EXISTENCE AND UNIQUENESS OF LINKING SYSTEMS: CHERMAK’S PROOF VIA OBSTRUCTION THEORY

BOB OLIVER

Abstract. We present a version of a proof by Andy Chermak of the existence and uniqueness of centric linking systems associated to arbitrary saturated fusion systems. This proof differs from the one in [Ch2] in that it is based on the computation of higher derived functors of certain inverse limits. This leads to a much shorter proof, but one which is aimed mostly at researchers familiar with homological algebra.

One of the central questions in the study of fusion systems is whether to each saturated fusion system one can associate a centric linking system, and if so, whether it is unique. This question was recently answered positively by Andy Chermak [Ch2], using direct constructions. His proof is quite lengthy, although some of the structures developed there seem likely to be of independent interest.

There is also a well established obstruction theory for studying this problem, involving higher derived functors of certain inverse limits. This is analogous to the use of group cohomology as an “obstruction theory” for the existence and uniqueness of group extensions. By using this theory, Chermak’s proof can be greatly shortened, in part because it allows us to focus on the essential parts of Chermak’s constructions, and in part by using results which are already established. The purpose of this paper is to present this shorter version of Chermak’s proof, a form which we hope will be more easily accessible to researchers with a background in topology or homological algebra.

A saturated fusion system over a finite p-group S is a category whose objects are the subgroups of S, and whose morphisms are certain monomorphisms between the subgroups. This concept is originally due to Puig (see [P2]), and one version of his definition is given in Section 1 (Definition 1.1). One motivating example is the fusion system of finite group G with S ∈ Syl_{p}(G): the category F_{S}(G) whose objects are the subgroups of S and whose morphisms are those group homomorphisms which are conjugation by elements of G.

For S ∈ Syl_{p}(G) as above, there is a second, closely related category which can be defined, and which supplies the “link” between F_{S}(G) and the classifying space BG of G. A subgroup P ≤ S is called p-centric in G if Z(P) ∈ Syl_{p}(C_{G}(P)); equivalently, if C_{G}(P) = Z(P) × C′_{G}(P) for some (unique) subgroup C′_{G}(P) of order prime to p. Let L_{S}^{c}(G) (the centric linking system of G) be the category whose objects are the subgroups of S which are p-centric in G, and where for each pair of objects P, Q:

2000 Mathematics Subject Classification. Primary 55R35. Secondary 20J05, 20N99, 20D20, 20D05.

Key words and phrases. Finite groups, fusion, fusion systems, derived functors, classifying spaces, F F -offenders.

B. Oliver was partially supported by the DNRF through a visiting professorship at the Centre for Symmetry and Deformation in Copenhagen; and also by UMR 7539 of the CNRS and by project ANR BLAN08-2 338236, HGRT.

Mor_{L_{S}^{c}(G)}(P, Q) = {g ∈ G |^{g}P ≤ Q}/C′_{G}(P) .

Such categories were originally defined by Puig in [P1].

To explain the significance of linking systems from a topologist’s point of view, we must first define the geometric realization of an arbitrary small category C. This is a space |C| built up of one vertex (point) for each object in C, one edge for each nonidentity morphism (with endpoints attached to the vertices corresponding to its source and target), one 2-simplex (triangle) for each commutative triangle in L, etc. (See, e.g., [AKO, § III.2.1–2] for more details.) By a theorem of Broto, Levi, and Oliver [BLO1, Proposition 1.1], for any G and S as above, the space |L_{S}^{c}(G)|, after p-completion in the sense of Bousfield and Kan, is homotopy equivalent to the p-completed classifying space BG_{p}^{∧} of G. Furthermore, many of the homotopy theoretic properties of the space BG_{p}^{∧}, such as its self homotopy equivalences, can be determined combinatorially by the properties (such as automorphisms) of the finite category L_{S}^{c}(G) [BLO1, Theorems B & C].

Abstract centric linking systems associated to a fusion system were defined in [BLO2] (see Definition 1.3). One of the motivations in [BLO2] for defining these categories was that it provides a way to associate a classifying space to a saturated fusion system. More precisely, if L is a centric linking system associated to a saturated fusion system F, then we regard the p-completion |L|_{p}^{∧} of its geometric realization as a classifying space for F. This is motivated by the equivalence |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧} noted above. To give one example of the role played by these classifying spaces, if L′ is another centric linking system, associated to a fusion system F′, and the classifying spaces |L|_{p}^{∧} and |L′|_{p}^{∧} are homotopy equivalent, then L ≅ L′ and F ≅ F′. We refer to [BLO2, Theorem A] for more details and discussion.

It is unclear from the definition whether there is a centric linking system associated to any given saturated fusion system, and if so, whether it is unique. Even when working with fusion systems of finite groups, which always have a canonical associated linking system, there is no simple reason why two groups with isomorphic fusion systems need have isomorphic linking systems, and hence equivalent p-completed classifying spaces. This question — whether F_{S}(G) ≅ F_{T} (H) implies L_{S}^{c}(G) ≅ L_{T}^{c} (H) and hence BG_{p}^{∧} ≃ BH_{p}^{∧} — was originally posed by Martino and Priddy, and was what first got this author interested in the subject.

The main theorem of Chermak described in this paper is the following.

Theorem A (Chermak [Ch2]). Each saturated fusion system has an associated centric linking system, which is unique up to isomorphism.

Proof. This follows immediately from Theorem 3.4 in this paper, together with [BLO2, Proposition 3.1]. □

In particular, this provides a new proof of the Martino-Priddy conjecture, which was originally proven in [O1, O2] using the classification of finite simple groups. Chermak’s theorem is much more general, but it also (indirectly) uses the classification in its proof.

Theorem A is proven by Chermak by directly and systematically constructing the linking system, and by directly constructing an isomorphism between two given linking systems. The proof given here follows the same basic outline, but uses as its main tool the obstruction theory which had been developed in [BLO2, Proposition 3.1] for dealing with this problem. So if this approach is shorter, it is only because we are able to profit by the results of [BLO2, § 3], and also by other techniques which have been developed more recently for computing these obstruction groups.

By [BLO3, Proposition 4.6], there is a bijective correspondence between centric linking systems associated to a given saturated fusion system F up to isomorphism, and homotopy classes of rigidifications of the homotopy functor O(F^{c}) ⟶ hoTop which sends P to BP. (See Definition 1.5 for the definition of O(F^{c}).) Furthermore, if L corresponds to a rigidification B̃, then |L| is homotopy equivalent to the homotopy direct limit of B̃. Thus another consequence of Theorem A is:

Theorem B. For each saturated fusion system F, there is a functor

B̃: O(F^{c}) ⟶ Top,

together with a choice of homotopy equivalences B̃(P) ≃ BP for each object P, such that for each [φ] ∈ Mor_{O(F^{c})}(P, Q), the composite

BP ≃ B̃(P) ⟶^{B̃([φ])} B̃(Q) ≃ BQ

is homotopic to Bφ. Furthermore, B̃ is unique up to homotopy equivalence of functors, and (hocolim(B̃))_{p}^{∧} is the (unique) classifying space for F.

We also want to compare “outer automorphism groups” of fusion systems, linking systems, and their classifying spaces. When F is a saturated fusion system over a p-group S, set

Aut(S, F) = {α ∈ Aut(S) |^{α}F = F} and Out(S, F) = Aut(S, F)/Aut_{F}(S) .

Here, for α ∈ Aut(S), ^{α}F is the fusion system over S for which Hom_{^{α}F}(P, Q) = α ∘Hom_{F}(α^{−1}(P), α^{−1}(Q)) ∘α^{−1}. Thus Aut(S, F) is the group of “fusion preserving” automorphisms of S.

When L is a centric linking system associated to F, then for each object P of L, there is a “distinguished monomorphism” δ_{P} : P ⟶ Aut_{L}(P) (Definition 1.3). An automorphism α of L (a bijective functor from L to itself) is called isotypical if it permutes the images of the distinguished monomorphisms; i.e., if α(δ_{P} (P)) = δ_{α(P)}(α(P)) for each P. We denote by Out_{typ}(L) the group of isotypical automorphisms of L modulo natural transformations of functors. See also [AOV, § 2.2] or [AKO, Lemma III.4.9] for an alternative description of this group.

By [BLO2, Theorem D], Out_{typ}(L) ≅ Out(|L|_{p}^{∧}), where Out(|L|_{p}^{∧}) is the group of homotopy classes of self homotopy equivalences of the space |L|_{p}^{∧}. This is one reason for the importance of this particular group of (outer) automorphisms of L. Another reason is the role played by Out_{typ}(L) in the definition of a tame fusion system in [AOV, § 2.2].

The other main consequence of the results in this paper is the following.

Theorem C. For each saturated fusion system F over a p-group S with associated centric linking system L, the natural homomorphism

Out_{typ}(L) ⟶^{μ_{L}} Out(S, F)

induced by restriction to δ_{S}(S) ≅ S is surjective, and is an isomorphism if p is odd.

Proof. By [AKO, III.5.12], Ker(μ_{L}) ≅ \underleftarrow{lim}^{1}(Z_{F}), and μ_{L} is onto whenever \underleftarrow{lim}^{2}(Z_{F}) = 0. (This was shown in [BLO1, Theorem E] when L is the linking system of a finite group.) So the result follows from Theorem 3.4 in this paper. □

I would like to thank Assaf Libman for very carefully reading this manuscript, pointing out a couple gaps in the arguments, and making other suggestions for improving it. I also thank the three referees who read very carefully different parts of the paper and made detailed suggestions for simplifying or clarifying several of the arguments. And, of course, I very much want to thank Andy Chermak for solving this problem, which has taken up so much of my time for the past ten years.

1. Notation and background

We first briefly recall the definitions of saturated fusion systems and centric linking systems. For any group G and any pair of subgroups H, K ≤ G, set

Hom_{G}(H, K) = {c_{g} = (x ↦ gxg^{−1}) | g ∈ G, ^{g}H ≤ K} ⊆ Hom(H, K).

