EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO

BOB OLIVER

Abstract. We prove here the Martino-Priddy conjecture for the prime 2: the 2completions of the classifying spaces of two groups G and G′ are homotopy equivalent if and only if there is an isomorphism between their Sylow 2-subgroups which preserves fusion. This is a consequence of a technical algebraic result, which says that for a finite group G, the second higher derived functor of the inverse limit vanishes for a certain functor Z_{G} on the 2-subgroup orbit category of G. The proof of this result uses the classification theorem for finite simple groups.

In their paper [MP], John Martino and Stewart Priddy conjectured that for any prime p and any pair G, G′ of finite groups, the p-completed classifying spaces BG_{p}^{∧} and BG′_{p}^{∧} are homotopy equivalent if and only if the p-local structures of G and of G′ are isomorphic in a sense to be made precise below. In an earlier paper [BLO] in collaboration with Carles Broto and Ran Levi, we identified the obstruction groups to constructing a homotopy equivalence between these spaces, given an isomorphism between the p-local structures. For odd primes p, these groups have already been shown [Ol] to vanish in all cases. The main technical result of this paper is that these obstruction groups also vanish when p = 2. The proof of this result (like the proof of the conjecture for odd primes) depends on the classification theorem for finite simple groups.

Fix a prime p and a finite group G. For any pair of subgroups P, Q ≤ G, let N_{G}(P, Q) denote the transporter:

N_{G}(P, Q) = {x ∈ G | xP x^{−1} ≤ Q}.

The p-subgroup orbit category of G is the category O_{p}(G) whose objects are the psubgroups of G, and where

Mor_{O_{p}(G)}(P, Q) = Q\N_{G}(P, Q) ≅ Map_{G}(G/P, G/Q).

A p-subgroup P ≤ G is called p-centric if Z(P) is a Sylow p-subgroup of C_{G}(P), or equivalently if C_{G}(P) = Z(P) × C′_{G}(P) for some subgroup C′_{G}(P) of order prime to p. Define the functor

Z_{G}: O_{p}(G)^{op} ⟶ Ab

by setting Z_{G}(P) = Z(P) if P is p-centric in G and Z_{G}(P) = 0 otherwise, and letting

c_{x} ∈ Mor_{O_{p}(G)}(P, Q) induce the morphism Z(Q) ⟶^{c_{x}^{−1}} Z(P) if P and Q are both pcentric. Our main algebraic result is the following.

Theorem A. For any finite group G,

\underleftarrow{lim}^{i}_{O_{2}(G)}(Z_{G}) = 0 for all i ≥ 2.

1991 Mathematics Subject Classification. Primary 55R37. Secondary 55P60, 20D05.

Key words and phrases. Classifying space, p-completion, finite simple groups.

Partially supported by UMR 7539 of the CNRS.

Proof. In Proposition 2.9, we show that \underleftarrow{lim}^{i}(Z_{G}) = 0 for all i ≥ 2 if each nonabelian simple group L which appears in the decomposition series of G belongs to a certain class 𝔏^{≥2}(2). We then show that 𝔏^{≥2}(2) contains all alternating groups (Theorem 5.1); all simple groups of Lie type in characteristic two including the Tits group (Theorem 6.3); all simple groups of Lie type in odd characteristic (Theorems 7.4 and 8.11); and all sporadic groups (Theorem 9.1). Theorem A then follows from the classification theorem for finite simple groups. □

Theorem A was motivated by studying equivalences between completed classifying spaces of finite groups. Let p be a prime, let G and G′ be finite groups, and let S ≤ G and S′ ≤ G′ be Sylow p-subgroups. An isomorphism φ: S ⟶^{≅} S′ is called fusion preserving if for all P, Q ≤ S and all P ⟶_{≅}^{α} Q, α is conjugation by an element of G

if and only if φ(P) ⟶_{≅}^{φαφ^{−1}} φ(Q) is conjugation by an element of G′.

The Martino-Priddy conjecture states that for any prime p, and any pair G, G′ of finite groups, BG_{p}^{∧} ≃ BG′_{p}^{∧} if and only if there is a fusion preserving isomorphism between Sylow p-subgroups of G and G′. The “only if” part of the conjecture was proved by Martino and Priddy [MP], and follows from the bijection

Rep(P, G) =^{def} Hom(P, G)/ Inn(G) ⟶^{≅} [BP, BG_{p}^{∧}]

for any p-group P and any finite group G [BL, Proposition 2.1]. Conversely, by [BLO, Proposition 6.1], given a fusion preserving isomorphism between Sylow p-subgroups of G and G′, the obstruction to extending it to a homotopy equivalence BG_{p}^{∧} ≃ BG′_{p}^{∧} lies in \underleftarrow{lim}^{2}(Z_{G}). Hence Theorem A implies:

Theorem B (Martino-Priddy conjecture at the prime 2). For any pair G and G′ of finite groups with Sylow 2-subgroups S ≤ G and S′ ≤ G′, BG_{2}^{∧} ≃ BG′_{2}^{∧} if and only if there is a fusion preserving isomorphism S ⟶^{≅} S′.

We next turn to the question of self equivalences of BG_{p}^{∧}. For any space X, let Out(X) denote the group of homotopy classes of self homotopy equivalences of X. For any finite group G, any prime p, and any Sylow p-subgroup S ≤ G, let Aut_{fus}(S) be the group of fusion preserving automorphisms of S, let Aut_{G}(S) be the group of automorphisms induced by conjugation by elements of G (i.e., elements of N_{G}(S)), and set

Out_{fus}(S) = Aut_{fus}(S)/ Aut_{G}(S).

Theorem A, when combined with [BLO, Theorem 6.2], gives the following description, up to extension, of Out(BG_{p}^{∧}).

Theorem C. For any finite group G with Sylow 2-subgroup S ≤ G, there is a short exact sequence

1 ⟶ \underleftarrow{lim}^{1}_{O_{2}(G)}(Z_{G}) ⟶ Out(BG_{2}^{∧}) ⟶ Out_{fus}(S) ⟶ 1.

In [Ol], we showed that when p is odd, the groups \underleftarrow{lim}^{i}(Z_{G}) vanish for all finite G and all i ≥ 1. In contrast, when p = 2, the groups \underleftarrow{lim}^{1}(Z_{G}) can be nonvanishing. Examples of this include the groups G = P SL_{2}(q) when q ≡ ±1 (mod 8), G = A_{n} when n ≡ 2, 3 (mod 4), and G = P SL_{4}(q) when q ≡ 3 (mod 4). A simple proof of this when G has dihedral Sylow 2-subgroup is given in Proposition 1.6, and other cases are discussed in Section 10.

If G is an arbitrary finite group, then Z_{G} can be filtered by subfunctors Z_{G}^{K}, defined for all K ◁ G by setting

Z_{G}^{K}(P) = \begin{cases}Z(P) ∩ K & if P is p-centric in G \\ 0 & otherwise. \end{cases}

The idea of the proof of Theorem A is to filter G by a maximal sequence 1 = K_{0} ≤ K_{1} ≤ ⋯ ≤ K_{n−1} ≤ K_{n} = G of normal subgroups, and then analyze higher limits of the Z_{G}^{K_{j}}/Z_{G}^{K_{j−1}}. In particular, \underleftarrow{lim}^{i}(Z_{G}) = 0 for all i ≥ 2 if the same holds for higher limits of Z_{G}^{K_{j}}/Z_{G}^{K_{j−1}} for all j.

We first reduce the general computation of \underleftarrow{lim}^{∗}(Z_{G}^{K_{j}}/Z_{G}^{K_{j−1}}) to the special case where L = K_{j} is quasisimple (i.e., L is perfect and L/Z(L) is simple) with p-group center A = K_{j−1} = Z(L). This is done in Lemmas 2.1 and 2.4. We then observe that in this case, \underleftarrow{lim}^{∗}(Z_{G}^{L}/Z_{G}^{A}) depends only on L and on Aut_{G}(L). This motivates the definition of new functors Y_{L}^{Γ} on O_{p}(Γ), defined for any quasisimple group L with p-group center and any Γ ≤ Aut(L) which contains Inn(L), with the property that

\underleftarrow{lim}^{∗}(Z_{G}^{L}/Z_{G}^{A}) ≅ \underleftarrow{lim}^{∗}(Y_{L}^{Γ})

for any A ◁ L ◁ G as above with Γ = Aut_{G}(L). For example, if L is simple (and identified with Inn(L) ◁ Γ), then Y_{L}^{Γ} = Z_{Γ}^{L}. The definition of Y_{L}^{Γ} in the general case is given in Definition 2.5.

To simplify notation in the rest of the paper, we then define 𝔏^{i}(p) to be the class of simple groups L for which \underleftarrow{lim}^{i}(YΓ ; L̃) = 0 for all choices of central extensions L̃ of L and Γ ≤ Aut(L̃). We are thus reduced to proving that all simple groups lie in 𝔏^{≥2}(2).

In Section 4, we define, for each nonabelian finite simple group L, certain sets ℜ^{i}(L; p) of p-subgroups which could “contribute” to \underleftarrow{lim}^{i}(−), in a way made precise in Definition 4.1. We then show (Proposition 4.2) that L ∈ 𝔏^{i}(p) if there is a p-centric subgroup Q ≤ L, which is weakly closed in a Sylow p-subgroup which contains it, and with the property that all subgroups in ℜ^{i}(L; p) contain Q up to conjugacy. This is the result which in almost all cases will be used to show L ∈ 𝔏^{≥2}(2) (simple groups of Lie type in characteristic two are handled in a different way). The last half of Section 4 then consists of a series of propositions, each of which gives some conditions to be used when proving that certain subgroups do not lie in ℜ^{≥2}(L; 2).

In this way, the paper splits into two halves. Sections 1–4 involve homological algebra, and reduce the problem to a series of criteria stated in purely group theoretic terms. These criteria are then applied to the individual groups in Sections 5–9, to show that L ∈ 𝔏^{≥2}(2) in all cases. Afterwards, in Section 10, some computations of \underleftarrow{lim}^{1}(Z_{G}) are listed (mostly without proof).

Clearly, as a topologist writing a paper which depends very heavily on the structure of the individual finite simple groups, I had a lot of assistance. Michael Aschbacher, Ron Solomon, and Richard Lyons all gave extensive help in answering my questions about the structure of certain simple groups. I am grateful to Sergey Shpektorov and Ulrich Meierfrankenfeld for sending me their manuscript listing the maximal 2-local subgroups of the monster and the baby monster — which allowed me to fill in the last step in the proof of Theorem A. I had several helpful discussions with George

4 BOB OLIVER

Glauberman and Jesper Grodal during my short visits to the University of Chicago. I am especially indebted to my earlier collaborator, Yoav Segev, who (while not involved in this project) taught me much of what I know about the classification theorem, and especially about the finite simple groups of Lie type. Finally, I would like to thank my colleagues at Northwestern University and the University of Wisconsin for their hospitality while working on many of the later stages of this project.

General notation: We list, for easy reference, the following notation which will be used throughout the paper.

• S_{p}(G) denotes the set of p-subgroups of G

• Syl_{p}(G) denotes the set of Sylow p-subgroups of G

• G_{p} denotes a Sylow p-subgroup of the group G, but only when it is abelian and a direct factor of G

• O_{p}(G) is the maximal normal p-subgroup of G

• C_{n}, D_{n}, and Q_{n} denote cyclic, dihedral, and quaternion groups of order n

• A_{n} and Σ_{n} are the alternating and symmetric groups on n elements

• A radical p-subgroup of G is a p-subgroup P ≤ G such that O_{p}(N_{G}(P)/P) = 1

• Ω_{n}(P) (for a p-group P) is the subgroup generated by all g ∈ P such that g^{p^{n}} = 1 • N_{G}(H, K) = {x ∈ G | xHx^{−1} ≤ K} (for H, K ≤ G)

• c_{x} denotes conjugation by x (g ↦ xgx^{−1})

• Hom_{G}(H, K) = {c_{x} ∈ Hom(H, K) | x ∈ N_{G}(H, K)} (for H, K ≤ G)

• Aut_{G}(H) = Hom_{G}(H, H) ≅ N_{G}(H)/C_{G}(H), and Out_{G}(H) = Aut_{G}(H)/ Inn(H)

• A functor F:C^{op} → Ab is called acyclic if \underleftarrow{lim}^{i}(F) = 0 for all i > 0.

In the later sections, we also use the following standard shorthand notation for referring to certain groups:

• 2^{n} = C_{2}^{n} is an elementary abelian 2-group;

• 2_{+}^{1+2k} is the central product of k copies of D_{8};

• 2_{−}^{1+2k} is the central product of Q_{8} with k−1 copies of D_{8};

• 2^{a+b} is a 2-group P such that Z(P) ≅ 2^{a} and P/Z(P) ≅ 2^{b};

• [2^{n}] is an unspecified group of order 2^{n}; and

• H:K, H·K, and H.K are extensions (split, unsplit, or indeterminate, respectively) with kernel H and quotient K.

Contents

1. Higher limits over orbit categories 5 2. Reduction to simple groups 12 3. A relative version of Λ-functors 19 4. Subgroups which contribute to higher limits 25 5. Alternating groups 33 6. Groups of Lie type in characteristic two 34 7. Classical groups of Lie type in odd characteristic 41 8. Exceptional groups of Lie type in odd characteristic 46 9. Sporadic groups 58 10. Computations of lim^{1}(Z_{G}) 69 References 70

1. Higher limits over orbit categories

We first fix our notation. For any prime p and any finite group G, the p-subgroup orbit category of G is the category O_{p}(G) whose objects are the p-subgroups of G, and where

Mor_{O_{p}(G)}(P, Q) = Q\N_{G}(P, Q) ≅ Map_{G}(G/P, G/Q).

Recall that N_{G}(P, Q) = {x ∈ G | xP x^{−1} ≤ Q} (the transporter). This can also be thought of as a category whose objects are orbits G/P of G and whose morphisms are G-maps, but for our purposes it is more convenient to let the objects be subgroups. For any homomorphism G ⟶^{ρ} G′ of groups, we let O_{p}(G) ⟶^{ρ_{#}} O_{p}(G′) denote the induced functor between orbit categories.

This section contains a variety of different results whose main point in common is that they all involve higher limits of functors over orbit categories.

\underline{1.1 The functor Λ^{∗}} In [JMO], certain graded ℤ_{(p)}-modules Λ^{∗}(G; M) are defined, for any prime p, any finite group G, and any ℤ_{(p)}[G]-module M, by setting

Λ^{∗}(G; M) = \underleftarrow{lim}^{∗}_{O_{p}(G)}(Φ_{M}^{G}) where Φ_{M}^{G}(P) = \begin{cases}M & if P = 1 \\ 0 & otherwise. \end{cases}

Note that these depend on the prime p, even though that has been suppressed from the notation. We first list some of the basic properties of these groups.

Proposition 1.1. The following hold for any prime p and any finite group G.

(a) Fix a p-subgroup P ≤ G, and let F: O_{p}(G) → ℤ_{(p)}-mod be any functor which vanishes except on subgroups conjugate to P. Then

\underleftarrow{lim}^{∗}_{O_{p}(G)}(F) ≅ Λ^{∗}(N_{G}(P)/P; F (P)).

(b) If H ◁ G is a normal subgroup which acts trivially on the ℤ_{(p)}[G]-module M, then

Λ^{∗}(G; M) ≅ \begin{cases}Λ^{∗}(G/H; M) & if (p, |H|) = 1 \\ 0 & otherwise. \end{cases}

(c) If O_{p}(G) ≠ 1 (if G contains a nontrivial normal p-subgroup), then Λ^{∗}(G; M) = 0 for all ℤ_{(p)}[G]-modules M.

(d) A short exact sequence 0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0 of ℤ_{(p)}[G]-modules induces a long exact sequence

⋯ ⟶ Λ^{i}(G; M′) ⟶ Λ^{i}(G; M) ⟶ Λ^{i}(G; M′′) ⟶ Λ^{i+1}(G; M′) ⟶ · · · .

Proof. See [JMO, Propositions 5.4, 5.5, & 6.1]. □

Proposition 1.1 is usually applied by filtering an arbitrary functor on the orbit category O_{p}(G) in such a way that all quotient functors vanish except on one conjugacy class. By Proposition 1.1(a), the higher limits of these quotient functors can then be described in terms of the graded groups Λ^{∗}. In most cases, this will be applied via the following lemma.

Lemma 1.2. The following hold for any prime p, any finite group G, and any functors F, F′: O_{p}(G)^{op} → ℤ_{(p)}-mod.

(a) Let φ: F → F′ be a natural morphism of functors such that for all P ∈ S_{p}(G),

Λ^{∗}(N_{G}(P)/P; Ker(φ(P))) = 0  and  Λ^{∗}(N_{G}(P)/P; Coker(φ(P))) = 0.

Then φ induces an isomorphism \underleftarrow{lim}^{∗}(F) ≅ \underleftarrow{lim}^{∗}(F′).

(b) Let C ⊆ O_{p}(G) be a full subcategory with the property that Mor_{O_{p}(G)}(P, Q) = ∅ whenever P ∈ Ob(C) and Q ∉ Ob(C). Assume Λ^{∗}(N_{G}(P)/P; F (P)) = 0 for all P not in C. Then

\underleftarrow{lim}^{∗}_{O_{p}(G)}(F) ≅ \underleftarrow{lim}_{C}^{∗}(F |_{C}).  (1)

Proof. In the situation of (a), \underleftarrow{lim}^{∗}(Ker(φ)) = 0 and \underleftarrow{lim}^{∗}(Coker(φ)) = 0 by Proposition 1.1(a), together with the obvious filtration of these functors and the exact sequences for higher limits of extensions of functors. So \underleftarrow{lim}^{∗}(F) ≅ \underleftarrow{lim}^{∗}(Im(φ)) ≅ \underleftarrow{lim}^{∗}(F′).

Now assume C ⊆ O_{p}(G) and F are as described in (b), and let F′ be the quotient functor F′(P) = F (P) if P ∈ Ob(C) and F′(P) = 0 otherwise. By (a), the projection F −↠ F′ induces an isomorphism \underleftarrow{lim}^{∗}(F) ≅ \underleftarrow{lim}^{∗}(F′). Also, any injective resolution of F′|_{C} can be extended to an injective resolution of F′ by assigning all functors the value zero on objects not in C, and this shows that \underleftarrow{lim}^{∗}(F′) ≅ \underleftarrow{lim}^{∗}(F |_{C}). □

Proposition 1.1(b) motivates the following definition.

Definition 1.3. For any finite group G and any ℤ_{(p)}[G]-module M, we say that G acts p-faithfully on M if the kernel of the action Ker[G → Aut(M)] has order prime to p.

The following lemma is an immediate consequence of Proposition 1.1(b,d).

Lemma 1.4. If the action of a finite group G on a finite ℤ_{(p)}[G]-module M is not p-faithful, then Λ^{∗}(G; M) = 0. □

There is also a connection between p-faithful actions and p-centric subgroups.

Lemma 1.5. Fix a finite group G and a p-subgroup P ≤ G. If N_{G}(P)/P acts pfaithfully on Z(P), then P is p-centric in G. More generally, if H ◁ G is a normal subgroup and N_{G}(P)/P acts p-faithfully on Z(P) ∩ H, then P ∩ H is p-centric in H.

Proof. We prove the second statement; the first then follows in the case H = G. Set K = C_{H}(P ∩ H). If P ∩ H is not p-centric in H, then Z(P ∩ H) = K ∩ P is not a Sylow p-subgroup of K, and hence N_{KP} (P)/P has order a multiple of p by Lemma 1.11. But N_{KP} (P)/P acts trivially on Z(P) ∩ H, and this contradicts the assumption that N(P)/P acts p-faithfully on Z(P) ∩ H. This shows that P ∩ H is p-centric in H. □

To give a quick illustration of how these techniques are applied, we describe the computation of \underleftarrow{lim}^{1}(Z_{G}) in certain very simple cases. In particular, the following proposition shows that these groups are nonvanishing when G is any of the simple groups A_{6}, A_{7}, or P SL_{2}(q) for q ≡ ±1 (mod 8).

Proposition 1.6. Fix a finite group G and a Sylow subgroup S ∈ Syl_{2}(G). Assume that S is a dihedral group of order ≥ 8, and let T_{1}, T_{2} be S-conjugacy class representatives for the subgroups isomorphic to C_{2}^{2}. Then

\underleftarrow{lim}^{1}(Z_{G}) ≅ ℤ/2

if Aut_{G}(T_{i}) = Aut(T_{i}) for i = 1, 2, and \underleftarrow{lim}^{1}(Z_{G}) = 0 otherwise.

Proof. Assume P ≤ S is such that Λ^{∗}(N_{G}(P)/P; Z_{G}(P)) ≠ 0. Then P is 2-centric in G, so P C_{G}(P)/P has odd order, and

Λ^{∗}(Out_{G}(P); Z_{G}(P)) ≅ Λ^{∗}(N_{G}(P)/P C_{G}(P); Z(P)) ≅ Λ^{∗}(N_{G}(P)/P; Z(P)) ≠ 0

by Proposition 1.1(b). Hence O_{2}(Out_{G}(P)) = 1 (Proposition 1.1(c)). If P is cyclic of order ≥ 4 or dihedral of order ≥ 8, then Out(P) is a nontrivial 2-group, so these cases cannot occur. So we are left only with the cases

Λ^{i}(Out_{G}(P); Z(P)) ≅ \begin{cases}ℤ/2 & if P = S, i = 0 \\ ℤ/2 & if P ≅ C_{2}^{2}, Aut_{G}(P) ≅ Σ_{3}, i = 1 \\ 0 & otherwise. \end{cases}

The computation Λ^{1}(Σ_{3}; (ℤ/2)^{2}) ≅ ℤ/2 follows from [JMO, Proposition 6.2(i)].

By Alperin’s fusion theorem (or the theorem of Burnside shown in [Gor, Theorem 7.1.1]), T_{1} and T_{2} cannot be G-conjugate. So using Proposition 1.1(a) and the obvious filtration of Z_{G} by subgroups, we obtain an exact sequence

0 ⟶ \underleftarrow{lim}^{0}_{O_{2}(G)}(Z_{G}) ⟶ ℤ/2 ⟶ (ℤ/2)^{k} ⟶ \underleftarrow{lim}^{1}_{O_{2}(G)}(Z_{G}) ⟶ 0,

where k is the number of i = 1, 2 such that Aut_{G}(T_{i}) ≅ Σ_{3}. If k ≥ 1, then Z(S) is G-conjugate to another subgroup of S, and this implies that \underleftarrow{lim}^{0}(Z_{G}) = 0. Thus \underleftarrow{lim}^{1}(Z_{G}) has rank k−1 in this case, and is trivial otherwise. □

\underline{1.2 Reduction to smaller orbit categories} We next list some conditions which allow us, in certain situations, to reduce the computation of higher limits of a functor on O_{p}(G) to those of another functor on O_{p}(N(Q)/Q) for some Q ≤ G.

Lemma 1.7. Fix a finite group G and a p-subgroup Q ≤ G. Then there is a well defined functor

Ψ_{Q}^{G}: O_{p}(N_{G}(Q)/Q) ⟶ O_{p}(G)

such that Ψ_{Q}^{G}(P/Q) = P for all P/Q ≤ N_{G}(Q)/Q. Let T be any set of p-subgroups of G with the property

P ∈ T ⟹ Q ◁ P, and Q ◁ xP x^{−1} for x ∈ G implies x ∈ N_{G}(Q). (∗)

Then for any functor F: O_{p}(G)^{op} → ℤ_{(p)}-mod which vanishes except on subgroups G-conjugate to elements of T , the induced homomorphism

\underleftarrow{lim}^{∗}_{O_{p}(G)}(F) ⟶_{≅}^{Ψ_{Q}^{G}^{∗}} \underleftarrow{lim}^{∗}_{O_{p}(N_{G}(Q)/Q)}(F ∘ Ψ_{Q}^{G})  (1)

is an isomorphism.

Proof. Set Ψ = Ψ_{Q}^{G} for short. Clearly, Ψ is well defined on objects. To see that it is well defined on morphisms, recall first that

Mor_{O_{p}(G)}(P, P′) = P′\N_{G}(P, P′),

where N_{G}(P, P′) is the set of all x ∈ G such that xP x^{−1} ≤ P′. Hence for any pair of objects P/Q and P′/Q in O_{p}(N_{G}(Q)/Q),

Mor_{O_{p}(N_{G}(Q)/Q)}(P/Q, P′/Q) = (P′/Q)\N_{N(Q)/Q}(P/Q, P′/Q) ≅ P′\N_{N(Q)}(P, P′) ⊆ P′\N_{G}(P, P′) = Mor_{O_{p}(G)}(P, P′);

and Ψ is defined on morphism sets to be this inclusion.

Composition with Ψ is natural in F and preserves short exact sequences of functors. Hence if F′ ⊆ F is a pair of functors from O_{p}(G) to ℤ_{(p)}-mod, and the lemma holds for F′ and for F/F′, then it also holds for F by the 5-lemma. It thus suffices to prove that (1) is an isomorphism when F vanishes except on the G-conjugacy class of one subgroup P ∈ T . When P = Q, then (1) is precisely the isomorphism \underleftarrow{lim}^{∗}(F) ≅ Λ^{∗}(N(Q)/Q; F (Q)) of Proposition 1.1(a).

Now let P ∈ T be arbitrary. By condition (∗), Q ◁ P, N_{G}(P) ≤ N_{G}(Q), and F ∘ Ψ vanishes except on the O_{p}(N_{G}(Q)/Q)-isomorphism class of P/Q. Let

Ψ′ = Ψ_{P/Q}^{N(Q)/Q}: O_{p}(N_{G}(P)/P) ⟶ O_{p}(N_{G}(Q)/Q)

be the functor Ψ′(R/P) = R/Q for p-subgroups R ≤ N_{G}(P) ≤ N_{G}(Q) containing P. Then the following square commutes

\underleftarrow{lim}^{∗}_{O_{p}(G)}(F)  \overline{^{Ψ^{∗}} →} \underleftarrow{lim}^{∗}_{O_{p}(N(Q)/Q)}(F ∘ Ψ)

Λ^{∗}(N_{G}(P)/P; F (P)) ↓^{(Ψ∘Ψ′)^{∗} ≅}  {= \atop =} Λ^{∗}(N_{G}(P)/P; F (P)) , ↓^{Ψ′^{∗} ≅}

and the vertical maps are isomorphisms by Proposition 1.1(a) (see the proof of [JMO, Lemma 5.4] for the precise description of the isomorphisms). This shows that Ψ^{∗} is an isomorphism. □

We next consider two special cases of Lemma 1.7.

Lemma 1.8. Fix a finite group G and a normal subgroup H ◁ G. Fix a p-subgroup Q ≤ H, and let F: O_{p}(G) → ℤ_{(p)}-mod be any functor such that F (P) = 0 whenever P ∩ H is not G-conjugate to Q. Then the induced homomorphism

\underleftarrow{lim}^{∗}_{O_{p}(G)}(F) ⟶_{≅}^{Ψ_{Q}^{G}} \underleftarrow{lim}^{∗}_{O_{p}(N(Q)/Q)}(F ∘ Ψ_{Q}^{G}),

is an isomorphism, where Ψ_{Q}^{G}: O_{p}(N(Q)/Q) ⟶ O_{p}(G) is the functor of Lemma 1.7.

Proof. This follows immediately from Lemma 1.7, upon letting T be the set of psubgroups P ≤ G such that P ∩ H = Q. Condition (∗) holds since if P ∩ H = Q and Q ◁ xP x^{−1}, then x^{−1}Qx ≤ P ∩ H = Q and hence x ∈ N_{G}(Q). □

In the next lemma, recall that for any finite group G and any S ∈ Syl_{p}(G), a subgroup P ≤ S is weakly closed in S with respect to G if P is not G-conjugate to any other subgroup of S.

Lemma 1.9. Fix a finite group G, a Sylow p-subgroup S ∈ Syl_{p}(G), and a subgroup Q ≤ S which is weakly closed in S with respect to G. Let F: O_{p}(G) → ℤ_{(p)}-mod be a functor such that F (P) = 0 whenever P does not contain a subgroup G-conjugate to Q. Then the induced homomorphism

\underleftarrow{lim}^{∗}_{O_{p}(G)}(F) ⟶_{≅}^{Ψ_{Q}^{G}} \underleftarrow{lim}^{∗}_{O_{p}(N(Q)/Q)}(F ∘ Ψ_{Q}^{G}),

is an isomorphism, where Ψ_{Q}^{G}: O_{p}(N(Q)/Q) ⟶ O_{p}(G) is the functor of Lemma 1.7.

Proof. We apply Lemma 1.7, with T = {P ∈ S_{p}(G) | Q ≤ P }. By hypothesis, P ∈ T for any p-subgroup P ≤ G such that F (P) ≠ 0.

For each P ∈ T , there is x ∈ G such that xP x^{−1} ≤ S; xQx^{−1} = Q since Q is weakly closed in S, and thus x ∈ N_{G}(Q). If yP y^{−1} ≥ Q, then xy^{−1} ∈ N_{G}(Q) by the same reasoning, and thus y ∈ N_{G}(Q). This also shows that Q ◁ P (the case y ∈ P), and thus that condition (∗) of Lemma 1.7 holds for all P ∈ T . Hence \underleftarrow{lim}^{∗}(F) ≅ \underleftarrow{lim}^{∗}(F ∘Ψ_{Q}^{G}). □

The following lemma is very similar in nature to Lemma 1.7.

Lemma 1.10. Fix a surjection φ: G −↠ G′ of finite groups. Then for any functor F: O_{p}(G′)^{op} ⟶ ℤ_{(p)}-mod,

\underleftarrow{lim}^{∗}_{O_{p}(G)}(F ∘ φ_{#}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(G′)}(F).

More generally, set H = Ker(φ), and let O_{p}^{∗}(G) ⊆ O_{p}(G) be the full subcategory whose objects are the p-subgroups P ≤ G such that P ∩ H ∈ Syl_{p}(H). Then

\underleftarrow{lim}^{∗}_{O_{p}(G)}(F) ≅ \underleftarrow{lim}^{∗}_{O_{p}(G′)}(F)  (1)

for any F: O_{p}(G′)^{op} ⟶ ℤ_{(p)}-mod and F: O_{p}(G)^{op} ⟶ ℤ_{(p)}-mod such that

(F ∘ φ_{#})|_{O_{p}^{∗}(G)} ≅ F |_{O_{p}^{∗}(G)},  (2)

and such that the action of N_{HP} (P)/P on F (P) is trivial for all P ∈ S_{p}(G).

Proof. It suffices to prove the last statement, since N_{HP} (P)/P acts trivially on F (φ(P)) for each P.

For all P ∈ S_{p}(G) such that P ∩ H ∉ Syl_{p}(H), p||N_{HP} (P)/P | (see Lemma 1.11), and N_{HP} (P)/P acts trivially on F (P). Hence by Lemma 1.4,

Λ^{∗}(N_{G}(P)/P; F (P)) = 0

for all such P. So by Lemma 1.2(b),

\underleftarrow{lim}^{∗}_{O_{p}(G)}(F) ≅ \underleftarrow{lim}^{∗}_{O_{p}^{∗}(G)}(F |_{O_{p}^{∗}(G)}).  (3)

Let φ′_{#} : O_{p}^{∗}(G) → O_{p}(G′) be the restriction of φ_{#}. This is a bijection on isomorphism classes of objects, since φ′_{#}(P) = P′ (for p-subgroups P ≤ G and P′ ≤ G′) if and only if P ∈ Syl_{p}(φ^{−1}P′). It is also surjective on all morphism sets, since for any pair of objects P, Q in O_{p}^{∗}(G), and any x ∈ N_{G}(P H, QH), xP x^{−1} ≤ QH is H-conjugate to a subgroup of Q ∈ Syl_{p}(QH), and hence x ∈ H·N_{G}(P, Q).

Now assume x, y ∈ N_{G}(P, Q) induce the same morphism in O_{p}(G′). After replacing y by an appropriate element of Qy, we can assume that y ∈ Hx = xH. Set y = xh (where h ∈ H). Then P and hP h^{−1} are both subgroups of x^{−1}Qx, so [h, P] ≤ x^{−1}Qx ∩ H = P ∩ H (since P ∩ H ∈ Syl_{p}(H)), and thus h ∈ N(P). In other words,

Mor_{O_{p}(G′)}(φ(P), φ(Q)) ≅ Mor_{O_{p}(G)}(P, Q)/(N_{HP} (P)/P) .

Also, N_{HP} (P)/P has order prime to p since P ∈ Syl_{p}(HP). The category O_{p}(G′) is thus equivalent to O_{p}^{∗}(G) after dividing out by the action of certain subgroups of automorphisms of order prime to p. So by [BLO, Lemma 1.3],

\underleftarrow{lim}^{∗}_{O_{p}(G′)}(F) ≅ \underleftarrow{lim}^{∗}_{O_{p}^{∗}(G)}(F ∘ φ′_{#}),

and (1) follows from this together with (2) and (3). □

\underline{1.3 Radical p-subgroups} Recall that a radical p-subgroup of a finite group G is a p-subgroup P ≤ G such that O_{p}(N_{G}(P)/P) = 1; i.e., such that N_{G}(P)/P has no nontrivial normal p-subgroups. By Proposition 1.1(c), if P is a p-subgroup which is not radical, then Λ^{∗}(N_{G}(P)/P; M) = 0 for all ℤ_{(p)}[N_{G}(P)/P]-modules M. Thus by Lemma 1.2, \underleftarrow{lim}^{∗}(F) = 0 for any functor F on O_{p}(G)^{op} which vanishes on all radical p-subgroups of G. This helps explain the important role played by radical p-subgroups when working with higher limits of functors on orbit categories. In this subsection, we prove some properties of radical p-subgroups which will be needed later.

We first note the following, very elementary, group theoretic lemma.

Lemma 1.11. Fix subgroups P, H ≤ G such that P is a p-subgroup which normalizes H, and P ∩ H ∉ Syl_{p}(H). Then P ∩ H ∉ Syl_{p}(N_{H}(P)), and hence

p||N_{HP} (P)/P | = |N_{H}(P)/(H∩P)|.

Proof. Consider the action of P on HP/P ≅ H/(P ∩H), whose order is a multiple of p by assumption. Then

(HP/P)^{P} = N_{HP} (P)/P ≅ N_{H}(P)/(P ∩H)

has order a multiple of p, and thus P ∩ H ∉ Syl_{p}(N_{H}(P)). □

The next lemma lists some conditions for a p-subgroup to be radical or not.

Lemma 1.12. Fix a finite group G and a prime p.

(a) If G splits as a product G = ∏_{i∈I} G_{i}, then each radical p-subgroup P ≤ G is of the form P = ∏_{i∈I} P_{i}, where P_{i} ≤ G_{i}.

(b) Let H ◁ G be any normal subgroup. Then for each radical p-subgroup P in G, P ∩ H is radical in H. Conversely, if Q is radical in H, then Q = P ∩ H for some radical p-subgroup P in G.

Proof. (a) If P ≤ G = ∏_{i∈I} G_{i} does not split as a product, then let P_{i} ≤ G_{i} be the image of P under the projection, and set P′ = ∏_{i∈I} P_{i} ≤ G. Then P′ ≩ P by assumption, and N_{G}(P′) ≥ N_{G}(P). Thus N_{P′}(P)/P is a nontrivial normal p-subgroup of N_{G}(P)/P, and P is not p-radical.

(b) Fix H ◁ G. Assume first that P ∩ H is not radical in H, and set Q = O_{p}(N_{H}(P ∩ H)) ≩ (P ∩ H). Then QP ≩ P (since Q ≤ H), and so N_{QP} (P)/P ≠ 1 by Lemma 1.11. Furthermore, N_{QP} (P) is normalized by N(P) since Q is, N_{QP} (P)/P is thus a nontrivial normal p-subgroup of N_{G}(P)/P, and so P is not radical in G.

Conversely, fix a radical p-subgroup Q in H, and set P = O_{p}(N_{G}(Q)). Then P ∩ H is a normal p-subgroup of N_{H}(Q) which contains Q, and hence P ∩ H = Q. Clearly, N_{G}(P) ≥ N_{G}(Q); they are equal since any x ∈ N_{G}(P) normalizes P ∩ H = Q; and thus P is radical. □

\underline{1.4 Fixed point and norm functors} Fix a finite group G. For any subgroup H ≤ G, set 𝔑_{H} = ∑_{h∈H} h ∈ ℤ[G]. For any prime p and any ℤ_{(p)}[G]-module M, we consider the functors

H^{0}M, 𝔑M: O_{p}(G)^{op} ⟶ ℤ_{(p)}-mod,

defined by setting

H^{0}M(P) = H^{0}(P; M) = M^{P} and 𝔑M(P) = 𝔑_{P} ·M.

The next proposition plays a central role — almost as important as that of the functors Λ^{∗}(G; M) — when computing higher limits over orbit categories.

