, 5, then P^{∗} ∈ ℐ_{1}(H_{i}^{∗}) by Lemma 4.5(b,c), and hence [S:P] = 2 since H_{i} ≥ S. So W = Z_{2}(S) in these cases by Lemmas 4.7(a) and 4.6.

If i = 4, then H_{i}^{∗} ≅ Ω_{8}^{−}(2), and the conditions P^{∗} ∈ E_{2}(H_{4}^{∗}) and rk(C_{V} (P^{∗})) ≥ 2 imply that N_{G}(W) ≅ 2^{8}.(2^{3+6}.(S_{4} × 3)) (the stabilizer of an isotropic line and plane in the projective space of V_{4}). Hence rk(W) = 2 and |P | = 2^{19}. By [W8, Table 15], there are only two classes of four-groups in G with centralizer large enough, one of type 2AAB (impossible since Aut(W) is not a 2-group), and the other Z_{2}(S) of type 2B^{2}. Thus Ẑ = {Z_{2}(S)}, and Ker(μ_{G}) = 1 by Proposition 4.3(e).

G ≅ F_{5} : By [NW, § 3.1], each 2-local subgroup of G is contained up to conjugacy in one of the subgroups

H_{1} ≅ 2_{+}^{1+8}.(A_{5} × A_{5}):2,  H_{3} ≅ 2^{3}.2^{2}.2^{6}.(3 × L_{3}(2)),  H_{6} ≅ 2^{6}·U_{4}(2),

K_{1} ≅ 2·HS:2, K_{2} ≅ (A_{4} × A_{8}):2 < A_{12}.

As usual, set Q_{i} = O_{2}(H_{i}) and V_{i} = Z(Q_{i}) for i = 1, 3, 6. Then V_{1} and V_{3} are 2B-pure, and O_{2}(K_{1}) and O_{2}(K_{2}) are 2A-pure. By [NW, § 3.1], for each elementary abelian 2-subgroup V ≤ G, there is a quadratic form 𝔮: V → 𝔽_{2} defined by sending 2A-elements to 1 and 2B-elements to 0.

Fix W ∈ Ẑ, and set P = C_{S}(W). Then W ≥ Z(S), so W ∩ 2B ≠ ∅. So either the quadratic form 𝔮 on W is nondegenerate and rk(W) ≥ 3, or there is a 2B-pure subgroup W_{0} ≤ W such that N_{G}(W) ≤ N_{G}(W_{0}). By [NW, § 3.1], in this last case, N_{G}(W_{0}) ≤ H_{i} for i = 1 or 3.

If N_{G}(W) ≤ N_{G}(W_{0}) ≤ H_{i} for i = 1, 3, then P ≥ O_{2}(H_{i}), so W ≤ V_{i}. In particular, i ≠ 1. If N_{G}(W) ≤ H_{3}, then P has index 2 in S since Aut_{G}(V_{i}) ≅ L_{3}(2), so W = Z_{2}(S) by Lemmas 4.6 and 4.7(a).

Now assume 𝔮 is nondegenerate as a quadratic form (and rk(W) ≥ 3). Let W^{∗} < W be a 2A-pure subgroup of rank 2, and identify C_{G}(W^{∗}) with (2^{2} × A_{8}) < A_{12} < G. If rk(W) = 3, then we can identify W with ⟨(1 2)(3 4), (1 3)(2 4), (5 6)(7 8)⟩, so C_{G}(W) ≅ 2^{2} × (2^{2} × A_{4}):2, P = C_{S}(W) ≅ 2^{2}×(2^{4}:2), Z(P) ≅ 2^{4}, which contradicts the assumption that W = Ω_{1}(Z(P)). If rk(W) ≥ 4, then it must be conjugate to one of the subgroups (1), (2), or (3) defined in [NW, p. 364] (or contains (2) or (3) if rk(W) = 5). Then C_{G}(W) ≅ E_{2^{6}} or E_{16} × A_{4}, so P = W ∼_{G} V_{6}, which is impossible since Aut_{G}(V_{6}) ≅ U_{4}(2) does not contain a strongly embedded subgroup.

Thus Ẑ = {Z_{2}(S)}, and μ_{G} is injective by Proposition 4.3(e).

G ≅ F_{3} : By [W11, Theorem 2.2], there are two classes of maximal 2-local subgroups of G, represented by H_{1} ≅ 2_{+}^{1+8}.A_{9} and H_{5} ≅ 2^{5}.SL_{5}(2). Set Q_{i} = O_{2}(H_{i}) and V_{i} = Z(Q_{i}) ≅ E_{2^{i}} (i = 1, 5).

Fix W ∈ Ẑ, set P = C_{S}(W), and let i = 1, 5 be such that N_{G}(W) ≤ H_{i}. Then P ≥ O_{2}(H_{i}) and W ≤ V_{i}, so i = 5. By Lemma 4.5(b), P/V_{5} ∈ E_{2}(H_{5}/V_{5}) (where H_{5}/V_{5} ≅ L_{5}(2)) and [S:P] = 2. Hence W ≤ Z_{2}(S) by Lemma 4.6. Since |Z_{2}(S)| = 4 by Lemma 4.7, this proves that Ẑ = {Z_{2}(S)}, and hence that Ker(μ_{G}) = 1 by Proposition 4.3(e).