A fusion system F over a finite p-group S is a category whose objects are the subgroups of S, and whose morphism sets Hom_{F}(P, Q) satisfy the following two conditions:

• Hom_{S}(P, Q) ⊆ Hom_{F}(P, Q) ⊆ Inj(P, Q) for all P, Q ≤ S.

• For each φ ∈ Hom_{F}(P, Q), φ^{−1} ∈ Hom_{F}(φ(P), P).

Two subgroups P, P′ ≤ S are called F-conjugate if they are isomorphic in the category F. Let P^{F} denote the set of subgroups F-conjugate to P.

The following is the definition of a saturated fusion system first formulated in [BLO2]. Other (equivalent) definitions, including the original one by Puig, are discussed and compared in [AKO, §§ I.2 & I.9].

Definition 1.1. Let F be a fusion system over a p-group S.

• A subgroup P ≤ S is fully centralized in F if |C_{S}(P)| ≥ |C_{S}(Q)| for all Q ∈ P^{F}.

• A subgroup P ≤ S is fully normalized in F if |N_{S}(P)| ≥ |N_{S}(Q)| for all Q ∈ P^{F}.

• A subgroup P ≤ S is F-centric if C_{S}(Q) ≤ Q for all Q ∈ P^{F}.

• The fusion system F is saturated if the following two conditions hold:

(I) For all P ≤ S which is fully normalized in F, P is fully centralized in F and Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)).

(II) If P ≤ S and φ ∈ Hom_{F}(P, S) are such that φ(P) is fully centralized, and if we set

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

then there is φ ∈ Hom_{F}(N_{φ}, S) such that φ|_{P} = φ.

The following technical result will be needed later.

Lemma 1.2 ([AKO, Lemma I.2.6(c)]). Let F be a saturated fusion system over a finite p-group S. Then for each P ≤ S, and each Q ∈ P^{F} which is fully normalized in F, there is φ ∈ Hom_{F}(N_{S}(P), S) such that φ(P) = Q.

For any fusion system F over S, let F^{c} ⊆ F be the full subcategory whose objects are the F-centric subgroups of S, and also let F^{c} denote the set of F-centric subgroups of S.

Definition 1.3 ([BLO2]). Let F be a fusion system over the p-group S. A centric linking system associated to F is a category L with Ob(L) = F^{c}, together with a functor π: L ⟶ F^{c} and distinguished monomorphisms P ⟶^{δ_{P}} Aut_{L}(P) for each P ∈ Ob(L), which satisfy the following conditions.

(A) π is the identity on objects and is surjective on morphisms. For each P, Q ∈ F^{c}, δ_{P} (Z(P)) acts freely on Mor_{L}(P, Q) by composition, and π induces a bijection

Mor_{L}(P, Q)/δ_{P} (Z(P)) ⟶^{≅} Hom_{F}(P, Q).

(B) For each g ∈ P ∈ F^{c}, π sends δ_{P} (g) ∈ Aut_{L}(P) to c_{g} ∈ Aut_{F}(P).

(C) For each P, Q ∈ F^{c}, ψ ∈ Mor_{L}(P, Q), and g ∈ P, ψ ∘ δ_{P} (g) = δ_{Q}(π(ψ)(g)) ∘ ψ in Mor_{L}(P, Q).

We next fix some notation for sets of subgroups of a given group. For any group G, let 𝒮 (G) be the set of subgroups of G. If H ≤ G is any subgroup, set

𝒮 (G)_{≥H} = {K ∈ 𝒮 (G) | K ≥ H}.

Definition 1.4. Let F be a saturated fusion system over a finite p-group S. An interval of subgroups of S is a subset R ⊆ 𝒮 (S) such that P < Q < R and P, R ∈ R imply Q ∈ R. An interval is F-invariant if it is invariant under F-conjugacy.

Thus, for example, an F-invariant interval R ⊆ 𝒮 (S) is closed under overgroups if and only if S ∈ R. Each F-invariant interval has the form R∖R_{0} for some pair of F-invariant intervals R_{0} ⊆ R which are closed under overgroups.

We next recall the obstruction theory to the existence and uniqueness of linking systems.

Definition 1.5. Let F be a saturated fusion system over a finite p-group S.

(a) Let O(F^{c}) be the centric orbit category of F: Ob(O(F^{c})) = Ob(F^{c}), and

Mor_{O(F^{c})}(P, Q) = Inn(Q)\Hom_{F}(P, Q).

(b) Let Z_{F}: O(F^{c})^{op} ⟶ Ab be the functor which sends P to Z(P) = C_{S}(P). If φ ∈ Hom_{F}(P, Q), and [φ] denotes its class in Mor(O(F^{c})), then Z_{F}([φ]) = φ^{−1} as a homomorphism from Z(Q) = C_{S}(Q) to Z(P) = C_{S}(P).

(c) For any F-invariant interval R ⊆ F^{c}, let Z_{F}^{R} be the subquotient functor of Z_{F} where Z_{F}^{R}(P) = Z(P) if P ∈ R and Z_{F}^{R}(P) = 0 otherwise.

(d) For each F-invariant interval R ⊆ F^{c}, we write for short

L^{∗}(F; R) = \underleftarrow{lim}^{∗}_{O(F^{c})}(Z_{F}^{R}) ;

i.e., the higher derived functors of the inverse limit of Z_{F}^{R}.

We refer to [AKO, § III.5.1] for more discussion of the functors \underleftarrow{lim}^{∗}(−).

Thus Z_{F} = Z_{F}^{F^{c}}, and \underleftarrow{lim}^{∗}(Z_{F}) = L^{∗}(F; F^{c}). By [BLO2, Proposition 3.1], the obstruction to the existence of a centric linking system associated to F lies in L^{3}(F; F^{c}), and the obstruction to uniqueness lies in L^{2}(F; F^{c}).

For any F and any F-invariant interval R, Z_{F}^{R} is a quotient functor of Z_{F} if S ∈ R (if R is closed under overgroups). If R_{0} ⊆ R are both F-invariant intervals, and P ∈ R_{0} and Q ∈ R∖R_{0} implies P ≱ Q, then Z_{F}^{R_{0}} is a subfunctor of Z_{F}^{R}.

Lemma 1.6. Fix a finite group Γ with Sylow subgroup S ∈ Syl_{p}(Γ), and set F = F_{S}(Γ). Let Q ⊆ F^{c} be an F-invariant interval such that S ∈ Q (i.e., Q is closed under overgroups).

(a) Let F: O(F^{c})^{op} ⟶ Ab be a functor such that F (P) = 0 for each P ∈ F^{c}∖Q. Let O(F_{Q}) ⊆ O(F^{c}) be the full subcategory with object set Q. Then

\underleftarrow{lim}^{∗}_{O(F^{c})}(F) ≅ \underleftarrow{lim}^{∗}_{O(F_{Q})}(F |_{O(F_{Q})}) .

(b) Assume Q = 𝒮 (S)_{≥Y} for some p-subgroup Y ⊴ Γ such that C_{Γ}(Y) ≤ Y. Then

L^{k}(F; Q) =^{def} \underleftarrow{lim}^{k}_{O(F^{c})}(Z_{F}^{Q}) ≅ \begin{cases}Z(Γ) & if k = 0 \\ 0 & if k > 0. \end{cases}

Proof. (a) Set C = O(F^{c}) and C_{0} = O(F_{Q}) for short. There is no morphism in C from any object of C_{0} to any object not in C_{0}. Hence for any functor F: C^{op} ⟶ Ab such that F (P) = 0 for each P ∉ Ob(C_{0}), the two chain complexes C^{∗}(C; F) and C^{∗}(C_{0}; F |_{C_{0}}) are isomorphic (see, e.g., [AKO, § III.5.1]). So \underleftarrow{lim}^{∗}(F) ≅ \underleftarrow{lim}^{∗}(F |_{C_{0}}) in this situation, and this proves (a). Alternatively, (a) follows upon showing that any C_{0}-injective resolution of F |_{C_{0}} can be extended to an C-injective resolution of F by assigning to all functors the value zero on objects not in C_{0}.

(b) To simplify notation, set H = H/Y for each H ∈ 𝒮 (Γ)_{≥Y}, and g = gY ∈ Γ for each g ∈ Γ. Let O_{S}(Γ) be the “orbit category” of Γ: the category whose objects are the subgroups of S, and where for P, Q ∈ Q,

Mor_{O_{S}(Γ)}(P , Q) = Q\{g ∈ Γ |^{g}P ≤ Q} .

There is an isomorphism of categories Ψ: O(F_{Q}) ⟶^{≅} O_{S}(Γ) which sends P ∈ Q to P = P/Y and sends [c_{g}] ∈ Mor_{O(F_{Q})}(P, Q) to Qg. Then Z_{F}^{Q} ∘ Ψ^{−1} sends P to Z(P) = C_{Z(Y)}(P). Hence for k ≥ 0,

\underleftarrow{lim}^{k}_{O(F^{c})}(Z_{F}^{Q}) ≅ \underleftarrow{lim}^{k}_{O(F_{Q})}(Z_{F}^{Q}|_{O(F_{Q})}) ≅ \underleftarrow{lim}^{k}_{O_{S}(Γ)}(Z_{F}^{Q} ∘ Ψ^{−1}) ≅ \begin{cases}C_{Z(Y)}(Γ) = Z(Γ) & if k = 0 \\ 0 & if k > 0 \end{cases}

where the first isomorphism holds by (a), and the last by a theorem of Jackowski and McClure [JM, Proposition 5.14]. We refer to [JMO, Proposition 5.2] for more details on the last isomorphism. □

More tools for working with these groups come from the long exact sequence of derived functors induced by a short exact sequence of functors.

Lemma 1.7. Let F be a saturated fusion system over a finite p-group S. Let Q and R be F-invariant intervals such that

(i) Q ∩ R = ∅,

(ii) Q ∪ R is an interval, and

(iii) Q ∈ Q, R ∈ R implies Q ≰ R.