Proposition 1.13. For any finite group G, any prime p, and any ℤ_{(p)}[G]-module M,

\underleftarrow{lim}^{i}_{O_{p}(G)}(H^{0}M) = \begin{cases}H^{0}(G; M) = M^{G} & if i = 0 \\ 0 & if i > 0 \end{cases} and \underleftarrow{lim}^{i}_{O_{p}(G)}(𝔑M) = \begin{cases}𝔑_{G}·M & if i = 0 \\ 0 & if i > 0 . \end{cases}

Proof. Set Ĥ^{0}M = H^{0}M/𝔑M; thus Ĥ^{0}M(P) = Ĥ^{0}(P; M) for all P. These functors are all proto-Mackey functors in the sense of [JM], and hence are acyclic by [JM, Proposition 5.14]. The complete description of the higher limits of H^{0}M and Ĥ^{0}M is shown in [JMO, Proposition 5.2], and that of 𝔑M follows immediately. □

\underline{1.5 Kan extensions and limits} If H ≤ G, and F: O_{p}(H)^{op} ⟶ Ab is any functor, then we let

F ↑_{H}^{G}: O_{p}(G)^{op} ⟶ Ab

denote the right Kan extension of F. To define this, fix an object P in O_{p}(G), let ι↓P be the overcategory whose objects are the morphisms Q → P in O_{p}(G) for Q ≤ H, and let κ_{P} : ι↓P → O_{p}(H) be the forgetful functor which sends (Q → P) to Q. Then

(F ↑_{H}^{G})(P) = \underleftarrow{lim}_{ι↓P}(F ∘ κ_{P}),

and a morphism P ⟶ P′ in O_{p}(G) induces the obvious map between inverse limits. We refer to [McL, §X.3] for more details.

Lemma 1.14. The following hold for any H ≤ G and any F: O_{p}(H)^{op} ⟶ Ab.

(a) \underleftarrow{lim}^{∗}_{O_{p}(G)}(F ↑_{H}^{G}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(H)}(F).

(b) We can identify

(F ↑_{H}^{G})(P) = (⨁_{g∈G} F (H ∩ gP g^{−1}))^{P ×H} ≅ ⨁_{HgP ∈H\G/P} F (H ∩ gP g^{−1});  (1)

where (x, y) ∈ P × H acts on the first sum by sending the summand for g to the summand for xgy^{−1} via F (y). When aP a^{−1} ≤ Q, the induced morphism

a^{∗} = (F ↑_{H}^{G})(a): (F ↑_{H}^{G})(Q) ⟶ (F ↑_{H}^{G})(P)

satisfies (a^{∗}(ξ))_{g} = F (incl)(ξ_{ga^{−1}}).

Proof. By [McL, §X.3], the right Kan extension (−)↑_{H}^{G} is a right adjoint to the restriction functor from O_{p}(G)-mod to O_{p}(H)-mod. From this, one easily sees that (−)↑_{H}^{G} preserves exact sequences, and sends injectives in O_{p}(H)-mod to injectives in O_{p}(G)-mod. Also, if \underline{ℤ} denotes the constant functors with value ℤ, then

\underleftarrow{lim}_{O_{p}(G)}(F ↑_{H}^{G}) ≅ Hom_{_{O_{p}(G)}-_{mod}} (\underline{ℤ}, F ↑_{H}^{G}) ≅ Hom_{_{O_{p}(H)}-_{mod}} (\underline{ℤ}, F) ≅ \underleftarrow{lim}_{O_{p}(H)}(F).

Hence

\underleftarrow{lim}^{∗}_{O_{p}(G)}(F ↑_{H}^{G}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(H)}(F).

The formulas in (b) follow immediately from the definition of (F ↑_{H}^{G})(P) as an inverse limit. In particular, the term for g ∈ G corresponds to the maximal object H ∩ gP g^{−1} ^{g^{−1}}→ P in the overcategory ι↓P. □

2. Reduction to simple groups

For any finite group G and any normal subgroup K ◁ G, we define a subfunctor Z_{G}^{K} of Z_{G} by setting

Z_{G}^{K}(P) = \begin{cases}Z(P) ∩ K & if P is p-centric in G \\ 0 & otherwise. \end{cases}

Our goal now is to study the higher limits of quotient functors Z_{G}^{K_{1}}/Z_{G}^{K_{2}}, when K_{1}/K_{2} is a minimal normal subgroup of G/K_{2}. We want to reduce the computation of higher limits of Z_{G}^{K_{1}}/Z_{G}^{K_{2}} to the case where K_{1}/K_{2} is a nonabelian simple group, K_{2} = Z(K_{1}) is an abelian p-group, and K_{1} is perfect (thus a quasisimple group). This involves a series of reductions.

Lemma 2.1. Fix a prime p, a finite group G, and subgroups K_{2} ≨ K_{1} both normal in G, such that K_{1}/K_{2} is a minimal normal subgroup of G/K_{2}. Set

H = {g ∈ G | [g, K_{1}] ≤ K_{2}} ◁ G

(thus H/K_{2} = C_{G/K_{2}}(K_{1}/K_{2})), choose Q ∈ Syl_{p}(H), and set

G′ = N_{G}(Q)  and  K′_{i} = C_{K_{i}}(Q).

Then K′_{2} = A × T, where A = K′_{2} ∩ Q = K_{2} ∩ Z(Q) is an abelian p-group and T has order prime to p. Set

G′′ = G′/T  and  K′′_{i} = K′_{i}/T.

Then either

(a) K′_{1} ≤ K_{2} and the functor Z_{G}^{K_{1}}/Z_{G}^{K_{2}} is acyclic; or

(b) K′′_{1}/K′′_{2} ≅ K′_{1}/K′_{2} ≅ K_{1}/K_{2}, and

\underleftarrow{lim}^{∗}_{O_{p}(G)}(Z_{G}^{K_{1}}/Z_{G}^{K_{2}}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(G′)}(Z_{G′}^{K′_{1}}/Z_{G′}^{K′_{2}}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(G′′)}(Z_{G′′}^{K′′_{1}}/Z_{G′′}^{K′′_{2}}).  (1)

Proof. Since K_{2} ◁ H by definition, Q′ =^{def} Q ∩ K_{2} ∈ Syl_{p}(K_{2}) is p-centric in K_{2}. Thus C_{K_{2}}(Q′) = Z(Q′) × U for some U of order prime to p. After taking fixed points of the Q/Q′-action, we get that C_{K_{2}}(Q) = A × T, where A = C_{Z(Q′)}(Q) = Z(Q) ∩ K_{2} and T = C_{U}(Q).

Let O_{p}^{∗}(G) ⊆ O_{p}(G) be the full subcategory whose objects are those p-subgroups P ≤ G such that P ∩ H ∈ Syl_{p}(H). If P ∩ H ∉ Syl_{p}(H), then p||N_{HP} (P)/P | by Lemma 1.11, N_{HP} (P)/P acts trivially on (Z_{G}^{K_{1}}/Z_{G}^{K_{2}})(P), and hence

Λ^{∗}(N_{G}(P)/P; (Z_{G}^{K_{1}}/Z_{G}^{K_{2}})(P)) = 0

by Lemma 1.4. So by Lemma 1.2(b),

\underleftarrow{lim}^{∗}_{O_{p}(G)}(Z_{G}^{K_{1}}/Z_{G}^{K_{2}}) ≅ \underleftarrow{lim}^{∗}_{O_{p}^{∗}(G)}((Z_{G}^{K_{1}}/Z_{G}^{K_{2}})|_{O_{p}^{∗}(G)}).  (2)

For any g ∈ G, gQg^{−1} = xQx^{−1} for some x ∈ H, since Q ∈ Syl_{p}(H), so x^{−1}g ∈ N_{G}(Q) = G′, and g ∈ HG′. Thus G = HG′. The subgroup K′_{1}K_{2} = C_{K_{1}}(Q)·K_{2} is clearly normalized by G′ = N_{G}(Q), and is normalized by H since [H, K_{1}] ≤ K_{2} by definition. Hence K′_{1}K_{2} ◁ HG′ = G; and the minimality assumption on K_{1}/K_{2} implies that K′_{1}K_{2} = K_{2} or K_{1}. If K′_{1}K_{2} = K_{2}, then K′_{1} = C_{K_{1}}(Q) ≤ K_{2}, so Z(P) ∩ K_{1} ≤ K_{2} for any P ≥ Q, (Z_{G}^{K_{1}}/Z_{G}^{K_{2}})|_{O_{p}^{∗}(G)} is the zero functor, and so \underleftarrow{lim}^{∗}(Z_{G}^{K_{1}}/Z_{G}^{K_{2}}) = 0 by (2). This proves case (a).

It remains to consider the case where K′_{1}K_{2} = K_{1}, and thus where K′′_{1}/K′′_{2} ≅ K′_{1}/K′_{2} ≅ K_{1}/K_{2}. By analogy with the definition of O_{p}^{∗}(G), let O_{p}^{∗}(G′) ⊆ O_{p}(G′) be the full subcategory on subgroups which contain Q. Consider the functor

O_{p}^{∗}(G′) ⟶^{α} O_{p}^{∗}(G)

induced by the inclusion G′ ≤ G. For any P in O_{p}^{∗}(G), P ∩H ∈ Syl_{p}(H), and hence P is conjugate to a subgroup containing Q ∈ Syl_{p}(H), and thus to an object in O_{p}^{∗}(G′). If P_{1} and P_{2} both contain Q, then N_{G}(P_{1}, P_{2}) = N_{G′}(P_{1}, P_{2}) (any element of the transporter sends Q = P_{1} ∩ H = P_{2} ∩ H to itself), and thus α is the inclusion of a full subcategory and an equivalence of categories.

For any p-subgroup P ≤ G containing Q, C_{G}(P) ≤ C_{G}(Q) ≤ G′, and thus P is p-centric in G if and only if it is p-centric in G′. Also, for i = 1, 2,

Z(P) ∩ K_{i} = Z(P) ∩ C_{K_{i}}(Q) = Z(P) ∩ K′_{i}.

So there is a natural isomorphism (Z_{G}^{K_{1}}/Z_{G}^{K_{2}})(P) ≅ (Z_{G′}^{K′_{1}}/Z_{G′}^{K′_{2}})(P) for all P in O_{p}^{∗}(G′). The first isomorphism in (1) thus follows from (2), together with the analogous isomorphism for G′. The second isomorphism in (1) follows from Lemma 1.10 (applied with H = T). □

As a first application of Lemma 2.1, we consider the case where the subquotient K_{1}/K_{2} is abelian.

Proposition 2.2. Let G be a finite group. If K_{2} ◁ K_{1} ◁ G are subgroups, both normal in G, such that K_{1}/K_{2} is abelian, then the functor Z_{G}^{K_{1}}/Z_{G}^{K_{2}} is acyclic. If G is solvable, then the functor Z_{G} is acyclic.

Proof. Assume first that K_{2} ◁ K_{1} ◁ G are normal subgroups such that K_{1}/K_{2} is abelian. It clearly suffices to prove the acyclicity of Z_{G}^{K_{1}}/Z_{G}^{K_{2}} when K_{1}/K_{2} is a minimal normal subgroup of G/K_{2}. If K_{1}/K_{2} has order prime to p, then Z_{G}^{K_{1}}/Z_{G}^{K_{2}} = 0, and so we can assume that K_{1}/K_{2} is a p-group.

By Lemma 2.1, we can also assume that there is a normal p-subgroup Q ◁ G such that K_{2} ≤ Q, [K_{1}, Q] = 1, and Q/K_{2} ∈ Syl_{p}(C_{G/K_{2}}(K_{1}/K_{2})). In particular, Q ≥ K_{1} since K_{1}/K_{2} is an abelian p-group. So

C_{G}(Q) ≤ C_{G}(K_{1}) ≤ {g ∈ G | [g, K_{1}] ≤ K_{2}},

and C_{G}(Q) ≤ Q since Q is a Sylow p-subgroup of the last group. Thus Q is p-centric.

Let O_{p}^{∗}(G) ⊆ O_{p}(G) be the full subcategory with objects the p-subgroups which contain Q. If P ≱ Q, then 1 ≠ N_{P Q}(P)/P ≤ O_{p}(N_{G}(P)/P), so P is not radical. Thus O_{p}^{∗}(G) contains all radical p-subgroups of G. So by Lemma 1.2 (and Proposition 1.1(c)), for any p-local functor F on O_{p}(G), \underleftarrow{lim}^{∗}(F) ≅ \underleftarrow{lim}^{∗}(F |_{O_{p}^{∗}(G)}).

Set M_{i} = Z(Q) ∩ K_{i}. For any p-subgroup P ≤ G containing Q, Z(P) = Z(Q)^{P/Q} and hence

(Z_{G}^{K_{1}}/Z_{G}^{K_{2}})(P) ≅ (Z(P) ∩ K_{1})/(Z(P) ∩ K_{2}) ≅ M_{1}^{P/Q}/M_{2}^{P/Q}.

Thus (Z_{G}^{K_{1}}/Z_{G}^{K_{2}})|_{O_{p}^{∗}(G)} ≅ (H^{0}M_{1}/H^{0}M_{2})|_{O_{p}^{∗}(G)}, and hence

\underleftarrow{lim}^{∗}_{O_{p}(G)}(Z_{G}^{K_{1}}/Z_{G}^{K_{2}}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(G)}(H^{0}M_{1}/H^{0}M_{2}).

The last functor is acyclic by Proposition 1.13, and so Z_{G}^{K_{1}}/Z_{G}^{K_{2}} is also acyclic.

Now assume G is solvable, and let 1 = K_{n} ◁ K_{n−1} ◁ ⋯ ◁ K_{1} ◁ K_{0} = G be its derived sequence; i.e., K_{i+1} = [K_{i}, K_{i}] for each i. We have just seen that Z_{G}^{K_{i}}/Z_{G}^{K_{i+1}} is acyclic for each i. So Z_{G}^{K_{i}} is acyclic for each i, and in particular Z_{G} = Z_{G}^{K_{0}} is acyclic. □

The following technical lemma will be useful in the later reductions.

Lemma 2.3. Let G be a finite group with normal subgroups A ◁ K ◁ G, such that A ≤ Z(K) is an abelian p-group. Let P ≤ G be a p-subgroup such that either A ≰ P, or (P ∩ K)/A is not p-centric in K/A. Then there is a subgroup H ≤ G such that p||N_{HP} (P)/P | and [H, P ∩ K] ≤ A; and furthermore

Λ^{∗}(N_{G}(P)/P; (Z_{G}^{K}/Z_{G}^{A})(P)) = 0.

Proof. Set H = {x ∈ K | [x, P ∩ K] ≤ A}. Then P normalizes H. Also, P ∩ H ∉ Syl_{p}(H): this is clear if A ≰ P, and follows by definition (of p-centric) if (P ∩ K)/A is not p-centric in K/A. Hence p||N_{HP} (P)/P | by Lemma 1.11. Since N_{HP} (P)/P acts trivially on \frac{Z(P)∩K}{Z(P)∩A}, the last statement now follows from Proposition 1.1(b). □

We next reduce the computation of \underleftarrow{lim}^{∗}(Z_{G}^{K_{1}}/Z_{G}^{K_{2}}) when K_{1}/K_{2} is nonabelian to the case where K_{1} is quasisimple; i.e., to the case where K_{1}/K_{2} is nonabelian and simple, K_{2} = Z(K_{1}), and K_{1} is perfect.

Lemma 2.4. Fix a finite group G, and subgroups A ≤ K ≤ G, both normal in G, such that A is a p-group, [A, K] = 1, and K/A is nonabelian and a minimal normal subgroup of G/A. Then A = Z(K), and we can write K/A = ∏_{j∈J} L_{j}, where each L_{j} is simple and a minimal normal subgroup of K/A. Let K_{j} ≤ K be such that L_{j} = K_{j}/A, and set L̃_{j} = [K_{j}, K_{j}], A_{j} = L̃_{j} ∩ A = Z(L̃_{j}), and G_{j} = N_{G}(L̃_{j}). Then for any given j ∈ J,

\underleftarrow{lim}^{i}_{O_{p}(G)}(Z_{G}^{K}/Z_{G}^{A}) ≅ \underleftarrow{lim}^{i}_{O_{p}(G_{j})}(ZL̃_{j} ; G_{j}/Z_{G_{j}}^{A_{j}})

for all i ≥ 2, and there is a surjection

\underleftarrow{lim}^{1}_{O_{p}(G_{j})}(ZL̃_{j} ; G_{j}/Z_{G_{j}}^{A_{j}}) −↠ \underleftarrow{lim}^{1}_{O_{p}(G)}(Z_{G}^{K}/Z_{G}^{A}).

Proof. Since K/A is nonabelian and a minimal normal subgroup of G/A, it must be a product of nonabelian simple groups isomorphic to each other (cf. [Gor, Theorem 2.1.5]). Since we also assume [A, K] = 1, this implies that A = Z(K). So write K/A = ∏_{j∈J} L_{j} where L_{j} = K_{j}/A as above. For any i ≠ j, K_{i} ∩ K_{j} = A, and hence [K_{i}, K_{j}] ≤ A. Since A is central, this means that the commutator map [−, −]: K_{i} × K_{j} ⟶ A is a homomorphism in each coordinate; e.g., [gg′, h] = [g, h][g′, h] for g, g′ ∈ K_{i} and h ∈ K_{j}. Thus

[K_{i}, K_{j}] = 1 for all i ≠ j,  (1)

since K_{i}/A and K_{j}/A are simple and centralize A.

Set Z_{j} = ZL̃_{j} ; G_{j}/Z_{G_{j}}^{A_{j}} for short. Define functors Φ_{1}, Φ_{2} : O_{p}(G)^{op} → Ab by setting

Φ_{1}(P) = (∏_{j∈J} Z_{j}(P ∩ G_{j}))^{P} and Φ_{2} = Z_{G}^{K}/Z_{G}^{A}.

Each factor Z_{j}(P ∩ G_{j}) in Φ_{1}(P) can be identified with a subgroup of L_{j} ≅ L̃_{j}/A_{j}, and hence Φ_{1}(P) can be identified with a subgroup of K/A. Under this identification, Φ_{1} sends the morphism in O_{p}(G) represented by x ∈ N_{Γ}(P, Q) to that given by restriction of x^{−1}.

Let Φ′_{1} and Φ′_{2} be the functors

Φ′_{i}(P) = \begin{cases}Φ_{i}(P) & if P ∩ K is p-centric in K \\ 0 & otherwise. \end{cases}

In Step 1, we show that

\underleftarrow{lim}^{∗}_{O_{p}(G)}(Φ_{1}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(G_{j})}(Z_{j}) for all j ∈ J;  (2)

and in Step 2 that

\underleftarrow{lim}^{∗}_{O_{p}(G)}(Φ′_{1}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(G)}(Φ_{1}) and  \underleftarrow{lim}^{∗}_{O_{p}(G)}(Φ′_{2}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(G)}(Φ_{2})  (3)

In Step 3, we identify Φ′_{1} as a subfunctor of Φ′_{2}, and prove in Step 4 that Φ′_{2}/Φ′_{1} is acyclic. The lemma then follows from the relative exact sequence for higher limits of the pair Φ′_{1} ⊆ Φ′_{2}.

Step 1: For each j ∈ J, there is an obvious morphism of functors on O_{p}(G_{j})

Φ_{1}|_{O_{p}(G_{j})} ⟶ Z_{j},

defined by projection to the j-th factor, which is adjoint to a natural morphism

ω: Φ_{1} ⟶ Z_{j}↑_{G_{j}}^{G}.

Since G acts transitively on the factors L_{j} ≤ K/A (otherwise K/A would not be a minimal normal subgroup), ω is an isomorphism of functors by the formula in Lemma 1.14(b) for Kan extensions. Hence (2) follows from Lemma 1.14(a).

Step 2: To prove (3), it suffices using Lemma 1.2(b) to show that

Λ^{∗}(N_{G}(P)/P; Φ_{i}(P)) = 0

whenever i = 1, 2 and P ∩ K is not p-centric in K. By Proposition 2.3, there is H ≤ G such that [H, P ∩ K] ≤ A and p||N_{HP} (P)/P |. Also, both groups Φ_{i}(P) can be identified with subgroups of K/A ≅ ∏_{j∈J} L̃_{j}/A_{j}, so N_{HP} (P)/P acts trivially on Φ_{i}(P), and Λ^{∗}(N_{G}(P)/P; Φ_{i}(P)) = 0 by Lemma 1.4.

Step 3: Set K̃ = ∏_{j∈J} L̃_{j} and Ã = ∏_{j∈J} A_{j}. We will write R_{p} for the Sylow psubgroup of a group R, but only in situations where it is a direct factor of R and abelian.

Fix a p-subgroup P ≤ G such that P ∩ K is p-centric in K, and note that

CK̃(P) ≅ [∏_{j∈J} CL̃_{j}(P ∩ G_{j})]^{P} and CÃ(P) ≅ [∏_{j∈J} C_{A_{j}}(P ∩ G_{j})]^{P} .

Furthermore, the action of any g ∈ P permutes the factors under each of these product decompositions, and is trivial whenever it sends a factor to itself. Hence

Φ_{1}(P) ≅ [∏_{j∈J} \frac{CL̃_{j}(P ∩ G_{j})_{p}}{C_{A_{j}}(P ∩ G_{j})} ]^{P} ≅ \frac{[∏_{j∈J} CL̃_{j}(P ∩ G_{j})_{p}]^{P}}{[∏_{j∈J} C_{A_{j}}(P ∩ G_{j})]^{P}} ≅ CK̃(P)_{p}/CÃ(P).

We can thus write

Φ′_{1}(P) ≅ CK̃(P)_{p}/CÃ(P) and Φ′_{2}(P) ≅ C_{K}(P)_{p}/C_{A}(P)

for all P such that P ∩ K is p-centric in K. The natural map from K̃ to K induces a natural morphism of functors Φ′_{1} → Φ′_{2}, and this is injective since Φ′_{1}(P) and Φ′_{2}(P) can both be identified as subgroups of K/A ≅ K̃/Ã.

Step 4: Set F = Φ′_{1}/Φ′_{2} for short; it remains to show that F is acyclic. Fix S ∈ Syl_{p}(K). Define functors

F_{0} : O_{p}(G)^{op} ⟶ ℤ_{(p)}-mod and F: O_{p}(N_{G}(S)/S)^{op} ⟶ ℤ_{(p)}-mod

by setting

F_{0}(P) = \begin{cases}F (P) & if P ∩ K ∈ Syl_{p}(K) \\ 0 & otherwise \end{cases} and F (P/S) = F (P) = F_{0}(P).

We claim that

\underleftarrow{lim}^{∗}_{O_{p}(G)}(F) ≅ \underleftarrow{lim}^{∗}_{O_{p}(G)}(F_{0}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(N_{G}(S)/S)}(F),  (4)

and that F is acyclic. The second isomorphism follows immediately from Lemma 1.8.

Recall that K/A = ∏_{j∈J} K_{j}/A, where the factors K_{j}/A are simple. Also, L̃_{j} = [K_{j}, K_{j}] is perfect (hence quasisimple), with center Z(L̃_{j}) = A_{j} = L̃_{j} ∩ A. To prove the first isomorphism in (4), it will suffice, by Lemma 1.2, to show that

Λ^{∗}(N_{G}(P)/P; F (P)) = 0 whenever P ∩ K ∉ Syl_{p}(K).  (5)

Let Q be the set of p-subgroups Q ≤ K such that Q ≥ A, and Q/A = ∏_{j∈J} Q_{j}/A for some Q_{j} ≤ K_{j}. Since A ≤ G is a normal p-subgroup and K/A = ∏_{j∈J} K_{j}/A, P ∩K ∈ Q for all radical p-subgroups P ≤ G by Lemma 1.12(a,b). Hence if P ∩ K ∉ Q, then P is not radical, and (5) holds by Proposition 1.1(c).

Now let P ≤ G be any p-subgroup such that Q = P ∩ K ∈ Q and is p-centric in K. The following sequence is exact

1 → CÃ(P) → CK̃(P)_{p} × C_{A}(P) → C_{K}(P)_{p} → F (P) → 1  (6)

since Φ′_{2}(P) ≅ CK̃(P)_{p}/CÃ(P) and (Z_{G}^{K}/Z_{G}^{A})(P) ≅ C_{K}(P)_{p}/C_{A}(P) by Step 3. We claim that the sequence

1 → Ã → CK̃(Q) × A → C_{K}(Q) → 1  (7)

is exact. This is obtained from the exact sequence 1 → Ã → K̃ × A → K → 1 by taking fixed subgroups of the conjugation action of Q. So the exactness of (7) will follow upon showing that CK̃(Q) × A surjects onto C_{K}(Q). Any g ∈ K can be written g = a· ∏_{j∈J} g_{j} for a ∈ A and g_{j} ∈ L̃_{j}, [g, Q] = 1 implies [g_{j}, Q_{j}] = 1 for each j (since [Q_{j}, L̃_{i}] ≤ [K_{j}, K_{i}] = 1 for i ≠ j by (1)), and thus ga^{−1} is the image of (g_{j})_{j∈J} ∈ CK̃(Q).

After taking fixed points of the P/Q-action on (7), we get an exact sequence

1 → CÃ(P) → CK̃(P)_{p} × C_{A}(P) → C_{K}(P)_{p} → H^{1}(P/Q; Ã).

A comparison with (6) shows that F (P) is contained in H^{1}(P/Q; Ã) as a module over N(P)/P. In particular, if Q = P ∩K ∉ Syl_{p}(K), then N_{KP} (P)/P has order a multiple of p (Lemma 1.11), and acts trivially on F (P) since it acts trivially on P/Q ≅ P K/K and on Ã. So (5) follows from Lemma 1.4 in this case.

Finally, when Q = P ∩ K = S, the first three terms in the exact sequence (6) are acyclic as functors on O_{p}(N_{G}(S)/S) by Proposition 1.13. (For example, CK̃(P) = CK̃(S)^{P/S} is the fixed subgroup of the P/S-action.) So F is also acyclic, and this finishes the proof that Φ′_{1}/Φ′_{2} is acylic. □

We have now reduced the computation of \underleftarrow{lim}^{∗}(Z_{G}^{K_{1}}/Z_{G}^{K_{2}}) to the case where K_{1} is quasisimple and K_{2} = Z(K_{1}) is an abelian p-group. It turns out that this only depends on K_{1} and Aut_{G}(K_{1}). To make this precise, we define certain functors Y_{L}^{Γ} as follows. As usual, c_{x} denotes conjugation by an element x.

Definition 2.5. Fix a finite group L and a group of automorphisms Γ ≤ Aut(L) which contains Inn(L). Define Y_{L}^{Γ} : O_{p}(Γ)^{op} → Ab by setting

Y_{L}^{Γ}(P) = \begin{cases}\{c_{x} ∈ Inn(L) | x ∈ C_{L}(P)\}_{p} & if P ∩ Inn(L) is p-centric in Inn(L) \\ 0 & otherwise. \end{cases}

For any P, Q ≤ Γ, Y_{L}^{Γ} sends the morphism represented by g ∈ N_{Γ}(P, Q) to conjugation by g^{−1}.

Here, we again write K_{p} for the Sylow p-subgroup of K in a situation where it is a direct factor of K and abelian. Thus whenever P ∩ Inn(L) is p-centric in Inn(L), Y_{L}^{Γ}(P) ≤ C_{Inn(L)}(P ∩ Inn(L))_{p} = Z(P ∩ Inn(L)).

Lemma 2.6. Fix a finite group G with quasisimple normal subgroup L ◁ G, and assume that A = Z(L) is a p-group. Set Γ = Aut_{G}(L). Then

\underleftarrow{lim}^{∗}_{O_{p}(G)}(Z_{G}^{L}/Z_{G}^{A}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(Γ)}(Y_{L}^{Γ}).

Proof. Let Z′: O_{p}(G)^{op} → Ab be the functor Z′(P) = (Z_{G}^{L}/Z_{G}^{A})(P) if P ≥ A and (P ∩ L)/A is p-centric in L, and Z′(P) = 0 otherwise. We regard this as a quotient functor of Z_{G}^{L}/Z_{G}^{A}. By Proposition 2.3, Λ^{∗}(N(P)/P; (Z_{G}^{L}/Z_{G}^{A})(P)) = 0 for any P such that Z′(P) = 0, and hence \underleftarrow{lim}^{∗}(Z_{G}^{L}/Z_{G}^{A}) ≅ \underleftarrow{lim}^{∗}(Z′) by Lemma 1.2(a).

It remains to show that \underleftarrow{lim}^{∗}(Z′) ≅ \underleftarrow{lim}^{∗}(Y_{L}^{Γ}). Let c: G −↠ Γ be the surjection which sends g ∈ G to c_{g} ∈ Aut(L). Set H = Ker(c) = C_{G}(L), and let O_{p}^{∗}(G) ⊆ O_{p}(G) be the full subcategory whose objects are the p-groups P ≤ G such that P ∩ H ∈ Syl_{p}(H). By Lemma 1.10, it suffices to show that

(a) (Y_{L}^{Γ} ∘ c_{#})|_{O_{p}^{∗}(G)} ≅ Z′|_{O_{p}^{∗}(G)}; and

(b) the action of N_{HP} (P)/P on Z_{G}^{L}/Z_{G}^{A} is trivial for all p-subgroups P ≤ G.

Point (b) is immediate from the definition of the functor Z_{G}^{K}. Also, for any P ≤ G such that P ∩ H ∈ Syl_{p}(H),

(Y_{L}^{Γ} ∘ c_{#})(P) = Y_{L}^{Γ}(Aut_{P} (L)) ≅ \begin{cases}C_{L}(P)_{p}/C_{L}(A) & if (P ∩ L)/A is p-centric in L/A \\ 0 & otherwise \end{cases}
≅ Z′(P),

and this proves (a). □

Lemma 2.6 finishes the process of reducing the general computation of \underleftarrow{lim}^{∗}(Z_{G}) to that of \underleftarrow{lim}^{∗}(Y_{L}^{Γ}) when L is quasisimple and Inn(L) ≤ Γ ≤ Aut(L). These results are now summarized in the following:

Proposition 2.7. Fix a finite group G and normal subgroups K_{2} ≤ K_{1} in G, such that K_{1}/K_{2} is a minimal subgroup of G/K_{2}. If K_{1}/K_{2} is abelian, or if there is a p-subgroup Q ≤ G such that [Q, K_{1}] ≤ K_{2} and C_{K_{1}}(Q) ≤ K_{2}, then the functor Z_{G}^{K_{1}}/Z_{G}^{K_{2}} is acyclic. Otherwise, there is a quasiperfect group L with p-group center such that K_{1}/K_{2} is isomorphic to a product of copies of L/Z(L), a subgroup Γ ≤ Aut(L) containing Inn(L), and (for each i) a homomorphism

\underleftarrow{lim}^{i}_{O_{p}(Γ)}(Y_{L}^{Γ}) ⟶ \underleftarrow{lim}^{i}_{O_{p}(G)}(Z_{G}^{K_{1}}/Z_{G}^{K_{2}})

which is onto when i = 1 and an isomorphism when i ≥ 2.

Proof. The abelian case was handled in Proposition 2.3. The cases where K_{1}/K_{2} is nonabelian follow from Lemmas 2.1, 2.4, and 2.6. □

In order to avoid repeating these conditions about central extensions and groups of automorphisms throughout the remaining sections, we define the following classes of finite simple groups.

Definition 2.8. For each prime p and each i ≥ 1, let 𝔏^{i}(p) be the class of finite nonabelian simple groups L with the property that

\underleftarrow{lim}^{i}_{O_{p}(Γ)}(YΓ ; L̃) = 0

for each quasisimple group L̃ such that Z(L̃) is a p-group and L̃/Z(L̃) ≅ L, and each subgroup Γ ≤ Aut(L̃) which contains Inn(L̃). Also, 𝔏^{≥i}(p) denotes the intersection of the classes 𝔏^{j}(p) for all j ≥ i.

The important consequence of Proposition 2.7 is:

Proposition 2.9. For any finite group G and any i ≥ 1, \underleftarrow{lim}^{i}(Z_{G}) = 0 if for each nonabelian simple group L which appears in the decomposition series for G, L ∈ 𝔏^{i}(p).

Our goal now, throughout the rest of the paper, is to show that every finite nonabelian simple group lies in 𝔏^{≥2}(2).

3. A relative version of Λ-functors

Throughout this section, p denotes a fixed prime. We first present a relative version of the functors Λ^{∗}(G; M) which will be useful when handling outer automorphisms of a quasisimple group L.

Definition 3.1. Fix a prime p, a pair H ◁ G of finite groups, and a ℤ_{(p)}[G]-module M. Let Φ_{M}^{G,H}: O_{p}(G)^{op} ⟶ ℤ_{(p)}-mod be the functor defined by

Φ_{M}^{G,H}(P) = \begin{cases}M^{P} & if P ∩ H = 1 \\ 0 & otherwise; \end{cases}

and define

Λ^{∗}(G, H; M) = \underleftarrow{lim}^{∗}_{O_{p}(G)}(Φ_{M}^{G,H}).

Note in particular that Φ_{M}^{G} = Φ_{M}^{G,G}, and hence Λ^{∗}(G; M) = Λ^{∗}(G, G; M).

The importance of these groups arises from the following generalization of [JMO, Proposition 5.4]. For any H ◁ G, we say that a functor F: O_{p}(G)^{op} → Ab is Hcontrolled if for all P, the inclusion of (P ∩H) in P induces an isomorphism F (P) ≅ F (P ∩H)^{P}.

Proposition 3.2. Fix a finite group G, a prime p, a normal subgroup H ◁ G, and a p-subgroup Q ≤ H. Let F: O_{p}(G) ⟶ ℤ_{(p)}-mod be any H-controlled functor which vanishes except on subgroups P ≤ G such that P ∩ H is G-conjugate to Q. (Thus F (P) = F (Q)^{P/Q} whenever P ∩ H = Q.) Then

\underleftarrow{lim}^{∗}_{O_{p}(G)}(F) ≅ Λ^{∗}(N_{G}(Q)/Q, N_{H}(Q)/Q; F (Q)).

Proof. This is a special case of Lemma 1.8. □

The idea now is to filter an arbitrary H-controlled functor F: O_{p}(G) → ℤ_{(p)}-mod by subfunctors F_{i} in such a way that for each quotient functor F_{i}/F_{i−1}, F_{i}/F_{i−1}(P) = 0 except for those P ≤ G such that P ∩ H lies in one conjugacy class of p-subgroups of H. Then each \underleftarrow{lim}^{∗}(F_{i}/F_{i−1}) is described via Proposition 3.2 and the Λ^{∗}(G, H; −). When L is quasisimple and Inn(L) ≤ Γ ≤ Aut(L), the functors Y_{L}^{Γ} need not be Inn(L)- controlled, but we will see that the same techniques can be used to compute higher limits of these functors.

For any G, and any short exact sequence of ℤ_{(p)}[G]-modules

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0,

the induced sequence of functors 0 ⟶ Φ_{M′}^{G} ⟶ Φ_{M}^{G} ⟶ Φ_{M′′}^{G} ⟶ 0 is also exact, and hence induces a long exact sequence of the groups Λ^{∗}(G; −). This is not in general the case for the relative groups Λ^{∗}(G, H; −) when H ◁ G is a proper normal subgroup, which is why the statement of point (b) in the next proposition is so detailed.

Most of the properties of the relative groups Λ^{∗}(G, H; M) listed in the following proposition generalize properties of the groups Λ^{∗}(G; M) proven in [JMO].

Proposition 3.3. Fix a prime p, a finite group G, a normal subgroup H ◁ G, and a ℤ_{(p)}[G]-module M. Then the following hold.

(a) If (p, |H|) = 1, then Λ^{0}(G, H; M) ≅ M^{G}, and Λ^{i}(G, H; M) = 0 for all i > 0. If p||H|, then Λ^{0}(G, H; M) = 0.

(b) Λ^{∗}(G, H; M) = 0 if the kernel of the action of H on M has order a multiple of p.

More generally, if M_{0} ⊆ M is a G-invariant submodule, then

• Λ^{∗}(G, H; M) ≅ Λ^{∗}(G, H; M_{0}) if the kernel of the H-action on M/M_{0} has order a multiple of p; and

• Λ^{∗}(G, H; M) ≅ Λ^{∗}(G, H; M/M_{0}) if the kernel of the H-action on M_{0} has order a multiple of p.

(c) Λ^{∗}(G, H; M) = 0 if O_{p}(H) ≠ 1.

(d) If K ◁ G is a normal subgroup which acts trivially on the ℤ_{(p)}[G]-module M, and H ∩ K has order prime to p, then

Λ^{∗}(G, H; M) ≅ Λ^{∗}(G/K, HK/K; M).

(e) Assume p||H|, let ∼ be the equivalence relation on Syl_{p}(H) generated by nontrivial intersection, and set G_{0} = {g ∈ G | gSg^{−1} ∼ S} for some fixed S ∈ Syl_{p}(H). Then Λ^{1}(G, H; M)≅M^{G_{0}}/M^{G}.

(f) Assume the Sylow p-subgroups of H are cyclic, or (if p = 2) quaternion. Then Λ^{i}(G, H; M) = 0 for all i ≥ 2.