G ≅ F_{2}, F_{1} : If G ≅ F_{1}, then by [MS, Theorem 1], there are maximal 2-local subgroups of the form

H_{1} ≅ 2^{1+24}.Co_{1}, H_{2} ≅ 2^{2}.[2^{33}].(M_{24} × S_{3}), H_{3} ≅ 2^{3}.[2^{36}].(L_{3}(2) × 3·S_{6}),
H_{5} ≅ 2^{5}.[2^{30}].(S_{3} × L_{5}(2)), H_{10} ≅ 2^{10+16}·Ω_{10}^{+}(2),

If G ≅ F_{2}, then by [MS, Theorem 2], there are maximal 2-local subgroups of the form

H_{1} ≅ 2^{1+22}.Co_{2}, H_{2} ≅ 2^{2}.[2^{30}].(M_{22}:2 × S_{3}), H_{3} ≅ 2^{3}.[2^{32}].(L_{3}(2) × S_{5}),
H_{5} ≅ 2^{5}.[2^{25}].L_{5}(2), H_{9} ≅ 2^{9+16}·Sp_{8}(2),

As usual, we set Q_{i} = O_{2}(H_{i}), and V_{i} = Z(Q_{i}) ≅ E_{2^{i}}. In both cases (G ≅ F_{1} or F_{2}), H_{1} = C_{G}(x) ≥ S for x ∈ 2B, and H_{i} > S (V_{i} ⊴ S) for each i.

Fix W ∈ Ẑ, and set P = C_{S}(W). Then W ≥ Z(S), and hence W contains 2B-elements. By [Mei, Lemma 2.2], W is “of 2-type”, in the sense that C_{G}(O_{2}(C_{G}(W))) is a 2-group, since the subgroup generated by a 2B-element is of 2-type. In particular, C_{G}(P) is a 2-group and hence C_{G}(P) = Z(P).

A 2B-pure elementary abelian 2-subgroup V ≤ G is called singular if V ≤ O_{2}(C_{G}(x)) for each x ∈ V^{#}. If G ≅ F_{1}, then by [MS, Proposition 9.1], applied with P in the role of Q and t = 1, there is a subgroup W_{0} ≤ W such that N_{G}(W) ≤ N_{G}(W_{0}), and either W_{0} is 2B-pure and singular or W = W_{0} ∼_{G} V_{10}. Since Aut_{G}(V_{10}) ≅ Ω_{10}^{+}(2) has no strongly embedded subgroup, W_{0} must be 2B-pure and singular, and hence N_{G}(W) ≤ H_{i} for some i = 1, 2, 3, 5 by [MS, Theorem 1]. Thus P ≥ Q_{i} and W ≤ V_{i}, so W is also 2B-pure and singular.

If G ≅ F_{2}, identify G = C_{M}(x)/x, where M ≅ F_{1} and x is a 2A-element in M. Let P̃ ≤ C_{M}(x) be such that x ∈ P̃ and P̃/⟨x⟩ = P, and set W̃ = Ω_{1}(Z(P̃)). Then x ∈ W̃ and W̃/⟨x⟩ ≤ W, and (W ∩ 2B) ⊆ W̃/⟨x⟩ since 2B-elements in G lift to pairs of involutions of classes 2A and 2B in M (coming from a subgroup of type 2BAA in Q_{1} < M). By [MS, Proposition 9.1] again, applied with P̃ in the role of Q and t = x, there is a subgroup W̃_{0} ≤ W̃ such that N_{M}(W̃) ≤ N_{M}(W̃_{0}), and either W̃_{0} is 2B-pure and singular or W̃ = W̃_{0} ∼_{M} V_{10}^{(M)}. In the latter case, W ∼_{G} V_{9}, which is impossible since Aut_{G}(V_{9}) ≅ Sp_{8}(2) has no strongly embedded subgroup. Again, we conclude that N_{G}(W) ≤ H_{i} for some i = 1, 2, 3, 5 [MS, Theorem 2], and that W ≤ V_{i} by Lemma 4.5(a) and hence is 2B-pure and singular.

By [MS, Lemma 4.2.2], applied with W = 1 (if G ≅ F_{1}) or W = ⟨x⟩ (G ≅ F_{2}), the automizer of a singular subgroup is its full automorphism group. Since GL_{n}(2) has no strongly embedded subgroup for n ≠ 2, this implies that rk(W) = 2. By [MS, Lemma 4.4], if we identify Q_{1}/V_{1} ≅ E_{2^{24}} with the mod 2 Leech lattice, then 2A-elements correspond to the 2-vectors and 2B-elements to the classes of 4-vectors, and hence H_{1}/Q_{1} ≅ Co_{1} acts transitively on each. So F_{1} contains a unique class of singular subgroups of rank 2. A similar argument, using [MS, Corollary 4.6], now shows that F_{2} also contains a unique class of singular subgroup of rank 2. Since H_{2} > S, each of these classes has a representative normal in S, so W ⊴ S, and W = Z_{2}(S) by Lemmas 4.7(a) and 4.6.

To conclude, we have now shown that Ẑ = {Z_{2}(S)} in both cases. Hence Ker(μ_{G}) = 1 by Proposition 4.3(e).

This finishes the proof of Proposition 5.1. □

By inspection in the above proof, in all cases where Z_{2}(S) ≅ E_{4} and its involutions are G-conjugate, we have Ẑ(F) = {Z_{2}(S)}. A general result of this type could greatly shorten the proof of Proposition 5.1, but we have been unable to find one. The following example shows that this is not true without at least some additional conditions.

Set G = 2^{4}:15:4 ≅ 𝔽_{16} ⋊ Γ L_{1}(16). Set E = O_{2}(G) ≅ E_{16}, fix S ∈ Syl_{2}(G), and let P ⊴ S be the subgroup of index 2 containing E. Then Z_{2}(S) = Z(P) ≅ E_{4} and Aut_{G}(Z_{2}(S)) ≅ S_{3}, while Ẑ(F_{S}(G)) = {Z_{2}(S), E}.

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32 BOB OLIVER

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