Then Z_{F}^{R} is a subfunctor of Z_{F}^{Q∪R}, Z_{F}^{Q∪R}/Z_{F}^{R} ≅ Z_{F}^{Q}, and there is a long exact sequence

0 ⟶ L^{0}(F; R) ⟶ L^{0}(F; Q∪R) ⟶ L^{0}(F; Q) ⟶ ⋯

⟶ L^{k−1}(F; Q) ⟶ L^{k}(F; R) ⟶ L^{k}(F; Q∪R) ⟶ L^{k}(F; Q) ⟶ · · · .

In particular, the following hold.

(a) If L^{k}(F; R) ≅ L^{k}(F; Q) = 0 for some k ≥ 0, then L^{k}(F; Q ∪ R) = 0.

(b) Assume F = F_{S}(Γ), where S ∈ Syl_{p}(Γ), and there is a normal p-subgroup Y ⊴ Γ such that C_{Γ}(Y) ≤ Y and Q ∪ R = 𝒮 (S)_{≥Y}. Then for each k ≥ 2, L^{k−1}(F; Q) ≅ L^{k}(F; R). Also, there is a short exact sequence

1 ⟶ C_{Z(Y)}(Γ) ⟶ C_{Z(Y)}(Γ^{∗}) ⟶ L^{1}(F; R) ⟶ 1,

where Γ^{∗} = ⟨g ∈ Γ |^{g}P ∈ Q for some P ∈ Q⟩.

Proof. Condition (iii) implies that Z_{F}^{R} is a subfunctor of Z_{F}^{Q∪R}, and it is then immediate from the definitions (and (i) and (ii)) that Z_{F}^{Q∪R}/Z_{F}^{R} ≅ Z_{F}^{Q}. The long exact sequence is induced by this short exact sequence of functors and the snake lemma. Point (a) now follows immediately.

Under the hypotheses in (b), by Lemma 1.6(b), L^{k}(F; Q ∪ R) = 0 for k > 0 and L^{0}(F; Q ∪ R) ≅ Z(Γ) = C_{Z(Y)}(Γ). The first statement in (b) thus follows immediately from the long exact sequence, and the second since L^{0}(F; Q) ≅ C_{Z(Y)}(Γ^{∗}) (by definition of inverse limits). □

We next consider some tools for making computations in the groups \underleftarrow{lim}^{∗}(−) for functors on orbit categories.

Definition 1.8. Fix a finite group G and a ℤ[G]-module M. Let O_{p}(G) be the category whose objects are the p-subgroups of G, and where Mor_{O_{p}(G)}(P, Q) = Q\{g ∈ G |^{g}P ≤ Q}. Define a functor F_{M}: O_{p}(G)^{op} ⟶ Ab by setting

F_{M}(P) = \begin{cases}M & if P = 1 \\ 0 & if P ≠ 1. \end{cases}

Here, F_{M}(1) = M has the given action of Aut_{O_{p}(G)}(1) = G. Set

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

These groups Λ^{∗}(G; M) provide a means of computing higher limits of functors on orbit categories which vanish except on one conjugacy class.

Proposition 1.9 ([BLO2, Proposition 3.2]). Let F be a saturated fusion system over a p-group S. Let

F: O(F^{c})^{op} ⟶ ℤ_{(p)}-mod

be any functor which vanishes except on the isomorphism class of some subgroup Q ∈ F^{c}. Then

\underleftarrow{lim}^{∗}_{O(F^{c})}(F) ≅ Λ^{∗}(Out_{F}(Q); F (Q)).

Upon combining Proposition 1.9 with the exact sequences of Lemma 1.7, we get the following corollary.

Corollary 1.10. Let F be a saturated fusion system over a p-group S, and let R ⊆ F^{c} be an F-invariant interval. Assume, for some k ≥ 0, that Λ^{k}(Out_{F}(P); Z(P)) = 0 for each P ∈ R. Then L^{k}(F; R) = 0.

What makes these groups Λ^{∗}(−; −) so useful is that they vanish in many cases, as described by the following proposition.

Proposition 1.11 ([JMO, Proposition 6.1(i,ii,iii,iv)]). The following hold for each finite group G and each ℤ_{(p)}[G]-module M.

(a) If p∤|G|, then Λ^{i}(G; M) = \begin{cases}M^{G} & if i = 0 \\ 0 & if i > 0. \end{cases}

(b) Let H = C_{G}(M) be the kernel of the G-action on M. Then Λ^{∗}(G; M) ≅ Λ^{∗}(G/H; M) if p∤|H|, and Λ^{∗}(G; M) = 0 if p||H|.

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

(d) If M_{0} ≤ M is a ℤ_{(p)}[G]-submodule, then there is an exact sequence

0 ⟶ Λ^{0}(G; M_{0}) ⟶ Λ^{0}(G; M) ⟶ Λ^{0}(G; M/M_{0}) ⟶ ⋯

⋯ ⟶ Λ^{n−1}(G; M/M_{0}) ⟶ Λ^{n}(G; M_{0}) ⟶ Λ^{n}(G; M) ⟶ · · · .

The next lemma allows us in certain cases to replace the orbit category for one fusion system by that for a smaller one. For any saturated fusion system F over S and any Q ≤ S, the normalizer fusion system N_{F}(Q) is defined as a fusion system over N_{S}(Q) (cf. [AKO, Definition I.5.3]). If Q is fully normalized, then N_{F}(Q) is always saturated (cf. [AKO, Theorem I.5.5]).

Lemma 1.12. Let F be a saturated fusion system over a p-group S, fix a subgroup Q ∈ F^{c} which is fully normalized in F, and set E = N_{F}(Q). Set E^{•} = F^{c} ∩ E^{c}, a full subcategory of E^{c}, and let O(E^{•}) ⊆ O(E^{c}) be its orbit category. Define

T = {P ≤ S | Q ⊴ P, and R ∈ Q^{F}, R ⊴ P implies R = Q} .

Let F: O(F^{c})^{op} ⟶ ℤ_{(p)}-mod be any functor which vanishes except on subgroups F-conjugate to subgroups in T , set F_{0} = F |_{O(E^{•})}, and let F_{1} : O(E^{c})^{op} ⟶ Ab be such that F_{1}|_{O(E^{•})} = F_{0} and F_{1}(P) = 0 for all P ∈ E^{c}∖E^{•}. Then restriction to E^{•} induces isomorphisms

\underleftarrow{lim}^{∗}_{O(F^{c})}(F) ⟶_{≅}^{R} \underleftarrow{lim}^{∗}_{O(E^{•})}(F_{0}) ⟵_{≅}^{R_{1}} \underleftarrow{lim}^{∗}_{O(E^{c})}(F_{1}) .  (1)

Proof. Since R_{1} is an isomorphism by Lemma 1.6(a), we only need to show that R is an isomorphism. If F′ ⊆ F is a pair of functors from O(F^{c})^{op} to ℤ_{(p)}-mod, and the lemma holds for F′ and for F/F′, then it also holds for F by the 5-lemma (and since R is natural with respect to functors on O(F^{c})^{op}). It thus suffices to prove that R is an isomorphism when F vanishes except on the F-conjugacy class of one subgroup in T .

Fix P ∈ T , and assume F (R) = 0 for all R ∉ P^{F}. Then Q ⊴ P by definition of T . If φ ∈ Hom_{F}(P, S) is such that Q ⊴ φ(P), then φ^{−1}(Q) ⊴ P, and φ(Q) = Q since P ∈ T . Thus Out_{F}(P) = Out_{E}(P), and

P^{E} = {R ∈ P^{F} | Q ⊴ R} .  (2)

Since P^{E} ⊆ T , we can assume P was chosen to be fully normalized in E.

Let F_{2} ⊆ F_{1} be the subfunctor on O(E^{c}) defined by setting F_{2}(R) = F_{1}(R) if R ≱ Q, and F_{2}(R) = 0 if R ≥ Q. Then P^{E} contains all subgroups in P^{F} (hence all objects in E^{c}) on which F_{1}/F_{2} is nonvanishing. If R ∈ E^{c} and R ≱ Q, then O_{p}(Out_{E}(R)) ≠ 1 (R is not E-radical) since Q ⊴ E and R ∈ E^{c} (cf. [AKO, Proposition I.4.5(b)]), so Λ^{∗}(Out_{E}(R); F_{1}(R)) = 0 by Proposition 1.11(c). Thus \underleftarrow{lim}^{∗}(F_{2}) = 0 by Corollary 1.10. Set F_{2}^{•} = F_{2}|_{O(E^{•})} ⊆ F_{0}; then \underleftarrow{lim}^{∗}(F_{2}^{•}) ≅ \underleftarrow{lim}^{∗}(F_{2}) = 0 by Lemma 1.6(a), so \underleftarrow{lim}^{∗}(F_{0}) ≅ \underleftarrow{lim}^{∗}(F_{0}/F_{2}^{•}). By (2), (F_{0}/F_{2}^{•})(R) = 0 for all R ∈ Ob(E^{•})∖P^{E}.

This yields the following diagram:

\xymatrix{\underleftarrow{lim}^{∗}_{O(F^{c})}(F) \ar[r]^{R} \ar[drr]_{≅}^{Φ^{∗}} & \underleftarrow{lim}^{∗}_{O(E^{•})}(F_{0}) \ar[rr]^{R_{2}}_{≅} &  & \underleftarrow{lim}^{∗}_{O(E^{•})}(F_{0}/F_{2}^{•}) & \underleftarrow{lim}^{∗}_{O(E^{c})}(F_{1}/F_{2}) \ar[l]_{R_{3}}^{≅} \ar@{-->}[dll]_{Φ_{1}^{∗}}^{≅} \\  &  & Λ^{∗}(Out_{F}(P); F (P))}

where Φ^{∗} and Φ_{1}^{∗} are the isomorphisms of Proposition 1.9, where R_{2} is induced by (F_{0} ↠ F_{0}/F_{2}^{•}), and where R_{3} is induced by restriction (and is an isomorphism by Lemma 1.6(a)).