Proof. (a) If (p, |H|) = 1, then P ∩ H = 1 for all p-subgroups H ≤ G, and hence Φ_{M}^{G,H} = H^{0}M in the notation of Proposition 1.13. So by that proposition, \underleftarrow{lim}^{0}(Φ_{M}^{G,H}) = M^{G} and its higher limits vanish. If p||H|, then Φ_{M}^{G,H} vanishes on the Sylow subgroups of G, and hence \underleftarrow{lim}^{0}(Φ_{M}^{G,H}) = 0.

(b) The first statement is a special case of either of the other two.

Assume first that K =^{def} Ker[H → Aut(M/M_{0})] ◁ G has order a multiple of p. Fix P ≤ G such that P ∩ H = 1, and choose Q ∈ Syl_{p}(P K) such that Q ≩ P. Then N_{Q}(P) ≩ P, so N_{Q}(P)/P ≠ 1 acts trivially on M^{P} /M_{0}^{P}, and hence

Λ^{∗}(N_{G}(P)/P; (Φ_{M}^{G,H}/Φ_{M_{0}}^{G,H})(P)) = 0

by Proposition 1.1(b). Since this holds for all such P, \underleftarrow{lim}^{∗}(Φ_{M}^{G,H}/Φ_{M_{0}}^{G,H}) = 0 by Lemma 1.2(a), and hence Λ^{∗}(G, H; M) ≅ Λ^{∗}(G, H; M_{0}).

Now assume that K =^{def} Ker[H → Aut(M_{0})] ◁ G has order a multiple of p. There is an exact sequence of functors on O_{p}(G)

0 ⟶ Φ_{M_{0}}^{G,H} ⟶ Φ_{M}^{G,H} ⟶ Φ_{M/M_{0}}^{G,H} ⟶ Ψ ⟶ 0,

where Ψ(P) ⊆ H^{1}(P; M_{0}) if P ∩ H = 1 and Ψ(P) = 0 otherwise. We have just seen that \underleftarrow{lim}^{∗}(Φ_{M_{0}}^{G,H}) = 0, and hence we will be done upon showing that \underleftarrow{lim}^{∗}(Ψ) = 0. For each P ≤ G such that P ∩ H = 1, N_{P K}(P)/P has order a multiple of p (Lemma 1.11) and acts trivially on Ψ(P) ⊆ H^{1}(P; M_{0}), and thus Λ^{∗}(N_{G}(P)/P; Ψ(P)) = 0 by Proposition 1.1(b). So all higher limits of Ψ vanish by Lemma 1.2(a).

(c) If O_{p}(H) ≠ 1, then for all p-subgroups P ≤ G such that P ∩ H = 1,

O_{p}(N_{G}(P)/P) ≥ N_{O_{p}(H)·P} (P)/P ≠ 1.

Hence Λ^{∗}(N_{G}(P)/P; M^{P}) = 0 for all such P by [JMO, Proposition 6.1(ii)], and so Λ^{∗}(G, H; M) = 0 by Lemma 1.2.

(d) Let φ: G ⟶ G/K be the projection, and let φ_{#} denote the induced functor between orbit categories. Let O_{p}^{∗}(G) ⊆ O_{p}(G) be the full subcategory whose objects are those P ≤ G such that P ∩ K ∈ Syl_{p}(K). We claim that

Φ_{M}^{G,H}|_{O_{p}^{∗}(G)} ≅ (Φ_{M}^{G/K,HK/K} ∘ φ_{#})|_{O_{p}^{∗}(G)}.  (1)

This is clear, once we have checked that for any p-subgroup P ≤ G such that P ∩ K ∈ Syl_{p}(K), P ∩ H = 1 if and only if P K/K ∩ HK/K = 1. If P K/K ∩ HK/K = 1, then P ∩ H ≤ K, and hence P ∩ H = 1 since P is a p-group and K ∩ H has order prime to p.

Conversely, if P ∩ H = 1, then P K ∩ H has order prime to p, since P ∈ Syl_{p}(P K) and thus any element of P K ∩ H of p-power order would be G-conjugate to an element of P ∩ H. Hence P K ∩ H ≤ K, since any element of P K∖K has order a multiple of p. It follows that P K ∩ HK ≤ K, and P K/K ∩ HK/K = 1.

This finishes the proof of (1). Lemma 1.10 now applies to show that

Λ^{∗}(G, H; M) = \underleftarrow{lim}^{∗}_{O_{p}(G)}(Φ_{M}^{G,H}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(G/K)}(Φ_{M}^{G/K,HK/K}) = Λ^{∗}(G/K, HK/K; M).

(e) Regard Φ_{M}^{G,H} as a subfunctor of the functor H^{0}M, which sends P to M^{P} for all P. By Proposition 1.13, H^{0}M is acyclic and \underleftarrow{lim}^{0}(H^{0}M) ≅ M^{G}. So there is a short exact sequence

0 ⟶ M^{G} ⟶ \underleftarrow{lim}^{0}(H^{0}M/Φ_{M}^{G,H}) ⟶ Λ^{1}(G, H; M) ⟶ 0.

It remains only to show that the middle term is isomorphic to M^{G_{0}}.

To see this, fix some T ∈ Syl_{p}(G) which contains our given S ∈ Syl_{p}(H), and restrict to the full subcategory O_{T} (G) ⊆ O_{p}(G) whose objects are the subgroups of T. Then

G_{0} = {g ∈ G | gSg^{−1} ∼ S} = ⟨g ∈ G | gT g^{−1} ∩ T ∩ H ≠ 1⟩
= ⟨g ∈ G | ∃ P, gP g^{−1} ≤ T with P ∩ H ≠ 1⟩;

and hence \underleftarrow{lim}^{0}(H^{0}M/Φ_{M}^{G,H}) ≅ M^{G_{0}}.

(f) Fix Ŝ ∈ Syl_{p}(G), set S = Ŝ∩ H ∈ Syl_{p}(H), and let Z ≤ S be the unique subgroup of order p. In particular, Z is weakly closed in Ŝ with respect to G.

Regard Φ_{M}^{G,H} as a subfunctor of the functor H^{0}M which sends P to M^{P} for all P. Then H^{0}M/Φ_{M}^{G,H} sends a p-subgroup P ≤ G to M^{P} if P contains a subgroup conjugate to Z and sends P to 0 otherwise. So by Lemma 1.9,

\underleftarrow{lim}^{∗}_{O_{p}(G)}(H^{0}M/Φ_{M}^{G,H}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(N_{G}(Z)/Z)}(P/Z ↦ M^{P} ),

and this last functor is acyclic by Proposition 1.13. Since H^{0}M is also acyclic by Proposition 1.13, the exact sequence for the extension of functors now shows that Λ^{i}(G, H; M) = \underleftarrow{lim}^{i}(Φ_{M}^{G,H}) = 0 for all i ≥ 2. □

The next proposition describes the role of these relative functors Λ^{∗}(G, H; M) when computing higher limits of the YΓ ; L̃. For these purposes, it is useful to consider the following subquotient functors of YΓ ; L̃, defined for any p-centric subgroup Q ≤ L:

(YΓ ; L̃)_{≥Q}(P) = \begin{cases}YΓ ; L̃(P) & if P ≥ Q′, some Q′ Γ-conjugate to Q \\ 0 & otherwise \end{cases}

(YΓ ; L̃)_{Q}(P) = \begin{cases}YΓ ; L̃(P) & if P ∩ L is Γ-conjugate to Q \\ 0 & otherwise \end{cases}

Thus (YΓ ; L̃)_{Q} is a subfunctor of (YΓ ; L̃)_{≥Q}, and this is a quotient functor of YΓ ; L̃. When Γ and L̃ are clear from context, we drop them, and just write Y_{Q} ⊆ Y_{≥Q}, etc.

Lemma 3.4. Fix a simple group L, a central extension L̃ −↠^{π} L such that L̃ is quasisimple and Ker(π) is a p-group, and a subgroup Γ ≤ Aut(L̃) which contains Inn(L̃) ≅ L. Then the following hold for any p-centric subgroup Q ≤ L.

(a) If Q ∉ Syl_{p}(L), then \underleftarrow{lim}^{∗}_{O_{p}(Γ)}((YΓ ; L̃)_{Q}) ≅ Λ^{∗}(N_{Γ}(Q)/Q, N_{L}(Q)/Q; YΓ ; L̃(Q)).

(b) If S ∈ Syl_{p}(L), and Q is p-centric in L and weakly closed in S with respect to Γ, then for all i ≥ 0,

\underleftarrow{lim}^{i}_{O_{p}(Γ)}((YΓ ; L̃)_{≥Q}) ≅ \begin{cases}(π^{−1}Q)^{N_{Γ}(Q)}/Z(L̃)^{Γ} & if i = 0 \\ 0 & if i > 0. \end{cases}

Proof. Let F^{1} ⊇ F^{0} be the following functors on O_{p}(Γ):

F^{1}(P) = \begin{cases}(π^{−1}(P ∩ L))^{P} & if P ∩ L is p-centric in L \\ Z(L̃)^{P} & otherwise \end{cases} and F^{0}(P) = Z(L̃)^{P} .

Here, we identify L = Inn(L̃) ◁ Γ. By definition, Y = YΓ ; L̃ ≅ F^{1}/F^{0}. Define quotient functors F_{≥Q}^{i} of F^{i} and subfunctors F_{Q}^{i} ⊆ F_{≥Q}^{i} in analogy with the definitions for Y. Thus

Y_{≥Q} ≅ F_{≥Q}^{1}/F_{≥Q}^{0} and Y_{Q} ≅ F_{Q}^{1}/F_{Q}^{0}

By Proposition 3.2,

\underleftarrow{lim}^{∗}_{O_{p}(Γ)}(F_{Q}^{i}) ≅ Λ^{∗}(N_{Γ}(Q)/Q, N_{L}(Q)/Q; F_{Q}^{i}(Q))

for i = 0, 1. Since the action of N_{L}(Q)/Q on F^{0}(Q) = Z(L̃) is trivial, Proposition 3.3(b) implies that if Q ∉ Syl_{p}(L), then

\underleftarrow{lim}^{∗}_{O_{p}(Γ)}(Y_{Q}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(Γ)}(F_{Q}^{1}/F_{Q}^{0}) ≅ Λ^{∗}(N_{Γ}(Q)/Q, N_{L}(Q)/Q; F^{1}(Q)/F^{0}(Q))

≅ Λ^{∗}(N_{Γ}(Q)/Q, N_{L}(Q)/Q; YΓ ; L̃(Q)).

Now assume that Q is p-centric in L, and weakly closed in some S ∈ Syl_{p}(L) with respect to Γ. By definition, for all P ≥ Q in Γ, F^{i}(P) = F^{i}(Q)^{P} for i = 0, 1. Hence by Lemma 1.9 and Proposition 1.13,

\underleftarrow{lim}^{∗}(F_{≥Q}^{i}) ≅ \begin{cases}0 & if ∗ > 0 \\ F^{i}(Q)^{N_{Γ}(Q)} & if ∗ = 0. \end{cases}

Thus Y_{≥Q} ≅ F_{≥Q}^{1}/F_{≥Q}^{0} is acyclic, and

\underleftarrow{lim}^{0}(Y_{≥Q}) ≅ (π^{−1}Q)^{N_{Γ}(Q)}/Z(L̃)^{Γ}.  □

We will need a much stronger vanishing theorem for the Λ^{∗}(G, H; M) than what was shown in Proposition 3.3. In the following lemma, for any set H_{1}, …, H_{m} ≤ G of subgroups, we write N_{G}(H_{1}, …, H_{m}) to denote the subgroup of elements of G which normalize all of them. A radical p-chain of length n in G is a sequence

O_{p}(G) = P_{0} ◁ P_{1} ◁ ⋯ ◁ P_{n}

of distinct p-subgroups of G such that P_{i} = O_{p}(N_{G}(P_{0}, …, P_{i})) for all i (in particular, P_{i} ◁ P_{n} for all i), and such that P_{n} ∈ Syl_{p}(N_{G}(P_{0}, …, P_{n−1})).

Proposition 3.5. Fix a finite group G, a normal subgroup H ◁ G, and a finite ℤ_{(p)}[G]- module M. Assume, for some n ≥ 1, that Λ^{n}(G, H; M) ≠ 0. Let 0 = M′_{0} ⊆ M′_{1} ⊆ ⋯ ⊆ M′_{n} = M/pM be any filtration by ℤ_{(p)}[G]-submodules. Then there is a radical p-chain

1 = P_{0} ≨ P_{1} ≨ P_{2} ≨ ⋯ ≨ P_{n}

of length n in H such that 𝔑_{N_{H}(P_{1},...,P_{n})}·(M′_{k}/M′_{k−1}) ≠ 0 for some k, and such that

rk_{p}(M) ≥ rk(M′_{k}/M′_{k−1}) ≥ |P_{n}| ≥ p^{n}.

Proof. For any ℤ_{(p)}[G]-module M, let M_{M}^{G} be the set of all sub-Mackey functors Ψ ⊆ H^{0}M; i.e., those functors on O_{p}(G) such that Ψ(P) ≤ M^{P} for all P, and such that the relative norm 𝔑_{Q}^{P}: M^{Q} ⟶ M^{P} sends Ψ(Q) into Ψ(P) for all p-subgroups Q ≤ P in G. Let N_{M}^{G,H} be the set of all functors on O_{p}(G) of the form Ψ ∩ Φ_{M}^{G,H} (objectwise intersection) for some Ψ ∈ M_{M}^{G}. We first show:

for any Φ ∈ N_{M}^{G,H} and any n ≥ 1 with \underleftarrow{lim}^{n}(Φ) ≠ 0, there exists a radical p-
subgroup P ≤ H such that 𝔑_{P} ·M ≠ 0, and a functor Φ′ ∈ N_{𝔑_{P} ·M}^{N_{G}(P)/P,N_{H}(P)/P}
such that \underleftarrow{lim}^{n−1}(Φ′) ≠ 0.

(1)

To see this, let Ψ_{0} ∈ M_{M}^{G} be such that Φ = Ψ_{0} ∩ Φ_{M}^{G,H}. Let Ψ ⊆ Ψ_{0} be the subfunctor Ψ(P) = Ψ_{0}(P) ∩ (𝔑_{P ∩H}·M); this still lies in M_{M}^{G}. Since Ψ is acyclic [JM, Proposition 5.14], \underleftarrow{lim}^{n−1}(Ψ/Φ) ≠ 0. By an appropriate filtration of Ψ/Φ, there is a p-subgroup 1 ≠ P ≤ H such that if we define Ψ_{P} by Ψ_{P} (Q) = Ψ(Q) if Q ∩ H is conjugate to P and Φ′(Q) = 0 otherwise (still as a functor on O_{p}(G)), then \underleftarrow{lim}^{n−1}(Ψ_{P}) ≠ 0. By Proposition 1.1(a,c), for some p-subgroup Q ≤ G such that Q ∩ H = P, Q is radical in G, and hence P is radical in H by Lemma 1.12(b). Furthermore, if we define Φ′ ⊆ Ψ′ on O_{p}(N_{G}(P)/P) by setting Ψ′(Q/P) = Ψ(Q) and Φ′(Q/P) = Ψ_{P} (Q), then Ψ′ ∈ M_{𝔑_{P} ·M}^{N_{G}(P)/P}, and so

Φ′ = Ψ′ ∩ Φ_{𝔑_{P} ·M}^{N_{G}(P)/P,N_{H}(P)/P} ∈ N_{𝔑_{P} ·M}^{N_{G}(P)/P,N_{H}(P)/P}.

Finally, by Lemma 1.8, the functors Ψ_{P} and Φ′ have the same higher limits, and in particular \underleftarrow{lim}^{n−1}(Φ′) ≠ 0. This finishes the proof of (1).

We next claim, by induction on n, that for any Φ ∈ N_{M}^{G,H} with \underleftarrow{lim}^{n}(Φ) ≠ 0, there exists a radical p-chain 1 = P_{0} ≨ P_{1} ≨ ⋯ ≨ P_{n} of length n such that 𝔑_{K}·M ≠ 0, where K = N_{H}(P_{1}, …, P_{n}).

(2)

To prove this, let P and Φ′ be as in (1). If n = 1, then

0 ≠ \underleftarrow{lim}^{0}(Φ′) ⊆ Λ^{0}(N_{G}(P)/P, N_{H}(P)/P; 𝔑_{P} ·M)

implies that (p, |N_{H}(P)/P |) = 1, hence that P ∈ Syl_{p}(H), and

(𝔑_{P} ·M)^{N_{H}(P)} = 𝔑_{N_{H}(P)}·M ≠ 0

by Proposition 3.3(a). Thus (0 ≨ P) is a radical p-chain, and (2) holds in this case. If n > 1, then by the induction hypothesis (applied to Φ′), there is a radical p-chain

1 ≠ P_{2}/P ≨ ⋯ ≨ P_{n}/P  in N_{H}(P)/P  (3)

such that 𝔑_{K/P} ·(𝔑_{P} ·M) = 𝔑_{K}·M ≠ 0, where K/P = N_{N_{H}(P)/P} (P_{2}/P, …, P_{n}/P). Since P is radical in H, O_{p}(N_{H}(P)/P) = 1, and the sequence 1 ≨ P ≨ P_{2} ≨ ⋯ ≨ P_{n} is a radical p-chain of length n in H. Also, K = N_{H}(P, P_{2}, …, P_{n}), and this finishes the proof of (2).

Now let M be any finite ℤ_{(p)}[G]-module such that Λ^{n}(G, H; M) = \underleftarrow{lim}^{n}(Φ_{M}^{G,H}) ≠ 0, and let 0 = M′_{0} ⊆ M′_{1} ⊆ ⋯ ⊆ M′_{n} = M/pM be a filtration by G-invariant submodules. Since each p^{k}M/p^{k+1}M is isomorphic to an 𝔽_{p}[G]-submodule of M/pM, we can lift this to a filtration of M by submodules M′′_{i} such that each M′′_{i}/M′′_{i−1} is isomorphic to a submodule of some M′_{j}/M′_{j−1}. Choose ℓ such that \underleftarrow{lim}^{n}(Φ_{M′′_{ℓ}}^{G,H}/Φ_{M′′_{ℓ−1}}^{G,H}) ≠ 0. Set V = M′′_{ℓ}/M′′_{ℓ−1} for short.

Consider the functor Ψ: O_{p}(G)^{op} ⟶ ℤ_{(p)}-mod defined by Ψ(P) = (M′′_{ℓ})^{P} /(M′′_{ℓ−1})^{P}. This is a sub-Mackey functor of H^{0}V, and thus an element of M_{V}^{G}. Hence

Φ =^{def} Ψ ∩ Φ_{V}^{G,H} = Φ_{M′′_{ℓ}}^{G,H}/Ψ_{M′′_{ℓ−1}}^{G,H} ∈ N_{V}^{G,H}.

By (2), since \underleftarrow{lim}^{n}(Φ) ≠ 0, there is a radical p-chain 1 = P_{0} ≨ P_{1} ≨ ⋯ ≨ P_{n} of length n such that 𝔑_{K}·V ≠ 0, where K = N_{H}(P_{1}, …, P_{n}).

In particular, 𝔑_{P_{n}}·V ≠ 0. Fix x ∈ V such that 𝔑_{P_{n}}·x ≠ 0, and consider the P_{n}-linear homomorphism 𝔽_{p}[P_{n}] ⟶^{α} V defined by setting α(g) = gx. Thus Ker(α) is an ideal in 𝔽_{p}[P_{n}] which does not contain 𝔑_{P_{n}}, and hence is the zero ideal (cf. [Se, §8.3, Prop. 26]). So α is injective, and V contains a copy of 𝔽_{p}[K]. Since V is isomorphic to a subgroup of some M′_{k}/M′_{k−1}, this module also contains a copy of 𝔽_{p}[P_{n}]. The lower bounds for rk_{p}(M) are now immediate. □

4. Subgroups which contribute to higher limits

As noted earlier, our remaining goal is to prove that every nonabelian finite simple group lies in the class 𝔏^{≥2}(2) (see Definition 2.8). Once we have shown this, then Theorem A will follow as a consequence of Proposition 2.9. The aim of this section is to prove a series of propositions, whose hypotheses are stated in purely group theoretic terms (without reference to higher limits or Λ^{∗}’s), which can then be used in later sections to carry out a case-by-case check that L ∈ 𝔏^{≥2}(2).

The following definition is mostly of interest when L is simple, but in a few cases (when carrying out inductive arguments) we need to also deal with almost simple groups. For simplicity in notation, for any group L with Z(L) = 1, we identify L = Inn(L) as a subgroup of Aut(L). Recall that for any p-group P and any n ≥ 1,

Ω_{n}(P) =^{def} ⟨g ∈ P | g^{p^{n}} = 1⟩.

Definition 4.1. Fix a centerfree group L and a prime p||L|.

(a) For each i ≥ 1, let ℜ^{i}(L; p) be the set of all p-subgroups P ≤ L with the property that for some Γ ≤ Aut(L) which contains Inn(L), and some N_{Γ}(P)-invariant subgroup Z_{0} ⊆ Z(P),

Λ^{i}(N_{Γ}(P)/P, N_{L}(P)/P; Z_{0}) ≠ 0.

(b) For each i ≥ 1, set

𝔈^{i}(L; p) = {Ω_{1}(Z(P)) | P ∈ ℜ^{i}(L; p)}.

(c) An elementary abelian p-subgroup E ≤ L is called pivotal if E = Ω_{1}(Z(P)) for some P ∈ Syl_{p}(C_{L}(E)), and O_{p}(Aut_{L}(E)) = 1.

We also write ℜ^{≥i}(L; p) = ⋃_{j≥i} ℜ^{j}(L; p), and similarly for 𝔈^{≥i}(L; p). In addition, for any p-subgroup Q ≤ L, we let ℜ^{i}(L; p)_{≱Q} and ℜ^{≥i}(L; p)_{≱Q} denote the set of subgroups in ℜ^{i}(L; p) and ℜ^{≥i}(L; p), respectively, which do not contain any subgroup conjugate to Q.

As will be seen in the next proposition (or in its proof), we can think of ℜ^{i}(L; p) as the set of p-subgroups of L which could “contribute” to \underleftarrow{lim}^{i}(YΓ ; L̃), for some quasicentric group L̃ with L̃/Z(L̃) ≅ L, and some Γ ≤ Aut(L̃) which contains Inn(L̃) ≅ L. Of course, the existence of an element of ℜ^{i}(L; p) does not mean that \underleftarrow{lim}^{i}(YΓ ; L̃) is nonvanishing for some Γ and L̃: a nonzero “contribution” to \underleftarrow{lim}^{i}(−) by a subgroup in ℜ^{i}(L; p) could be cancelled by a contribution to \underleftarrow{lim}^{i±1}(−).

Proposition 4.2. Fix a prime p, a finite simple group L, S ∈ Syl_{p}(L), and i ≥ 1. Assume there is a subgroup Q ≤ S which is p-centric in L and weakly closed in S with respect to Aut(L), such that ℜ^{i}(L; p)_{≱Q} = ∅. Then L ∈ 𝔏^{i}(p). In particular, L ∈ 𝔏^{i}(p) if ℜ^{i}(L; p) = ∅.

Proof. Fix Γ and L̃, and let Y_{≱Q} be the subfunctor of YΓ ; L̃ defined by setting Y_{≱Q}(P) = YΓ ; L̃(P) if P does not contain a subgroup conjugate to Q and Y_{≱Q}(P) = 0 otherwise.

For each P ≤ L ≅ Inn(L̃) which does not contain a subgroup conjugate to Q, we defined YΓ ; L̃(P) to be a certain N_{Γ}(P)/P -invariant subgroup of Z(P) (Definition 2.5). Thus P ∉ ℜ^{i}(L; p) implies that

Λ^{i}(N_{Γ}(P)/P, N_{L}(P)/P; YΓ ; L̃(P)) = 0.

Hence \underleftarrow{lim}^{i}(Y_{≱Q}) = 0 by Lemma 3.4(a) and an appropriate filtration of Y_{≱Q}. Furthermore, \underleftarrow{lim}^{i}(YΓ ; L̃/Y_{≱Q}) = 0 by Lemma 3.4(b), and this finishes the proof of the proposition.

The last statement is just the case Q ∈ Syl_{p}(L). □

In the course of the next five sections, we will prove that all simple groups lie in 𝔏^{≥2}(2), and most cases this will be done using Proposition 4.2. The following proposition will, however, be needed when proving that simple groups of Lie type in characteristic two lie in 𝔏^{≥2}(2) (and in fact, with the exception of L_{3}(2), lie in 𝔏^{≥1}(2)).

Proposition 4.3. Let L be a simple group. Fix S ∈ Syl_{p}(L), and let Q ◁ S be a p-subgroup with the following properties:

(a) Q is p-centric in L and weakly closed in S with respect to Aut(L).

(b) for each P ≤ S which does not contain Q, and which is p-centric and radical in L, P is weakly closed in S with respect to L, and Z(N_{L}(P))_{p} = Z(N_{L}(P Q))_{p}.

Then L ∈ 𝔏^{≥1}(p).

Proof. Let P be the set of p-subgroups of S which are radical and p-centric in L. For any P ∈ P, let P̃ denote its inverse image in L̃. We first check that for each P ∈ P,

Z(NL̃(P̃))_{p} = Z(NL̃(P̃ Q))_{p}.  (1)

By assumption, Z(N_{L}(P))_{p} = Z(N_{L}(P Q))_{p}. Also,

Z(NL̃(P̃))/Z(L̃) = Ker[Z(N_{L}(P)) ⟶^{ψ_{P}} Hom(N_{L}(P), Z(L̃))] ,

and similarly Z(NL̃(P̃ Q))/Z(L̃) = Ker(ψ_{P Q}). Here, ψ_{P} and ψ_{P Q} are defined by lifting to L̃ and taking commutators. Thus ψ_{P} and ψ_{P Q} have the same domain. Since Z(L̃) is a p-group and N_{L}(P) ≥ N_{L}(P Q) ≥ S (P and Q being weakly closed in S), their target groups both inject into Hom(S, Z(L̃)). Thus Ker(ψ_{P}) = Ker(ψ_{P Q}), and this proves (1).

For each R ∈ P, define functors Y_{R} and Y_{≥R} on O_{p}(Γ) by setting

Y_{≥R}(P) = \begin{cases}YΓ ; L̃(P) & if P ∩ L ≥ R′, some R′ Γ-conjugate to R \\ 0 & otherwise \end{cases}

and

Y_{R}(P) = \begin{cases}YΓ ; L̃(P) & if P ∩ L is Γ-conjugate to R \\ 0 & otherwise. \end{cases}

Thus Y_{≥R} is a quotient functor of YΓ ; L̃, and Y_{R} is a subfunctor of Y_{≥R}.

We claim that \underleftarrow{lim}^{∗}(Y_{R}) = 0 for each R ∈ P which does not contain Q. It suffices to prove this when N_{Γ}(R)·L = Γ (otherwise replace Γ by N_{Γ}(R)·L without changing the higher limits). In this case, R and RQ are both weakly closed in S with respect to Γ. Hence by Lemma 3.4(b), Y_{≥R} and Y_{≥RQ} are acyclic, and by (1),

\underleftarrow{lim}^{0}(Y_{≥R}) ≅ Z(NL̃(R̃))_{p}^{Γ/L}/Z(L̃)^{Γ} ≅ Z(NL̃(R̃Q))_{p}^{Γ/L}/Z(L̃)^{Γ} ≅ \underleftarrow{lim}^{0}(Y_{≥RQ}). (2)

Thus if we set Y_{≥R}^{•} = Ker[Y_{≥R} → Y_{≥RQ}], then \underleftarrow{lim}^{∗}(Y_{≥R}^{•}) = 0. We can assume inductively that \underleftarrow{lim}^{∗}(Y_{P}) = 0 for all P ∈ P such that P ≩ R and P ≱ Q. Via the obvious filtration of Y_{≥R}^{•}, we now see that \underleftarrow{lim}^{∗}(Y_{R}) = 0.

Thus \underleftarrow{lim}^{∗}(YΓ ; L̃) ≅ \underleftarrow{lim}^{∗}(Y_{≥Q}), and Y_{≥Q} is acyclic by Lemma 3.4(b) again. Since this holds for all Γ and L̃, L ∈ 𝔏^{≥1}(2). □

It remains to develop some tools for determining which subgroups belong to the sets ℜ^{i}(L; p). The first step is to study their connection with the sets 𝔈^{i}(L; p) and the pivotal subgroups of L, also defined in 4.1.

Proposition 4.4. Fix a prime p, a finite centerfree group L, and i ≥ 1. Then the following hold for any p-subgroup P ≤ L and any elementary abelian p-subgroup E ≤ L.

(a) If P ∈ ℜ^{i}(L; p) and E = Ω_{1}(Z(P)), then P ∈ Syl_{p}(C_{L}(E)). If E ∈ 𝔈^{i}(L; p) and P ∈ Syl_{p}(C_{L}(E)), then P ∈ ℜ^{i}(L; p) and E = Ω_{1}(Z(P)). In other words, there are inverse bijections

ℜ^{i}(L; p)/(conj) \overrightarrow{⟵}_{𝔯𝔞𝔡}^{𝔢𝔩𝔢𝔪} 𝔈^{i}(L; p)/(conj),

where 𝔢𝔩𝔢𝔪(P) = Ω_{1}(Z(P)) and 𝔯𝔞𝔡(E) ∈ Syl_{p}(C_{L}(E)).

(b) Assume E = Ω_{1}(Z(P)) and P ∈ Syl_{p}(C_{L}(E)). Then the natural map

N_{L}(P)/P −↠ Aut_{L}(E)

induced by restriction is a surjection, its kernel has order prime to p, and the action of N_{L}(P)/P on Z(P) factors through Aut_{L}(E). Furthermore, for any Γ ≤ Aut(L) which contains Inn(L), the natural map N_{Γ}(P)/P −↠ Aut_{Γ}(E) is a surjection.

(c) If E ∈ 𝔈^{i}(L; p), then E is a pivotal subgroup.

(d) If E is a pivotal subgroup and P ∈ Syl_{p}(C_{L}(E)), then P is a radical p-subgroup of L.

(e) Fix S ∈ Syl_{p}(L). Each pivotal p-subgroup of L is L-conjugate to a subgroup E ≤ S such that E = Ω_{1}(Z(C_{S}(E))), and hence such that E ≥ Ω_{1}(Z(S)).

(f) Let E ≤ L be an elementary abelian p-subgroup, and let 𝔛 ⊆ C_{L}(E)∖E be a C_{L}(E)-conjugacy class of elements of order p such that |𝔛| is prime to p. Then no elementary abelian subgroup E′ ≥ E such that E′ ∩ 𝔛 = ∅ is pivotal in L.

Proof. (a) Assume P ∈ ℜ^{i}(L; p), and let Γ ≤ Aut(L) and Z′ ⊆ Z(P) be such that

Λ^{i}(N_{Γ}(P)/P, N_{L}(P)/P; Z′) ≠ 0.

Set H = C_{L}(E) for short. Then P ≤ H by definition of E. Since N_{H}(P)/P acts trivially on E, it acts trivially on Ω_{n}(Z′)/Ω_{n−1}(Z′) for each n, and hence must have order prime to p by Proposition 3.3(b). So by Lemma 1.11, P ∈ Syl_{p}(H).

Now assume E ∈ 𝔈^{i}(L; p). By definition, E = Ω_{1}(Z(P′)) for some P′ ∈ ℜ^{i}(L; p), and we just saw that P′ ∈ Syl_{p}(C_{L}(E)). Hence if P ∈ Syl_{p}(C_{L}(E)), then P = xP′x^{−1} for some x ∈ C_{L}(E), so P ∈ ℜ^{i}(L; p), and Ω_{1}(Z(P)) = xEx^{−1} = E.

(b) Fix Inn(L) ≤ Γ ≤ Aut(L) (and we identify L with Inn(L) ◁ Γ). Since E = Ω_{1}(Z(P)), we have N_{Γ}(P) ≤ N_{Γ}(E). Since P is a Sylow p-subgroup of C_{L}(E) ◁ N_{Γ}(E), conjugation by any x ∈ N_{Γ}(E) sends P to hP h^{−1} for some h ∈ C_{L}(E), and so h^{−1}x ∈ N_{Γ}(P). This shows that N_{Γ}(E) = C_{L}(E)·N_{Γ}(P), and hence that N_{Γ}(P)/P surjects onto Aut_{Γ}(E) ≅ N_{Γ}(E)/C_{Γ}(E). When Γ = L, then the kernel of this surjection is N_{C_{L}(E)}(P)/P, which has order prime to p since P ∈ Syl_{p}(C_{L}(E)).

It remains to show that N_{C_{L}(E)}(P)/P acts trivially on Z(P). It acts trivially on E = Ω_{1}(Z(P)) by definition, and hence on Ω_{n}(Z(P))/Ω_{n−1}(Z(P)) for each n. By [Gor, Corollary 5.3.3], any group of automorphisms of Z(P) with this property must be a p-group; and since N_{C_{L}(E)}(P)/P has order prime to p, it must act trivially.

(c) Assume E ∈ 𝔈^{i}(L; p), and fix P ∈ Syl_{p}(C_{L}(E)). Then E = Ω_{1}(Z(P)) by (a). So to show E is pivotal, it remains to show that O_{p}(Aut_{L}(E)) = 1.

Let Z′ ≤ Z(P) and Γ ≤ Aut(L) be such that

Λ^{i}(N_{Γ}(P)/P, N_{L}(P)/P; Z′) ≠ 0.

By (b), N_{L}(P)/P surjects onto Aut_{L}(E) with kernel K/P =^{def} N_{C_{L}(E)}(P)/P of order prime to p, and K acts trivially on Z(P) (hence on Z′). So by Proposition 3.3(d),

Λ^{i}(N_{Γ}(P)/K, Aut_{L}(E); Z′) ≠ 0,

and hence O_{p}(Aut_{L}(E)) = 1 by Proposition 3.3(c).

(d) Assume E is pivotal and P ∈ Syl_{p}(C_{L}(E)). By (b), N_{L}(P)/P surjects onto Aut_{L}(E) with kernel of order prime to p. Since O_{p}(Aut_{L}(E)) = 1 by definition of a pivotal subgroup, O_{p}(N(P)/P) = 1, and so P is radical.

(e) Fix S ∈ Syl_{p}(L). Any pivotal subgroup of L is L-conjugate to a subgroup E such that some P ∈ Syl_{p}(C_{L}(E)) is contained in S. Thus P = C_{S}(E), and so E = Ω_{1}(Z(C_{S}(E))) by definition of pivotal. Also, Z(S)·P ≤ C_{S}(E) = P, so Z(S) ≤ P. Hence Z(S) ≤ Z(P), so that Ω_{1}(Z(S)) ≤ Ω_{1}(Z(P)) = E.

(f) Fix an elementary abelian p-subgroup E ≤ L, and a C_{L}(E)-conjugacy class 𝔛 ⊆ C_{L}(E) of elements of order p such that |𝔛| is prime to p. Assume E′ ≥ E is such that E′ ∩ 𝔛 = ∅. Then C_{L}(E′) ∩ 𝔛 is the fixed point set of an E′/E-action on 𝔛, hence also has order prime to p, and contains a C_{L}(E′)-conjugacy class 𝔛′ of order prime to p. Fix x ∈ 𝔛′. Then [C_{L}(E′):C_{L}(E′, x)] is prime to p, so there is P ∈ Syl_{p}(C_{L}(E′)) such that x ∈ Z(P), Ω_{1}(Z(P)) ≩ E′, and thus E′ is not pivotal. □

It remains to list some more conditions which can be used later to prove that certain subgroups are not in 𝔈^{≥2}(L; 2) or ℜ^{≥2}(L; 2). Almost all of these will be based on the following, very general, proposition. Recall (from Section 3) that a radical p-chain in a group G is a sequence

O_{p}(G) = P_{0} ◁ P_{1} ◁ ⋯ ◁ P_{n}

of distinct p-subgroups of G such that P_{i} = O_{p}(N_{G}(P_{0}, …, P_{i})) for all i (in particular, P_{i} ◁ P_{n} for all i), and such that P_{n} ∈ Syl_{p}(N_{G}(P_{0}, …, P_{n−1})).

Proposition 4.5. Fix a finite centerfree group L and an elementary abelian p-subgroup E ≤ L. Assume, for some m ≥ 1, that E ∈ 𝔈^{m}(L; p). Then for any sequence 1 = E_{0} ≤ E_{1} ≤ ⋯ ≤ E_{k} = E of N_{Aut(L)}(E)-invariant subgroups, there is some 1 ≤ i ≤ k, and a radical p-chain 1 = P_{0} ≨ P_{1} ≨ ⋯ ≨ P_{m} in Aut_{L}(E), such that E_{i}/E_{i−1} (when regarded as a 𝔽_{p}[P_{m}]-module) contains a copy of the free module 𝔽_{p}[P_{m}].

Proof. Fix P ∈ Syl_{p}(C_{L}(E)); then P ∈ ℜ^{m}(L; 2) by Proposition 4.4(a). Let Γ ≤ Aut(L) and Z′ ≤ Z(P) be such that Λ^{m}(N_{Γ}(P)/P, N_{L}(P)/P; Z′) ≠ 0. By Proposition 4.4(b), the natural map N_{L}(P)/P ⟶ Aut_{L}(E) is surjective with kernel of order prime to p, and the action of N_{L}(P)/P on Z(P) (hence on Z′) factors through Aut_{L}(E).

Since each quotient Ω_{n}(Z′)/Ω_{n−1}(Z′) is isomorphic as an 𝔽_{p}[N_{Γ}(P)/P]-module to a submodule of E, and since the E_{i} are all Aut_{Γ}(E)-invariant (hence N_{Γ}(P)/P -invariant) by assumption, there is a sequence 1 = Z_{0} ≤ Z_{1} ≤ ⋯ ≤ Z_{k} = Z′ of N_{Γ}(P)/P -invariant subgroups such that each Z_{i}/Z_{i−1} is isomorphic to a subgroup of some E_{j}/E_{j−1}. Hence it suffices to show that at least one of the quotients Z_{i}/Z_{i−1} contains a free module 𝔽_{p}[P_{m}] for some radical p-chain ending in P_{m}, and this follows from Proposition 3.5. □

The next proposition gives some special cases of Proposition 4.5.

Proposition 4.6. Fix a finite centerfree group L and an elementary abelian 2-subgroup E ≤ L. Let 1 = E_{0} ≤ E_{1} ≤ ⋯ ≤ E_{k} = E be any sequence of N_{Aut(L)}(E)-invariant subgroups such that at least one of the following conditions holds for each i:

(a) The Aut_{L}(E)-action on E_{i}/E_{i−1} is not p-faithful; i.e., its kernel has order a multiple of p.

(b) rk(E_{i}/E_{i−1}) ≤ 3.

(c) rk(E_{i}/E_{i−1}) ≤ 7 and the Sylow 2-subgroups of Aut_{L}(E) are neither dihedral nor semidihedral.

(d) rk(E_{i}/E_{i−1}) < 2·|R| for each radical 2-subgroup 1 ≠ R ≤ Aut_{L}(E).

(e) rk(E_{i}/E_{i−1}) < |R_{2}| for each sequence 1 ≠ R_{1} ≨ R_{2} ≤ L such that R_{1} is a radical 2-subgroup of L and R_{2} is a radical 2-subgroup of N_{L}(R_{1}).

Then E ∉ 𝔈^{≥2}(L; 2).

Proof. Assume otherwise: that E ∈ 𝔈^{m}(L; 2) for some m ≥ 2. By Proposition 4.5, there is some i, and a radical 2-chain 1 = P_{0} ≨ P_{1} ≨ ⋯ ≨ P_{m} in Aut_{L}(E), such that E_{i}/E_{i−1} contains 𝔽_{p}[P_{m}] as a summand. In particular, rk(E_{i}/E_{i−1}) ≥ |P_{m}| ≥ 2^{m}, and this is impossible if any of the conditions (b), (d), or (e) hold.

If rk(E_{i}/E_{i−1}) ≤ 7, then |P_{m}| ≤ 4, so m = 2, P_{1} = ⟨x⟩ for some involution x, and P_{2} = C_{S}(x) for some S ∈ Syl_{2}(Aut_{L}(E)). By [Hp, III.14.23], |C_{S}(x)| = 4 implies that S has maximal class (its central series has length k − 1 if |S| = 2^{k}). By [Gor, Theorem 5.4.5], each 2-group of maximal class is dihedral, quaternion, or semidihedral. Together with Proposition 3.3(f), this shows that S must be dihedral or semidihedral.

It remains to consider case (a). Let H ◁ Aut_{L}(E) be the kernel of the action on E_{i}/E_{i−1}, and assume p||H|. By Lemma 1.11, for any p-subgroup P ≤ Aut_{L}(E), either P ∩ H ∈ Syl_{p}(H) or p||N_{HP} (P)/P |; in either case N(P) ∩ H has order a multiple of p. Upon applying this inductively, we see that N(P_{1}, …, P_{m−1}) ∩ H has order a multiple of p. Since P_{m} is assumed to be a Sylow p-subgroup of this intersection of normalizers, this shows that P_{m} ∩ H ≠ 1, which contradicts the assumption that Z_{i}/Z_{i−1} contains a summand isomorphic to 𝔽_{p}[P_{m}]. □

We next look at some cases where Aut_{L}(E) is a symmetric group or general linear group. Similar results involving other classical groups in characteristic two will be shown in Proposition 6.4.

Proposition 4.7. Fix a finite centerfree group L, a pivotal 2-subgroup E ≤ L, and N_{Aut(L)}(E)-invariant subgroups 1 = E_{0} ≤ E_{1} ≤ ⋯ ≤ E_{k} = E. Let Aut_{L}(E_{i}/E_{i−1}) denote the image of Aut_{L}(E) in Aut(E_{i}/E_{i−1}). Assume, for each 1 ≤ i ≤ k, that E_{i}/E_{i−1} either satisfies one of the conditions (a–e) in Proposition 4.6, or satisfies one of the following conditions: either

(a) Aut_{L}(E_{i}/E_{i−1}) ≅ GL_{n}(2) for some n and rk(E_{i}/E_{i−1}) < 2^{n}; or

(b) Aut_{L}(E_{i}/E_{i−1}) ≅ Σ_{n} or A_{n}, and E_{i}/E_{i−1} is the permutation representation on (ℤ/2)^{n} or (ℤ/2)^{n−1}; or

(c) (Aut_{L}(E_{i}/E_{i−1}), E_{i}/E_{i−1}) ≅ (GL_{m}(2) ≀ Σ_{n}, (ℤ/2)^{mn}) for some m ≥ 3 and n ≥ 2.

Then E ∉ 𝔈^{≥2}(L; 2).

Proof. Set E′ = E_{i}/E_{i−1} and G = Aut_{L}(E_{i}/E_{i−1}) for short. By Propositions 4.5 and 4.6, it suffices to show that there is no radical 2-chain 1 ≨ P_{1} ≨ ⋯ ≨ P_{k} of length k ≥ 2 in G such that E′ contains a copy of 𝔽_{2}[P_{k}] as a summand.

When G ≅ GL_{n}(2), then the smallest radical 2-subgroups of G have order 2^{n−1} (see Lemma 6.1 for a description of the radical 2-subgroups). Hence by Proposition 4.6(d), E ∈ 𝔈^{≥2}(L; 2) implies rk(E_{i}/E_{i−1}) ≥ 2^{n}.

Now assume G ≅ Σ_{n} or A_{n}, regarded as acting on a set X of n elements, such that E_{i}/E_{i−1} is isomorphic to a submodule or quotient module of 𝔽_{2}(X). Then for any 2-subgroup P ≤ G, E_{i}/E_{i−1} contains a free submodule 𝔽_{2}[P] only if the action of P on X has at least one free orbit. Furthermore, no radical 2-subgroup P ≤ G can have m > 1 free orbits: this is clear if |P | = 2, m = 2, and G = A_{n}, and holds in the other cases since O_{2}(N_{Σ_{n}}(P)/P) contains a copy of P^{m−1}. Thus there is no radical 2-chain of length ≥ 2 such that P_{2} has a free orbit, and so (b) follows from Proposition 4.5.

The third case now follows since for any radical 2-chain {P_{i}} in GL_{m}(2) ≀ Σ_{n} of length ≥ 2, either P_{2} ∩ (GL_{m}(2))^{n} ≠ 1 and contains a nontrivial radical 2-subgroup of (GL_{m}(2))^{n} (Lemma 1.12), in which case the module contains no summand 𝔽_{2}[P_{2}]; or P_{2} ∩ (GL_{m}(2))^{n} = 1, and an argument similar to that used for G ≅ Σ_{n} shows that the module contains no summand 𝔽_{2}[P_{2}]. □

It is important to note that there is an action of Σ_{5} on (ℤ/2)^{4} with the property that Λ^{2}(Σ_{5}; (ℤ/2)^{4}) ≅ ℤ/2. This is the action obtained by identifying Σ_{5} ≅ ΣL_{2}(4)

— the group SL_{2}(4) ≅ A_{5} extended by its field automorphism — and (ℤ/2)^{4} ≅ (𝔽_{4})^{2}. Note that this action is transitive on the nonzero elements in the module, unlike the permutation action.

The following variant of Proposition 4.7 will be needed when handling classical groups in odd characteristic.

Proposition 4.8. Fix a finite centerfree group L and a pivotal 2-subgroup E ≤ L, and set Γ = Aut_{L}(E) for short. Assume there is a finite set X with Γ-action such that E is isomorphic to a Γ-submodule of the permutation representation 𝔽_{2}(X). Let X_{1}, …, X_{k} ⊆ X be the Γ-orbits, and assume that the Γ-action contains all permutations whose restriction to each X_{i} is an even permutation. Then E ∉ 𝔈^{≥2}(L; 2).

Proof. Assume otherwise. By Proposition 4.5, there is a radical 2-chain 1 ≨ P_{1} ≨ ⋯ ≨ P_{k} of length k ≥ 2 in G such that X contains a free orbit of P_{k}.

Let X_{i} ⊆ X be a Γ-orbit which contains a free P_{k}-orbit. Let m be the number of free orbits of P_{1} in X_{i} (thus m ≥ [P_{2}:P_{1}] ≥ 2), and let Y ⊆ X_{i} be the union of those orbits. Let H ≅ P_{1} ≀ Σ_{m} be the group of all permutations of X which centralize P_{1} and are the identity on X∖Y. Since Γ contains all even permutations of X_{i} (hence of Y) by assumption, Γ ∩ H ◁ C_{Γ}(P_{1}) has index at most 2 in H. Also, O_{2}(H) ≅ P_{1}^{m} if m > 2 and O_{2}(H) = H if m = 2. In either case, O_{2}(Γ ∩ H) ≰ P_{1}, so O_{2}(N_{Γ}(P_{1})) ≠ P_{1}, and this contradicts the assumption that P_{1} is a radical 2-subgroup of Γ. □

The following proposition will be useful when working with the sporadic groups.

Proposition 4.9. Let L be any finite centerfree group, let E ≤ L be a pivotal psubgroup, and fix P ∈ Syl_{p}(C_{L}(E)). In particular, E = Ω_{1}(Z(P)). Let H ≤ L be a subgroup which contains N_{L}(E) ≥ N_{L}(P).

(a) Assume O_{p}(H) ≠ 1, and set E_{0} = E ∩ Z(O_{p}(H)). Then P ≥ O_{p}(H), [E, O_{p}(H)] = 1, E_{0} ≠ 1, and E_{0} is an Aut_{L}(E)-invariant submodule of E. In particular, in this case, E ≤ Z(O_{p}(H)) if E is Aut_{L}(E)-irreducible or if O_{p}(H) is centric in H.

(b) Assume P ∈ ℜ^{i}(L; p). Let H′ ◁ H be a characteristic subgroup which is quasisimple, and set K = C_{H}(H′). Assume that N_{Aut(L)}(E) ≤ N_{Aut(L)}(H), and that P ∩ H′ ∉ Syl_{p}(H′). Then P K/K ∈ ℜ^{i}(H/K; p).

Proof. (a) Since E is pivotal, P is a radical p-subgroup of L (Proposition 4.4(d)), and hence a radical p-subgroup of H (recall N_{L}(P) = N_{H}(P)). So P ≥ O_{p}(H), and E ≤ Z(P) centralizes O_{p}(H). In particular,

E_{0} =^{def} E ∩ Z(O_{p}(H)) = Ω_{1}(Z(P) ∩ Z(O_{p}(H))),

and so E_{0} ≠ 1 since Z(P) ∩ Q ≠ 1 for any 1 ≠ Q ◁ P. Also, E_{0} is Aut_{L}(E)-invariant, since N_{L}(E) = N_{H}(E) normalizes both E and O_{p}(H).

(b) We now assume that P ∈ ℜ^{i}(L; p) (hence E ∈ 𝔈^{i}(L; p)), that H′ ◁ H is a characteristic subgroup which is quasisimple, that P ∩ H′ ∉ Syl_{p}(H′), and that N_{Aut(L)}(E) ≤ N_{Aut(L)}(H). Let Γ ≤ Aut(L) and Z′ ≤ Z(P) be such that

Λ^{i}(N_{Γ}(P)/P, N_{L}(P)/P; Z′) ≠ 0.

Set K = C_{H}(H′), and consider the groups

Z_{0} = Z′ ∩ K = Z′ ∩ C_{H}(H′) and Z′′ = Z′/Z_{0} ≅ Z′K/K ≤ H/K.

By assumption, N_{Γ}(P) ≤ N_{Γ}(H), and N_{Γ}(H) ≤ N_{Γ}(H′) since H′ is characteristic in H. Hence Z_{0} is an N_{Γ}(P)-invariant subgroup of Z′. Since P ∩ H′ ∉ Syl_{p}(H′), N_{P H′}(P)/P has order a multiple of p by Lemma 1.11. This group acts trivially on Z_{0} ≤ K, and thus Λ^{i}(N_{Γ}(P)/P, N_{L}(P)/P; Z′′) ≠ 0 by Proposition 3.3(b).

Now set

K̂ = C_{Γ}(H′) and K = C_{L}(H′), so that K = H ∩ K̂ and K = L ∩ K̂.

The action of N_{L}(P)/P on Z′′ must be p-faithful by Proposition 4.6(a). Since P ≤ H normalizes H′ and hence K and K, N_{P K}(P)/P acts trivially on Z′′ = Z′/(Z′ ∩ K) = Z′/(Z′ ∩ K). Hence N_{P K}(P)/P has order prime to p, and P ∈ Syl_{p}(P K) by Lemma 1.11 again.

Set

P′ = P K/K and Γ′ = Aut_{Γ}(H′) ≅ N_{Γ}(H′)/C_{Γ}(H′).

Recall that K = K̂ ∩ H. We can identify

P′_{≅P K̂/K̂} ≤ H/K_{≅HK̂/K̂} ≤ Γ′ and Z′′_{≅Z′K̂/K̂} ≤ Z(P′),

as subgroups of Aut(H′). Since P ∈ Syl_{p}(P K) (hence P ∈ Syl_{p}(P K)), and N_{Γ}(P) ≤ N_{Γ}(H) normalizes K and K,

N_{H}(P K) = N_{L}(P)·K and N_{Γ}(P K) = N_{Γ}(P)·K.

We can thus identify

N_{H/K}(P′)/P′ ≅ N_{H}(P K)/P K = (N_{L}(P)·K)/P K ≅ \frac{N_{L}(P)/P}{N_{P K}(P)/P} ,

where N_{P K}(P)/P has order prime to p. Also, since N_{Γ}(P) ≤ N_{Γ}(H) ≤ N_{Γ}(H′),

N_{Γ}(P)·K̂ ≤ N_{Γ}(H′) ∩ N_{Γ}(P K̂) ≤ N_{Γ}(P K) = N_{Γ}(P)·K ≤ N_{Γ}(P)·K̂;

so these are all equal, and

N_{Γ′}(P′)/P′ ≅ N_{N_{Γ}(H′)}(P K̂)/P K̂ = (N_{Γ}(P)·K̂)/P K̂ ≅ \frac{N_{Γ}(P)/P}{N_{P K̂}(P)/P} .

Hence by Proposition 3.3(d),

Λ^{i}(N_{Γ′}(P′)/P′, N_{H/K}(P′)/P′; Z′′) ≅ Λ^{i}(N_{Γ}(P)/P, N_{L}(P)/P; Z′′) ≠ 0,

and this proves that P′ = P K/K ∈ ℜ^{i}(H/K; p). □

5. Alternating groups

We are now ready to go through the list of all finite nonabelian simple groups L, to show in each case that L ∈ 𝔏^{≥2}(2). When L is of Lie type in characteristic 2, this will be done using Proposition 4.3. In all other cases, we prove L ∈ 𝔏^{≥2}(2) with the help of Proposition 4.2. This means either showing that ℜ^{≥2}(L; 2) = ∅ (equivalently 𝔈^{≥2}(L; 2) = ∅); or else choosing a 2-centric subgroup Q ≤ L which is weakly closed in a Sylow subgroup of L, and then showing that ℜ^{≥2}(L; 2)_{≱Q} = ∅ (equivalently 𝔈^{≥2}(L; 2)_{≰Z(Q)} = ∅).

We will be constantly referring to Proposition 4.4 for some of the most basic properties of subgroups P ∈ ℜ^{i}(L; 2) and E = Ω_{1}(Z(P)) ∈ 𝔈^{i}(L; 2), as well as the correspondence between these two sets. Propositions 4.5, 4.6, 4.7, and 4.8, all of which give restrictions on the pairs (E, Aut_{L}(E)) for E ∈ 𝔈^{i}(L; 2), will then be used to further eliminate subgroups from these sets; while Proposition 4.9 will be used in some cases to reduce to a problem involving a smaller group. Together with Lemma 1.12, these will be the only results from earlier sections used in this part of the proof.

The easiest case to consider is that of the alternating groups.

Theorem 5.1. For any n ≥ 5, A_{n} ∈ 𝔏^{≥2}(2).

Proof. Set L = A_{n}. When n ≤ 7, then rk_{2}(L) = 2, so ℜ^{≥2}(L; 2) = ∅ by Proposition 4.6(b), and L ∈ 𝔏^{≥2}(2) by Proposition 4.2. So we can assume that n ≥ 8, and hence that Aut(L) = Σ_{n} (cf. [Sz, 3.2.17]).

We regard Σ_{n} as the group of permutations of the set n = {1, …, n}. For each i ≥ 1, let E_{2^{i}} be the set of subgroups E ≤ Σ_{2^{i}}, such that E ≅ (C_{2})^{i}, and the action of E on n contains one free orbit and is otherwise fixed. These subgroups (for each i) clearly make up one conjugacy class in Σ_{n}.

Fix some S ∈ Syl_{2}(L), and let Q ≤ S be the subgroup generated by all E ≤ S such that E ∈ E_{4}. For any E, E′ ∈ E_{4} with overlapping support, either E = E′ (if they have the same support), or ⟨E, E′⟩ is not a 2-group (hence not contained in S). Thus Q is the direct product of [n/4] subgroups in E_{4} with disjoint support, and is weakly closed in S with respect to Σ_{n} by construction. By Proposition 4.2, it remains only to show that ℜ^{≥2}(A_{n}; 2)_{≱Q} = ∅.

Fix P ∈ ℜ^{≥2}(A_{n}; 2)_{≱Q}, and set E = Ω_{1}(Z(P)). Then P ∈ Syl_{2}(C_{A_{n}}(E)) by Lemma 4.4(a). Each union of k orbits of E of order q (orbits under the action on n) which are isomorphic as E-orbits contributes a factor E_{q} ≀ Σ_{k} (for E_{q} ∈ E_{q}) to C_{Σ_{n}}(E). Hence P is the intersection with A_{n} of a product of subgroups of the form E_{q} ≀ C_{2} ≀ ⋯ ≀ C_{2} for E_{q} ∈ E_{q}. If N(P) does not act transitively on n, then E is reducible as a N(P)/P - module. By Proposition 4.6(a), N(P)/P must act p-faithfully on at least one of the Σ_{n}-irreducible composition factors of E, and this implies that all but one of the factors of N(P)/P must have odd order. Hence P must have the form

P = A_{n} ∩ ((E_{q} \underbrace{≀ C_{2} ≀ ⋯ ≀ C_{2}}_{k times})^{×r} × S′),

where q ≥ 2, E_{q} ∈ E_{q}, k ≥ 0, and S′ ∈ Syl_{2}(Σ_{n−q2^{k}r}). Since P does not contain any subgroup conjugate to Q, we must have either q ≥ 8, or q = 2 and k = 0.

Set E_{0} = E ∩ S′. Then Aut_{L}(E) acts trivially on E_{0}, and (Aut_{L}(E), E/E_{0}) is isomorphic to one of the pairs (GL_{s}(2) ≀ Σ_{r}, (Z/2)^{rs}) (if q = 2^{s}, s ≥ 3), or (Σ_{r}, (ℤ/2)^{r}) or (Σ_{r}, (ℤ/2)^{r−1}) (if q = 2). In either case, Proposition 4.7(b,c) applies to show that E ∉ 𝔈^{≥2}(A_{n}; 2) (and hence P ∉ ℜ^{≥2}(A_{n}; 2)). □

By extending the above arguments, one can show that ℜ^{≥3}(A_{n}; 2) = ∅ for all n, and

ℜ^{2}(A_{n}; 2) = \{(E_{4} \underbrace{≀ C_{2} ≀ ⋯ ≀ C_{2}}_{k times})^{×2} × S′ | E_{4} ∈ E_{4}, 2^{k+3} ≤ n, S′ ∈ Syl_{2}(A_{n−2^{k+3}})\}.

6. Groups of Lie type in characteristic two

We next consider simple groups of Lie type in characteristic two. We first summarize the structures in these groups which will be needed (in arbitrary characteristic p). We refer to [Ca1] and [GLS3] as general references.

Assume first that L is a Chevalley group: L = 𝔾(q), where 𝔾 is one of the groups A_{n}, B_{n}, etc., defined over the finite field 𝔽_{q} (q = p^{a}). For example, A_{n}(q) ≅ P SL_{n+1}(𝔽_{q}). Let Φ ⊆ 𝕍 denote the root system of 𝔾, where 𝕍 is a real vector space with inner product. Let Φ_{+} be the set of positive roots; thus Φ = {±r | r ∈ Φ_{+}}. Let I denote the set of primitive roots, an ℝ-basis of 𝕍. Then rk(𝔾) =^{def} dim(𝕍) = |I|.

To each root r ∈ Φ corresponds a root subgroup X_{r} ≅ 𝔽_{q} in L = 𝔾(q), and U =^{def} ∏_{r∈Φ_{+}} X_{r} is a Sylow p-subgroup of L. Also, B =^{def} N_{L}(U) = U⋊H (the Borel subgroup), where H ≤ (𝔽_{q})^{rk(𝔾)} (of index dividing q−1) is the subgroup of diagonal elements. Set N = N_{L}(H); then W ≅ N/H is the Weyl group of 𝔾 (and of the root system Φ).

For example, when L = A_{n}(q) = P SL_{n+1}(q), then we can take

𝕍 = {x ∈ ℝ^{n+1} | ∑ x_{i} = 0}, Φ = {ϵ_{i} − ϵ_{j} | i ≠ j}

(where {ϵ_{1}, …, ϵ_{n+1}} is the standard basis of ℝ^{n+1}),

Φ_{+} = {ϵ_{i} − ϵ_{j} | i < j}, and I = {ϵ_{i} − ϵ_{i+1}}.

Then X_{ϵ_{i}−ϵ_{j}} is the subgroup of matrices which have 1’s along the diagonal and are zero elsewhere except at entry (i, j), and hence U is the group of upper triangular matrices with 1’s along the diagonal, and B is the group of all upper triangular matrices. Diagonal elements are those represented by diagonal matrices, N is the image of the subgroup of monomial matrices, and W ≅ Σ_{n+1}.

For any subset J ⊆ I, let ⟨J⟩ be the set of positive linear combinations of elements of J contained in Φ_{+}. Set

𝔓_{J} = ⟨U·H, X_{−r} | r ∈ ⟨J⟩⟩, and U_{J} = ⟨X_{r} | r ∈ Φ_{+}∖⟨J⟩⟩.

The 𝔓_{J} are the parabolic subgroups of 𝔾(q). (We use this instead of the usual notation P_{J} to avoid confusion with our use of P for an arbitrary p-group.) Also, U_{J} = O_{p}(𝔓_{J}), and 𝔓_{J} = N_{L}(U_{J}) (see [GLS3, Theorem 2.6.5(d,e)]).

We next fix the notation for the twisted groups ^{t}𝔾(q). The index t is the order of an automorphism σ ∈ Aut(𝔾(q′)), which is the composite of a graph automorphism induced by τ ∈ Aut(𝕍, Φ, Φ_{+}) with a field automorphism (induced by an automorphism of 𝔽_{q′}). In most cases, q′ = q^{t}, and so 𝔽_{q} is the fixed subfield of the automorphism of order t on 𝔽_{q′}. In all cases, τ ∈ Aut(Φ_{+}) can be seen as an automorphism of the Dynkin diagram (and t = 2, 3). Also, σ(X_{r}) = X_{τ(r)} for each r ∈ Φ, and thus σ leaves invariant the subgroups U, H, and N. The twisted group L = ^{t}𝔾(q) is defined to be the subgroup O^{p′}(C_{𝔾(q′)}(σ)) generated by the Sylow p-subgroups of C_{𝔾(q′)}(σ) (recall p = char(𝔽_{q})), or alternatively as the subgroup of 𝔾(q′) generated by U^{1} =^{def} C_{U}(σ) and the analogous subgroup for the root groups of negative roots. Its Borel subgroup is defined to be

B^{1} = N_{L}(U^{1}) = U^{1}⋊H^{1} where H^{1} =^{def} C_{H}(σ) ∩ L.

Its parabolic subgroups are the groups 𝔓_{J}^{1} =^{def} L ∩ C_{𝔓_{J}}(σ), for all τ-invariant subsets J ⊆ I. In this case, we again have U_{J}^{1} =^{def} C_{U_{J}} (σ) = O_{p}(𝔓_{J}^{1}), and 𝔓_{J}^{1} = N_{L}(U_{J}^{1}) (again by [GLS3, Theorem 2.6.5(d,e)]).

Lemma 6.1. Let L = O^{p′}(C_{𝔾(q)}(σ)) be group of Lie type in characteristic p (possibly σ = Id and L = 𝔾(q)). Fix a root system (𝕍, Φ, Φ_{+}) for 𝔾, let I ⊆ Φ_{+} be the set of primitive roots, and let τ denote the action on Φ induced by σ. Let U be the product of the root subgroups X_{r} for r ∈ Φ_{+}, and set S = U^{1} ∈ Syl_{p}(L). Then a subgroup P ≤ S is radical in L if and only if P = U_{J}^{1} for some τ-invariant subset J ⊆ I. Furthermore, for each such J, U_{J}^{1} is weakly closed in S with respect to L, and more generally with respect to any subgroup Γ ≤ Aut(L) whose action on the set of primitive roots leaves J invariant.

Proof. By [BW], each radical p-subgroup of L is L-conjugate to some U_{J}^{1}. More precisely, this is shown in [BW] for radical p-subgroups of Inndiag(L): the extension of L by its diagonal automorphisms. But since the radical p-subgroups of L are precisely the intersections of L with radical p-subgroups of Inndiag(L) (Lemma 1.12(b)), the same result holds for radical p-subgroups of L.

It remains to show, for any Γ ≤ Aut(L) containing Inn(L) and any Γ-invariant subset J ⊆ I, that U_{J}^{1} is weakly closed in S = U^{1} with respect to Γ. The following argument is based on the proof of [Gro, Lemma 4.2]. Let B = 𝔓_{∅}^{1} = N_{L}(S) be the Borel subgroup, and let N be the normalizer of the group of diagonal elements in L. Then L = BNB by, e.g., [Ca1, 8.2.2]. Hence, since U_{J}^{1} ◁ B, if U_{J}^{1} ≠ gU_{J}^{1}g^{−1} ≤ S for any g ∈ L, then this occurs for g ∈ N.

Now, g permutes the root subgroups via the action of w = gH ∈ W^{τ} on Φ, and so w(Φ_{+}∖⟨J⟩) ⊆ Φ_{+}. Write Δ = Φ_{+}∖⟨J⟩ for short; this is closed in the sense that any r ∈ Φ which is a positive linear combination of elements of Δ also lies in Δ. So

w(Δ) = Φ_{+}∖w(±⟨J⟩) = Φ_{+}∖w(⟨J⟩_{ℝ})

has the same property, where ⟨J⟩_{ℝ} denotes the ℝ-linear subspace of 𝕍 generated by J. Hence w(±⟨J⟩) is generated by the set J′ ⊆ I of primitive roots which it contains, and thus w(Δ) = Φ_{+}∖⟨J′⟩. After replacing w by its product with some element in the Weyl group of ±⟨J′⟩, we can assume that it sends positive roots to positive roots, and hence must be the identity. But this contradicts the assumption that U_{J}^{1} ≠ gU_{J}^{1}g^{−1}. □

We next list, for future reference, the notation used here for the root systems of the simple Lie groups.

Notation 6.2. The following notation will be used for the root systems of the simple Lie groups. When listed, H ⊆ 𝕍 denotes a half-space which contains the positive roots, and {ϵ_{1}, …, ϵ_{n}} denotes the canonical basis of ℝ^{n}.

• 𝔾 = A_{n} : 𝕍 = {x ∈ ℝ^{n+1} | ∑ x_{i} = 0}, H = {x | ∑ (n − i)x_{i} > 0},

Φ_{+} = {ϵ_{i} − ϵ_{j} | i < j}, and I = {ϵ_{1}−ϵ_{2}, ϵ_{2} − ϵ_{3}, …, ϵ_{n}−ϵ_{n+1}}.

• 𝔾 = B_{n} : 𝕍 = ℝ^{n}, H = {x | ∑ (n + 1 − i)x_{i} > 0},

Φ_{+} = {ϵ_{i}, ϵ_{i} ± ϵ_{j} | i < j}, and I = {ϵ_{1}−ϵ_{2}, ϵ_{2} − ϵ_{3}, …, ϵ_{n−1}−ϵ_{n}, ϵ_{n}}.

• 𝔾 = C_{n} : 𝕍 = ℝ^{n}, H = {x | ∑ (n + 1 − i)x_{i} > 0},

Φ_{+} = {2ϵ_{i}, ϵ_{i} ± ϵ_{j} | i < j}, and I = {ϵ_{1}−ϵ_{2}, ϵ_{2} − ϵ_{3}, …, ϵ_{n−1}−ϵ_{n}, 2ϵ_{n}}.

• 𝔾 = D_{n} : 𝕍 = ℝ^{n}, H = {x | ∑ (n + 1 − i)x_{i} > 0},

Φ_{+} = {ϵ_{i} ± ϵ_{j} | i < j}, and I = {ϵ_{1}−ϵ_{2}, ϵ_{2} − ϵ_{3}, …, ϵ_{n−1}−ϵ_{n}, ϵ_{n−1}+ϵ_{n}}.

• 𝔾 = G_{2} : 𝕍 = {x ∈ ℝ^{3} | ∑ x_{i} = 0}, H = {(x_{1}, x_{2}, x_{3}) | 2x_{1} + x_{2} + 4x_{3} > 0},

Φ = {±(ϵ_{i} − ϵ_{j}), ±(2ϵ_{i} − ϵ_{j} − ϵ_{k}) | {i, j, k} = {1, 2, 3}}
I = {(1, −1, 0), (−2, 1, 1)}.

• 𝔾 = F_{4} : 𝕍 = ℝ^{4}, H = {x | 7x_{1} + 3x_{2} + 2x_{3} + x_{4} > 0},

Φ_{+} = \{Φ_{+}(B_{4}), \frac{1}{2}(ϵ_{1}±ϵ_{2}±ϵ_{3}±ϵ_{4})\}, and I = \{ϵ_{2}−ϵ_{3}, ϵ_{3}−ϵ_{4}, ϵ_{4}, \frac{1}{2}(ϵ_{1}−ϵ_{2}−ϵ_{3}−ϵ_{4})\}.

• 𝔾 = E_{8} : 𝕍 = ℝ^{8},

Φ_{+} = \{ϵ_{j}±ϵ_{i} | 1≤i<j≤8\} ∪ \{\frac{1}{2} ∑_{i=1}^{7}(−1)^{ν(i)}ϵ_{i} + ϵ_{8} | ∑ ν(i) even\}

I = \{\frac{1}{2}(ϵ_{1}−ϵ_{2}−ϵ_{3}−ϵ_{4}−ϵ_{5}−ϵ_{6}−ϵ_{7}+ϵ_{8}), ϵ_{2}−ϵ_{1}, ϵ_{2}+ϵ_{1}, ϵ_{3}−ϵ_{2}, ϵ_{4}−ϵ_{3},

ϵ_{5}−ϵ_{4}, ϵ_{6}−ϵ_{5}, ϵ_{7}−ϵ_{6}}.

• 𝔾 = E_{7} : 𝕍 = {x ∈ ℝ^{8} | x_{7} + x_{8} = 0}, Φ_{+} is the set of positive roots of E_{8} which lie in 𝕍, and

I = \{\frac{1}{2}(ϵ_{1}−ϵ_{2}−ϵ_{3}−ϵ_{4}−ϵ_{5}−ϵ_{6}−ϵ_{7}+ϵ_{8}), ϵ_{2}−ϵ_{1}, ϵ_{2}+ϵ_{1}, ϵ_{3}−ϵ_{2}, ϵ_{4}−ϵ_{3}, ϵ_{5}−ϵ_{4}, ϵ_{6}−ϵ_{5}\}.

• 𝔾 = E_{6} : 𝕍 = {x ∈ ℝ^{8} | x_{6} = x_{7} = −x_{8}}, Φ_{+} is the set of positive roots of E_{8} which lie in 𝕍, and

I = \{\frac{1}{2}(ϵ_{1}−ϵ_{2}−ϵ_{3}−ϵ_{4}−ϵ_{5}−ϵ_{6}−ϵ_{7}+ϵ_{8}), ϵ_{2}−ϵ_{1}, ϵ_{2}+ϵ_{1}, ϵ_{3}−ϵ_{2}, ϵ_{4}−ϵ_{3}, ϵ_{5}−ϵ_{4}\}.

To see that these do, in fact, describe root systems of the given types, we refer to Bourbaki [Bb, §VI.4].

Theorem 6.3. Let L be a finite simple group of Lie type in characteristic two, or the Tits group ^{2}F_{4}(2)′. Then L ∈ 𝔏^{≥2}(2), and L ∈ 𝔏^{≥1}(2) if L ≅ P̸ SL_{3}(2).

Proof. Write L ◁ G = O^{2′}(C_{𝔾(q)}(σ)), where q is a power of 2 (and possibly σ = Id). Thus L = G, except when L is the Tits group (in which case [G:L] = 2). Fix a root system Φ = Φ_{+} ∪ Φ_{−} for 𝔾, let I ⊆ Φ_{+} be the set of primitive roots, and let τ denote the action on Φ induced by σ. Set S = U^{1} = C_{U}(σ) ∈ Syl_{p}(G), where U is the product of the root subgroups X_{r} for r ∈ Φ_{+}. By Lemma 6.1, a subgroup P ≤ S is radical in G if and only if P = U_{J}^{1} for some τ-invariant subset J ⊆ I, each such subgroup is weakly closed in S with respect to G, and is weakly closed with respect to Aut(G) if J is invariant under all permutations of I induced by graph automorphisms.

Let Φ_{Z} ⊆ Φ_{+} and J_{0} ⊆ I be the subsets defined according to the following table:

\begin{matrix}𝔾 & Φ_{Z} & J_{0} \\ A_{n} (n ≥ 1) & \{ϵ_{1} − ϵ_{n+1}\} & \{ϵ_{1} − ϵ_{2}, ϵ_{n} − ϵ_{n+1}\} \\ B_{n} (n ≥ 2) & \{ϵ_{1} + ϵ_{2}, ϵ_{1}\} & \{ϵ_{2} − ϵ_{3}, ϵ_{1} − ϵ_{2}\} \\ C_{n} (n ≥ 3) & \{2ϵ_{1}, ϵ_{1} + ϵ_{2}\} & \{ϵ_{1} − ϵ_{2}, ϵ_{2} − ϵ_{3}\} \\ D_{n} (n ≥ 4) & \{ϵ_{1} + ϵ_{2}\} & \{ϵ_{2} − ϵ_{3}\} \\ G_{2} & \{(−1, −1, 2)\} & \{(−2, 1, 1)\} \\ F_{4} & \{ϵ_{1} + ϵ_{2}, ϵ_{1}\} & \{ϵ_{2} − ϵ_{3}, \frac{1}{2}(ϵ_{1} − ϵ_{2} − ϵ_{3} − ϵ_{4})\} \\ E_{6} & \{\frac{1}{2}(1, 1, 1, 1, 1, −1, −1, 1)\} & \{ϵ_{2} + ϵ_{1}\} \\ E_{7} & ϵ_{8} − ϵ_{7} & \{\frac{1}{2}(ϵ_{1} − ϵ_{2} − ⋯ − ϵ_{7} + ϵ_{8})\} \\ E_{8} & ϵ_{7} + ϵ_{8} & \{ϵ_{7} − ϵ_{6}\} \end{matrix}

Then by [GLS3, Theorem 3.3.1],

Z(U) = ∏_{r∈Φ_{Z}} X_{r}.  (1)

Furthermore, by the same reference, when L = ^{2}A_{n}(2^{k}), ^{2}D_{n}(2^{k}), ^{3}D_{4}(2^{k}), or ^{2}E_{6}(2^{k}), then

Z(S) = Z(U^{1}) = C_{Z(U)}(σ).  (2)

Both of these are also consequences of Chevalley’s commutator formula, to be discussed below.

Case 1: We consider here the cases

(i) L = 𝔾(2^{k}) for some 𝔾 and some k ≥ 2; or

(ii) L is one of the groups ^{2}A_{n}(2^{k}), ^{2}D_{n}(2^{k}), ^{3}D_{4}(2^{k}), or ^{2}E_{6}(2^{k}) for k ≥ 2; or

(iii) L is a Suzuki group ^{2}B_{2}(2^{2k+1}) or a Ree group ^{2}F_{4}(2^{2k+1}) for some k ≥ 1.