Let O_{Out_{S}(P)}(Out_{F}(P)) ⊆ O_{p}(Out_{F}(P)) be the full subcategory whose objects are the subgroups of Out_{S}(P) ∈ Syl_{p}(Out_{F}(P)). (Recall that P is fully normalized in E and Out_{F}(P) = Out_{E}(P).) By the proof of [BLO2, Proposition 3.2], Φ^{∗} and Φ_{1}^{∗} are both induced by restriction via an embedding of O_{Out_{S}(P)}(Out_{F}(P)) into O(E^{•}): the embedding which sends Out_{R}(P) to R (for P ≤ R ≤ N_{S}(P)), and sends a morphism (the coset of some γ ∈ Out_{F}(P)) to the class of the appropriate extension of γ. Hence the above diagram commutes, and R is an isomorphism. □

The following lemma can also be stated and proven as a result about extending automorphisms from a linking system to a group [Ch2, Lemma 4.15].

Lemma 1.13 ([Ch2, 4.15]). Fix a pair of finite groups H ⊴ G, together with S ∈ Syl_{p}(G) and T = S ∩ H ∈ Syl_{p}(H). Set F = F_{S}(G) and E = F_{T} (H). Assume Y ≤ T is such that Y ⊴ G and C_{G}(Y) ≤ Y. Let Q be an F-invariant interval in 𝒮 (S)_{≥Y} such that S ∈ Q, and such that Q ∈ Q implies H ∩ Q ∈ Q. Set Q_{0} = {Q ∈ Q | Q ≤ H}. Then restriction induces an injective homomorphism

L^{1}(F; Q) ⟶^{R} L^{1}(E; Q_{0}).

Proof. Since E^{c} need not be contained in F^{c}, we must first check that there is a well defined “restriction” homomorphism. Set E^{•} = E^{c} ∩ F^{c}: a full subcategory of E^{c}. Since the functor Z_{E}^{Q_{0}} vanishes on all subgroups in E^{c} not in Q_{0} ⊆ E^{•}, the higher limits are the same whether taken over O(E^{•}) or O(E^{c}) (Lemma 1.6(a)). Thus R is defined as the restriction map to \underleftarrow{lim}^{1}(Z_{E}^{Q_{0}}|_{E^{•}}) ≅ L^{1}(E; Q_{0}).

We work with the bar resolutions for O(F^{c}) and O(E^{•}), using the notation of [AKO, § III.5.1]. Fix a cocycle η ∈ Z^{1}(O(F^{c}); Z_{F}^{Q}) such that [η] ∈ Ker(R). Thus η is a function from Mor(O(F^{c})) to Z(Y) which sends the class [φ] of φ ∈ Hom_{G}(P, Q) to an element of Z(P) if P ∈ Q, and to 1 if P ∉ Q. We can assume, after adding an appropriate coboundary, that η(Mor(O(E^{•}))) = 1.

Define η̂ ∈ Z^{1}(N_{G}(T)/T; Z(T)) to be the restriction of η to Aut_{O(F^{c})}(T) = N_{G}(T)/T. For g ∈ N_{G}(T), let ḡ be its class in N_{G}(T)/T. Set γ = η([incl_{T}^{S}]) ∈ Z(T), so dγ ∈ Z^{1}(N_{G}(T)/T; Z(T)) is the cocycle dγ(ḡ) = γ^{g}·γ^{−1}. For each g ∈ S, [incl_{T}^{S}]∘[c_{g}] = [incl_{T}^{S}] in O(F^{c}), so γ^{g}·η([c_{g}]) = γ, and thus η̂(ḡ) = η([c_{g}]) = (dγ(ḡ))^{−1}. In other words, η̂|_{S/T} is a coboundary, and since S/T ∈ Syl_{p}(N_{G}(T)/T), [η̂] = 1 ∈ H^{1}(N_{G}(T)/T; Z(T)) (cf. [CE, Theorem XII.10.1]). Hence there is β ∈ Z(T) such that η̂ = dβ. Since η([c_{h}]) = 1 for all h ∈ N_{H}(T), [β, h] = 1 for all h ∈ N_{H}(T), and thus β ∈ Z(N_{H}(T)).

Let G^{∗} ≤ G be the subgroup generated by all g ∈ G such that for some Q ∈ Q, gQ ∈ Q. Define H^{∗} ≤ H similarly. Since S ≤ N_{G}(T) ≤ G^{∗} and N_{H}(T) ≤ H^{∗}, S ∈ Syl_{p}(G^{∗}), T ∈ Syl_{p}(H^{∗}), and HG^{∗} ≥ HN_{G}(T) = G by the Frattini argument (Lemma 1.14(b)). If g = ha where h ∈ H, a ∈ N_{G}(T), and ^{g}Q ∈ Q for some Q ∈ Q, then ^{a}(Q∩H) and ^{g}(Q∩H) = ^{g}Q∩H are both in Q_{0}, and thus h ∈ H^{∗}. Since N_{G}(T) normalizes H^{∗}, this shows that G^{∗} = H^{∗}N_{G}(T). So G^{∗} ∩ H = (H^{∗}N_{G}(T)) ∩ H = H^{∗}N_{H}(T) = H^{∗}. In particular, H^{∗} ⊴ G^{∗} and G^{∗}/H^{∗} ≅ G/H.

For each φ ∈ Hom_{H}(P, Q) (where Y ≤ P, Q ≤ T), and each g ∈ N_{G}(T), set gφ = c_{g}φc_{g}^{−1} ∈ Hom_{H}(^{g}P,^{g}Q). Since η([φ]) = η([^{g}φ]) = 1, we have φ^{−1}(η̂(ḡ)) = η̂(ḡ). Thus for each g ∈ N_{G}(T), η̂(ḡ) = β^{g}β^{−1} is invariant under the action of H^{∗}; i.e., β^{g}β^{−1} ∈ Z(H^{∗}). So the class [β] ∈ Z(N_{H}(T))/Z(H^{∗}) is fixed under the action of N_{G}(T) on this quotient.

Since p∤[H^{∗}:N_{H}(T)], and since N_{G}(T) normalizes H^{∗} and N_{H}(T), the inclusion of Z(H^{∗}) = C_{Z(Y)}(H^{∗}) into Z(N_{H}(T)) = C_{Z(Y)}(N_{H}(T)) is N_{G}(T)-equivariantly split by the trace homomorphism for the actions of H^{∗} ≥ N_{H}(T) on Z(Y). So the fixed subgroup for the N_{G}(T)-action on the quotient group Z(N_{H}(T))/Z(H^{∗}) is Z(N_{G}(T))/Z(G^{∗}). Thus β ∈ Z(N_{G}(T))Z(H^{∗}), and we can assume β ∈ Z(H^{∗}) without changing dβ = η̂.

Define a 0-cochain β̂ ∈ C^{0}(O(F^{c}); Z_{F}^{Q}) by setting β̂(P) = β if P ∈ Q_{0} and β̂(P) = 1 otherwise. Then η([φ]) = dβ̂([φ]) for all φ ∈ Mor(E^{•}) (since both vanish) and also for all φ ∈ Aut_{G}(T). Since G = HN_{G}(T), each morphism in F between subgroups of T is the composite of a morphism in E and the restriction of a morphism in Aut_{F}(T). Hence η([φ]) = dβ̂([φ]) for all such morphisms φ (since η and dβ̂ are both cocycles). Upon replacing η by η(dβ̂)^{−1}, we can assume η vanishes on all morphisms in F between subgroups of T.

For each P ∈ Q, set P_{0} = P ∩ T and let i_{P} ∈ Hom_{G}(P_{0}, P) be the inclusion. Then η([i_{P}]) ∈ Z(P_{0}) (and η([i_{P}]) = 1 if P ∉ Q). For each g ∈ P, the relation [i_{P}] = [i_{P}] ∘ [c_{g}] in O(F^{c}) (where [c_{g}] ∈ Aut_{O(F^{c})}(P_{0})) implies that η([i_{P}]) is c_{g}- invariant. Thus η([i_{P}]) ∈ Z(P). Let ρ ∈ C^{0}(O(F^{c}); Z_{F}^{Q}) be the 0-cochain ρ(P) = η([i_{P}]) when P ∈ Q and ρ(P) = 1 if P ∈ F^{c}∖Q. Thus ρ(P) = 1 if P ≤ T by the initial assumptions on η. Then dρ([i_{P}]) = η([i_{P}]) for each P, and dρ(φ) = 1 = η(φ) for each φ between subgroups of T. For each φ ∈ Hom_{G}(P, Q) in F^{c}, let φ_{0} ∈ Hom_{G}(P_{0}, Q_{0}) be its restriction; the relation [φ] ∘ [i_{Q}] = [i_{P}] ∘ [φ_{0}] in O(F^{c}) implies that η([φ]) = dρ([φ]) since this holds for [φ_{0}] and the inclusions. Thus η = dρ, and so [η] = 1 in L^{1}(F; Q). □

We end the section by recalling a few elementary results about finite groups.

Lemma 1.14. (a) If Q > P are p-groups for some prime p, then N_{Q}(P) > P.

(b) (Frattini argument) If H ⊴ G are finite groups and T ∈ Syl_{p}(H), then G = HN_{G}(T).

Proof. See, for example, [Sz1, Theorems 2.1.6 & 2.2.7]. □

As usual, for any finite p-group P, Ω_{1}(P) = ⟨g ∈ P | g^{p} = 1⟩.

Lemma 1.15. Let G be a finite group such that O_{p}(G) = 1, and assume G acts faithfully on an abelian p-group D. Then G acts faithfully on Ω_{1}(D).

Proof. The subgroup C_{G}(Ω_{1}(D)) is a normal p-subgroup of G (cf. [G, Theorem 5.2.4]), and hence is contained in O_{p}(G) = 1. □

2. The Thompson subgroup and offenders

The proof of the main theorem is centered around the Thompson subgroup of a p-group, and the F F -offenders for an action of a group on an abelian p-group. We first fix the terminology and notation which will be used.