By [Ca1, Theorem 6.3.1], for any r ∈ Φ, ⟨X_{r}, X_{−r}⟩ is the image of a homomorphism ϕ_{r} defined on SL_{2}(q), which sends (strict) upper and lower triangular matrices to the root subgroups X_{r} and X_{−r}, respectively. Let D ≤ SL_{2}(q) be the subgroup of diagonal matrices. The images ϕ_{r}(D) commute with each other (for all r ∈ Φ), ϕ_{r}(D) normalizes X_{s} for all r and s, the ϕ_{r}(D) generate the abelian subgroup H of diagonal elements of 𝔾(q), and N_{𝔾(q)}(U) = UH (cf. [Ca1, §7.1]). In particular, when q = 2^{k} for k ≥ 2, C_{X_{r}}(ϕ_{r}(D)) = 1.

Thus when L = 𝔾(2^{k}) for k ≥ 2, then H acts on U with trivial fixed subgroup, and hence Z(U)^{N_{L}(U)} = Z(N_{L}(U)) = 1. If L is one of the groups ^{2}A_{n}(2^{k}), ^{2}D_{n}(2^{k}), ^{3}D_{4}(2^{k}), or ^{2}E_{6}(2^{k}) for k ≥ 2, then Φ_{Z} = {s} has order 1, Z(U^{1}) = C_{X_{s}}(σ) ≅ 𝔽_{2^{k}}, ϕ_{s}(SL_{2}(2^{k})) is contained in L and contains C_{X_{s}}(σ), and hence Z(N_{L}(U^{1})) = 1. Similarly, if L is a Suzuki group ^{2}B_{2}(2^{2k+1}) or a Ree group ^{2}F_{4}(2^{2k+1}) for some k ≥ 1, then the center of the Borel subgroup is trivial: this follows from the description (cf. [Ca1, Theorem 13.7.4]) of the diagonal elements in these groups; or (more explicitly) from [HB3, §XI.3] for the Suzuki groups and from [Sh, §II] for the Ree groups.

Since N_{L}(U_{J}^{1}) ≥ N_{L}(U^{1}) for each τ-invariant subset J ⊆ I, this shows that in all of the cases (i), (ii), (iii) above,

Z(N_{L}(P))_{p} = 1 for all p-centric radical p-subgroups P ≤ S.  (3)

So in these cases, the theorem follows from Proposition 4.3, applied with Q = S.

Case 2: The groups ^{2}B_{2}(2) ≅ C_{5}⋊C_{4} and ^{2}A_{2}(2) = P SU_{3}(2) are solvable, and none of the groups B_{2}(2) ≅ C_{2}(2) ≅ Σ_{6}, or G_{2}(2) ≅ Aut(P SU_{3}(3)) is simple. When L = A_{2}(2) = P SL_{3}(2), then rk_{2}(L) = 2 (its Sylow 2-subgroups are dihedral of order 8), so ℜ^{≥2}(L; 2) = ∅ by Proposition 4.6(b), and L ∈ 𝔏^{≥2}(2) by Proposition 4.2.

So among the simple groups of Lie type, it remains only to consider those where L = 𝔾(2) with rk(𝔾) ≥ 3, or where L is one of the groups ^{2}A_{n}(2) (n ≥ 3), ^{2}D_{n}(2), 3D_{4}(2), or ^{2}E_{6}(2). In all of these cases, we claim (with reference to the above table) that

(b) J_{0} is invariant under any permutation of I induced by an automorphism of the root system;

(c) U_{J_{0}}^{1} is 2-centric in L; and

(d) for each J ⊆ I, Z(𝔓_{J}^{1}) = Z(𝔓_{J∩J_{0}}^{1}).

Once we have proven these points, then the theorem follows from Proposition 4.3 again, this time applied with Q = U_{J_{0}}^{1}.

Point (b) is clear in all cases. To see the other two points, we refer to Chevalley’s commutator formula (cf. [Ca1, Theorem 5.2.2]). This formula describes, for any r, s ∈ Φ_{+}, the commutator of a pair of elements in X_{r} and X_{s} as a product of elements in the root subgroups X_{ir+js} for all i, j ≥ 1 such that ir + js ∈ Φ. Upon examining more closely the coefficients in the formula (this is done explicitly in [GLS3, Theorem 1.12.1(b)]), we see that

[X_{r}, X_{s}] = 1 ⟺ r + s ∉ Φ_{+};

or 𝔾 ∈ {B_{n}, C_{n}, F_{4}}, r, s short roots, r⊥s, r+s a long root
[X_{r}, X_{s}] ≠ 1, 1 ≠ x ∈ X_{r}, 1 ≠ y ∈ X_{s} ⟹ [x, y] ≠ 1.

(4)

When L = 𝔾(2) (and rk(𝔾) ≥ 3), points (c) and (d) now follow easily from (4). To see (c), since ⟨J_{0}⟩ = J_{0} in all cases, it suffices to check that for each r ∈ J_{0} there is s ∈ Φ_{+}∖J_{0} such that [X_{r}, X_{s}] ≠ 1, and this is clear. To see (d), it suffices to check that the actions on Z(S) (as described above) by the groups ∏_{r∈⟨J⟩} X_{−r} and ∏_{r∈⟨J∩J_{0}⟩} X_{−r} have the same fixed subgroup. More precisely, in all cases, there is a map of sets ω: J_{0} → Φ_{Z} with the following property: Z(𝔓_{J}) is the product of the root subgroups X_{r} for r ∈ ω(J ∩ J_{0}). This is easily checked for the classical groups and F_{4}(2). In the remaining cases E_{6}(2), E_{7}(2), and E_{8}(2), the argument is simplified by using the descriptions in [Bb, pp.250–270] of the positive roots as linear combinations of primitive roots.

In all of the remaining (twisted) cases, Φ_{Z} = {s} has just one element, hence Z(S) = X_{s}^{1}, and so (d) can be checked using directly the formulas in (4) again. Similarly, |J_{0}| = 1 (and point (c) is easily checked), except when L = ^{2}A_{n}(2) ≅ P SU_{n+1}(2) for some n ≥ 3. In this case, if we take r = ϵ_{1} − ϵ_{2} ∈ J_{0} and s = ϵ_{2} − ϵ_{n} (so τ(s) = s), then by (4), [X_{r}, X_{τs}] = 1 and [X_{τr}, X_{s}] = 1, and hence no nontrivial element of C_{X_{r}·X_{τr}}(σ) centralizes C_{X_{s}·X_{τs}}(σ) ≤ U_{J_{0}}^{1}. Thus U_{J_{0}}^{1} is 2-centric in S, and (c) holds in this case.

Case 3: Now assume L is the Tits group ^{2}F_{4}(2)′. By Lemma 1.12(b), the radical 2-subgroups of L are precisely the subgroups L ∩ P for 2-radical subgroups P ≤ ^{2}F_{4}(2). Hence by Lemma 6.1, the subgroups of L ∩ C_{U}(σ) ∈ Syl_{2}(L) which are radical in L are exactly the subgroups L ∩ C_{U_{J}} (σ) for τ-invariant subsets J ⊆ I. Since I has just two τ-orbits, this means that there are exactly two such proper subgroups.

By [P1] (and in the notation used there), there are 2-subgroups J, K ≤ L, with normalizers H = N_{L}(J) and N = N_{L}(K), such that H and N both contain T ∈ Syl_{2}(L), and

O_{2}(H) = J, H/J ≅ C_{5}⋊C_{4}, O_{2}(N) = K, N/K ≅ Σ_{3}.

Thus J and K are the two radical subgroups of L which are proper subgroups of T. Since |J| ≠ |K|, this implies that J and K are both weakly closed in T with respect to Aut(L). Also, by [P1] again, Z(H) = Z(T) has order 2 (H is chosen to be the centralizer of an involution), while Z(K) ≅ C_{2}^{2}. Hence L ∈ 𝔏^{≥1}(2) by Proposition 4.3, applied with Q = K. □

Some more precise results will be needed when working with some of the other simple groups.

Proposition 6.4. Fix a finite centerfree group L, a pivotal 2-subgroup E ≤ L, and P ∈ Syl_{p}(C_{L}(E)). Fix N_{Aut(L)}(E)-invariant subgroups 1 = E_{0} ≤ E_{1} ≤ ⋯ ≤ E_{k} = E. Assume, for some 1 ≤ j ≤ k, that E_{i}/E_{i−1} satisfies one of the conditions (a–e) in Proposition 4.6 for each i ≠ j, and that one of the following conditions holds for E_{j}/E_{j−1}: either

(a) (Aut_{L}(E_{j}/E_{j−1}), E_{j}/E_{j−1}) ≅ (Sp_{2n}(2); (ℤ/2)^{2n}) for n ≥ 2; or

(b) (Aut_{L}(E_{j}/E_{j−1}), E_{j}/E_{j−1}) ≅ (Ω_{n}^{±}(2); (ℤ/2)^{n}) for n ≥ 5; or

(c) (Aut_{L}(E_{j}/E_{j−1}), E_{j}/E_{j−1}) ≅ (SO_{2n}^{±}(2); (ℤ/2)^{2n}) for n ≥ 3; or

(d) (Aut_{L}(E_{j}/E_{j−1}), E_{j}/E_{j−1}) ≅ (G_{2}(2), (ℤ/2)^{6}).

Here, Aut_{L}(E_{j}/E_{j−1}) is the image of Aut_{L}(E) in Aut(E_{j}/E_{j−1}). Then E ∉ 𝔈^{≥2}(L; 2).

Proof. This is closely related to a theorem of Grodal [Gro, Theorem 4.1], but does not seem to follow from that result (at least not easily) in the generality we need it here.

Set E′ = E_{j}/E_{j−1} and G = Aut_{L}(E_{j}/E_{j−1}) for short. By Proposition 4.5, it suffices to show that there is no radical 2-chain 1 ≨ P_{1} ≨ ⋯ ≨ P_{k} of length k ≥ 2 in G such that E′ contains a copy of 𝔽_{2}[P_{k}] as a summand.

When G ≅ G_{2}(2), the smallest nontrivial radical 2-subgroups of G are those U_{J} for |J| = 1, and have order 2^{5}.

When G ≅ B_{n}(2) ≅ Ω_{2n+1}(2) or C_{n}(2) ≅ Sp_{2n}(2) (for n ≥ 2), then using Lemma 6.1 and the descriptions of the root systems above, we see that the smallest nontrivial radical subgroup of G is U_{J} for

J = {ϵ_{2} − ϵ_{3}, ϵ_{3} − ϵ_{4}, …, ϵ_{n−1}−ϵ_{n}, ϵ_{n}} or J = {ϵ_{2} − ϵ_{3}, ϵ_{3} − ϵ_{4}, …, ϵ_{n−1}−ϵ_{n}, 2ϵ_{n}},

respectively. These correspond to the cases 𝔓_{J}/U_{J} ≅ B_{n−1}(2) or C_{n−1}(2). In both cases, U_{J} is the product of the root groups for positive roots ϵ_{1} ± ϵ_{i} (2 ≤ i ≤ n), and ϵ_{1} or 2ϵ_{1}. So |U_{J}| = 2^{2n−1}. Hence the second term of any radical 2-chain in G has order at least 2^{2n} > 2n + 1.

Now assume G ≅ Ω_{2n}^{±}(2) for n ≥ 3. When n ≥ 4, the smallest radical subgroup occurs for

J = {ϵ_{2}−ϵ_{3}, ϵ_{3} − ϵ_{4}, …, ϵ_{n−1}−ϵ_{n}, ϵ_{n−1}+ϵ_{n}},

U_{J} is the product of root groups for roots ϵ_{1} ± ϵ_{i}, and hence has order |2^{2n−2}|. Since 2^{2n−1} > 2n for n ≥ 3, the result again follows from Proposition 4.5. When n = 3, the smallest radical subgroups have order 2^{3} (for G ≅ Ω_{6}^{+}(2)) or 2^{4} (for G ≅ Ω_{6}^{−}(2)); and both of these have order > 6.

Now assume G ≅ SO_{2n}^{±}(2) (n ≥ 3), and set G′ = [G, G] ≅ Ω_{2n}^{±}(2). For any radical 2-subgroup 1 ≠ P ≤ G, P ∩ G′ is a radical 2-subgroup of G′ by Lemma 1.12(b). Then either P ∩ G′ ≠ 1, in which case |P | ≥ 2^{2n−2} (or |P | ≥ 2^{3} if n = 3); or |P | = 2 and P ∩ Ω_{2n}^{±}(2) = 1. In this last case, one can show by directly studying centralizers of involutions in SO_{2n}^{±}(2) that P must be generated by an orthogonal transvection: an involution which fixes a codimension one subspace of 𝔽_{2}^{2n}, and such that N_{G}(P) = C_{G}(P) ≅ C_{2} ×Sp_{2n}(2). Thus in either case, the second term of a radical 2-chain has order at least 2^{2n−2} > 2n. □

We finish the section with the following, stronger result, which involves one special case of Theorem 6.3 which will be needed when handling some of the sporadic groups.

Lemma 6.5. Set L = P SU_{6}(2). Then for any Γ ≤ Aut(L) containing Inn(L), ℜ^{≥2}(Γ ; 2) = ∅.

Proof. A general description of the outer automorphism group of a finite simple group of Lie type is given in [GLS3, Theorem 2.5.12]. In the notation of that theorem, when L = P SU_{6}(2) = ^{2}A_{5}(2), then Outdiag(L) ≅ C_{3}, Φ_{L} ≅ C_{2} and acts on Outdiag(L) via x ↦ x^{2}, and Γ_{L} = 1. Thus Out(L) ≅ Σ_{3}.

Fix P ∈ ℜ^{≥2}(Γ ; 2), and set E = Ω_{1}(Z(P)) ∈ 𝔈^{≥2}(Γ ; 2). In particular, P is a radical 2-subgroup of Γ (Proposition 4.4(c,d)), and so P_{0} =^{def} P ∩ L is a radical 2-subgroup of L by Lemma 1.12(b). Also, P_{0} ≠ 1, since P ∈ Syl_{p}(C_{Γ}(E)) is 2-centric. Thus N_{L}(P_{0}) ≨ L is a proper parabolic subgroup.

There are three parabolic subgroups H ≤ L for which the corresponding radical subgroup O_{2}(H) has center of rank ≥ 4:

N_{L}(P_{1}) = H_{1} ≅ 2^{9}:L_{3}(4), N_{L}(P_{2}) = H_{2} ≅ 2^{4+8}:(Σ_{3} × A_{5}),

N_{L}(P_{3}) = H_{3} = H_{1} ∩ H_{2} ≅ 2^{4+5+4}:A_{5}.

These correspond to the sets of primitive roots J_{1} = {ϵ_{i} − ϵ_{i+1} | i = 1, 2, 4, 5}, J_{2} = {ϵ_{i} − ϵ_{i+1} | i = 1, 3, 5}, and J_{3} = {ϵ_{i} − ϵ_{i+1} | i = 1, 5}, respectively. Their normalizers in Aut(L) have the form

N_{Aut(L)}(P_{1}) = N_{Aut(L)}(H_{1}) ≅ 2^{9}:L_{3}(4):Σ_{3},

N_{Aut(L)}(P_{2}) = N_{Aut(L)}(H_{2}) ≅ 2^{4+8}:(Σ_{3} × ΓL_{2}(4)),

N_{Aut(L)}(P_{3}) = N_{Aut(L)}(H_{3}) ≅ 2^{4+5+4}:ΓL_{2}(4).

Here, ΓL_{2}(4) ≅ (3×A_{5}):2 is the extension of GL_{2}(4) by the field automorphism. These are the only possibilities when P = P_{0} ≤ L; if P ≰ L then there is the additional possibility

N_{Aut(L)}(P_{4}) ≅ 2^{6+4}:L_{3}(2).

Since the smallest nontrivial radical subgroups of L_{3}(4) have order 16, Proposition 4.6(d,e) shows that P_{1} ∉ ℜ^{≥2}(Γ ; 2) for any choice of Γ. In all of the remaining cases, rk(E) = rk(Z(P)) ≤ 7, and in almost all of the cases Aut_{Γ}(E) (for any choice of Γ) contains no radical p-chain of length 2 ending with a subgroup of order 4 (so E ∉ 𝔈^{≥2}(Γ ; 2) by Proposition 4.5).

The only exceptions to this occur when P = P_{2} or P_{3} and Γ/L has even order. In these cases, Aut_{Γ}(E) ≅ Σ_{5}. Identify GU_{6}(2) ≤ GL_{6}(4) as the subgroup of matrices A such that Â = A^{−1}, where A ↦ Â is the “backward transposition” (a_{ij} ↔ a_{7−j,7−i}) followed by the field automorphism. In either case (P = P_{2} or P_{3}), E is the group of matrices of the form ( \begin{matrix}I & 0 & X \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix} ), where each entry is a 2 × 2 block and X = X̂. Then E contains two different conjugacy classes of involutions — five transvections (5-dimensional fixed subspace) and ten involutions with 4-dimensional fixed subspace — and this is possible only if Σ_{5} acts via the permutation action. So by Proposition 4.7(b), P_{2} and P_{3} cannot lie in ℜ^{≥2}(Γ ; 2). □

7. Classical groups of Lie type in odd characteristic

We next show that when q is an odd prime power, the simple classical groups P SL_{n}(q), P SU_{n}(q), P Sp_{2n}(q), and P Ω_{n}^{±}(q) are all in 𝔏^{≥2}(2). We refer to [Di] or [Ca1, §1] for definitions and descriptions of these groups.

Recall that the modular character χ_{V} of an 𝔽_{q}[G]-module V is defined by identifying 𝔽_{q}^{∗} with a subgroup of ℂ^{∗}, and then letting χ_{V} (g) ∈ ℂ be the sum of the eigenvalues

of V ^{g}→ V lifted to ℂ. We always consider this in the case where G has order prime to q, and hence when two representations with the same character are isomorphic. See [Se, §18] for more details.

Lemma 7.1. Assume G = Aut(V, 𝔟), where V is a finite dimensional vector space over a finite field K of odd characteristic p, and 𝔟 is a symplectic, quadratic, hermitian, or trivial form. Fix H ≤ G of order prime to p, let χ: H → ℂ be the character of V as an H-representation, and let Aut_{χ}(H) be the group of automorphisms α ∈ Aut(H) such that χ ∘ α = χ. Then

(a) Aut_{χ}(H) = Aut_{G}(H) if G is a linear or unitary group;

(b) Aut_{χ}(H) = Aut_{G}(H) if there is no irreducible H-representation W ⊆ V such that 𝔟|_{W} ≠ 0; and

(c) [Aut_{χ}(H): Aut_{G}(H)] ≤ 2 if G is an orthogonal or symplectic group and V ≅ W^{k} for some irreducible H-representation W.

Proof. We give here a direct proof; this can also be shown as a consequence of Proposition 8.5 in the next section.

Clearly, Aut_{G}(H) ≤ Aut_{χ}(H). If H ≤ Aut(V) has order prime to p, and α ∈ Aut_{χ}(H), then the two representations of H on V induced by the inclusion H ≤ Aut(V) and by its composite with α are isomorphic. Thus Aut_{χ}(H) ≤ Aut_{Aut(V)}(H).

We now study how this can be done while preserving a hermitian, symplectic, or symmetric form 𝔟. For any irreducible summand W ⊆ V, either 𝔟|_{W} is nonsingular (hence V = W ⊕W^{⊥}), or 𝔟|_{W} = 0 and V splits as a direct sum of some W ⊕W′ (where 𝔟 defines an isomorphism W′ ≅ W^{∗}) with its orthogonal complement. We can thus decompose V as orthogonal direct sums

V = (⨁_{i=1}^{k} W_{i}) ⊕ V′ where V′ = ⨁_{j=1}^{ℓ}(U_{j} ⊕ U′_{j}),  (1)

where each W_{i}, U_{j}, and U′_{j} is an irreducible H-representation, all irreducible subrepresentations of V′ are isotropic (𝔟 vanishes on them), and 𝔟 defines an isomorphism U_{j} ≅ (U′_{j})^{∗}. These conditions determine V′ uniquely, and determine the W_{i}, U_{j}, and U′_{j} up to isomorphism. Hence for any α ∈ Aut_{χ}(H), there is an α-linear automorphism φ ∈ Aut(V) which permutes these summands. Also, φ can be chosen to preserve 𝔟 on each U_{j} ⊕ U′_{j}, since otherwise we could compose it by an appropriate H-linear endomorphism of U_{j} (and the identity on U′_{j}). This shows that α ∈ Aut_{G}(H) if V = V′; i.e., if 𝔟|_{W} = 0 for all irreducible W ⊆ V.

Next assume that 𝔟 is a symplectic or symmetric form, and that V ≅ W^{k} for some irreducible H-representation W. We must show that [Aut_{χ}(H): Aut_{G}(H)] ≤ 2. By the decomposition (1), it suffices to do this when V is the orthogonal direct sum of copies of W; in particular when W supports a form of the same type as 𝔟. From this, we are easily reduced to the case where k = 1, and hence V = W is irreducible.

Set K̂ = End_{K[H]}(V), a field. Let K′ ≤ K̂ be the subfield of all elements r ∈ K̂ such that 𝔟(rx, y) = 𝔟(x, ry) for x, y ∈ V. For any automorphism φ ∈ N_{Aut(V)}(H), there is a unique K-linear automorphism σ = ω(φ) of V such that

𝔟(φ(x), φ(y)) = 𝔟(x, σ(y))

for all x, y ∈ V; and one easily sees that σ is H-linear and hence σ ∈ K̂^{∗}. By the (anti-)symmetry of 𝔟, 𝔟(σ(x), y) = 𝔟(x, σ(y)) for all x, y, and thus σ ∈ (K′)^{∗}. This defines a homomorphism

ω: N_{Aut(V)}(H) ⟶ (K′)^{∗},

where ω(u· Id) = u^{2} for all u ∈ (K′)^{∗}. The subgroup T =^{def} ω(C_{Aut(V)}(H)) thus contains all squares in (K′)^{∗}, and so

Aut_{G}(H) = Ker[N_{Aut(V)}(H)/C_{Aut(V)}(H) ⟶ (K′)^{∗}/T ]

has index at most two in N_{Aut(V)}(H)/C_{Aut(V)}(H) ≅ Aut_{χ}(H).

It remains to consider the case where 𝔟 is a hermitian form; we claim that Aut_{G}(H) = Aut_{χ}(H). Let K_{0} ⊆ K be the subfield of index 2: the fixed field of the involution r ↦ r̄ used to define 𝔟. Using the decomposition (1), it suffices to show, for any irreducible K[H]-representation W which supports a hermitian form, and any two Hinvariant forms 𝔟, 𝔟′ on W, that there is an K[H]-linear map φ: W ⟶ W such that 𝔟′(x, y) = 𝔟(φ(x), φ(y)) for all x, y ∈ W.

Let σ ∈ End_{K[H]}(W) be such that 𝔟′(x, y) = 𝔟(x, σ(y)). Since both forms are hermitian, 𝔟(x, σ(y)) = 𝔟(σ(x), y) for all x and y. Let K̂_{0} ≤ End_{K[H]}(W) be the subfield generated by σ and K_{0}, and let K̂ be the subfield generated by σ and K. We regard W as a K̂[H]-module. Then 𝔟(x, ry) = 𝔟(rx, y) for all r ∈ K̂_{0}, so K̂_{0} ∩ K = K_{0}. Thus [K̂_{0}:K_{0}] = [K̂:K] is odd, and [K̂:K̂_{0}] = 2. Let r ↦ r̄ be the automorphism of K̂ of order 2. Then 𝔟(rx, y) = 𝔟(x, r̄y) for all x, y and all r ∈ K̂. So if we choose r such that rr̄ = σ, then 𝔟′(x, y) = 𝔟(rx, ry), and this finishes the proof. □

The next lemma will be needed later as an explicit way of constructing automorphisms.

Lemma 7.2. Assume G = Aut(V, 𝔟), where V is a finite dimensional vector space over a finite field K of odd characteristic p, and 𝔟 is a symplectic, quadratic, hermitian, or trivial form. Fix H ≤ G of order prime to p, and let T ≤ Z(H) be an elementary abelian 2-subgroup. Set

Δ = {α ∈ Aut_{G}(H) | α|_{T} = Id, α ≡ Id (mod {± Id})}.

For α ∈ Δ and x ∈ T, define α_{x} ∈ Aut(H) by setting α_{x}(g) = g if α(g) = g and α_{x}(g) = xg otherwise. Then α_{x} ∈ Aut_{G}(H) for all α ∈ Δ and x ∈ T.

Proof. Let T ≤ Z(H) and Δ ≤ Aut_{G}(H) be as described. Fix x ∈ T and α ∈ Δ, and let V_{±} be the ±1-eigenspaces for x. Then V = V_{+} ⊕ V_{−} is an orthogonal direct sum of H-invariant subspaces.

By assumption, there is an α-linear automorphism φ ∈ Aut(V, 𝔟), and φ preserves the V_{±} since α(x) = x. Define ψ ∈ Aut(V, 𝔟) by setting ψ|_{V_{+}} = Id and ψ|_{V_{−}} = φ|_{V_{−}}. Then ψ is α_{x}-linear, and hence α_{x} ∈ Aut_{G}(H). □

Some more information is needed about elementary abelian 2-subgroups of the projective classical groups.

Lemma 7.3. Assume G = Aut(V, 𝔟), where V is a finite dimensional vector space over a finite field K of odd characteristic, and 𝔟 is a symplectic, quadratic, hermitian, or trivial form. Set Z = Z(G), and let Ẽ ≤ G be a subgroup containing Z such that Ẽ/Z is an elementary abelian 2-group. Set Ẽ_{0} = Z·Ω_{1}(Z(Ẽ)). Let χ: Ẽ → ℂ be the character of V as an Ẽ-representation. Then

(a) χ(g) = 0 for all g ∈ Ẽ∖Ẽ_{0}; and

(b) Δ =^{def} {α ∈ Aut(Ẽ) | α|Ẽ_{0} = Id, α ≡ Id (mod {± Id})} ≤ Aut_{G}(Ẽ).

Proof. Write

Ẽ ≅ T × (Z′ ×_{C_{2}} X),

where T is elementary abelian, X is an extraspecial 2-group (or X ≅ C_{2} if Ẽ is abelian), and Z′ is a cyclic group with [Z′:Z] ≤ 2. Under this identification, Z(Ẽ) = T × Z′ and Ẽ_{0} = T × Z. For any g ∈ Ẽ, we let −g denote g·(− Id). Since V splits as an orthogonal direct sum indexed by the characters of T, we are immediately reduced to the case where T = 1; i.e., when Ẽ = Z′ ×_{C_{2}} X and Ẽ_{0} = Z.

Point (a) follows from (b) since χ(g) = 0 whenever there is α ∈ Aut_{G}(Ẽ) such that α(g) = −g. For any α ∈ Δ, α ∈ Hom_{G}(Ẽ) if α|_{Z′} = Id since it is an inner automorphism of Ẽ. This proves (a) and (b) when Z′ = Z.

Now assume [Z′:Z] = 2. By the above remarks, χ(g) = 0 for g ∈ Ẽ∖Z′. If V splits as a sum of 1-dimensional irreducible Z′-representations (this occurs whenever 𝔟 is hermitian, or is symplectic or symmetric and 4||K^{∗}|), then none of these irreducible summands supports a form of the same type as 𝔟. Hence in this case, V splits as a sum of of Ẽ-representations (U ⊕ U′), whose characters χ_{U} and χ_{U′} are such that χ_{U}(g) = χ_{U′}(g) if g ∈ Z and χ_{U}(g) = −χ_{U′}(g) if g ∈ Z′∖Z. This shows that χ = χ_{V} vanishes on Z′∖Z, and hence that (a) holds in this case. Also, since Δ ≤ Aut_{χ}(Ẽ),

(b) holds by Lemma 7.1(b).

Now assume that the Z′-irreducible summands of V are 2-dimensional (and still [Z′:Z] = 2). Since the character of any such irreducible representation vanishes on Z′∖Z, this finishes the proof of (a). Hence Δ ≤ Aut_{χ}(Ẽ), and so (b) follows from Lemma 7.1(a) when G is a linear group (this case does not occur when G is unitary). So assume 𝔟 is symplectic or symmetric, and hence that |Z′| = 4 (and |K^{∗}| ≡ 2 (mod 4)). In these cases, each irreducible summand of V extends to irreducible representations of D_{8} ×_{C_{2}} X and of Q_{8} ×_{C_{2}} X, one of which supports a symplectic form and the other a symmetric (quadratic) form. Since any such form on an irreducible summand is unique up to scalar multiple, this shows that each irreducible summand of V can be extended to an action of one of the groups D_{8} ×_{C_{2}} X or Q_{8} ×_{C_{2}} X which preserves the form 𝔟. In particular, the involution α ∈ Δ which is the identity on X and sends α(g) = −g for g ∈ Z′∖Z does lie in Aut_{G}(Ẽ); and this finishes the proof of (b). □

We are now ready to show that all classical groups of Lie type in odd characteristic lie in 𝔏^{≥2}(2).

Theorem 7.4. Let q be an odd prime power, and let L be one of the simple groups P SL_{n}(q) (n ≥ 2), P SU_{n}(q) (n ≥ 3), P Sp_{2n}(q) (n ≥ 2), or P Ω_{n}^{±}(q) (n ≥ 5). Then 𝔈^{≥2}(L; 2) = ∅, and hence L ∈ 𝔏^{≥2}(2).

Proof. Write L = [G, G], where G = PAut(V, 𝔟), V is a vector space of dimension n or 2n over the field K = 𝔽_{q} or 𝔽_{q^{2}} and 𝔟 is a symplectic, quadratic, hermitian, or trivial form. Since P SL_{4}(q) ≅ P Ω_{6}^{+}(q), P SU_{4}(q) ≅ P Ω_{6}^{−}(q), and P Sp_{4}(q) ≅ P Ω_{5}(q), we can assume that dim_{K}(V) ≠ 4.

Set G = Aut(V, 𝔟), Z = Z(G) and L̃ = [G, G]. Thus G is one of the groups GL_{n}(q), GU_{n}(q), Sp_{2n}(q), or GO_{n}^{±}(q). Also, G = G/Z and L = L̃/(Z ∩ L̃). Let π: G → G be the projection.

Fix E ∈ 𝔈^{≥2}(L; 2), and set Ẽ = π^{−1}(E). Define

Δ = {α ∈ Aut_{G}(Ẽ) | α ≡ Id (mod {± Id})}.

Set

Ẽ_{0} = Z·Ω_{1}(Z(Ẽ)),  Ẽ_{1} = {g ∈ Ẽ_{0} | α(g) = g ∀α ∈ Δ},

and E_{i} = π(Ẽ_{i}).

By Lemma 7.3, together with the definition of Ẽ_{1},

Δ = {α ∈ Aut(Ẽ) | α|Ẽ_{1} = Id, α ≡ Id (mod {± Id})} ≅ Hom(E/E_{1}, {±1}). (1)

So if we let χ: Ẽ → ℂ be the character of V as an Ẽ-representation, then

χ(g) = 0 for all g ∈ Ẽ∖Ẽ_{1}.  (2)

Let A ≤ Aut(E) be the subgroup of all automorphisms which induce the identity on E_{1}, E_{0}/E_{1}, and E/E_{0}. Then A ≤ Aut_{G}(E) by (1) and Lemma 7.2, and is a normal subgroup since E_{0} and E_{1} are both Aut_{G}(E)-invariant subgroups of E. Also, A is a 2-group (see [Gor, Corollary 5.3.3]), so A ≤ O_{2}(Aut_{G}(E)), and A ∩ L = 1 since O_{2}(Aut_{L}(E)) = 1 (E is pivotal by Lemma 4.4(c)). Either |G/L| ≤ 4 (if G is orthogonal), or G/L is cyclic. Hence A is cyclic or of order 4. Also, rk(E) ≥ 4 by Proposition 4.6(b), and this is easily seen to imply that two of the three groups E_{1}, E_{0}/E_{1}, and E/E_{0} must vanish. So we are reduced to considering the three cases (E_{0}, E_{1}) = (1, 1), (E, 1), or (E, E).

Case 1: Assume first that E_{0} = 1. Then Ẽ = P̃ ≅ Z′ ×_{C_{2}} X, where Z′ is cyclic, [Z′:Z] ≤ 2, and X is an extraspecial 2-group. Set 2m = rk(X/Z(X)). Since rk(E) ≥ 4, we must have 2m ≥ 4.

Let Γ ≤ Aut(E) be the group of automorphisms which lift to automorphisms of Ẽ which are the identity on Z. Thus Γ ≅ Sp_{2m}(2) if |Z′| ≥ 4, and Γ ≅ SO_{2m}^{±}(2) otherwise. By (2), the character χ of Ẽ on V vanishes on Ẽ∖Z, and hence Γ ≤ Aut_{χ}(Ẽ). So by Lemma 7.1(c), Aut_{G}(E) has index at most two in Γ. Hence if m ≥ 3, then Aut_{L}(E) is isomorphic to Sp_{2m}(2), SO_{2m}^{±}(2), or Ω_{2m}^{±}(2), with the usual action on Ẽ/Z′ ≅ (ℤ/2)^{2m}; and by Proposition 6.4, E cannot lie in 𝔈^{≥2}(L; 2). If m = 2, and Γ = Sp_{4}(2) ≅ Σ_{6} or SO_{4}^{−}(2) ≅ Σ_{5}, then again, Aut_{L}(E) has index at most two in Γ, and E ∉ 𝔈^{≥2}(L; 2) by Proposition 4.7(b). (The orthogonal action of SO_{4}^{−}(2) ≅ Σ_{5} on (ℤ/2)^{4} is the permutation action modulo its fixed component.)

It remains to consider the case where Γ ≅ SO_{4}^{+}(2) ≅ Σ_{3} ≀ C_{2}, and thus where Ẽ ≅ D_{8} ×_{C_{2}} D_{8} ≅ Q_{8} ×_{C_{2}} Q_{8}. This group has a unique irreducible representation W on which its central involution acts via (− Id), a representation of dimension four. Since we are assuming dim(V) ≠ 4, we must have V ≅ W^{k} for some k ≥ 2. Then C_{L}(E) = (CL̃(Ẽ)/Z)×E, and hence P ∈ Syl_{2}(C_{L}(E)) has the form P = P′×E for some 2-group P′. Also, E = Ω_{1}(Z(P)) by Proposition 4.4(a), so P′ = 1, and CL̃(Ẽ)/Z has odd order. But if k > 1, then either V splits nontrivially as an orthogonal direct sum, in which case P′ contains the automorphisms which are the identity on one summand and − Id on the others; or k = 2, G = Sp_{8}(q), and C_{G}(Ẽ) ≅ Sp_{2}(q) ≅ SL_{2}(q). Thus P′ ≠ 1 in all cases, and hence no subgroup of this type can be in 𝔈^{≥2}(L; 2).

Case 2: Assume E_{1} = 1 and E_{0} = E. Write Ẽ = Z×T where T is elementary abelian. By Lemma 7.2, applied with Δ as in (1), any transvection α ∈ Aut(E) (any involution which fixes an index two subgroup) lies in Aut_{G}(E). Hence Aut_{G}(E) = Aut(E), since the transvections generate the full automorphism group (cf. [Di, §§6,10,30]). Since Aut(E) ≅ GL_{n}(2) is a perfect group for n ≥ 3, this shows that Aut_{L}(E) = Aut(E). Hence E ∉ 𝔈^{≥2}(L; 2) by Proposition 4.7(a).

Case 3: Now assume E_{1} = E. In this case, C_{L}(E) is the image in L of CL̃(Ẽ), and hence Ẽ must contain the center of CL̃(Ẽ). Write Ẽ = Z × T where T is elementary abelian, and let V = ⨁_{i=1}^{k} V_{i} be the decomposition into eigenspaces for the characters of T. This is an orthogonal decomposition, and C_{G}(Ẽ) is the product of the groups Aut(V_{i}, 𝔟|_{V_{i}}). In particular, since E = Ω_{1}(Z(P)) for some P ∈ Syl_{2}(C_{L}(E)) (Proposition 4.4(a)), Ẽ contains the center of C_{G}(Ẽ), and thus

Ẽ ≥ {φ ∈ L̃ | φ|_{V_{i}} = ± Id ∀i}.

It follows that Aut_{G}(Ẽ) (or its image in Aut(E)) is a product of symmetric groups: one for each isomorphism class of pairs (V_{i}, 𝔟|_{V_{i}}). But this is impossible for E ∈ 𝔈^{≥2}(L; 2) by Proposition 4.8. □

8. Exceptional groups of Lie type in odd characteristic

In addition to the five families of Chevalley groups G_{2}(q), F_{4}(q), and E_{n}(q), it remains to consider the twisted groups ^{2}G_{2}(3^{2k+1}), ^{3}D_{4}(q), and ^{2}E_{6}(q) for odd q. The following cases are easy.

Proposition 8.1. Assume L is one of the groups G_{2}(q) or ^{3}D_{4}(q) for any odd prime power q, or ^{2}G_{2}(3^{2k+1}) for some k ≥ 1. Then L ∈ 𝔏^{≥2}(2).