Definition 2.1. (a) For any p-group S, set d(S) = sup{|A| | A ≤ S abelian}, let A(S) be the set of all abelian subgroups of S of order d(S), and set J(S) = ⟨A(S)⟩.

(b) Let G be a finite group which acts faithfully on the abelian p-group D. A best offender in G on D is an abelian subgroup A ≤ G such that |A||C_{D}(A)| ≥ |B||C_{D}(B)| for each B ≤ A. (In particular, |A||C_{D}(A)| ≥ |D|.) Let A_{D}(G) be the set of best offenders in G on D, and set J_{D}(G) = ⟨A_{D}(G)⟩.

(c) Let Γ be a finite group, and let D ⊴ Γ be a normal abelian p-subgroup. Let J(Γ, D) ≤ Γ be the subgroup such that J(Γ, D)/C_{Γ}(D) = J_{D}(Γ/C_{Γ}(D)).

Note, in the situation of point (c) above, that

D ≤ C_{Γ}(D) ≤ J(Γ, D) ≤ Γ and J(J(Γ, D), D) = J(Γ, D).

The relation between the Thompson subgroup J(−) and best offenders is described by the next lemma and corollary.

Lemma 2.2. (a) Assume G acts faithfully on a finite abelian p-group D. If A is a best offender in G on D, and U is an A-invariant subgroup of D, then A/C_{A}(U) is a best offender in N_{G}(U)/C_{G}(U) on U.

(b) Let S be a finite p-group, let D ⊴ S be a normal abelian subgroup, and set G = S/C_{S}(D). Assume A ∈ A(S). Then the image of A in G is a best offender on D.

Proof. We give here the standard proofs.

(a) Set A = A/C_{A}(U) for short. For each B = B/C_{A}(U) ≤ A,

|C_{U}(B)||C_{D}(A)| = |C_{U}(B)C_{D}(A)||C_{U}(B) ∩ C_{D}(A)| ≤ |C_{D}(B)||C_{U}(A)|.

Also, |B||C_{D}(B)| ≤ |A||C_{D}(A)| since A is a best offender on D, and hence

|B||C_{U}(B)| = \frac{|B||C_{U}(B)|}{|C_{A}(U)|} ≤ \frac{|B||C_{D}(B)|}{|C_{D}(A)|}·\frac{|C_{U}(A)|}{|C_{A}(U)|} ≤ |A|·\frac{|C_{U}(A)|}{|C_{A}(U)|} = |A||C_{U}(A)|.

Thus A is a best offender on U.

(b) Set A = A/C_{A}(D), identified with the image of A in G. Fix some B = B/C_{A}(D) ≤ A, and set B^{∗} = C_{D}(B)B. This is an abelian group since D and B are abelian, and hence |B^{∗}| ≤ |A| since A ∈ A(S). Since B ∩ C_{D}(B) ≤ C_{D}(A),

|B||C_{D}(B)| = \frac{|B||C_{D}(B)|}{|C_{A}(D)|} = \frac{|B^{∗}||B ∩ C_{D}(B)|}{|C_{A}(D)|} ≤ \frac{|A||C_{D}(A)|}{|C_{A}(D)|} = |A||C_{D}(A)|.

Since this holds for all B ≤ A, A is a best offender on D. □

The following corollary reinterprets Lemma 2.2 in terms of the groups J(Γ, D) defined above.

Corollary 2.3. Let Γ be a finite group, and let D ⊴ Γ be a normal abelian psubgroup.

(a) If U ≤ D is also normal in Γ, then J(Γ, U) ≥ J(Γ, D).

(b) If Γ is a p-group, then J(Γ) ≤ J(Γ, D).

An action of a group G on a group D is quadratic if [G, [G, D]] = 1. If D is abelian and G acts faithfully, then a quadratic best offender in G on D is an abelian subgroup A ≤ G which is a best offender and whose action is quadratic.

Lemma 2.4. Let G be a finite group which acts faithfully on an elementary abelian p-group V. If the action of G on V is quadratic, then G is also an elementary abelian p-group.

Proof. We write V additively for convenience; thus [g, v] = gv − v for g ∈ G and v ∈ V. By an easy calculation, and since the action is quadratic, [gh, v] = [g, v] + [h, v] for each g, h ∈ G and v ∈ V. Thus g ↦ (v ↦ [g, v]) is a homomorphism from G to the additive group End(V), and is injective since the action is faithful. Since End(V) is an elementary abelian p-group, so is G. □

We will also need the following form of Timmesfeld’s replacement theorem.

Theorem 2.5. Let A ≠ 1 and V ≠ 1 be abelian p-groups. Assume A acts faithfully on V and is a best offender on V. Then there is 1 ≠ B ≤ A such that B is a quadratic best offender on V. More precisely, we can take B = C_{A}([A, V]) ≠ 1, in which case |A||C_{V} (A)| = |B||C_{V} (B)| and C_{V} (B) = [A, V] + C_{V} (A) < V.

Proof. We follow the proof given by Chermak in [Ch1, § 1]. Set m = |A||C_{V} (A)|. Since A is a best offender,

m = sup{|B||C_{U}(B)| | B ≤ A, U ≤ V }.  (1)

For each U ≤ V, consider the set

M_{U} = {B ≤ A | |B||C_{U}(B)| = m} .

By the maximality of m in (1),

B ∈ M_{U} ⟹ |C_{U}(B)| = |C_{V} (B)| ⟹ C_{V} (B) ≤ U .  (2)

Step 1: (Thompson’s replacement theorem) For each x ∈ V, set

V_{x} = [A, x] =^{def} ⟨[a, x] = ax − x | a ∈ A⟩ and A_{x} = C_{A}(V_{x}).

Note that V_{x} is A-invariant. We will show that

|A_{x}||C_{V} (A_{x})| = |A||C_{V} (A)| = m and C_{V} (A_{x}) = V_{x} + C_{V} (A).  (3)

Define Φ: A ⟶ V_{x} (a map of sets) by setting Φ(a) = [a, x] = ax − x for each a ∈ A. We first claim that Φ induces an injective map of sets

ϕ: A/A_{x} ⟶ V_{x}/C_{V_{x}}(A)

between these quotient groups. Since A is abelian, [a, [b, x]] = abx − bx − ax + x = [b, [a, x]] for all a, b ∈ A. Hence for all g, h ∈ A,

Φ(g) − Φ(h) =gx − hx ∈ C_{V_{x}}(A) ⟺ h([h^{−1}g, x]) ∈ C_{V_{x}}(A)
⟺ 1 = [A, [h^{−1}g, x]] = [h^{−1}g, [A, x]] = [h^{−1}g, V_{x}]
⟺ h^{−1}g ∈ C_{A}(V_{x}) = A_{x} .

Thus ϕ is well defined and injective.

Now,

|V_{x}||C_{V} (A)| = |C_{V_{x}}(A)||V_{x} + C_{V} (A)| ≤ |C_{V_{x}}(A)||C_{V} (A_{x})|,  (4)

since V_{x} ≤ C_{V} (A_{x}) by definition of A_{x}. Together with the injectivity of ϕ, this implies that

\frac{|A|}{|A_{x}|} ≤ \frac{|V_{x}|}{|C_{V_{x}}(A)|} ≤ \frac{|C_{V} (A_{x})|}{|C_{V} (A)|},

and so m = |A||C_{V} (A)| ≤ |A_{x}||C_{V} (A_{x})|. The opposite inequality holds by (1), so A_{x} ∈ M_{V} and the inequality in (4) is an equality. Thus |V_{x} + C_{V} (A)| = |C_{V} (A_{x})|, finishing the proof of (3).

Step 2: Assume, for some U ≤ V, that B_{0}, B_{1} ∈ M_{U}. Then m = |B_{0}||C_{U}(B_{0})| ≥ |B_{0}B_{1}||C_{U}(B_{0}B_{1})| by (1), and hence

\frac{|B_{1}|}{|B_{0} ∩ B_{1}|} = \frac{|B_{0}B_{1}|}{|B_{0}|} ≤ \frac{|C_{U}(B_{0})|}{|C_{U}(B_{0}B_{1})|} = \frac{|C_{U}(B_{0}) + C_{U}(B_{1})|}{|C_{U}(B_{1})|} ≤ \frac{|C_{U}(B_{0} ∩ B_{1})|}{|C_{U}(B_{1})|}.

So m = |B_{1}||C_{U}(B_{1})| ≤ |B_{0} ∩ B_{1}||C_{U}(B_{0} ∩ B_{1})| with equality by (1) again, and we conclude that B_{0} ∩ B_{1} ∈ M_{U}.

Step 3: Set B = C_{A}([A, V]) and U = [A, V] + C_{V} (A). For each x ∈ V, (3) implies that C_{V} (A_{x}) = [A, x] + C_{V} (A) ≤ U and A_{x} ∈ M_{U}. Hence B = ⋂_{x∈V} A_{x} ∈ M_{U} by Step 2, so B is a best offender on V, and is quadratic since [B, [A, V]] = 1 by definition. Also, C_{V} (B) ≤ U by (2). Since U = [A, V] + C_{V} (A) ≤ C_{V} (B) by definition, we conclude that U = C_{V} (B).

If U = V, then V = [A, V] ⊕ W is an A-invariant splitting for some W ≤ C_{V} (A). But this would imply [A, V] = [A, [A, V]] + [A, W] = [A, [A, V]], which is impossible since [A, X] < X for each finite nontrivial p-group X on which A acts. We conclude that U = C_{V} (B) < V, and hence that B ≠ 1. □

We finish the section with two lemmas which were suggested to us by one of the referees, and which will be very useful in Section 4.

Lemma 2.6. Let A ≠ 1 be a finite abelian p-group, and let V ≠ 0 be an 𝔽_{p}[A]- module such that A acts freely on a basis of V and quadradically on V. Then p = 2 and |A| = 2. If A is a best offender on V, then rk(V) = 2.