Proof. By [Gr2, Theorem 6.1], rk_{2}(G_{2}(𝔽_{q})) = 3. Hence G_{2}(q) and ^{2}G_{2}(q) have 2-rank at most 3 (in fact, equal to three in all cases). Furthermore, the tables of orders of the groups of Lie type in [Ca1] or [GLS3] show that for odd q,

[^{3}D_{4}(q):G_{2}(q)] = q^{6}(q^{8} + q^{4} + 1)

is odd, and thus rk_{2}(^{3}D_{4}(q)) = rk_{2}(G_{2}(q)). So 𝔈^{≥2}(L; 2) = ∅ by Proposition 4.6(b), and hence L ∈ 𝔏^{≥2}(2) by Proposition 4.2. □

The groups F_{4}(q) are almost as easy to handle.

Proposition 8.2. Assume, for some odd prime power q, that L = F_{4}(q). Then ℜ^{≥2}(L; 2) = ∅, and L ∈ 𝔏^{≥2}(2).

Proof. We regard L as a subgroup of G = F_{4}(𝔽_{q}). There are two conjugacy classes of involutions in G, denoted in [Gr2] as being of type 2A or 2B (see [Gr2, Table VI]). By [Gr2, Theorem 7.3], G contains a unique conjugacy class of maximal elementary abelian 2-subgroups, represented by E_{5} = T_{(2)} × ⟨θ⟩, where T is a maximal torus, T_{(2)} is its 2-torsion subgroup, and θ ∈ N_{G}(T) is an element which inverts T. Furthermore, the elements of type 2B in E_{5} form (together with the identity) a subgroup E_{2} ≤ E_{5} of rank 2.

Thus for any elementary abelian 2-subgroup E ≤ L, there is a subgroup E′ ≤ E such that all involutions in E′ are of type 2B in G and all involutions in E∖E′ are of type 2A, and such that rk(E′) ≤ 2 and rk(E/E′) ≤ 3. In particular, E′ is N_{Aut(G)}(E)- invariant, and thus N_{Aut(L)}(E)-invariant. Hence 𝔈^{≥2}(L; 2) = ∅ by Proposition 4.6(b), and so L ∈ 𝔏^{≥2}(2) by Proposition 4.2. □

In order to deal with the remaining cases, we need to look more closely at the algebraic groups over 𝔽_{q} of which they are subgroups. Our general references for the properties of algebraic groups are [Hum] and [Ca2]. Note in particular that connected algebraic groups always have maximal tori (products of copies of F^{∗}) which are unique up to conjugacy [Hum, §21.3]. For an arbitrary algebraic group G, we let G^{0} denote the connected component of the identity: a normal subgroup of finite index [Hum, §7.3].

A connected algebraic group over an algebraically closed field F is reductive if it has no nontrivial normal unipotent subgroup; this is equivalent to its being the central product over a finite group of a semisimple group and a torus [Hum, §§19.5 & 27.5]. We first note the following very elementary and well known results about centralizers and normalizers of subgroups of a maximal torus in a reductive group.

Lemma 8.3. Let G be an algebraic group over an algebraically closed field F whose identity component G^{0} is reductive. Fix a maximal torus T ≤ G, and set W = N_{G}(T)/T (regarded as a group acting on T). Let Φ be the set of roots of G, regarded as elements of Hom(T, F^{∗}). For each α ∈ Φ, let X_{α} ≤ G denote the corresponding root subgroup (X_{α} ≅ F). Then for any subgroup H ≤ T,

C_{G}(H)^{0} = ⟨T, X_{α} | α(s) = 1, all s ∈ H⟩

is a reductive group, with root system {α ∈ Φ | α(s) = 1, all s ∈ H}. Also,

C_{G}(H) = C_{G}(H)^{0}·{wT ∈ W | w(s) = s all s ∈ H};

Aut_{G}(H) = Aut_{W} (H); and two elements x, y ∈ T are C_{G}(H)-conjugate if and only if they are C_{W} (H)-conjugate.

Proof. The description of C_{G}(H)^{0} in (a) is shown in [Ca2, Theorem 3.5.3] when G is connected and H = ⟨s⟩ is cyclic, and the proof given there also applies in the more general case.

Since T is a maximal torus in C_{G}(H)^{0}, and all maximal tori are conjugate, each element of N_{G}(H) conjugates T to another maximal torus, and hence C_{G}(H) = C_{G}(H)^{0}·C_{N_{G}(T)}(H) and N_{G}(H) = C_{G}(H)^{0}·N_{N_{G}(T)}(H). This proves the descriptions of C_{G}(H) and Aut_{G}(H). Similarly, for any x, y ∈ T and g ∈ C_{G}(H) such that y = gxg^{−1}, T and gT g^{−1} are two maximal tori of the reductive group C_{G}(H, y)^{0}, hence conjugate in C_{G}(H, y)^{0}, so there is a ∈ C_{G}(H, y) such that a(gT g^{−1})a^{−1} = T, and ag ∈ C_{N_{G}(T)}(H) conjugates x into y. □

We adopt the terminology used in [GLS3], and write Steinberg endomorphism to mean a surjective algebraic endomorphism of an algebraic group whose fixed subgroup is finite. All finite simple groups of Lie type can be constructed as (commutator subgroups of) fixed subgroups of Steinberg endomorphisms. The following result is one of the key properties of these endomorphisms.

Proposition 8.4. (Lang-Steinberg theorem) Fix a prime p and a connected algebraic group G over 𝔽_{p}. Let σ be any Steinberg endomorphism of G. Then every element of G is of the form x^{−1}σ(x) for some x ∈ G.

Proof. See [St, Theorem 10.1]. The proof is also sketched in [Ca2, §1.17]. □

The following proposition is a special case of [GLS3, Theorem 2.1.5]. It describes, in many cases, the relationship between conjugacy classes and normalizers in a connected algebraic group with those in the subgroup fixed by a Steinberg endomorphism.

Proposition 8.5. Let G be any connected algebraic group over 𝔽_{q}. Fix a Steinberg endomorphism σ of G. Let H ≤ C_{G}(σ) be any subgroup, and let H be the set of C_{G}(σ)-conjugacy classes of subgroups G-conjugate to H. Let N_{G}(H) act on C_{G}(H) =^{def} π_{0}(C_{G}(H)) by sending g to xgσ(x)^{−1} (for x ∈ N_{G}(H)). Let C_{G}(H)/N and C_{G}(H)/C denote sets of orbits of the N_{G}(H)- and C_{G}(H)-actions, respectively. Then there is a bijection

ω: H ⟶^{≅} C_{G}(H)/N,

where ω(xHx^{−1}) = [x^{−1}σ(x)] for x ∈ G such that xHx^{−1} ≤ C_{G}(σ). Also, for any H′ ∈ H, Aut_{C_{G}(σ)}(H′) is the stabilizer of ω(H′) under the action of Aut_{G}(H) on the set C_{G}(H))/C of C_{G}(H)-orbits.

Proof. Note first that N_{G}(H) does act on C_{G}(H) and hence on π_{0}(C_{G}(H)). If g ∈ N_{G}(H) and x ∈ C_{G}(H), then for all h ∈ H,

(gxσ(g)^{−1})·h·(gxσ(g)^{−1})^{−1} = gxσ(g^{−1}hg)x^{−1}g^{−1} = g(g^{−1}hg)g^{−1} = h

(since g^{−1}hg ∈ H ≤ C_{G}(σ)), and hence gxσ(g)^{−1} ∈ C_{G}(H).

If x ∈ G is such that xHx^{−1} ≤ C_{G}(σ), then c_{σ(x)} is equal to c_{x} on H, and thus x^{−1}σ(x) ∈ C_{G}(H). We first check that ω is well defined; i.e., that the N_{G}(H)-orbit of x^{−1}σ(x) depends only on the C_{G}(σ)-conjugacy class of xHx^{−1}.

Assume first that xHx^{−1} = yHy^{−1} ≤ C_{G}(σ), and set g = y^{−1}x ∈ N_{G}(H). Thus x = yg,

x^{−1}σ(x) = g^{−1}(y^{−1}σ(y))σ(g),

and so [x^{−1}σ(x)] = [y^{−1}σ(y)] ∈ C_{G}(H)/N. Finally, if x ∈ G is such that xHx^{−1} ≤ C_{G}(σ), and g ∈ C_{G}(σ), then

(gx)^{−1}σ(gx) = x^{−1}(g^{−1}σ(g))σ(x) = x^{−1}σ(x),

and thus ω(xHx^{−1}) = ω(gxHx^{−1}g^{−1}).

This shows that ω is well defined. To see that ω is onto, fix some z ∈ C_{G}(H). By the Lang-Steinberg theorem, there is x ∈ G such that x^{−1}σ(x) = z. Then conjugation by x and by σ(x) are equal on H (since x^{−1}σ(x) centralizes H), so xHx^{−1} ≤ C_{G}(σ), and [z] = ω(xHx^{−1}) ∈ Im(ω).

To see that ω is injective, it now suffices to consider the case where x, y ∈ G are such that x^{−1}σ(x) ≡ y^{−1}σ(y) (mod C_{G}(H)^{0}). Let z ∈ C_{G}(H)^{0} be such that x^{−1}σ(x) = y^{−1}σ(y)·z. Then

x^{−1}σ(x) = y^{−1}(σ(y)zσ(y)^{−1})σ(y)

so that

(xy^{−1})^{−1}σ(xy^{−1}) = σ(y)zσ(y)^{−1} ∈ C_{G}(yHy^{−1})^{0}.

By the Lang-Steinberg theorem, (xy^{−1})^{−1}σ(xy^{−1}) = g^{−1}σ(g) for some element g ∈ C_{G}(yHy^{−1})^{0}. Then xy^{−1}g^{−1} ∈ C_{G}(σ), and so xHx^{−1} and yHy^{−1} are C_{G}(σ)-conjugate.

It remains to describe Aut_{C_{G}(σ)}(H′) for any H′ = xHx^{−1} ∈ H. Consider the monomorphism

γ: Aut_{C_{G}(σ)}(xHx^{−1}) ⟶ Aut_{G}(H) ≅ N_{G}(H)/C_{G}(H)  (1)

which sends c_{xgx^{−1}} to c_{g}. We must show that Im(γ) is the stabilizer of [x^{−1}σ(x)] ∈ C_{G}(H)/C under the action of Aut_{G}(H).

Assume first that c_{g} ∈ Aut_{G}(H) stabilizes [x^{−1}σ(x)]. Upon replacing g by ag for some appropriate a ∈ C_{G}(H), we can assume that g fixes the class of x^{−1}σ(x) in C_{G}(H) = π_{0}(C_{G}(H)) itself. Thus

g(x^{−1}σ(x))σ(g)^{−1} = (xg^{−1})^{−1}σ(xg^{−1}) ≡ x^{−1}σ(x) (mod C_{G}(H)^{0}),

and hence (as already shown above) there is z ∈ C_{G}(xHx^{−1}) such that x(xg^{−1})^{−1}z^{−1} ∈ C_{G}(σ). Thus, xgx^{−1}z^{−1} ∈ N_{C_{G}(σ)}(xHx^{−1}), and γ sends conjugation by this element to c_{g}.

Conversely, for any g ∈ N_{C_{G}(σ)}(xHx^{−1}),

(x^{−1}gx)·(x^{−1}σ(x))·σ(x^{−1}gx)^{−1} = x^{−1}gσ(g)^{−1}σ(x) = x^{−1}σ(x);

and hence x^{−1}gx stabilizes [x^{−1}σ(x)]. □

For example, if H = T_{(2)} is the 2-torsion in a σ-invariant maximal torus T ≤ G (and H ≤ C_{G}(σ)), then C_{G}(H) is generated by T, together with the element of W = N(T)/T which inverts T if there is such an element. Hence π_{0}(C_{G}(H)) has at most two elements. So either there are two C_{G}(σ)-conjugacy classes of subgroups G-conjugate to H (if some element of W inverts T), or there is just one such class (if no element of W inverts T).

Proposition 8.6. Let p be any prime, and let G be a connected reductive algebraic group over an algebraically closed field 𝔽 of characteristic ≠ p. Then for any finite p-subgroup P ≤ G, π_{0}(C_{G}(P)) is a p-group, and the identity connected component C_{G}(P)^{0} of the centralizer is also a reductive algebraic group over 𝔽.

Proof. Let P′ ◁ P be a subgroup of index p; we can assume inductively that the lemma holds for P′. Thus C_{G}(P′)^{0} is reductive and has p-power index in C_{G}(P′). Fix any g ∈ P ∖P′. Then g acts on C_{G}(P′) as an algebraic automorphism, and we must show that the fixed point set of its action on C_{G}(P′)^{0} has reductive identity component, and has p-power index in the full fixed point set. In other words, it suffices to consider the case where P = ⟨α⟩ is cyclic, but where instead of α being an element of G we only assume that it is an algebraic automorphism of G of p-power order. Since p is prime to char(𝔽), α is a semisimple automorphism.

Let T ◁ G be the maximal normal toral subgroup, and let H ◁ G be the maximal normal semisimple subgroup. Let H̃ be the universal cover of H. Set K = Ker[T × H̃ ⟶^{ψ} G], where ψ is the sum of the inclusion of the torus and the projection of H̃ onto H. By [St, 9.16], α lifts to a unique automorphism of H̃, which can be assumed also to have p-power order (otherwise, just replace it by an appropriate power). We thus get an exact sequence

1 ⟶ C_{K}(α) ⟶ C_{T} (α) × CH̃(α) ⟶ C_{G}(α) ⟶ H^{1}(⟨α⟩; K).  (1)

The last group is a finite p-group, since K is finite and ⟨α⟩ is a p-group. Also, CH̃(α) is connected and reductive by [St, Theorem 8.1] (and this is the deep result which lies behind the proposition). The identity component of C_{T} (α) is, of course, a torus; and is the image of the norm map N_{⟨α⟩} (since this image is connected and has finite index in the centralizer). Thus

π_{0}(C_{T} (α)) = (T^{α})/(N_{⟨α⟩}·T) ≅ Ĥ^{0}(⟨α⟩; T)

is a p-group. Together with the exact sequence (1), this finishes the proof that π_{0}(C_{G}(α)) is a p-group, and that C_{G}(α)^{0} is reductive. □

The next proposition will be used to show that certain elementary abelian subgroups are not pivotal. Note that the condition on σ^{2} in the statement holds in all situations which occur for finite groups of Lie type, except those involving the triality automorphism and ^{3}D_{4}(q). In general, when T is a maximal torus in an algebraic group G, we let T_{(2)} denote the subgroup of elements of order 2.

Proposition 8.7. Let G be a connected reductive algebraic group over 𝔽_{q}, where q is odd, and fix a maximal torus T of G. Let σ be a Steinberg endomorphism of G such that σ(T) = T, and such that σ^{2} is the identity on W = N_{G}(T)/T and on T_{(2)}. Let E ≤ C_{T} (σ) be an elementary abelian 2-subgroup. Assume there is an involution x ∈ T ∖E such that the orbit of x under the C_{W} (E)-action on T has odd order. Then no subgroup of C_{G}(σ) which is G-conjugate to E is pivotal in C_{G}(σ). More generally, if E ≥ E is also elementary abelian, and is such that x is not C_{G}(E)-conjugate to any element of E, then for any L ◁ C_{G}(σ) which contains {gxg^{−1} | g ∈ G} ∩ C_{G}(σ), no subgroup of L which is G-conjugate to E is pivotal in L.

Proof. We first show, for any E′ ≤ L ◁ C_{G}(σ) which is G-conjugate to E, that there are g ∈ G and k ≥ 1 which satisfy the following conditions:

(a) E′ = gEg^{−1} and g^{−1}σ^{2^{k}}(g) ∈ T;

(b) σ^{2^{k}} leaves T′ =^{def} gT g^{−1} invariant and acts via the identity on W′ =^{def} N_{G}(T′)/T′ and on T′_{(2)};

(c) Aut_{C_{G}(σ^{2^{k}})}(T′) = Aut_{G}(T′); and

(d) there is a C_{L}(E′)-conjugacy class 𝔛_{0} of odd order whose elements are all Gconjugate to x′ =^{def} gxg^{−1}.

Assume these have been shown, and let E′ ≤ L be a subgroup G-conjugate to E. Choose h ∈ G such that E′ = hEh^{−1}, and set E′ = hEh^{−1}. By (c), we can replace g (the element satisfying (a) and (b)) by ag for an appropriate element a ∈ N_{C_{G}(σ^{2^{k}})}(T′),

to arrange that h ∈ C_{G}(E′)·g. Then E′ and gEg^{−1} are C_{G}(E′)-conjugate; and E′ contains no elements C_{G}(E′)-conjugate to x′ = gxg^{−1} since E contains no elements C_{G}(E)-conjugate to x. Hence E′ ∩ 𝔛_{0} = ∅ by (d), and E′ cannot be pivotal in L by Proposition 4.4(f).

It remains to show points (a–d). Consider the homomorphism

C_{W} (E) −↠^{ρ} π_{0}(C_{G}(E)),

which is surjective by Lemma 8.3. Since σ acts on W with order 2, and since C_{W} (E) has an odd number of Sylow 2-subgroups, we can choose S_{0} ∈ Syl_{2}(C_{W} (E)) which is σ-invariant. Since π_{0}(C_{G}(E)) is a 2-group (Proposition 8.6), ρ|_{S_{0}} is also surjective. Using Proposition 8.5, we can now choose g such that E′ = gEg^{−1}, and such that g^{−1}σ(g) ∈ aT for some aT ∈ S_{0}. Then for k sufficiently large,

g^{−1}σ^{2^{k}}(g) ∈ (a·σ(a)·σ^{2}(a)⋯σ^{2^{k}−1}(a))T = (a·σ(a))^{2^{k−1}}T = T.

This proves (a), and (b) then follows easily. Furthermore, upon replacing k by k + 1, we can assume that σ^{2^{k}} acts via the identity on the subgroup H ≤ T′ generated by elements of order 4 in T′.

Set G_{k} = C_{G}(σ^{2^{k}}) for short. Let 𝔛′ be the C_{G_{k}}(E′)-orbit of x′ =^{def} gxg^{−1}. By Lemma 8.3, any two elements of 𝔛′ ∩ T′ are C_{W′}(E′)-conjugate. Also, by the Lang-Steinberg theorem, each coset of T′ in N_{G}(T′) has representatives in G_{k}, and thus 𝔛′ ∩ T′ is the full C_{W′}(E′)-orbit of x′. Since the C_{W} (E)-orbit of x has odd order by assumption, this shows that 𝔛′ ∩ T′ has odd order. Since 𝔛′ ∩ T′ = 𝔛′ ∩ C_{G}(H) is the fixed set of an action of the 2-group H on 𝔛′, 𝔛′ must also have odd order.

Note also that Aut_{G}(E′) = Aut_{W′}(E′) by Lemma 8.3, and hence Aut_{G}(E′) = Aut_{G_{k}}(E′) since each coset of T′ in N_{G}(T′) has representatives in G_{k}. This proves (c).

Since |𝔛′| is odd, and since σ acts on 𝔛′ ⊆ C_{G_{k}}(E′) with order 2^{k}, the fixed point set 𝔛′ ∩ C_{G}(σ) of this action also has odd order. Also, by assumption, 𝔛′ ∩ C_{G}(σ) ⊆ L ◁ C_{G}(σ), and is a union of C_{L}(E′)-conjugacy classes. So there is a C_{L}(E′)-conjugacy class 𝔛_{0} ⊆ 𝔛′ ∩ L of odd order, and this proves (d). □

We are now ready to consider the remaining exceptional groups E_{n}(q), and ^{2}E_{6}(q). We adopt some of the notation used by Griess in [Gr2]. In particular, 2A and 2B will be used to denote conjugacy classes of elements of order 2.

In all cases, we let E_{6}(𝔽_{q}), E_{7}(𝔽_{q}), or E_{8}(𝔽_{q}) denote the adjoint (centerfree) forms of these groups, and let Ẽ_{n}(𝔽_{q}) denote their universal covers. Thus Ẽ_{6}(𝔽_{q}) is a 3-fold cover of E_{6}(𝔽_{q}) (when q is not a power of 3), and Ẽ_{7}(𝔽_{q}) is a 2-fold cover of E_{7}(𝔽_{q}). In all cases, by [Gr2, Lemma 2.16], if T ≤ Ẽ_{n}(𝔽_{q}) is a maximal torus, there is a quadratic form 𝔮: T_{(2)} → 𝔽_{2} which sends one of the conjugacy classes to 1 and the other to 0, and the Weyl group acts on T_{(2)} as the full orthogonal group for the form 𝔮.

Proposition 8.8. Assume, for some odd prime power q, that L is one of the simple groups E_{6}(q) or ^{2}E_{6}(q). Then ℜ^{≥2}(E_{6}(q) ; 2) = ∅, and hence L ∈ 𝔏^{≥2}(2).

Proof. Set G = E_{6}(𝔽_{q}) (in adjoint form, with trivial center). Let ψ^{q} ∈ Aut(G) be the field automorphism induced by (x ↦ x^{q}), and let τ ∈ Aut(G) be the graph automorphism with fixed subgroup F_{4}(𝔽_{q}). Set σ = ψ^{q} if L ≅ E_{6}(q) or σ = ψ^{q} ∘ τ if L ≅ ^{2}E_{6}(q). In either case, we let L be the commutator subgroup of C_{G}(σ). Note that C_{G}(σ) = Inndiag(L) contains L with index (3, q − 1) if L = E_{6}(q), or with index (3, q + 1) if L = ^{2}E_{6}(q).

Fix a maximal torus T ≤ G upon which σ acts via (x ↦ x^{q}) if L = E_{6}(q), or via (x ↦ x^{−q}) if L = ^{2}E_{6}(q). In the latter case, this can be found using the Lang-Steinberg theorem: if T′ is the “standard” torus upon which ψ^{q} acts via (x ↦ x^{q}) and τ acts with fixed subtorus of rank 4, then there is wT′ ∈ N(T′)/T′ such that c_{w} ∘ τ(x) = x^{−1} for all x ∈ T′, w = g^{−1}σ(g) for some g ∈ G, and we can take T = gT′g^{−1}.

Thus in either case, T_{(2)} ≤ L. By [Gr2, Lemma 2.16], there is a nonsingular quadratic form 𝔮: T_{(2)} ⟶ 𝔽_{2} such that 𝔮(x) = 1 for x of type 2A and 𝔮(x) = 0 for x of type 2B, and such that the Weyl group action on T_{(2)} is that of the orthogonal group O(T_{(2)}, 𝔮) ≅ SO_{6}^{−}(2). Since the fundamental group of G has order prime to 2, [Ca2, Proposition 3.7.3] applies to show that L has the same two classes of involutions.

By [Gr2, Theorem 8.2], there is a unique maximal elementary abelian 2-subgroup W_{5} ≤ E_{6}(𝔽_{q}) which is not contained in a maximal torus, and W_{5} is contained in a subgroup F_{4}(𝔽_{q}) ≤ E_{6}(𝔽_{q}). By [Gr2, Lemma 2.16(i)], involutions in F_{4}(𝔽_{q}) of types 2A and 2B are sent under this embedding to involutions of types 2A and 2B in E_{6}(𝔽_{q}). So by [Gr2, Theorem 7.3(ii)], the elements of type 2B in W_{5} (together with the identity) form a subgroup W_{2} ≤ W_{5} of rank 2. Thus for any elementary abelian 2subgroup E ≤ L which is not contained in a maximal torus of G, there is an N_{Aut(L)}(E)- invariant subgroup E′ ≤ E such that rk(E′) ≤ 2 and rk(E/E′) ≤ 3, and E ∉ 𝔈^{≥2}(L; 2) by Proposition 4.6(b).

Now fix some E ∈ 𝔈^{≥2}(L; 2). Since E cannot be G-conjugate to a subgroup of W_{5}, it must be G-conjugate to a subgroup E′ ≤ T_{(2)}. Also, E is pivotal (Proposition 4.4(c)), and rk(E) ≥ 4 by Proposition 4.6(b). By Lemma 8.3(b), Aut_{G}(E) ≅ Aut_{G}(E′) ≅ SO(E, 𝔮), since every element of Aut_{G}(E′) is the restriction of the action of an element of the Weyl group.

Assume rk(E) = 4. If 𝔮|_{E} is singular, then E ∩ E^{⊥} is a proper subgroup of E which is N_{Aut(L)}(E)-invariant, so E ∉ 𝔈^{≥2}(L; 2) by Proposition 4.6(b). So we assume 𝔮 is nonsingular on E, and hence on E′ ≤ T_{(2)}. Then T_{(2)} = E′ × E′^{⊥}, E′^{⊥} is C_{W} (E′)- invariant, and hence there is 1 ≠ x ∈ E′^{⊥} whose C_{W} (E′)-orbit has odd order. By Proposition 8.7, no subgroup of L which is G-conjugate to E′ is pivotal in L. In particular, E ∉ 𝔈^{≥2}(L; 2) in this case.

Finally, if rk(E) ≥ 5, then we are in one of the following situations: either

(i) rk(E) = 6 and Aut_{G}(E) ≅ SO_{6}^{−}(2); or

(ii) rk(E) = 5, 𝔮|_{E∩E^{⊥}} = 0, and Aut_{G}(E) ≅ 2^{4}:SO_{4}^{−}(2); or

(iii) rk(E) = 5, 𝔮|_{E∩E^{⊥}} ≠ 0, and Aut_{G}(E) ≅ Ω_{5}(2) ≅ Sp_{4}(2) ≅ Σ_{6}.

If any element of W fixes a rank five subgroup E′ ≤ T_{(2)}, then it must be an orthogonal transvection, and hence E′ must be of type (iii) above (see [Di, §19]). Thus the kernel of any root of E_{6} (regarded as an element of Hom(T, 𝔽_{q}^{∗})), when restricted to T_{(2)}, must also be a subgroup of type (iii). So by Lemma 8.3, C_{G}(E) = T if E has type (i) or (ii), and C_{G}(E) is connected if E has type (iii). In all cases, Aut_{C_{G}(σ)}(E) = Aut_{G}(E) by Proposition 8.5, and hence Aut_{L}(E) = Aut_{G}(E) since Aut_{G}(E) has no normal subgroup of index 3. So E ∉ 𝔈^{≥2}(L; 2): by Proposition 4.6(c) in the first and third cases (the Sylow 2-subgroups of Aut_{L}(E) are neither dihedral nor semidihedral), or by Proposition 4.4(c) in the second case (E is not pivotal). □

Let Ẽ_{7}(q) ≤ Ẽ_{7}(𝔽_{q}) denote the universal groups, with center Z of order 2, and set E_{7}(𝔽_{q}) = Ẽ_{7}(𝔽_{q})/Z and E_{7}(q) = Ẽ_{7}(q)/Z.

Proposition 8.9. Assume, for some odd prime power q, that L = E_{7}(q). Then ℜ^{≥2}(L; 2) = ∅, and hence L ∈ 𝔏^{≥2}(2).

Proof. Set G̃ = Ẽ_{7}(𝔽_{q}), let z ∈ Z(G̃) be the central involution, and set G = G̃/⟨z⟩ = E_{7}(𝔽_{q}). Fix a Steinberg endomorphism σ of G̃ such that CG̃(σ) = L̃ =^{def} Ẽ_{7}(q). We also let σ denote the induced endomorphism of G. Note, however, that C_{G}(σ) = Inndiag(L) ≅ L:2 is the extension of L ≅ E_{7}(q) by its diagonal automorphisms. Let T̃ ≤ G̃ be a σ-invariant maximal torus, and set T = T̃/⟨z⟩.

By [Gr2, Table VI], the group Ẽ_{7}(𝔽_{q}) has two conjugacy classes of noncentral involutions, referred to as types 2B and 2C, which are exchanged under multiplication by z. Define 𝔮: T̃_{(2)} ⟶ 𝔽_{2} by setting 𝔮(x) = 1 if x = z or x is of type 2B, and 𝔮(x) = 0 if x = 0 or x is of type 2C. Then 𝔮 is a quadratic form [Gr2, Lemma 2.16], and

W (E_{7}) = SO(T_{(2)}, 𝔮) × C_{2} ≅ SO_{7}(2) × C_{2}.

In particular, this means that no automorphism of L̃ can switch the two classes 2B and 2C — since 𝔮 would then no longer be a quadratic form.

The symplectic form associated to 𝔮 induces a symplectic form 𝔟 on the subgroup T′_{(2)} ≤ T_{(2)} of elements which lift to involutions. To describe this, for each 1 ≠ x ∈ T′_{(2)}, let x̃ ∈ T̃_{(2)} be the (unique) lifting of x of type 2C (and set ˜1 = 1). Then 𝔟(x, y) = 0 if x̃·ỹ = x̃y, and 𝔟(x, y) = 1 if x̃·ỹ = z·x̃y. It will be convenient to extend this definition to any elementary abelian subgroup E ≤ L all of whose elements lift to involutions; although of course 𝔟 will no longer be bilinear in general if E is not toral.

We note that no element of C_{G}(σ)∖L lifts to an involution in G̃. Since if there were such an element, lifting to x ∈ G̃, then x and xz would be exchanged by σ, and hence σ would send all involutions of type 2B to involutions of type 2C and vice versa. That would imply that the only involution in L̃ ≅ Ẽ_{7}(q) is the central element, which is clearly not true.

By [Gr2, Theorem 9.8], G contains two conjugacy classes of maximal elementary abelian 2-subgroups, both nontoral: M_{8} of rank 8 and M_{7} of rank 7. These lift to centric subgroups M̃_{8} ≅ Q_{8} × 2^{6} and M̃_{7} ≅ Q_{8} × 2^{5} in G̃. Also, M_{8} = T_{(2)}·⟨θ⟩, the extension of 2-torsion in a maximal torus by an involution in N_{G}(T) which inverts the torus; while M̃_{7} ≤ Q_{8} × F_{4}(𝔽_{q}) ≤ G̃.

Fix E ∈ 𝔈^{≥2}(L; 2), and P ∈ Syl_{p}(C_{L}(E)). Thus P ∈ ℜ^{≥2}(L; 2) and E = Ω_{1}(Z(P)). Let E′ ≤ E be the subgroup of elements of E which lift to involutions in L̃ (a subgroup by the above remarks, and clearly N_{Aut(L)}(E)-invariant). We now consider the different possibilities.

Case 1: Assume E is G-conjugate to a subgroup of M_{7}. As noted above, M̃_{7} = Q×V, where Q ≅ Q_{8}, and V ≅ 2^{5} is a subgroup of F_{4}(𝔽_{q}) ≤ Ẽ_{7}(𝔽_{q}). There is a subgroup V_{2} ≤ V such that the involutions in V_{2} lie in one class in F_{4}(𝔽_{q}) and those in V ∖V_{2} in the other ([Gr2, Theorem 7.3]). Also, any index two subgroup of V containing V_{2} is toral in F_{4}(𝔽_{q}), hence in G, so the function 𝔮 as defined above is quadratic on this subgroup. Since T̃_{(2)} contains no rank four isotropic subspace, this is possible only if the involutions in V_{2} have type 2C in G and those in V ∖V_{2} have type 2B. Thus 𝔟: V × V → 𝔽_{2} takes the form 𝔟(x, y) = 1 if x, y generate a rank two subgroup of V/V_{2} and 𝔟(x, y) = 0 otherwise.

Now set E′′ = {x ∈ E′ | 𝔟(x, E′) = 0}; this is again a N_{Aut(L)}(E)-invariant subgroup of E. Upon identifying E′ with a subgroup of V, we see that either E′′ = E′ and rk(E′) ≤ 3, or rk(E′′) ≤ 2 and rk(E′/E′′) ≤ 3. In both cases, E ∉ 𝔈^{≥2}(L; 2) by Proposition 4.6(b).

Case 2: Assume E is G-conjugate to a subgroup of M_{8} = T_{(2)}·⟨θ⟩ as above. In particular, E′ is a toral subgroup, and hence 𝔟 is a symplectic form on E′ which is invariant under the action of N_{Aut(L)}(E). By Proposition 4.6(b) again, rk(E′) ≥ 4; and if rk(E′) = 4 then 𝔟 must be nonsingular on E′.

By [Gr2, Theorem 9.8], Aut_{G}(M_{8}) ≅ 2^{7}:Sp_{6}(2). The normal subgroup 2^{7} is the group of all automorphisms which are the identity on T_{(2)} (induced by conjugation by elements of order 4 in T). Thus Aut_{G}(E) ≅ Aut(E′, 𝔟) if E is toral (use Lemma 8.3). If E is not toral, then Aut_{G}(E) surjects onto Aut(E′, 𝔟) with kernel an elementary abelian 2-group.

Case 2a: If rk(E′) = 6, then Aut_{G}(E) surjects onto Aut_{G}(E′, 𝔟) ≅ Sp_{6}(2). Also, E′ is conjugate to T̃_{(2)}/⟨z⟩, and C_{G}(T̃_{(2)}/⟨z⟩) = T ·⟨θ⟩ by Lemma 8.3. Hence either

• E is toral, C_{G}(E) is a maximal torus extended by the Weyl element which inverts it, and Aut_{G}(E) ≅ Sp_{6}(2); or

• E is not toral, C_{G}(E) ≅ 2^{8}, and Aut_{G}(E) ≅ 2^{k}:Sp_{6}(2) where k = rk(E) − 1.

In the first case, Aut_{L}(E) ≅ Sp_{6}(2) by Proposition 8.5.

Assume E is not toral. We look at the action of Aut_{G}(E) on C_{G}(E), where x acts by g ↦ xgσ(x)^{−1}. We assume that T is the “standard” torus, on which σ acts by x ↦ x^{q}. Thus either σ fixes all x ∈ T of order 4, or σ(x) = x^{−1} for all such x. Also, all Weyl group elements (cosets of T in N_{G}(T)) contain elements in C_{G}(σ).

We must determine the isotropy subgroups of this action, and it suffices to do this when E ≤ M_{8} = T_{(2)}·⟨θ⟩. We have C_{G}(E) = M_{8}, and O_{2}(Aut_{G}(E)) is the group of automorphisms which are induced by conjugation by elements of order 4 in T. Thus O_{2}(Aut_{G}(E)) acts freely on one of the cosets T_{(2)} or T_{(2)}·θ (with one or two orbits), and on the other coset with the fixed action. Hence for each isotropy subgroup H ≤ Aut_{G}(E) of this action, either O_{2}(H) ≠ 1, or (since Sp_{6}(2) has no subgroup of index 2) H ≅ Sp_{6}(2).

Thus by Proposition 8.5 again, either O_{2}(Aut_{L}(E)) ≠ 1, and E is not pivotal (contradicting Proposition 4.4(c)); or Aut_{L}(E) ≅ Sp_{6}(2), and E ∉ 𝔈^{≥2}(L; 2) by Proposition 6.4.

Case 2b: If rk(E′) = 5, then E′′ =^{def} E′∩E′^{⊥} has rank one and is N_{Aut(L)}(E)-invariant, and N_{L}(E) (and N_{L}(P)) are contained in

C_{L}(E′′) ≅ SL_{2}(q) ×_{C_{2}} SSpin_{12}^{+}(q)

(see [Gr2, Table VI] for the centralizer of an involution of type 2B or 2C). Also, N_{Aut(L)}(E) ≤ C_{Aut(L)}(E′′). Hence by Proposition 4.9(b), applied with H = C_{L}(E′′), H′ = SSpin_{12}^{+}(q), and K = SL_{2}(q), P K/K ∈ ℜ^{2}(P Ω_{12}^{+}(q) ; 2). But this is impossible, since ℜ^{2}(P Ω_{12}^{+}(q) ; 2) = ∅ by Theorem 7.4.

Case 2c: Assume rk(E′) = 4. Let V ⊆ T_{(2)} be the subgroup of elements which lift to involutions in G̃, and let V_{4} ≤ V be a rank 4 subgroup on which 𝔟 is nonsingular; thus G-conjugate to E′. Fix an involution x ∈ V_{4}^{⊥} ⊆ V. The orbit of x, under the action of those Weyl group elements which fix V_{4}, has order 3; and thus the group of Weyl group elements which fix ⟨V_{4}, x⟩ has index 3 in the group of those which fix V_{4}. By construction, x is not G-conjugate to any element of E∖E′. Also, as noted above, x is not conjugate to any element of C_{G}(σ)∖L since it lifts to an involution in G̃. So by Proposition 8.7, E is not a pivotal 2-subgroup in L. □

It remains only to consider the groups E_{8}(q).

Proposition 8.10. Assume, for some odd prime power q, that L = E_{8}(q). Then ℜ^{≥2}(L; 2) = ∅, and hence L ∈ 𝔏^{≥2}(2).

Proof. Set G = E_{8}(𝔽_{q}). Fix a Steinberg endomorphism σ of G such that C_{G}(σ) = L. Let T ≤ G be a σ-invariant maximal torus.