Proof. Assume |A| > 2, and choose g, h ∈ A such that g ≠ h and g ≠ 1 ≠ h. Let B be a basis permuted freely by A. For b ∈ B, (1 − g)(1 − h)b = b − gb − hb + ghb ≠ 0 since the elements b, gb, and hb are independent in V, contradicting the assumption that A acts quadratically. Thus |A| = 2 (and hence p = 2).

If A is a best offender on V, then |A||C_{V} (A)| ≥ |V |, so rk(C_{V} (A)) = rk(V) − 1. But rk(C_{V} (A)) = \frac{1}{2}|B| = \frac{1}{2}rk(V) since B is permutted freely by A, so rk(V) = 2. □

Lemma 2.7. Let G be a nontrivial finite group, and let V be a faithful 𝔽_{p}[G]-module. Fix p-subgroups Q ⊴ P ≤ G, where Q < P, and |P/Q| ≥ 4 if p = 2. Assume C_{V} (Q), with its induced action of P/Q, contains a copy of the free module 𝔽_{p}[P/Q]. Then for each quadratic best offender A ≤ P on V, A ≤ Q.

Proof. Set A_{0} = A ∩ Q, and assume A > A_{0}. By assumption, there is an 𝔽_{p}[P/Q]- submodule 1 ≠ W ≤ C_{V} (Q) with a basis on which P/Q acts freely. Thus rk(W) ≥ |P/Q|, |P/Q| ≥ 3 by assumption, and A/A_{0} permutes freely the basis. By Lemma 2.2(a), A/A_{0} is a quadratic best offender on W. Hence by Lemma 2.6, |A/A_{0}| = 2, p = 2, and rk(W) = 2, which is a contrdiction. Thus A = A_{0} ≤ Q. □

3. Proof of the main theorem

The following terminology will be very useful when carrying out the reduction procedures used in this section.

Definition 3.1 ([Ch2, 6.3]). A general setup is a triple (Γ, S, Y), where Γ is a finite group, S ∈ Syl_{p}(Γ), Y ⊴ Γ is a normal p-subgroup, and C_{Γ}(Y) ≤ Y (Y is centric in Γ). A reduced setup is a general setup (Γ, S, Y) such that Y = O_{p}(Γ), C_{S}(Z(Y)) = Y, and O_{p}(Γ/C_{Γ}(Z(Y))) = 1.

The next proposition, which will be shown in Section 4, is the key technical result needed to prove the main theorem. Its proof uses the classification by Meierfrankenfeld and Stellmacher [MS] of F F -offenders, and through that depends on the classification of finite simple groups.

Proposition 3.2 (Compare [Ch2, 6.10]). Let (Γ, S, Y) be a reduced setup, set D = Z(Y), and assume Γ/C_{Γ}(D) is generated by quadratic best offenders on D. Set F = F_{S}(Γ), and let R ⊆ F^{c} be the set of all R ≥ Y such that J(R, D) = Y. Then L^{2}(F; R) = 0 if p = 2, and L^{1}(F; R) = 0 if p is odd.

Since this distinction between the cases where p = 2 or where p is odd occurs throughout this section and the next, it will be convenient to define

k(p) = \begin{cases}2 & if p = 2 \\ 1 & if p is an odd prime. \end{cases}

Thus under the hypotheses of Proposition 3.2, we claim that L^{k(p)}(F; R) = 0.

Proposition 3.2 seems very restricted in scope, but it can be generalized to the following situation.

Proposition 3.3 (Compare [Ch2, 6.11]). Let (Γ, S, Y) be a general setup. Set F = F_{S}(Γ) and D = Z(Y). Let R ⊆ 𝒮 (S)_{≥Y} be an F-invariant interval such that for each Q ∈ 𝒮 (S)_{≥Y}, Q ∈ R if and only if J(Q, D) ∈ R. Then L^{k}(F; R) = 0 for all k ≥ k(p).

Proof. Assume the proposition is false. Let (Γ, S, Y, R, k) be a counterexample for which the 4-tuple (k, |Γ|, |Γ/Y |, |R|) is the smallest possible under the lexicographical ordering.

We will show in Step 1 that R = {P ≤ S | J(P, D) = Y }, in Step 2 that k = k(p), in Step 3 that (Γ, S, Y) is a reduced setup, and in Step 4 that Γ/C_{Γ}(D) is generated by quadratic best offenders on D. The result then follows from Proposition 3.2.

Step 1: Let R_{0} ∈ R be a minimal element of R which is fully normalized in F. Since J(R_{0}, D) ∈ R by assumption (and J(R_{0}, D) ≤ R_{0}), J(R_{0}, D) = R_{0}. Let R_{0} be the set of all R ∈ R such that J(R, D) is F-conjugate to R_{0}, and set Q_{0} = R∖R_{0}. Then R_{0} and Q_{0} are both F-invariant intervals, and satisfy the conditions Q ∈ R_{0} (Q ∈ Q_{0}) if and only if J(Q, D) ∈ R_{0} (J(Q, D) ∈ Q_{0}). Since L^{k}(F; R) ≠ 0, Lemma 1.7(a) implies L^{k}(F; R_{0}) ≠ 0 or L^{k}(F; Q_{0}) ≠ 0. Hence Q_{0} = ∅ and R = R_{0} by the minimality assumption on |R| (and since R_{0} ≠ ∅).

Set Γ_{1} = N_{Γ}(R_{0}), S_{1} = N_{S}(R_{0}), F_{1} = N_{F}(R_{0}) = F_{S_{1}}(Γ_{1}) (see [AKO, Proposition I.5.4]), and R_{1} = {R ∈ R | J(R, D) = R_{0}}. Since R_{0} is fully normalized, each subgroup in R is F-conjugate to a subgroup in R_{1} (Lemma 1.2). Also, for P ∈ R_{1}, if R ∈ R_{0}^{F} and R ⊴ P, then J(P, D) ≥ J(R, D) = R implies R = R_{0}. The hypotheses of Lemma 1.12 are thus satisfied, and so L^{k}(F_{1}; R_{1}) ≅ L^{k}(F; R) ≠ 0. For R ∈ 𝒮 (S_{1})_{≥Y}, R ∈ R_{1} if and only if J(R, D) = R_{0}. Thus (Γ_{1}, S_{1}, Y, R_{1}, k) is another a counterexample to the proposition. By the minimality assumption, Γ_{1} = Γ, and thus R_{0} ⊴ Γ.

We have now shown that there is a p-subgroup R_{0} ⊴ Γ such that R = {R ≤ S | J(R, D) = R_{0}}. Set Y_{1} = R_{0} ≥ Y and D_{1} = Z(Y_{1}) ≤ D. For each R ≤ S such that R ≥ R_{0} and R ∉ R, J(R, D_{1}) ≥ J(R, D) by Corollary 2.3(a), J(R, D) ∉ R by the remarks just after Definition 2.1, and hence J(R, D_{1}) ∉ R. Thus (Γ, S, Y_{1}, R, k) is a counterexample to the proposition, and so Y = Y_{1} = R_{0} by the minimality assumption on |Γ/Y |. We conclude that R = {R ≤ S | J(R, D) = Y }.

Step 2: Let Q be the set of all overgroups of Y in S which are not in R. Equivalently, Q = {Q ≤ S | J(Q, D) > Y }. If k ≥ 2, then L^{k−1}(F; Q) ≅ L^{k}(F; R) ≠ 0 by Lemma 1.7(b). Since k was assumed to be the smallest degree ≥ k(p) for which the proposition is not true, we conclude that k = k(p).

Step 3: Assume (Γ, S, Y) is not a reduced setup. Let K ⊴ Γ be such that K ≥ C_{Γ}(D) and K/C_{Γ}(D) = O_{p}(Γ/C_{Γ}(D)), and set Y_{2} = S ∩K ⊴ S. Then Y_{2} > Y, since either Y_{2} ≥ O_{p}(Γ) > Y, or Y_{2} ≥ C_{S}(D) > Y, or p||K/C_{Γ}(D)| and hence Y_{2} > C_{S}(D) ≥ Y. Set Γ_{2} = N_{Γ}(Y_{2}), and set R_{2} = {P ∈ R | P ≥ Y_{2}}. Note that S ∈ Syl_{p}(Γ_{2}), and also that R_{2} is an F-invariant interval since Y_{2} is strongly closed in S with respect to Γ. Set F_{2} = F_{S}(Γ_{2}) = N_{F}(Y_{2}) [AKO, Proposition I.5.4].

Assume P ∈ R∖R_{2}. Then P ≱ Y_{2}, so P Y_{2} > P, and hence N_{P Y_{2}}(P) > P (Lemma 1.14(a)). Set G = Out_{Γ}(P) and G_{0} = Out_{K}(P). Then G_{0} ⊴ G since K ⊴ Γ, and C_{G_{0}}(Z(P)) = Out_{C_{K}(Z(P))}(P) ≥ Out_{C_{Γ}(D)}(P) since K ≥ C_{Γ}(D) and Z(P) ≤ Z(Y) = D. Hence G_{0}/C_{G_{0}}(Z(P)) is a p-group since K/C_{Γ}(D) is a p-group. For any g ∈ N_{P Y_{2}}(P)∖P, Id ≠ [c_{g}] ∈ Out_{K}(P) = G_{0} since Y_{2} ≤ K (and since C_{Γ}(P) ≤ C_{Γ}(Y) ≤ Y ≤ P). Thus Out_{K}(P) = G_{0} ⊴ G contains a nontrivial element of p-power order, and its action on Z(P) factors through the p-group G_{0}/C_{G_{0}}(Z(P)). Proposition 1.11(b,c) now implies that Λ^{∗}(Out_{Γ}(P); Z(P)) = 0.