By [Gr2, Table VI], the group E_{8}(𝔽_{q}) has two conjugacy classes of involutions, denoted types 2A and 2B. Define 𝔮: T_{(2)} ⟶ 𝔽_{2} by setting 𝔮(x) = 1 if x has type 2A and 𝔮(x) = 0 otherwise. Then by [Gr2, Lemma 2.16], 𝔮 is a quadratic form. Also, if θ ∈ N_{G}(T) is an involution which inverts T, then

W/⟨θT ⟩ = Aut_{G}(T)/{± Id} ≅ O(T_{(2)}, 𝔮) ≅ SO_{8}^{+}(2).

By [Gr2, Theorem 2.17], G contains two conjugacy classes of maximal elementary abelian 2-subgroups, represented by M_{9} = T_{(2)}×⟨θ⟩ of rank 9 (where θ again inverts T), and M_{8} of rank 8. All elements in M_{9}∖T_{(2)} are of type 2B. Also, there are subgroups V, W ≤ M_{8} of rank five, with intersection V ∩ W = X of rank two, such that the elements of type 2A in M_{8} are precisely those in (V ∪ W)∖X.

Assume E ∈ 𝔈^{≥2}(L; 2). Fix P ∈ Syl_{2}(C_{L}(E)), so that P ∈ ℜ^{≥2}(L; 2) and E = Ω_{1}(Z(P)) by Proposition 4.4(a). Also, E is pivotal (Proposition 4.4(c)), and thus O_{2}(Aut_{L}(E)) = 1. By Proposition 4.4(e), E must contain at least one involution of type 2B, since these are in the center of any Sylow 2-subgroup of L (there is an odd number of them in any maximal torus since there are exactly 135 nontrivial isotropic elements of T_{(2)}). Also, there must be at least one N_{Aut(L)}(E)-irreducible subquotient of E which has rank ≥ 4 (Proposition 4.6(b)). We will show that this is impossible, and thus that 𝔈^{≥2}(L; 2) = ∅.

Case 1: Assume first that there is an N_{Aut(L)}(E)-submodule E_{0} ≤ E of rank one. Then N_{Aut(L)}(E) ≤ C_{Aut(L)}(E_{0}), and the centralizer is described as follows in [Gr2, Table VI]:

type  C_{G}(E_{0})  C_{L}(E_{0})
2B SSpin_{16}(𝔽_{q}) SSpin_{16}^{+}(q)
2A 𝔽_{q}^{∗} ×_{C_{2}} Ẽ_{7}(𝔽_{q}) 𝔽_{q}^{∗} ×_{C_{2}} Ẽ_{7}(q)

We apply Proposition 4.9(b), with H = C_{L}(E_{0}), H′ = SSpin_{16}^{+}(q) or Ẽ_{7}(q), and K = C_{H}(H′) ≅ C_{2} or 𝔽_{q}^{∗}. Then P K/K ∈ ℜ^{≥2}(H/K; 2). But we have already seen that ℜ^{≥2}(P Ω_{16}^{+}(q) ; 2) and ℜ^{≥2}(E_{7}(q) ; 2) are both empty (Theorem 7.4 and Proposition 8.9), and so this is impossible.

Case 2: Assume that E is not G-conjugate to a subgroup of M_{9}. Hence E is contained in M_{8} up to G-conjugacy; and we assume (replacing M_{8} if necessary) that E ≤ M_{8}. Also, this means E contains involutions of both types.

Let V, W ≤ M_{8}, with intersection X = V ∩W of rank two, be as above. If E ∩X ≠ 0, then (since rk(E) ≥ 4) it is characterized as the largest type 2B pure subgroup of E each of whose cosets is all type 2B or all type 2A. Hence E ∩X is N_{Aut(L)}(E)-invariant. Thus since E has no rank one invariant submodules (case 1), either E ∩ X = 0, or E ≥ X. Also, since E must have some irreducible component of rank ≥ 4, we see that either E/(E ∩ X) has rank 6, or it has rank 4 and intersects each of V and W with rank 2. In the latter case, the map 𝔮: E ⟶ 𝔽_{2} which sends involutions of type 2A to 1 and other elements to zero is a quadratic form, hence E is toral by [Gr2, Theorem 9.2], in contradiction to our assumption.

If E ≥ X and rk(E/X) = 6, then E = M_{8}, and by [Gr2, Theorem 2.17], C_{G}(E) = E and O_{2}(Aut_{G}(E)) ≅ 2^{12}. Hence O_{2}(H) ≠ 1 for each isotropy subgroup H of the action of Aut_{G}(E) on C_{G}(E), so O_{2}(Aut_{L}(E)) ≠ 1 by Proposition 8.5, and E is not pivotal in this case.

If E ∩ X = 0 and rk(E) = 6, then E = E_{1} × E_{2} where the elements of type 2A are precisely those in (E_{1}∪E_{2})∖1. Let E_{0} ≤ E be a subgroup of rank 4 whose intersections with E_{1} and E_{2} each have rank two. Then 𝔮|_{E_{0}} is quadratic, hence E_{0} is toral by [Gr2, Theorem 9.2], and is a maximal toral subgroup of E. Choose E′_{0} ≤ T_{(2)} which is G-conjugate to E_{0}, and let E′ ≥ E′_{0} be such that the pair (E′, E′_{0}) is G-conjugate to (E, E_{0}). Then C_{W} (E′_{0}) leaves E′_{0}^{⊥} invariant, and we can choose 1 ≠ x ∈ E′_{0}^{⊥} whose C_{W} (E′_{0})-orbit has odd order. Since E′_{0} is a maximal toral subgroup of E′, x is not C_{G}(E′_{0})-conjugate to any element of E′. Hence by Proposition 8.7, no subgroup of L which is G-conjugate to E′ can be pivotal, and in particular E ∉ 𝔈^{≥2}(L; 2).

Case 3: Next assume E is G-conjugate to a subgroup of M_{9} = T_{(2)} ×⟨θ⟩ and contains involutions of both types. Let E_{0} ≤ E be the subgroup generated by all involutions of type 2A; this is clearly N_{Aut(L)}(E)-invariant, and is a toral subgroup since all elements of M_{9}∖T_{(2)} are of type 2B.

By inspection, one sees that for any V ≤ T_{(2)} which contains elements of both types, either V is generated by its elements of type 2A; or rk(V) ≤ 5 and the elements of type 2A generate a subgroup V_{0} ≤ V of index 2 within which the involutions of type 2B (with the identity) form a subgroup of index 2. Thus, either E = E_{0}, or [E:E_{0}] = 2 and E is not toral, or rk(E_{0}) ≤ 4 and there is an N_{Aut(L)}(E)-invariant subgroup E′_{0} ≤ E_{0} of index 2. In particular, no group of this last type contains an irreducible component of rank ≥ 4.

We can thus assume [E:E_{0}] ≤ 2, and that either E = E_{0} or E is not toral. Consider the quadratic form 𝔮 (as defined above) on E_{0}, and set E_{1} = E_{0}^{⊥} ∩ E_{0}. If rk(E_{1}) = 4, then E_{0} = E_{1}, 𝔮 is linear on E_{0}, and (since we are assuming E contains elements of type 2A) E_{2} =^{def} Ker(𝔮|_{E_{0}}) is an N_{Aut(L)}(E)-invariant subgroup of E of rank three. But this would contradict Proposition 4.6(b).

Thus rk(E_{1}) < 4, and so (by Proposition 4.6(b) again) we can assume rk(E_{0}/E_{1}) ≥ 4. Hence either the bilinear form on E_{0} is nonsingular (E_{1} = 0); or (since E has no invariant submodule of rank one) rk(E_{1}) = 2, rk(E_{0}/E_{1}) = 4, and all involutions in E_{1} are of type 2B.

Case 3a: Assume the bilinear form associated to 𝔮 is nonsingular on E_{0} (E_{1} = 0), and rk(E) < 8. Let E′_{0} ≤ T_{(2)} be G-conjugate to E_{0}, and let E′ ≥ E′_{0} be such that the pair (E′, E′_{0}) is G-conjugate to (E, E_{0}). Then C_{W} (E′_{0}) leaves E′_{0}^{⊥} invariant, and we can choose 1 ≠ x ∈ E′_{0}^{⊥} whose C_{W} (E′_{0})-orbit has odd order. If E ≩ E_{0}, then E′ is not toral and ⟨E′_{0}, x⟩ is, so x is not C_{G}(E′_{0})-conjugate to any element of E′. Hence by Proposition 8.7, no subgroup of L which is G-conjugate to E′ can be pivotal; and in particular, E ∉ 𝔈^{≥2}(L; 2).

Case 3b: Assume rk(E_{0}) = 8; i.e., E is G-conjugate to T_{(2)} or M_{9}. We apply Proposition 8.5 to determine the possibilities for Aut_{L}(E). Since π_{0}(C_{G}(T_{(2)})) ≅ C_{2}, T_{(2)} restricts to two conjugacy classes in L, both of which have Aut_{L}(E_{0}) ≅ SO_{8}^{+}(2).

Now, C_{G}(M_{9}) = M_{9} and Aut_{G}(M_{9}) ≅ 2^{8}:SO_{8}^{+}(2), and O_{2}(Aut_{G}(M_{9})) acts freely on one of the cosets T_{(2)} or T_{(2)}·θ and trivially on the other. Thus if E is G-conjugate to M_{9}, then either Aut_{L}(E) ≅ SO_{8}^{+}(2), or O_{2}(Aut_{L}(E)) ≠ 1 (has 2-rank at least 8).

Thus, since O_{2}(Aut_{L}(E)) = 1 (E is assumed pivotal), we have Aut_{L}(E) ≅ SO_{8}^{+}(2), and E ∉ 𝔈^{≥2}(L; 2) by Proposition 6.4.

Case 3c: Assume rk(E_{1}) = 2, rk(E_{0}/E_{1}) = 4, and all involutions in E_{1} are of type 2B. Each automorphism of E_{0} which is the identity on E_{1} and on E_{0}/E_{1} preserves 𝔮, and hence lies in Aut_{G}(E_{0}). Thus |O_{2}(Aut_{G}(E_{0}))| ≥ 2^{8}. Also, since E_{0} contains a 2Bpure subgroup of rank 4, C_{G}(E_{0})^{0} is a maximal torus (see Case 4), and π_{0}(C_{G}(E_{0})) ≅ 2^{2} (generated by the unique involution in SO_{8}^{+}(2) which fixes a rank 6 subgroup of this type and the Weyl group element which inverts the torus). So by Proposition 8.5, O_{2}(Aut_{L}(E_{0})) ≠ 1, and E_{0} is not pivotal.

Thus E ≩ E_{0}. Set T′ = C_{G}(E_{0})^{0}: a maximal torus of G. Since π_{0}(C_{G}(E_{0})) ≅ 2^{2} and the elements of E∖E_{0} invert T′, we have |C_{G}(E)| ≤ 2^{10}. Also, Aut_{G}(E) surjects onto Aut_{G}(E_{0}), since each coset of T′ in N_{G}(T′) contains an element which centralizes any given element of E∖E_{0}. For any x ∈ T′ of order 4 such that x^{2} ∈ E_{0}, xgx^{−1} = g for g ∈ E_{0} and xgx^{−1} = x^{2}g for g ∈ E∖E_{0}. Thus Aut_{G}(E) contains all automorphisms of E which are the identity on E_{0}, and these form a normal subgroup of order 2^{6}. Since |O_{2}(Aut_{G}(E_{0}))| ≥ 2^{8}, it now follows that |O_{2}(Aut_{G}(E))| ≥ 2^{14} > |C_{G}(E)|. So by Proposition 8.5 again, O_{2}(Aut_{L}(E)) ≠ 1, and E is not pivotal.

Case 4: Assume E is type 2B pure of rank 4. Any such subgroup is toral [Gr2, Theorem 9.2], and thus G-conjugate to a maximal isotropic subgroup E′ ≤ T_{(2)}.

Any element of W which fixes an index two subgroup E ≤ T_{(2)} must be an orthogonal transvection, and hence 𝔮|_{E∩E^{⊥}} ≠ 0 (see [Di, §19]). So the kernel of any root of G = E_{8}(𝔽_{q}), when regarded as an element of Hom(T, 𝔽_{q}^{∗})) and restricted to T_{(2)}, is also of this type, and thus does not contain any maximal isotropic subgroup of T_{(2)}. Thus C_{G}(E)^{0} is a maximal torus by Lemma 8.3. Since the stabilizer in W (E_{8})/C_{2} ≅ SO_{8}^{+}(2) of a maximal isotropic subspace is 2^{6}:L_{4}(2) ≅ 2^{6}:Ω_{6}^{+}(2), we also get that π_{0}(C_{G}(E)) ≅ 2_{+}^{1+6}, and Aut_{G}(E) = Aut(E) ≅ L_{4}(2).

Since each coset of T in N_{G}(T) contains elements of L, the action of Aut_{L}(E) on the conjugacy classes of π_{0}(C_{G}(E)) is that induced by conjugation. This action of Ω_{6}^{+}(2) on conjugacy classes in 2_{+}^{1+6} has four orbits: the orbits of the two central elements, and those of isotropic and nonisotropic elements. Hence there are four L-conjugacy classes of rank four 2B-pure subgroups E, with Aut_{L}(E) ≅ L_{4}(2), Σ_{6}, or 2^{4}:(Σ_{3} × Σ_{3}). Since none of these automorphism groups have dihedral or semidihedral Sylow 2-subgroup, E ∉ 𝔈^{≥2}(L; 2) by Proposition 4.6(c).

Case 5: Assume E is type 2B pure of rank 5. By [CG, Proposition 3.8], when G = E_{8}(ℂ), there is only one conjugacy class of such subgroups, and they have centralizer of the form C_{G}(E) ≅ 2^{5+10}. Hence by [GR, Theorem A.12], there is just one conjugacy class of such subgroups in G ≅ E_{8}(𝔽_{q}).

In particular, E is G-conjugate to E′ = ⟨E′_{0}, θ⟩, where E′_{0} ≤ T_{(2)} is a maximal isotropic subgroup. By Case 4, Aut_{G}(E′_{0}) ≅ L_{4}(2), and since this holds for any index 2 subgroup of E′, we see that Aut_{G}(E′) = Aut(E′) ≅ L_{5}(2). Also, C_{G}(E′_{0})/T ≅ 2_{+}^{1+6}: generated by θ, and those orthogonal transvections of T_{(2)} which fix E′_{0}. Hence C_{G}(E′) ≅ T_{(2)}⋊2_{+}^{1+6} has order 2^{15}. A closer check (or a comparison with E_{8}(ℂ) using [GR, Theorem A.12] again) shows that Z(C_{G}(E′)) = E′ ≅ 2^{5} and C_{G}(E′)/E′ ≅ 2^{10}; and also that C_{L}(E′) = C_{G}(E′). So under the action of N_{G}(E′) on C_{G}(E′) as defined in Proposition 8.5, C_{G}(E′) acts via conjugation, hence N_{G}(E′) leaves E′ invariant, and thus (since L_{5}(2) has no subgroup of index 32) must leave a point fixed. In other words, by Proposition 8.5, for some E′′ which is G-conjugate to E and E′, N_{L}(E′′) = N_{G}(E′′).

Thus by Proposition 8.5 again, Aut_{L}(E) is the stabilizer of the conjugation action of N_{G}(E′′)/C_{G}(E′′) ≅ L_{5}(2) on some conjugacy class in C_{G}(E′′) ≅ 2^{5+10}. Also, L_{5}(2) acts on C_{G}(E′′)/E′′ ≅ 2^{10} with two nonzero orbits having stabilizers 2^{6}:(Σ_{3} × L_{3}(2)) and 2^{4}:Σ_{6}. The point stabilizers of elements in C_{G}(E′′)∖E′′ thus have index ≤ 32 in one of the groups 2^{6}:(Σ_{3} × L_{3}(2)) or 2^{4}:Σ_{6}, while the point stabilizers of elements in E′′ are 2^{4}:L_{4}(2) and L_{5}(2). Hence either O_{2}(Aut_{L}(E)) ≠ 1, or Aut_{L}(E) ≅ A_{6}, Σ_{6}, or L_{5}(2). In all of these cases, E ∉ 𝔈^{≥2}(L; 2): either since E is not pivotal; or by Proposition 4.6(c)

(since the Sylow 2-subgroups of Σ_{6} and L_{5}(2) are neither dihedral nor semidihedral); or by Proposition 4.6(d) (since A_{6} has no radical 2-subgroup of order 2). □

The results of this section are now summarized in the following theorem.

Theorem 8.11. Fix an odd prime power q. Assume L is a simple group, isomorphic to one of the groups G_{2}(q), ^{2}G_{2}(q), F_{4}(q), ^{3}D_{4}(q), E_{6}(q), ^{2}E_{6}(q), E_{7}(q), or E_{8}(q). Then L ∈ 𝔏^{≥2}(2).

9. Sporadic groups

It remains to consider the sporadic simple groups.

Theorem 9.1. If L is one of the simple sporadic groups, then L ∈ 𝔏^{≥2}(2).

Proof. When L is one of the groups M_{11}, M_{12}, J_{1}, or O′N, then rk_{2}(L) ≤ 3 [GLS3, §5.6]. Hence ℜ^{≥2}(L; 2) = ∅ by Proposition 4.6(b), and so L ∈ 𝔏^{≥2}(2) in all of these cases by Proposition 4.2.

The remaining sporadic groups are considered individually. We recall now (without repeating it each time when used in the proof) that rk(E) ≥ 4.

L = M_{22} or M_{23}: We have the following inclusions with odd index:

M_{22} ≤ M_{23} ≥ M_{21}:2 ≅ P ΣL_{3}(4),

where P ΣL_{3}(4) is the extension of P SL_{3}(4) by the field automorphism. Hence all three of these groups have isomorphic Sylow 2-subgroups. Any elementary abelian 2-subgroup of rank 4 in P ΣL_{3}(4) is contained in P SL_{3}(4), and any Sylow 2-subgroup of P SL_{3}(4) contains exactly two such subgroups.

Identify L as a subgroup of M_{24}: the subgroup of elements which fix one or two points under the action on a set X of order 24. Fix S ∈ Syl_{2}(L), and let V_{1}, V_{2} ≤ S be the two elementary abelian subgroups of rank four. We take V_{1} to be the subgroup whose normalizer in M_{24} is the octad group V_{1}:A_{8}, where V_{1} acts freely on 16 points in X and A_{8} permutes the remaining 8 points in the obvious way (cf. [Gr3, 6.8]). Restriction to the subgroups fixing one or two points shows that

Aut_{M_{22}}(V_{1}) ≅ A_{6} and Aut_{M_{23}}(V_{1}) ≅ A_{7}.

Also, V_{2} is contained in O_{2}(H) ≅ 2^{6}, where H is the sextet subgroup of M_{24}, and the O_{2}(H)-action on X has orbits of order 4 (cf. [Gr3, §5]). Thus V_{1} and V_{2} are not Aut(L)-conjugate, and hence are both weakly closed in S with respect to Aut(L).

Since Aut_{L}(V_{1}) has no radical subgroup of order 2, V_{1} ∉ 𝔈^{≥2}(L; 2) by Proposition 4.6(d). Thus ℜ^{≥2}(L; 2)_{≱V_{2}} = ∅, and hence L ∈ 𝔏^{≥2}(2) by Proposition 4.2. (In fact, V_{2} ∈ ℜ^{2}(L; 2) for L = M_{22} or M_{23}.)

L = M_{24} or He: We refer to [A2, §39–42] for details of the structure of these groups. In both cases, there is an involution z ∈ L such that C_{L}(z) ≅ 2_{+}^{1+6}:L_{3}(2), the centralizer of a transvection in L_{5}(2), and this centralizer has odd index in L. To handle elements in this group, we fix V ≅ (𝔽_{2})^{5} with basis {v_{1}, …, v_{5}}, and set V_{i} = ⟨v_{1}, …, v_{i}⟩. We identify H = C_{L}(z) with the group of automorphisms of V which leave V_{1} and V_{4} invariant.

For 1 ≤ i < j ≤ 5, let e_{ij} ∈ H be the element which sends v_{j} ↦ v_{i} + v_{j} and is the identity on the other basis elements. Thus z = e_{15}. Let S be the subgroup generated by all e_{ij} for i < j; this is a Sylow 2-subgroup of H and hence of L. For each 1 ≤ i ≤ 4, let U_{i} ≤ S be the subgroup of automorphisms which are the identity on V_{i} and on V/V_{i}. By [A2, Lemma 39.1(3)] (or by the argument given below), U_{2} and U_{3} are the only subgroups of S of rank six.

Let E ≤ H be any elementary abelian subgroup of rank ≥ 4. For each involution u ∈ E, we write K(u) = Ker(u − Id) and I(u) = Im(u − Id); thus I(u) ≤ K(u) ≤ V. In particular, I(z) = V_{1} and K(z) = V_{4}.

Assume first that for some u, v ∈ E, I(v) ≰ K(u); i.e., (u − Id)(v − Id) ≠ 0. Then u|_{I(v)} and v|_{I(u)} must be (nonidentity) involutions, which implies that dim(I(u)) = dim(I(v)) = 2 and dim(I(u) ∩ I(v)) = 1. Similarly, dim(K(u) + K(v)) = 4, so dim(K(u) ∩ K(v)) = 2; and the three subspaces K(u) ∩ K(v), I(u), I(v) are linearly independent modulo I(u) ∩ I(v). Also, since both of these commute with z, we have V_{1} ≤ K(u) ∩ K(v) and I(u) + I(v) ≤ V_{4}. By an appropriate choice of basis elements, we are reduced (up to conjugacy in H) to one of the following situations:

• u = e_{12}e_{35}, v = e_{13}e_{25}, E ≤ C_{H}(⟨u, v⟩) = ⟨u, v, z, e_{14}, e_{45}⟩ ≅ 2^{2} × D_{8}

• u = e_{12}e_{34}, v = e_{13}e_{24}, E = C_{H}(⟨u, v⟩) = ⟨u, v, z, e_{14}⟩ ≅ 2^{4}

• u = e_{23}e_{45}, v = e_{24}e_{35}, E = C_{H}(⟨u, v⟩) = ⟨u, v, z, e_{25}⟩ ≅ 2^{4}

In the first case, for any choice of E of rank 4, Aut_{S}(E) contains all automorphisms which are the identity on z and on E/⟨z⟩, and thus contains a rank three subgroup. In the other two cases, Aut_{S}(E) contains all automorphisms which are the identity on the last two generators and modulo the last two generators, and thus has rank ≥ 4. Hence E ∉ 𝔈^{≥2}(L; 2) by Proposition 4.6(c), since the Sylow subgroups of Aut_{L}(E) are neither dihedral nor semidihedral.

We are left with the case where W =^{def} ⟨I(u) | u ∈ E⟩ is contained in W′ =^{def} ⋂_{u∈E} K(u). If W = W′, then either W = V_{1} or V_{4} and E = U_{1} or U_{4}, or dim(W) = 2, 3 and (up to H-conjugacy) E ≤ U_{i} for i = 2, 3. A straightforward argument shows that C_{L}(E) = C_{H}(E) = U_{i} in this last case, and hence E = U_{i} since E is pivotal (Proposition 4.4(c)). By [A2, Lemma 40.5], Aut_{L}(E) ≅ L_{4}(2) or 2^{3}:L_{3}(2) when E = U_{1} or U_{4} (rk(E) = 4), and Aut_{L}(E) ≅ L_{3}(2)×Σ_{3} or 3·Σ_{6} when E = U_{2} or U_{3} (rk(E) = 6). Since none of these automorphism groups has dihedral or semidihedral Sylow 2-subgroups, Proposition 4.6(c) shows that none of the U_{i} can lie in 𝔈^{≥2}(L; 2).

The only remaining case is where dim(W) = 2 and dim(W′) = 3, and hence E is H-conjugate to U_{2} ∩ U_{3}. Thus the only subgroup of S which could lie in 𝔈^{≥2}(L; 2) is U_{2} ∩ U_{3}. It follows that ℜ^{≥2}(L; 2)_{≱U_{2}U_{3}} = ∅. Since U_{2}U_{3} is weakly closed in S with respect to Aut(L) (U_{2} and U_{3} are the only rank six subgroups of S), Proposition 4.2 now implies that L ∈ 𝔏^{≥2}(2). (In fact, in all three cases L = M_{24}, L = He, and L = L_{5}(2), one can show that U_{2} ∩ U_{3} ∈ 𝔈^{2}(L; 2) and U_{2}U_{3} ∈ ℜ^{2}(L; 2).)

L = J_{2}: This group contains two conjugacy classes of involutions, of which those of type 2A are in the centers of Sylow subgroups. By [FR1, §3], the elements of type 2A in any elementary abelian E ≤ L form a subgroup, and there are no 2A- or 2B-pure subgroups of rank 3. Thus any elementary abelian 2-subgroup E ≤ L contains an N_{Aut(L)}(E)-invariant subgroup E_{0} ≤ E (generated by the elements of type 2A) such that rk(E_{0}) ≤ 2 and rk(E/E_{0}) ≤ 2. Hence 𝔈^{≥2}(L; 2) = ∅ by Proposition 4.6(c), and L ∈ 𝔏^{≥2}(2) by Proposition 4.2.

L = Co_{3} or L = HS: By [Fi, §4], there are two conjugacy classes of involutions in Co_{3}, of which those in the center of a Sylow 2-subgroup are of type 2A with centralizer 2Sp_{6}(2), and those of type 2B have centralizer 2×M_{12}. By [Fi, Lemma 4.7], this group 2Sp_{6}(2) has two conjugacy classes of noncentral involutions, whose centralizers have different orders. In other words, if x, y are commuting involutions and x has type 2A, then y and xy are conjugate in L (since their centralizers in C_{L}(x) are isomorphic), and thus have the same type. This shows that in any elementary abelian 2-subgroup E ≤ L, the elements of type 2A together with the identity form a subgroup of E.

By [PW, Lemma 2.2 & §4], there are two conjugacy classes of involutions in HS, of which those in the center of a Sylow 2-subgroup have type 2A and centralizer (2_{+}^{1+4} ×_{C_{2}} C_{4})·Σ_{5}, and the others have type 2B with centralizer 2 × Aut(A_{6}). Also, HS is also contained as a subgroup of Co_{3} (see [A2, §§23–24]), and a comparison of involution centralizers shows that the inclusion sends involutions of type 2A and 2B in HS to involutions of type 2A and 2B, respectively, in Co_{3}.

Now assume E ∈ 𝔈^{≥2}(L; 2), for L = Co_{3} or HS, and let E_{0} ≤ E be the subgroup generated by type 2A involutions. Then E_{0} ≠ 1, since E contains the center of a Sylow 2-subgroup (Proposition 4.4(e)), and it is clearly N_{Aut(L)}(E)-invariant. Thus rk(E_{0}) ≥ 4 or rk(E/E_{0}) ≥ 4 by Proposition 4.6(b). Since rk_{2}(Co_{3}) = 4 [GLS3, p.305], this means that E = E_{0} is a rank 4 type 2A-pure subgroup. If L = Co_{3}, then by [Fi, Lemma 5.9], L has a unique class of such subgroups, and Aut_{L}(E) ≅ A_{8} ≅ GL_{4}(2) for any such E. If L = HS, then by [PW, Lemma 4.1], Aut_{L}(E) ≅ Σ_{6} for any such E. In both cases, the Sylow 2-subgroups of Aut_{L}(E) are neither dihedral nor semidihedral, and hence E ∉ 𝔈^{≥2}(L; 2) by Proposition 4.6(c). Thus ℜ^{≥2}(L; 2) = ∅, and L ∈ 𝔏^{≥2}(2) by Proposition 4.2.

L = McL or L = Ly: By [GLS3, p.308], rk_{2}(L) = 4 (see also the discussion in [Fi, §5] when L = McL). By [Fi, Lemma 5.2] (when L = McL) or [W5, §2] (when L = Ly), Aut_{L}(E) ≅ A_{7} for every elementary abelian 2-subgroup E ≤ L of rank 4. Since A_{7} contains no radical 2-subgroup of order 2, such E cannot be in 𝔈^{≥2}(L; 2) by Proposition 4.6(d). Thus 𝔈^{≥2}(L; 2) = ∅, and hence L ∈ 𝔏^{≥2}(2).

L = F_{5} = HN: We refer to [NW, §3.1] for the following information about L. There are two conjugacy classes of involutions in L, types 2A and 2B. For any elementary abelian 2-subgroup E ≤ L, the function 𝔮: E ⟶ 𝔽_{2}, defined by 𝔮(v) = 1 if v is of type 2A and 𝔮(v) = 0 otherwise, is quadratic.

Assume E ∈ 𝔈^{≥2}(L; 2). Set E′ = E ∩ E^{⊥} (with respect to the quadratic form 𝔮), and E′′ = Ker(𝔮|_{E′}). Clearly, these subgroups are both N_{Aut(L)}(E)-invariant, and rk(E′/E′′) ≤ 1. So either rk(E′′) ≥ 4 or rk(E/E′) ≥ 4.

There are two conjugacy classes of 2B-pure subgroups of rank 2 in L, and each such subgroup is either contained in a unique 2B-pure subgroup of rank 3, or in a unique extraspecial subgroup X = 2_{+}^{1+8} with N_{L}(X)/X ≅ (A_{5} × A_{5}):2. Thus if rk(E′′) ≥ 4, then N_{L}(E) ≤ N_{L}(E′′) ≤ N_{L}(X) (up to conjugacy), so E ≤ Z(X) ≅ C_{2} by Proposition 4.9(a), and this contradicts the assumption rk(E′′) ≥ 4.

We are left with the possibility rk(E/E′) ≥ 4. Choose E_{0} ≤ E such that E = E_{0}×E′; then 𝔮|_{E_{0}} is nonsingular and rk(E_{0}) ≥ 4. Also, E_{0} contains a 2A-pure subgroup E_{1} of rank two, and C_{L}(E_{1}) ≅ 2^{2} × A_{8} ≤ A_{12} ≤ L. Inside A_{12}, the involutions whose support have order 4 or 12 are of type 2A, while those whose support have order 8 are of type 2B. Using this, one sees that either rk(E_{0}) = 6 and C_{L}(E_{0}) = E_{0}, or rk(E_{0}) = 4 and C_{L}(E_{0}) ≅ 2^{6} or 2^{4} × A_{4}. Thus in all of these cases, E is contained in a unique subgroup E of rank 6 (on which 𝔮 is nonsingular), and N_{L}(E) ≤ N_{L}(E). Also, Aut_{L}(E) = Ω(E, 𝔮) ≅ Ω_{6}^{−}(2), and so E ∉ 𝔈^{≥2}(L; 2) by Proposition 4.6(c) (Sylow 2subgroups of Aut_{L}(E) are neither dihedral nor semidihedral). If rk(E) = 5, then either 𝔮|_{E′} ≠ 0 and Aut_{L}(E) ≅ Ω_{5}(2) ≅ Σ_{6}, which again contradicts Proposition 4.6(c); or 𝔮|_{E′} = 0 and O_{2}(Aut_{L}(E)) ≠ 1, in which case E is not pivotal. Finally, if rk(E) = 4 and 𝔮 is nonsingular on E, then we have just seen that the Sylow 2-subgroups of C_{L}(E) are isomorphic to 2^{6}, and so E cannot be pivotal.

This shows that 𝔈^{≥2}(L; 2) = ∅, and thus that L ∈ 𝔏^{≥2}(2) by Proposition 4.2.

In all of the remaining cases, the proof that L ∈ 𝔏^{≥2}(2) will be based on a list of maximal 2-local subgroups of L — or in some cases, a list of proper subgroups of L (not necessarily 2-local) which contain all 2-local subgroups up to conjugacy. We label these subgroups H_{n} for n = 1, 2, …, and set V_{n} = Z(O_{2}(H_{n})). The goal is to show that ℜ^{≥2}(L; 2) = ∅; unless we set Q = V_{n} for some n, in which case we show that Q is weakly closed in some (any) Sylow 2-subgroup which contains it, and that ℜ^{≥2}(L; 2)_{≱Q} = ∅. In either case, Proposition 4.2 then implies that L ∈ 𝔏^{≥2}(2).

We fix a subgroup P ∈ ℜ^{≥2}(L; 2), and set E = Ω_{1}(Z(P)). If N_{L}(E) ≤ N_{L}(P) ≤ H_{n}, then P ≥ O_{2}(H_{n}), and so E ≤ V_{n} if O_{2}(H_{n}) is centric in H_{n}. In all cases, E ∩ V_{n} ≠ 1 by Proposition 4.9(a). If rk(V_{n}) = 2, then we can assume E ≥ V_{n} if the centralizers of involutions in E appear elsewhere in the list.

We use the standard notation 2A, 2B, etc. for the conjugacy classes of involutions in L. Recall that by Proposition 4.4(e), E ≥ Ω_{1}(Z(S)) for some S ∈ Syl_{2}(L), and thus E contains elements from each conjuacy class of involutions represented in Z(S).

L = J_{3}: By [FR2, §2], L contains three conjugacy classes of maximal 2-local subgroups.

• N_{L}(E) ≤ H_{1} ≅ 2_{−}^{1+4}:A_{5}, E ≤ V_{1} ≅ 2. Impossible since rk(V_{1}) = 1.

• N_{L}(E) ≤ H_{2} ≅ 2^{4}:GL_{2}(4), E ≤ V_{2} ≅ 2^{4}. Then E = V_{4} and Aut_{L}(E) ≅ GL_{2}(4) ≅ C_{3} × A_{5}. Since A_{5} has no radical subgroup of order 2, this contradicts Proposition 4.6(d).

• N_{L}(E) ≤ H_{3} ≅ 2^{2+4}:(3 × Σ_{3}), E ≤ V_{3} ≅ 2^{2}. Impossible since rk(V_{3}) = 2.

L = Suz: We refer to [W2] for the following information. There are two conjugacy classes of elements of order 2 in L, of which those of type 2A are in the centers of Sylow 2-subgroups. Also, in any elementary abelian subgroup E ≤ L, the involutions of type 2A together with the identity form a subgroup of E. Hence N_{L}(E) is contained in the normalizer of some 2A-pure subgroup; and hence by [W2, §2.4], in one of the groups H_{n} in the following list.

• N_{L}(E) ≤ H_{1} ≅ 2_{−}^{1+6}.U_{4}(2), E ≤ V_{1} ≅ 2. Impossible since rk(V_{1}) = 1.

• N_{L}(E) ≤ H_{2} ≅ 2^{2+8}:(A_{5} × Σ_{3}), E ≤ V_{2} ≅ 2^{2}. Impossible since rk(V_{2}) = 2.

• N_{L}(E) ≤ H_{3} ≅ 2^{4+6}:3A_{6}, E ≤ V_{3} ≅ 2^{4}. Then E = V_{3} and Aut_{L}(E) ≅ A_{6}, which contradicts Proposition 4.6(d) (A_{6} contains no radical 2-subgroup of order 2).

L = Ru: There are two conjugacy classes of involutions, of which those of type 2A lie in the centers of Sylow 2-subgroups. Thus all pivotal 2-subgroups of L contain elements of type 2A (Proposition 4.4(e)). By [W3, §2.4], the involutions of type 2A in E together with the identity form a subgroup E_{0} ≤ E, and thus N_{L}(E) ≤ N_{L}(E_{0}).

By [W3, §2.4–2.5], the normalizer of each 2A-pure elementary abelian subgroup of L is conjugate to a subgroup of one of the subgroups H_{n} listed below. In all cases, V_{i} is 2A-pure, and thus E = E_{0}. For V_{3} ≅ 2^{6} this is shown in [W3, Lemma 1] (where V_{3} is denoted R_{1}).

• N_{L}(E) ≤ H_{1} ≅ 2·2^{4+6}:Σ_{5}, E ≤ V_{1}. From the description in [A1, 12.12], we get that V_{1} = Z(O_{2}(H_{1})) has rank 1. Alternatively, this follows directly from the commutator relations listed in [P2, Lemma 12] (where O_{2}(H_{1}) is the subgroup generated by the eleven elements z, t, v, w, w_{1}, x_{1}, x_{2}, a, b, c, d). Thus this case is impossible.

• N_{L}(E) ≤ H_{2} ≅ 2^{3+8}:L_{3}(2), E ≤ V_{2} ≅ 2^{3}. Impossible since rk(V_{2}) = 3.

• N_{L}(E) ≤ H_{3} ≅ 2^{6}:G_{2}(2), E ≤ V_{3} ≅ 2^{6}. Assume here that N_{L}(E) is not conjugate to a subgroup of H_{1} or H_{2}. Let T be the conjugacy class of the subgroups of rank 2 in V_{2}. Let D be the “diagram” of E in the sense of [W3]: the graph with one node for each involution in E, and an edge connecting two nodes whenever the elements generate a subgroup in T . The automorphism group Aut_{L}(E) acts on D (and Out(L) = 1 in this case). There cannot be any Aut_{L}(E)-invariant node or triangle in D, since this would imply an Aut_{L}(E)-invariant element or subgroup in T , hence that N_{L}(E) ≤ H_{1} or H_{2}, contradicting our assumption on E.

By [W3, §2-4–2.5], for each E ≤ V_{3} of rank ≥ 2, either the diagram D contains an Aut_{L}(E)-invariant node or triangle; or D is a disjoint union of two or more triangles and isolated nodes in which case C_{L}(E) = V_{3} ≅ 2^{6}; or E = V_{3} and Aut_{L}(E) ≅ G_{2}(2) ≅ U_{3}(3):2; or rk(E) = 2 and N_{L}(E)/V_{3} ≅ (Σ_{3} × Σ_{3}). Since rk(E) ≥ 4, this shows that C_{L}(E) ≅ V_{3}, and hence (since E is pivotal) that E is conjugate to V_{3}. Thus (E, Aut_{L}(E)) ≅ (2^{6}, G_{2}(2)), and this contradicts Proposition 6.4.

L = F_{3} = Th: By [W7, Theorem 2.2], each 2-local subgroup of L is conjugate to a subgroup of H_{1} or H_{2} as listed here.

• N_{L}(E) ≤ H_{1} ≅ 2_{+}^{1+8}·A_{9}, E ≤ V_{1} ≅ 2. Impossible since rk(V_{1}) = 1.

• N_{L}(E) ≤ H_{2} ≅ 2^{5}·L_{5}(2), E ≤ V_{2} ≅ 2^{5}. Then Aut_{L}(E) = Aut(E), which contradicts Proposition 4.7(a).

L = J_{4}: By [KW, §2], there are four conjugacy classes of maximal 2-local subgroups, as listed below.

• N_{L}(E) ≤ H_{1} ≅ 2^{10}:L_{5}(2), E ≤ V_{1} ≅ 2^{10}. If N_{L}(E) ≤ H_{1} and E ≤ V_{1}, then P =^{def}

P/V_{1} is a radical 2-subgroup of H_{1}/V_{1} ≅ L_{5}(2), and N_{L}(P)/P ≅ N_{L_{5}(2)}(P)/P. Also, L_{5}(2) acts irreducibly on V_{1} [Ja, Theorem A], and hence acts as Λ^{2}(V), where V is one of the standard 5-dimensional representations. The only radical subgroups which have fixed subspace on V_{1} of rank ≥ 4 are the trivial subgroup, and two subgroups 2^{4} with normalizer 2^{4}:L_{4}(2) with fixed subspaces of rank 4 or 6. By Proposition 4.6(c), no such subgroup can be in ℜ^{≥2}(L; 2).

• N_{L}(E) ≤ H_{2} ≅ 2^{3+12}.(Σ_{5} × L_{3}(2)), E ≤ V_{2} ≅ 2^{3}. Impossible since rk(V_{2}) = 3.

• N_{L}(E) ≤ H_{3} ≅ 2_{+}^{1+12}·(3M_{22}:2), E ≤ V_{3} ≅ 2. Impossible since rk(V_{3}) = 1.

• N_{L}(E) ≤ H_{4} ≅ 2^{11}:M_{24}, E ≤ Q = V_{4} ≅ 2^{11}. By [KW, Lemma 1.1.2], Q = V_{4} is the unique subgroup of N_{L}(Q) ≅ 2^{11}:M_{24} of rank 11; and this group contains a Sylow 2-subgroup S ∈ Syl_{2}(L). Hence Q = V_{4} is weakly closed in S with respect to Aut(L); and we have just shown that ℜ^{≥2}(L; 2)_{≱Q} = ∅.

L = Co_{1}: By [Cu, Theorem 2.1], the normalizer of each elementary abelian 2-subgroup E ≤ L is contained in one of the seven subgroups listed below. We set Q = V_{2}, and show that ℜ^{≥2}(L; 2)_{≱Q} = ∅.

• N_{L}(E) ≤ H_{1} ≅ 2_{+}^{1+8}·Ω_{8}^{+}(2), E ≤ V_{1} ≅ 2. Impossible since rk(V_{1}) = 1.

• N_{L}(E) ≤ H_{2} ≅ 2^{4+12}.(3S_{6} × Σ_{3}), E ≤ V_{2} ≅ 2^{4}. Then Aut_{L}(E) = Sp_{4}(2) ≅ Σ_{6}, and this contradicts Proposition 4.6(c) (Sylow subgroups are neither dihedral nor semidihedral).

• N_{L}(E) ≤ H_{3} ≅ 2^{2+12}.(Σ_{3} × A_{8}), E ≤ V_{3} ≅ 2^{2}. Impossible since rk(V_{3}) = 2.

• N_{L}(E) ≤ H_{4} ≅ Co_{2}. Upon examination of the proof of [Cu, Theorem 2.1], one sees that the only case where N_{L}(E) is not contained in one of the other subgroups H_{n} occurs when E = ⟨(4), σ⟩ (in the notation of p.419) is a certain “BD-pure” subgroup (all involutions have type 2A in the notation of the atlas) of rank 4 (rank 5 in 2Co_{1}). As noted by Wilson in [W1, p.112], this subgroup is contained in a unique rank 5 subgroup ⟨(7), σ⟩ whose involutions all have type 2A, and hence its normalizer is contained in the normalizer of that subgroup, which is contained in some subgroup conjugate to H_{1}. We can thus ignore the case N_{L}(E) ≤ Co_{2}.

• N_{L}(E) ≤ H_{5} ≅ (A_{4} × G_{2}(4)).2. Then by Lemma 1.12(a,b), P ∩ (A_{4} × G_{2}(4)) = P_{1}×P_{2} where P_{1} ≅ 2^{2} ≤ A_{4} and P_{2} ≤ G_{2}(4) are radical 2-subgroups. By examination of the two maximal parabolic subgroups 2^{2+8}:(3 × A_{5}) and 2^{4+6}:(A_{5} × 3) of G_{2}(4), we see that the only possibility for E with an irreducible component of rank ≥ 4 (see Proposition 4.6(b)) is

E = 2^{2} × 2^{4} ≤ A_{4} × G_{2}(4).

In this case, E lifts to a subgroup of 2Co_{1} isomorphic to Q_{8} × 2^{4}; N_{L}(E) is contained in the normalizer of the second factor, whose elements are not of type 2B (since they lift to involutions in 2Co_{1}); and hence by [Cu, Lemmas 2.2 & 2.5], this normalizer is contained in one of the subgroups H_{n} for n = 1, 2, 3, 4 above.

• N_{L}(E) ≤ H_{6} ≅ (A_{6} × U_{3}(3)).2. Since rk_{2}(A_{6}) = rk_{2}(U_{3}(3)) = 2, E has a filtration by N_{L}(E)-invariant subgroups for which the quotients all have rank ≤ 2, and this contradicts Proposition 4.6(b).

• N_{L}(E) ≤ H_{7} ≅ 2^{11}·M_{24}, E ≤ Q = V_{7} ≅ 2^{11}. By [A1, (30.3) & (31.11)], Q is the unique subgroup of H_{7} of rank 11, and hence weakly closed (with respect to Aut(L)) in any Sylow 2-subgroup which contains it. We have now shown that ℜ^{≥2}(L; 2)_{≱Q} = ∅.

L = Co_{2}: By [W1, §3], the normalizer of each elementary abelian 2-subgroup of L is contained in one of the subgroups H_{n} in the following list.

• N_{L}(E) ≤ H_{1} = 2^{4+10}·(Σ_{5} × Σ_{3}), E ≤ V_{1} ≅ 2^{4}. Then E = V_{4} and Aut_{L}(E) ≅ Σ_{5}, and an examination of the first diagram in [W1, p.113] shows that Aut_{L}(E) acts via the permutation representation (with two orbits of lengths 5 and 10). This contradicts Proposition 4.7(b).

• N_{L}(E) ≤ H_{2} = 2_{+}^{1+8}:Sp_{6}(2), E ≤ V_{2} ≅ 2. Impossible since rk(V_{2}) = 1.

• N_{L}(E) ≤ H_{3} = 2_{+}^{1+6}.2^{4}A_{8}, E ≤ V_{3} ≅ 2. Impossible since rk(V_{3}) = 1.

• N_{L}(E) ≤ H_{4} = M_{23}. This subgroup arises as (one possible) intersection of a subgroup 2^{11}·M_{24} ≤ Co_{1} with Co_{2}. From the analysis in [Cu, §2], we see that each time the normalizer of an elementary abelian subgroup E ≤ Co_{1} was shown to be contained in a subgroup K ≅ 2^{11}:M_{24}, it was contained in such a way that E intersects nontrivially with the rank 11 subgroup. Hence if N_{L}(E) ≤ Co_{2}, then K ∩ Co_{2} cannot be isomorphic to M_{23}, and so we can ignore this case.

• N_{L}(E) ≤ H_{5} = U_{6}(2):2. Then P ∈ ℜ^{≥2}(H_{5} ; 2) by Proposition 4.9(b), which is empty by Lemma 6.5. (Note that Out(L) = 1.)

• N_{L}(E) ≤ H_{6} = McL. Then rk(E) = rk_{2}(McL) = 4, and Aut_{L}(E) ≅ A_{7} as described above. This is impossible by Proposition 4.6(d), since A_{7} contains no radical subgroups of order 2.

• N_{L}(E) ≤ H_{7} = 2^{10}:M_{22}:2, E ≤ Q = V_{7} ≅ 2^{10}. By [A1, (30.3) & (31.11)], V_{7} is the unique rank 10 subgroup of H_{7}, and hence weakly closed in any Sylow subgroup which contains it. We have just shown that ℜ^{≥2}(L; 2)_{≱Q} = ∅.

L = Fi_{22}: By [A3, (25.7)], for any S ∈ Syl_{2}(L), the set of involutions in S of type 2A generates a subgroup 2^{10}, which thus is weakly closed in S with respect to Aut(L). We fix S, and let Q ≅ 2^{10} denote this subgroup. We will show that ℜ^{≥2}(L; 2)_{≱Q} = ∅, and also that ℜ^{≥2}(Aut(L) ; 2)_{≱Q} = ∅. The latter will be needed later, when working with the group F i′_{24}. We set Γ = Aut(L) = F i_{22}:2 for short [A3, (37.2)]. Throughout the following discussion, we use the term “transposition” to refer to involutions of type 2A; these all have the property that the product of any two of them has order 2 or 3.

By [W4, Proposition 4.4] or [Fl], for each elementary abelian 2-subgroup 1 ≠ E ≤ L, N_{L}(E) ≤ H_{n} and N_{Γ}(E) ≤ H_{n} for some H_{n} and H_{n} (n = 1, …, 5) as described in the following list:

• N_{L}(E) ≤ H_{1} = 2U_{6}(2), N_{Γ}(E) ≤ H_{1} = 2U_{6}(2).2, E ≥ V_{1} ≅ 2. If N_{L}(E) ≤ H_{1} and N_{Γ}(E) ≤ H_{1}, then P/V_{1} is in ℜ^{≥2}(U_{6}(2) ; 2) or ℜ^{≥2}(U_{6}(2):2 ; 2) by Proposition 4.9(b); and these sets are empty by Lemma 6.5.

• N_{L}(E) ≤ H_{2} = (2 × 2_{+}^{1+8}:U_{4}(2)):2, N_{Γ}(E) ≤ H_{2} = (2 × 2_{+}^{1+8}:U_{4}(2):2):2, E ≤ V_{2} ≅ 2^{2}. Impossible since rk(V_{2}) = 2.

• N_{L}(E) ≤ H_{3} = 2^{5+8}:(Σ_{3} × A_{6}), N_{Γ}(E) ≤ H_{3} = 2^{5+8}:(Σ_{3} × Σ_{6}), E ≤ V_{3} ≅ 2^{5}. Thus Aut_{L}(E) is the stabilizer of the action of A_{6} or Σ_{6}. As described in [W4, §§3–4], V_{3} ≅ 2^{5} is generated by a unique hexad of transpositions, and hence Aut_{L}(V_{3}) ≅ A_{6} and Aut_{Γ}(V_{3}) ≅ Σ_{6} act on V_{3} via the permutation action on 𝔽_{2}^{6}/𝔽_{2}. Since neither A_{6} nor Σ_{6} has a radical subgroup of order 2, Proposition 4.5 shows that E cannot be equal to V_{3}, nor equal to the index two subgroup of V_{3} containing no transpositions (since that is also stabilized by A_{6}). The only remaining possibility is for E to have rank 4, containing exactly three or four transpositions (if it only has one or two its normalizer is contained in H_{1} or H_{2}). In these cases, either Aut_{L}(E)/O_{2}(Aut_{L}(E)) ≅ Σ_{3} (three transpositions) or Aut_{L}(E) = Aut_{Γ}(E) ≅ Σ_{4} (four transpositions); and in neither case can E be contained in 𝔈^{≥2}(L; 2) (Proposition 4.6).

• N_{L}(E) ≤ H_{4} = 2^{6}:Sp_{6}(2), N_{Γ}(E) ≤ H_{4} = 2^{7}:Sp_{6}(2), E ≤ V_{4} ≅ 2^{6}. By [W4, §§3–4], the only situation where N_{L}(E) ≤ H_{4} and is not contained in any of the other subgroups H_{n} occurs when all involutions in E have type 2B (in particular, E contains no transpositions), and E supports a nonsingular symplectic form 𝔟 such that Aut_{L}(E) = Aut(E, 𝔟). Thus by Proposition 6.4 (assuming rk(E) ≥ 4), E is in neither 𝔈^{≥2}(L; 2) nor 𝔈^{≥2}(Γ ; 2). Similarly, if E ≤ Γ and E_{0} =^{def} E∩L is 2B-pure and supports a symplectic form as described in (e), then E ≤ H_{4} ≅ 2^{7}:Sp_{6}(2), so E ≤ O_{2}(H_{4}) by Proposition 4.9(a), hence either Aut_{Γ}(E) ≅ Aut_{L}(E_{0}) or O_{2}(Aut_{Γ}(E)) ≠ 1; and the same reasoning shows that E ∉ 𝔈^{≥2}(Γ ; 2).

• N_{L}(E) ≤ H_{5} = 2^{10}:M_{22}, N_{Γ}(E) ≤ H_{5} = 2^{10}:M_{22}:2, E ≤ Q = V_{5} ≅ 2^{10}. We have already seen that Q is weakly closed in any Sylow 2-subgroup which contains it; and we have just shown that ℜ^{≥2}(L; 2)_{≱Q} = ∅.

L = Fi_{23}: Note that Out(L) = 1 [A3, (37.2)]. Fix S ∈ Syl_{2}(L). Then Z(S) ≅ 2^{2}, and contains a representative from each of the three classes of involutions in L. Thus by Proposition 4.4(e), each critical 2-subgroup of L contains involutions from each of the three classes. By [A3, (25.7)], the set of transpositions in S (involutions of type 2A) generates a subgroup 2^{11}, which thus is weakly closed in S. We let Q ≅ 2^{11} denote this subgroup, and will show that ℜ^{≥2}(L; 2)_{≱Q} = ∅.

By [Fl, §2], the normalizer of each elementary abelian 2-subgroup of L which contains transpositions is contained in one of the subgroups in the following list.

• N_{L}(E) ≤ H_{1} ≅ 2Fi_{22}, E ≥ V_{1} ≅ 2. By Proposition 4.9(b) (and since Out(L) = 1), P/V_{1} ∈ ℜ^{≥2}(F i_{22} ; 2). We have already seen that this implies that P/V_{1} contains the subgroup 2^{10} generated by the involutions of type 2A in some Sylow 2-subgroup of F i_{22}, and hence that P ≥ Q = V_{5} (up to conjugacy).

• N_{L}(E) ≤ H_{2} ≅ 2^{2}·U_{6}(2).2, E ≥ V_{2} ≅ 2^{2}. Since Out(L) = 1, Proposition 4.9(b) implies that P/V_{2} ∈ ℜ^{≥2}(U_{6}(2):2 ; 2), and this set is empty by Lemma 6.5.

• N_{L}(E) ≤ H_{3} ≅ (2^{2} × 2_{+}^{1+8}).(3 × U_{4}(2)).2, E ≤ V_{3} ≅ 2^{3}. Impossible since rk(V_{3}) = 3.

• N_{L}(E) ≤ H_{4} ≅ 2^{6+8}:(Σ_{3} × A_{7}), E ≤ V_{4} ≅ 2^{6}. The subgroup V_{4} contains (and is generated by) exactly seven transpositions, which are permuted in the obvious way by Aut_{L}(V_{4}) ≅ A_{7}. Hence Aut_{L}(E) is the stabilizer of this permutation action of A_{7}. Since A_{7} has no radical 2-subgroup of order 2, V_{4} ∉ 𝔈^{≥2}(L; 2) by Proposition 4.6(d)), and thus E ≨ V_{4}. Thus P/O_{2}(H_{4}) ≅ P_{1} × P_{2}, where P_{1} ≤ Σ_{3} and 1 ≠ P_{2} ≤ A_{7} are radical 2-subgroups (Lemma 1.12(a)); E is the fixed subgroup of the P_{2}-action on V_{4}; and this is impossible since the nontrivial radical 2-subgroups of A_{7} all have fixed subgroup on V_{4} of rank ≤ 3.

• N_{L}(E) ≤ H_{5} ≅ 2^{11}·M_{23}, E ≤ Q = V_{5} ≅ 2^{11}. We have already seen that Q is weakly closed in any Sylow 2-subgroup which contains it; and we have just shown that ℜ^{≥2}(L; 2)_{≱Q} = ∅.

L = Fi′_{24}: By [A3, (37.1)], Out(L) ≅ C_{2}, and Aut(L) = F i_{24}. We write Γ = F i_{24} for short. The group Γ is generated by transpositions: elements in a conjugacy class of involutions in Γ∖L the product of any two of which has order 2 or 3. By [A3, (37.4)], L has two conjugacy classes of involutions: each element of type 2A is a product of a unique pair of commuting transpositions (its factors), while each element of type 2B is a product of four commuting transpositions (but not uniquely).

Fix Ŝ ∈ Syl_{2}(Γ), and set S = Ŝ ∩ L ∈ Syl_{2}(L). By [A3, (25.7)], the set of transpositions in Ŝ generates a subgroup Q̂ ≅ 2^{12}. Set Q = Q̂∩L ≅ 2^{11}. By [A3, (34.9)], Q is the Todd module for N_{L}(Q)/Q ≅ M_{24}. Hence by [A1, 31.11], Q is the unique elementary abelian 2-subgroup of N_{L}(Q) of rank 11. Since Q ◁ S, N_{L}(Q) ≥ S, and thus Q is weakly closed in S with respect to Aut(L). We will show that ℜ^{≥2}(L; 2)_{≱Q} = ∅.

By [W6, Theorems D & E] (with corrections in [LW, §2]), each 2-local subgroup of L is contained up to conjugacy in one of the subgroups in the following list.

• N_{L}(E) ≤ H_{1} = N(2A) ≅ 2Fi_{22}:2, E ≥ V_{1} ≅ 2. Then N_{Γ}(E) ≤ N_{Γ}(H_{1}), since the factors of the generator of V_{1} normalize E. By Proposition 4.9(b) again, P/V_{1} ∈ ℜ^{≥2}(F i_{22}:2 ; 2). We have already shown that this implies that P/V_{1} contains (up to conjugacy) the subgroup 2^{10} generated by the involutions of type 2A in any Sylow 2-subgroup of F i_{22}. Since all involutions in F i_{22} lift to involutions in 2F i_{22} [A3, (23.8)], this shows that P contains a subgroup 2^{11}, which must be conjugate to Q.

• N_{L}(E) ≤ H_{2} = N(2B) ≅ 2_{+}^{1+12}·3U_{4}(3):2, E ≤ V_{2} ≅ 2. Impossible since rk(V_{2}) = 1.

• N_{L}(E) ≤ H_{3} ≅ 2^{2}·U_{6}(2):Σ_{3}, E ≥ V_{3} ≅ 2^{2}. In this case, V_{3} ≤ Q (up to conjugacy), and all of its involutions are of type 2A. The factors of the involutions in V_{3} all lie in Q̂, and generate a rank three subgroup with just three transpositions, which are permuted by the conjugation action of E and hence normalize E. Thus N_{Γ}(E) ≤ N_{Γ}(H_{3}), and so P/V_{3} ∈ ℜ^{≥2}(U_{6}(2):Σ_{3} ; 2) by Proposition 4.9(b). But this set is empty by Lemma 6.5.

• N_{L}(E) ≤ H_{4} ≅ 2^{6+8}:(Σ_{3} × A_{8}), E ≤ V_{4} ≅ 2^{6}. Here, Aut_{L}(V_{4}) acts on V_{4} via the natural action of A_{8} ≅ Ω_{6}^{+}(2). This follows from the discussion in [W6, p.91], where it is shown that V_{4} is the intersection with F i′_{24} of the subgroup of rank 7 in F i_{24} generated by the eight transpositions in an octad. Hence E ≨ V_{4} by Proposition 4.6(c), since the Sylow 2-subgroups of A_{8} are neither dihedral nor semidihedral. Hence P/V_{4} is a nontrivial radical 2-subgroup of A_{8}, and E is the fixed subgroup if its action on V_{4}. But the fixed subgroup of any such radical 2-subgroup has rank ≤ 3, and so this case is not possible.

• N_{L}(E) ≤ H_{5} ≅ 2^{3+12}.(A_{6} × L_{3}(2)), E ≤ V_{5} ≅ 2^{3}. Impossible since rk(V_{5}) = 3.

• N_{L}(E) ≤ H_{6} ≅ (A_{4} × Ω_{8}^{+}(2):3):2, E ≥ V_{6} ≅ 2^{2}. Then P ≤ A_{4} × Ω_{8}^{+}(2), and is a radical 2-subgroup of this product. So by Lemma 1.12(b), P = V_{6} × P′, where P′ is a radical subgroup of Ω_{8}^{+}(2), and hence of the form O_{2}(𝔓) for some parabolic subgroup 𝔓 ≤ Ω_{8}^{+}(2) for which rk(Z(O_{2}(𝔓))) ≥ 4. The only such subgroup is of the form 𝔓 ≅ 2^{6}:Ω_{6}^{+}(2), which would imply that

(E, Aut_{L}(E)) ≅ (2^{2} × 2^{6}, (3 × Ω_{6}^{+}(2)):2).

By Proposition 6.4, no such group E can be in 𝔈^{≥2}(L; 2).

• N_{L}(E) ≤ H_{7} ≅ 2^{8}:Ω_{8}^{−}(2), E ≤ V_{7} ≅ 2^{8}. The maximal parabolic subgroups of Ω_{8}^{−}(2) are 𝔓_{1} ≅ 2^{6}:U_{4}(2) (O_{2}(𝔓_{1}) has 1-dimensional fixed subgroup on V_{7} ≅ 2^{8}), 𝔓_{2} ≅ 2_{+}^{1+8}:(Σ_{3} × A_{5}) (2-dimensional fixed subgroup), and 𝔓_{3} ≅ 2^{3+6}:(L_{3}(2) × 3) (3-dimensional fixed subgroup). Hence P ≩ V_{7} would imply rk(E) < 4; while Proposition 6.4 shows that V_{7} ∉ 𝔈^{≥2}(L; 2).

• N_{L}(E) ≤ H_{8} ≅ 2^{11}·M_{24}, E ≤ Q = V_{8} ≅ 2^{11}. We have already seen that Q is weakly closed in S with respect to Aut(L); and have now shown that ℜ^{≥2}(L; 2)_{≱Q} = ∅.

L = F_{2}: This group contains four conjugacy classes of involutions. By [Gr1, Theorem 3], or by [Sg, Theorem 5.6], Out(L) = 1. By [MS], every 2-local subgroup of L is contained in one of the subgroups in the following list.

• N_{L}(E) ≤ H_{1} = N(2B) ≅ 2^{1+22}.Co_{2}, E ≤ V_{1} ≅ 2. Impossible since rk(V_{1}) = 1.

• N_{L}(E) ≤ H_{2} ≅ 2^{2+10+20}(M_{22}:2 × Σ_{3}), E ≤ V_{2} ≅ 2^{2}. Impossible since rk(V_{2}) = 2.

• N_{L}(E) ≤ H_{3} ≅ [2^{35}](Σ_{5} × L_{3}(2)), E ≤ V_{3} ≅ 2^{3}. Thus rk(E) ≤ 3. The subgroup H_{3} is described in [A1, p.33], and its construction in [A1, pp.219–220]. In particular, H_{3} = N_{L}(W) for a certain subgroup W ≤ O_{2}(H_{1}) ≅ 2^{1+22} of rank 3 (and with Z(H_{1}) ≤ W), and H_{3}/C_{L}(W) ≅ L_{3}(2). Since H_{1} is the centralizer of an element of W, this shows that C_{O_{2}(H_{1})}(W) ◁ O_{2}(H_{3}) ≤ H_{1}. Since O_{2}(H_{1}) is centric in H_{1}, this shows that Z(O_{2}(H_{3})) ≤ C_{O_{2}(H_{1})}(W), and hence (since O_{2}(H_{1}) is extraspecial) that W = V_{3}.

• N_{L}(E) ≤ H_{4} ≅ 2^{5+5+10+10}L_{5}(2), E ≤ V_{4} ≅ 2^{5}. Then Aut_{L}(E) = Aut(E), and this contradicts Proposition 4.6(b,c) (the Sylow 2-subgroups of Aut_{L}(E) are neither dihedral nor semidihedral).

• N_{L}(E) ≤ H_{5} ≅ 2^{9+16}.Sp_{8}(2), E ≤ V_{5} ≅ 2^{9}. Then E ≤ V_{5} ≅ 2^{9}, and P/O_{2}(H_{5}) is a 2-radical subgroup of H_{5}/O_{2}(H_{5}) ≅ Sp_{8}(2). The only radical 2-subgroups of Sp_{8}(2) whose fixed subgroup on 2^{8} has rank ≥ 4 are the trivial subgroup (and (Aut_{L}(E), E) ≅ (Sp_{8}(2), 2^{9}) implies E ∉ 𝔈^{≥2}(L; 2) by Proposition 6.4); and 2^{10}:A_{8} ≅ 2^{10}:L_{4}(2) (impossible by Proposition 4.6(c), since the Sylow subgroups are neither dihedral nor semidihedral).

• N_{L}(E) ≤ H_{6} = N(2A) ≅ 2.^{2}E_{6}(2):2. Then |V_{6}| = 2, and P/V_{6} ∈ ℜ^{≥2}(H_{6}/V_{6} ; 2) by Proposition 4.9(b). In particular, (P/V_{6})∩^{2}E_{6}(2) is a radical 2-subgroup of ^{2}E_{6}(2). From the list of maximal parabolic subgroups of^{2}E_{6}(2), we see that the only radical 2subgroup whose center has rank ≥ 4 has the form 2^{8+16}, with normalizer 2^{8+16}:SO_{8}^{−}(2) in ^{2}E_{6}(2):2. Hence the only possibilities are

(E/V_{6}, Aut_{L}(E/V_{6})) ≅ (2^{8}, SO_{8}^{−}(2)) or (2^{7}, Sp_{6}(2)).

(The second case corresponds to the extension of 2^{8+16} by a transvection in SO_{8}^{−}(2).) By Proposition 6.4, neither of this situations can occur for P/V_{6} ∈ ℜ^{≥2}(^{2}E_{6}(2):2 ; 2), and so this case is impossible.

• N_{L}(E) ≤ H_{7} = N(2C) ≅ (2^{2} × F_{4}(2)):2. From the list of maximal parabolic subgroups of F_{4}(2), we see that only two radical 2-subgroups U = O_{2}(𝔓) have centers of rank ≥ 4. In both cases, this would mean P = U of order 2^{20}, E = Z(P) ≅ 2^{5}, and Aut_{L}(E) ≅ 𝔓/U ≅ Σ_{3} × L_{3}(2); with a rank 2 subgroup of E_{0} ≤ E which is normal in 𝔓. Which contradicts Proposition 4.6(b).

• N_{L}(E) ≤ H_{8} = N(2C^{2}) ≅ Σ_{4} ×^{2}F_{4}(2). The radical 2-subgroups of ^{2}F_{4}(2) have centers of rank one or two (see [W3] or [P1]), and hence E ∉ 𝔈^{≥2}(L; 2) and P ∉ ℜ^{≥2}(L; 2).

L = F_{1}: By [Gr1, Theorem 3], or by [GMS, Theorem 5.10], Out(L) = 1. By [MS], every 2-local subgroup of L is contained in one of the subgroups in the following list.

• N_{L}(E) ≤ H_{1} = N(2A) ≅ 2·F_{2}, E ≥ V_{1} ≅ 2. By Proposition 4.9(b), P/V_{1} ∈ ℜ^{≥2}(F_{2} ; 2), and we have already shown that this last set is empty.

• N_{L}(E) ≤ H_{2} = N(2A^{2}) ≅ 2^{2}·^{2}E_{6}(2):Σ_{3}, E ≥ V_{2} ≅ 2^{2}. Then (P/V_{2}) ∩^{2}E_{6}(2) is a radical 2-subgroup of ^{2}E_{6}(2). All involutions in ^{2}E_{6}(2) lift to involutions in its 2-fold cover, so any elementary abelian 2-subgroup lifts to an elementary abelian 2-subgroup in the 2-fold cover. From the list of maximal parabolic subgroups of ^{2}E_{6}(2), we see that the only radical 2-subgroup whose center has rank ≥ 4 has the form 2^{8+16}, with normalizer 2^{8+16}:Ω_{8}^{−}(2). Also, the only involutions z ∈ SO_{8}^{−}(2)∖Ω_{8}^{−}(2) such that ⟨z⟩ is radical in SO_{8}^{−}(2) are the orthogonal transvections, with C_{Ω_{8}^{−}(2)}(z) ≅ Sp_{6}(2). Hence either P/V_{2} ≅ 2^{8+16}, E/V_{2} ≅ 2^{8}, and Aut_{L}(E/V_{2}) ≅ Ω_{8}^{−}(2); or P/V_{2} ≅ 2^{8+16}.2, E/V_{2} ≅ 2^{7}, and Aut_{L}(E/V_{2}) ≅ Sp_{6}(2). Both of these cases contradict Proposition 6.4.

• N_{L}(E) ≤ H_{3} = N(2B) ≅ 2^{1+24}.Co_{1}, E ≤ V_{3} ≅ 2. Impossible since rk(V_{3}) = 1.

• N_{L}(E) ≤ H_{4} ≅ 2^{2+11+22}(M_{24} × Σ_{3}), E ≤ V_{4} ≅ 2^{2}. Impossible since rk(V_{4}) = 2.

• N_{L}(E) ≤ H_{5} ≅ 2^{3+6+12+18}.(L_{3}(2) × 3S_{6}), E ≤ V_{5} ≅ 2^{3}. Impossible since rk(V_{5}) = 3.

• N_{L}(E) ≤ H_{6} ≅ 2^{5+10+20}.(L_{5}(2) × Σ_{3}), E ≤ V_{6} ≅ 2^{5}. Then Aut_{L}(E) = Aut(E), which contradicts Proposition 4.7(a).

• N_{L}(E) ≤ H_{7} ≅ 2^{10+16}.Ω_{10}^{+}(2), E ≤ V_{7} ≅ 2^{10}. Then P ≥ O_{2}(H_{7}), and P/O_{2}(H_{7}) is a 2-radical subgroup of H_{7}/O_{2}(H_{7}) ≅ Ω_{10}^{+}(2). By examination of the root system of the Lie group D_{5}, one sees that the only radical 2-subgroups of Ω_{10}^{+}(2) whose fixed subgroup on 2^{10} has rank ≥ 4 are the trivial subgroup (and (Aut_{L}(E), E) ≅ (Ω_{10}^{+}(2), 2^{10}) implies E ∉ 𝔈^{≥2}(L; 2) by Proposition 6.4); two conjugacy classes of subgroups 2^{10} with normalizers 2^{10}:L_{5}(2) (5-dimensional fixed subspace), and one conjugacy class of subgroup 2^{6+8} with normalizer 2^{6+8}:L_{4}(2) (4-dimensional fixed subspace). Hence in both of these cases, Aut_{L}(E) = Aut(E), which implies E ∉ 𝔈^{≥2}(L; 2) by Proposition 4.7(a).

This finishes the proof of Theorem 9.1. □

10. Computations of lim^{1}(Z_{G})

In this section, we summarize what we know about the groups \underleftarrow{lim}^{1}(YΓ ; L̃), with indications of the proof in some cases. In all cases, \underleftarrow{lim}^{0}(YΓ ; L̃) = 0 by Glauberman’s Z^{∗}-theorem [Gl]. Using this, the following proposition follows by essentially the same proof as that of Proposition 4.2.

Proposition 10.1. Fix a finite simple group L and S ∈ Syl_{2}(L). Assume Q ≤ S is 2centric in L and weakly closed in S with respect to Aut(L), and that it has the property that ℜ^{≥2}(L; p)_{≱Q} = ∅, and that no subgroup in ℜ^{1}(L; p)_{≱Q} is contained in any other subgroup in this set. Let L̃ be a quasisimple group such that Z(L̃) is a 2-group and L̃/Z(L̃) ≅ L, and let Γ ≤ Aut(L̃) be a subgroup which contains Inn(L̃) ≅ L. Then for any set P_{1}, …, P_{k} of Γ-conjugacy class representatives for subgroups in ℜ^{1}(L; p)_{≱Q},

\underleftarrow{lim}^{1}(YΓ ; L̃) ≅ Ker[⨁_{i=1}^{k} Λ^{1}(N_{Γ}(P_{i}), N_{L}(P_{i}); YΓ ; L̃(P_{i})) −↠ H^{0}(N_{Γ}(Q); YΓ ; L̃(Q))]

for some surjection between these two groups.

When L ≅ A_{n}, then

\underleftarrow{lim}^{1}_{O_{2}(Γ)}(YΓ ; L̃) ≅ \begin{cases}ℤ/2 & if L̃ ≅ A_{n}, Γ ≤ Σ_{n}, n ≡ 2, 3 (mod 4) \\ 0 & otherwise; \end{cases}

When Γ ≤ Σ_{n}, this is shown via an easy modification of the proof of Theorem 5.1. (The case L = A_{6} ≅ P SL_{2}(9) and Γ ≰ Σ_{6} must be handled separately.) Recall that we write E_{2^{k}} ≤ Σ_{n} for the elementary abelian group 2^{k} acting with one free orbit, and that Q ≤ A_{n} denotes the product of [n/4] copies of E_{4}. Then ℜ^{1}(A_{n}; 2)_{≱Q} is empty

if n ≡ 0, 1 (mod 4); and contains the conjugacy class of the group A_{n} ∩ (S′ × E_{2}^{×3}) for some S′ ∈ Syl_{2}(Σ_{n−6}) if n ≡ 2, 3 (mod 4). So A_{n} ∈ 𝔏^{1}(2) in the first case by Proposition 4.2, and one gets the above computations using Proposition 10.1 in the second case.

Note, however, that \underleftarrow{lim}^{1}(Z_{Σ_{n}}) = 0 for all n, even in those cases where \underleftarrow{lim}^{1}(Y_{A_{n}}^{Σ_{n}}) ≅ ℤ/2. This follows from the observation that \underleftarrow{lim}^{0}(Z_{Σ_{n}}/Y_{A_{n}}^{Σ_{n}}) ≅ ℤ/2 when n ≡ 2, 3 (mod 4).

When L is of Lie type in characteristic two, then by Theorem 6.3, L ∈ 𝔏^{1}(2) except when L = L_{3}(2) ≅ L_{2}(7). When L is of Lie type in odd characteristic, and not isomorphic to E_{7}(q) or E_{8}(q), then L ∈ 𝔏^{1}(2) except for the following cases:

• \underleftarrow{lim}^{1}(Y_{L}^{Γ}) ≅ ℤ/2 if L ≅ P SL_{2}(q) ≅ Ω_{3}(q), q ≡ ±1 (mod 8), Γ ≤ Aut_{fg}(L)

• \underleftarrow{lim}^{1}(Y_{L}^{Γ}) ≅ ℤ/2 if L ≅ P SL_{4}(q), q ≡ 3 (mod 4), Γ ≤ Aut_{fg}(L)

• \underleftarrow{lim}^{1}(Y_{L}^{Γ}) ≅ ℤ/2 if L ≅ P SU_{4}(q), q ≡ 1 (mod 4), Γ ≤ Aut_{fg}(L)

• \underleftarrow{lim}^{1}(Y_{L}^{Γ}) ≅ ℤ/2 if L ≅ Ω_{2n}^{ns}(q), n ≥ 3, Γ ≤ Aut_{fg}(L).

Here, Aut_{fg}(L) denotes the group generated by inner, field, and graph automorphisms of L. Also, Ω_{2n}^{ns}(q) = Ω(𝔽_{q}^{2n}, 𝔮) where 𝔮 is a quadratic form with nonsquare discriminant. The case L = P SL_{2}(q) is shown using Proposition 1.6 (or an easy modification of the proposition and its proof). In all other cases, there is a weakly closed subgroup Q such that ℜ^{1}(L; 2)_{≱Q} contains at most one conjugacy class, or in certain cases two L-conjugacy classes which are Aut(L)-conjugate. The results then follow from Propositions 4.2 and 10.1.

The sporadic groups all lie in 𝔏^{1}(2), with the possible exception of the baby monster and the monster. This is shown, either by modifying and extending the arguments used in Section 9, or by using lists of radical 2-subgroups of these groups such as those published by Yoshiara [Y1] [Y2] and by An & O’Brien [AO].

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LAGA, Institut Galilée, Av. J-B Clément, 93430 Villetaneuse, France

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