Since this holds for all P ∈ R∖R_{2}, L^{∗}(F; R∖R_{2}) = 0 by Corollary 1.10. Hence L^{∗}(F; R_{2}) ≅ L^{∗}(F; R) by the exact sequence in Lemma 1.7. Also, the hypotheses of Lemma 1.12 hold for the functor Z_{F}^{R_{2}} on O(F^{c}) (with Q = Y_{2}) since Y_{2} is strongly closed. So L^{∗}(F; R_{2}) ≅ L^{∗}(F_{2}; R_{2}). Since L^{k}(F; R) ≠ 0 by assumption, L^{k}(F_{2}; R_{2}) ≠ 0.

Set D_{2} = Z(Y_{2}) ≤ D. For each P ∈ 𝒮 (S)_{≥Y_{2}},

P ≥ J(P, D_{2}) ≥ J(P, D) ≥ C_{P} (D) ≥ Y  (1)

by Corollary 2.3(a) and by definition of J(P, −). We must show that for all P ≥ Y_{2}, P ∈ R_{2} if and only if J(P, D_{2}) ∈ R_{2}. If P ∈ R_{2} ⊆ R, then J(P, D) ∈ R by assumption, so J(P, D_{2}) ∈ R by (1) since R is an interval, and J(P, D_{2}) ∈ R_{2} since J(P, D_{2}) ≥ C_{P} (D_{2}) ≥ Y_{2}. If P ∉ R_{2}, then P ∉ R, so J(P, D) ∉ R, and J(P, D_{2}) ∉ R (hence J(P, D_{2}) ∉ R_{2}) by (1) again and since R is an interval containing Y.

Thus (Γ_{2}, S, Y_{2}, R_{2}, k) is a counterexample to the proposition. So Γ_{2} = Γ and Y_{2} = Y by the minimality assumption, which contradicts the above claim that Y_{2} > Y. We conclude that (Γ, S, Y) is a reduced setup.

Step 4: It remains to prove that Γ/C_{Γ}(D) is generated by quadratic best offenders on D; the result then follows from Proposition 3.2.

Let Γ_{3} ⊴ Γ be such that Γ_{3} ≥ C_{Γ}(D) and Γ_{3}/C_{Γ}(D) is generated by all quadratic best offenders on D. If Γ_{3} = Γ we are done, so assume Γ_{3} < Γ. Set S_{3} = Γ_{3} ∩ S and F_{3} = F_{S_{3}}(Γ_{3}). Set

Q = 𝒮 (S)_{≥Y} ∖R, Q_{3} = Q ∩ 𝒮 (S_{3})_{≥Y} , and R_{3} = R ∩ 𝒮 (S_{3})_{≥Y} .

Since L^{k}(F; R) ≠ 0, R ⫋ 𝒮 (S)_{≥Y} by Lemma 1.6(b), and Q ≠ ∅. The proposition holds for (Γ_{3}, S_{3}, Y, R_{3}, k) by the minimality assumption, and thus L^{k}(F_{3}; R_{3}) = 0.

For Q ∈ Q, J(Q, D) > Y, so Q/Y has nontrivial best offenders on D, hence has nontrivial quadratic best offenders on D by Theorem 2.5, and thus J(Q∩Γ_{3}, D) > Y. So Q ∈ Q implies Q ∩ Γ_{3} ∈ Q_{3} by Step 1. In particular, S_{3} ∈ Q_{3}.

If k = 2 (i.e., if p = 2), then L^{1}(F; Q) ≅ L^{2}(F; R) ≠ 0 and L^{1}(F_{3}; Q_{3}) ≅ L^{2}(F_{3}; R_{3}) = 0 by Lemma 1.7(b), which is impossible by Lemma 1.13.

If k = 1 (if p is odd), set

Γ^{∗} = ⟨g ∈ Γ |^{g}P ∈ Q for some P ∈ Q⟩ ≤ Γ
Γ_{3}^{∗} = ⟨g ∈ Γ_{3} |^{g}P ∈ Q_{3} for some P ∈ Q_{3}⟩ ≤ Γ_{3} .

Then Γ_{3}^{∗} ≤ Γ^{∗} since Γ_{3} ≤ Γ and Q_{3} ⊆ Q. By Lemma 1.7(b), there are exact sequences

\begin{matrix}1 ⟶ C_{Z(Y)}(Γ) ⟶ C_{Z(Y)}(Γ^{∗}) ⟶ L^{1}(F; R) ≠ 1 \\ 1 ⟶ C_{Z(Y)}(Γ_{3}) ⟶ C_{Z(Y)}(Γ_{3}^{∗}) ⟶ L^{1}(F_{3}; R_{3}) = 1. \end{matrix}  (2)

Also, Γ^{∗}Γ_{3} ≥ N_{Γ}(S_{3})Γ_{3} = Γ since S_{3} ∈ Q_{3}, where the equality follows from the Frattini argument (Lemma 1.14(b)), so

C_{Z(Y)}(Γ) = C_{Z(Y)}(Γ^{∗}Γ_{3}) = C_{Z(Y)}(Γ^{∗}) ∩ C_{Z(Y)}(Γ_{3}).

But this is impossible, since C_{Z(Y)}(Γ) < C_{Z(Y)}(Γ^{∗}) ≤ C_{Z(Y)}(Γ_{3}^{∗}) = C_{Z(Y)}(Γ_{3}) by the exactness in (2). □

We now have the tools needed to prove the main vanishing result.

Theorem 3.4. For each saturated fusion system F over a p-group S, \underleftarrow{lim}^{k}_{Q(F^{c})}(Z_{F}) = 0

for all k ≥ 2, and for all k ≥ 1 if p is odd.

Proof. As in [Ch2, § 6], we choose inductively subgroups X_{0}, X_{1}, …, X_{N} ∈ F^{c} and F-invariant intervals ∅ = Q_{−1} ⊆ Q_{0} ⊆ ⋯ ⊆ Q_{N} = F^{c} as follows. Assume Q_{n−1} has been defined (n ≥ 0), and Q_{n−1} ⫋ F^{c}. Consider the sets of subgroups

U_{1} = U_{1}^{(n)} = {P ∈ F^{c}∖Q_{n−1} | d(P) maximal}
U_{2} = U_{2}^{(n)} = {P ∈ U_{1} | |J(P)| maximal}
U_{3} = U_{3}^{(n)} = {P ∈ U_{2} | J(P) ∈ F^{c}}
U_{4} = U_{4}^{(n)} = \begin{cases}\{P ∈ U_{3} | |P | minimal\} & if U_{3} ≠ ∅ \\ \{P ∈ U_{2} | |P | maximal\} & otherwise. \end{cases}

(See Definition 2.1(a) for the definition of d(P).) Let X_{n} be any subgroup in U_{4} such that X_{n} and J(X_{n}) are both fully normalized in F.

We first check that there is such an X_{n}. For each X ∈ U_{4} and each Y ∈ J(X)^{F} which is fully normalized in F, there is φ ∈ Hom_{F}(N_{S}(J(X)), N_{S}(Y)) such that φ(J(X)) = Y (Lemma 1.2), and φ(X) is also fully normalized since N_{S}(X) ≤ N_{S}(J(X)). Since U_{4} is invariant under F-conjugacy, this shows that X_{n} ∈ U_{4} can be chosen as required.

Let Q_{n} be the union of Q_{n−1} with the set of all overgroups of subgroups Fconjugate to X_{n}. Set R_{n} = Q_{n}∖Q_{n−1} for each 0 ≤ n ≤ N. Thus the sets Q_{n} are all closed under overgroups, and the R_{n} are intervals. By definition of U_{4}, X_{n} = J(X_{n}) if J(X_{n}) ∈ F^{c}, while R_{n} = X_{n}^{F} if J(X_{n}) ∉ F^{c}. Note also that X_{0} = J(S) and R_{0} = Q_{0} = 𝒮 (S)_{≥J(S)}.

We will show, for each n, that

L^{k}(F; R_{n}) = 0 for all k ≥ k(p).  (3)

Then by Lemma 1.7(a), for all k ≥ k(p), L^{k}(F; Q_{n−1}) = 0 implies L^{k}(F; Q_{n}) = 0. The theorem now follows by induction on n.

Case 1: Assume n is such that J(X_{n}) ∉ F^{c}, and hence that R_{n} = X_{n}^{F}. Since J(X_{n}) is fully normalized and not F-centric, C_{S}(J(X_{n})) ≰ J(X_{n}). Then X_{n}C_{S}(J(X_{n})) > X_{n}, since J(X_{n}) is centric in X_{n}. Hence N_{X_{n}C_{S}(J(X_{n}))}(X_{n}) > X_{n} by Lemma 1.14(a), so there is g ∈ N_{S}(X_{n})∖X_{n} such that [g, J(X_{n})] = 1. Then g acts trivially on Z(X_{n}) ≤ J(X_{n}), so the kernel of the Out_{F}(X_{n})-action on Z(X_{n}) has order a multiple of p, and Λ^{∗}(Out_{F}(X_{n}); Z(X_{n})) = 0 by Proposition 1.11(b). Hence (3) holds by Proposition 1.9.

Case 2: Assume n is such that J(X_{n}) ∈ F^{c}, and hence X_{n} = J(X_{n}) by definition of U_{4}. By definition of U_{1} and U_{2}, for each P ≥ X_{n} in R_{n}, d(P) = d(X_{n}) and J(P) = X_{n}. Hence

P ∈ R_{n} ⟹ J(P) is the unique subgroup of P F-conjugate to X_{n}. (4)

Set T = N_{S}(X_{n}) and E = N_{F}(X_{n}). Then E is a saturated fusion system over T (cf. [AKO, Theorem I.5.5]), and contains X_{n} as normal centric subgroup. Hence there is a model for E (cf. [AKO, Theorem III.5.10]): a finite group Γ such that T ∈ Syl_{p}(Γ), X_{n} ⊴ Γ, C_{Γ}(X_{n}) ≤ X_{n}, and F_{T} (Γ) = E.

Let R be the set of all P ∈ R_{n} such that P ≥ X_{n}. Then (Γ, T, X_{n}) is a general setup, and R is an E-invariant interval containing X_{n}. If P ∈ R and Y ≤ P is F-conjugate to X_{n}, then Y = X_{n} by (4). Also, each subgroup in R_{n} is F-conjugate to a subgroup in R by (4) and Lemma 1.2 (recall X_{n} is fully normalized). The hypotheses of Lemma 1.12 thus hold, and hence

L^{∗}(F; R_{n}) ≅ L^{∗}(E; R).  (5)

Set D = Z(X_{n}). We claim that for each P ∈ 𝒮 (T)_{≥X_{n}},

P ∈ R ⟺ J(P, D) ∈ R .  (6)

Fix such a P. By Corollary 2.3(b), J(P, D) ≥ J(P), and X_{n} ≥ J(X_{n}, D) ≥ J(X_{n}) = X_{n}. If P ∈ R, then J(P, D) ∈ R since X_{n} = J(X_{n}, D) ≤ J(P, D) ≤ P and R is an interval. If P ∉ R, then P ∈ R_{i} for some 0 ≤ i ≤ n−1. By definition of U_{1}^{(i)} and U_{2}^{(i)}, either d(P) = d(X_{i}) > d(X_{n}), or d(P) = d(X_{i}) = d(X_{n}) and J(P) > J(X_{n}) = X_{n}, or J(P) = X_{n} ∈ J(X_{i})^{F}. The latter is not possible since by definition of U_{4}^{(i)}, either J(X_{i}) = X_{i} or J(X_{i}) ∉ F^{c}. If d(P) > d(X_{n}), then d(J(P, D)) = d(P) > d(X_{n}) since J(P) ≤ J(P, D) ≤ P, and J(P, D) ∉ R since d(R) = d(X_{n}) for all R ∈ R. If J(P) > X_{n}, then J(P) ∉ R since J(R) = X_{n} for all R ∈ R, and hence J(P, D) ∉ R since J(P, D) ≥ J(P) and R is an interval. This proves (6).

Thus by Proposition 3.3, L^{k}(E; R) = 0 for all k ≥ k(p). Together with (5), this finishes the proof of (3), and hence of the theorem. □

4. Proof of Proposition 3.2

It remains to prove Proposition 3.2, which we restate here as:

Proposition 4.1. Let (Γ, S, Y) be a reduced setup, set D = Z(Y), and assume Γ/C_{Γ}(D) is generated by quadratic best offenders on D. Set F = F_{S}(Γ), and let R ⊆ F^{c} be the set of all R ≥ Y such that J(R, D) = Y. Then L^{k(p)}(F; R) = 0.

It is in this section that we use the classification of offenders by Meierfrankenfeld and Stellmacher [MS], and through that the classification of finite simple groups. The following theorem is a summary of those parts of [MS, Theorems 1 & 2] which we need here. The complete results in [MS] give a much more precise description of all representations of groups containing elementary abelian best offenders.

We adopt the notation in [GLS3], and let 𝔏𝔦𝔢(p) denote the class of groups G = O^{p′}(C_{G}(σ)), where G is a connected, (quasi-)simple algebraic group over 𝔽_{p}, and σ ∈ End(G) is an algebraic endomorphism with finite fixed subgroup. Most of these groups are quasisimple, with a few exceptions such as SL_{2}(2), SL_{2}(3), and G_{2}(2). Note that SO_{2m}^{±}(2^{k}) ∉ 𝔏𝔦𝔢(2) (m ≥ 3), since SO_{2m}(𝔽_{2}) is not connected.

When G ≅ A_{n} or Σ_{n}, the “natural module for G” in characteristic two is the simple 𝔽_{2}[G]-module of rank n − 1 (n odd) or n − 2 (n even) which is a subquotient of the permutation module of rank n.

A CK-group is a finite group all of whose composition factors are known simple groups.

Theorem 4.2. Fix a finite CK-group G such that O_{p}(G) = 1, and a faithful finite dimensional 𝔽_{p}[G]-module V. Assume G is generated by elementary abelian p-groups which are best offenders on V. Let J be the set of all subgroups 1 ≠ K ⊴ G which are minimal with the property that [K, G] = K. Set W = [J , V]C_{V} (J)/C_{V} (J). Then

(a) O^{p}(G) = ⟨J ⟩ = ×J;

(b) W is a faithful, semisimple 𝔽_{p}[G]-module; and

(c) each elementary abelian best offender on V is a best offender on W.

(d) If W is a simple 𝔽_{p}[G]-module, then either

(1) G ∈ 𝔏𝔦𝔢(p) (possibly one of the non-quasisimple groups G ≅ SL_{2}(2), SL_{2}(3), Sp_{4}(2), or G_{2}(2)); or

(2) G ≅ SO_{2m}^{±}(2^{k}), where p = 2, m ≥ 3, and V is the natural module for G; or

(3) G ≅ 3A_{6} or A_{7} and p = 2; or

(4) G ≅ A_{n} (n ≥ 6, n even) or Σ_{n} (n ≥ 3, n ≠ 4), p = 2, and V is the natural module for G.

Proof. Points (a)–(c) are points (c)–(e) in [MS, Theorem 1], while (d) follows from [MS, Theorem 2] (which gives a much more explicit list). Note that we have dropped the group SO_{4}^{−}(2) ≅ Σ_{5} from point (d.2), since its natural module is isomorphic to that of Σ_{5} (case (d.4)). □

When H_{1} ≤ H_{2} ≤ ⋯ ≤ H_{k} are subgroups of a group G, we let N_{G}(H_{1}, …, H_{k}) denote the intersection of their normalizers.

Definition 4.3. Let G be a finite group.

(a) A radical p-subgroup of G is a p-subgroup P ≤ G such that O_{p}(N_{G}(P)) = P; i.e., O_{p}(N_{G}(P)/P) = 1.

(b) A radical p-chain of length k in G is a sequence of p-subgroups P_{0} < P_{1} < ⋯ < P_{k} ≤ G such that P_{0} is radical in G, P_{i} is radical in N_{G}(P_{0}, …, P_{i−1}) for each i ≥ 1, and P_{k} ∈ Syl_{p}(N_{G}(P_{0}, …, P_{k−1})).

The reason for defining this here is the following vanishing result, which involves only radical p-chains with P_{0} = 1.

Proposition 4.4 ([AKO, Lemma III.5.27] and [O2, Proposition 3.5]). Fix a finite group G, a finite 𝔽_{p}[G]-module M, and k ≥ 1 such that Λ^{k}(G; M) ≠ 0. Then there is a radical p-chain 1 = P_{0} < P_{1} < ⋯ < P_{k} of length k such that M contains a copy of the free module 𝔽_{p}[P_{k}].

Since the trivial subgroup is a radical p-subgroup of G only if O_{p}(G) = 1, Proposition 4.4 includes the statement that Λ^{k}(G; M) = 0 if O_{p}(G) ≠ 1 (Proposition 1.11(c)). The reason for defining radical p-chains more generally here — to also allow chains where P_{0} ≠ 1 — will be seen in the proof of the next proposition and in that of Proposition 4.1.

Proposition 4.5. Let G be a nontrivial finite group with O_{p}(G) = 1, and let V be a faithful 𝔽_{p}[G]-module. Let U be the set of quadratic best offenders in G on V, and assume G = ⟨U⟩. Set G_{0} = O^{p}(G) and W = C_{V} (G_{0})[G_{0}, V]/C_{V} (G_{0}). Assume, for some p-subgroup P_{0} ≤ G, some 𝔽_{p}[N_{G}(P_{0})/P_{0}]-submodule X ≤ C_{W} (P_{0}), and some k ≥ k(p), that Λ^{k}(N_{G}(P_{0})/P_{0}; X) ≠ 0. Then each U ∈ U is G-conjugate to a subgroup of P_{0}.

Proof. Quadratic offenders with faithful action on V are elementary abelian by Lemma 2.4. So Theorem 4.2 ([MS, Theorems 1 & 2]) applies. Then O^{p}(G) = ⟨J ⟩ by Theorem 4.2(a), and hence W as defined here is the same as W defined in that theorem.

Case 1: Assume V is a simple 𝔽_{p}[G]-module. Thus V = W. Set H_{0} = C_{N_{G}(P_{0})}(C_{W} (P_{0})). Thus P_{0} ⊴ H_{0}, and p ∤ |H_{0}/P_{0}| by Proposition 1.11(b). By Proposition 1.11(c), P_{0} is radical in G. By Proposition 4.4, there is a radical p-chain 1 < R_{1}/H_{0} < ⋯ < R_{k}/H_{0} of length k in N_{G}(P_{0})/H_{0} such that X, and hence C_{W} (P_{0}), contains a copy of 𝔽_{p}[R_{k}/H_{0}]. Hence by Lemma A.4 (applied with H_{0}/P_{0} ⊴ N_{G}(P_{0})/P_{0} in the role of H ⊴ G), there is a radical p-chain 1 < P_{1}/P_{0} < ⋯ < P_{k}/P_{0} in N_{G}(P_{0})/P_{0} such that P_{i}H_{0} = R_{i} for each i. Then P_{0} < P_{1} < ⋯ < P_{k} is a radical p-chain in G. Also, P_{k}/P_{0} ≅ R_{k}/H_{0}, so C_{W_{i}}(P_{0}) contains a copy of 𝔽_{p}[P_{k}/P_{0}].

By Lemma 4.6, P_{k} ∈ Syl_{p}(G). Hence each quadratic best offender in G on W is G-conjugate to some U ≤ P_{k}. By Lemma 2.7 (and since C_{W} (P_{0}) contains a copy of 𝔽_{p}[P_{k}/P_{0}]), U ≤ P_{0}.

Case 2: Now assume V is arbitrary. By Theorem 4.2(b,c), W is a semisimple 𝔽_{p}[G]- module, and each U ∈ U is a quadratic best offender on W. Set W = W_{1}⊕⋯⊕W_{m}, where each W_{i} is a simple 𝔽_{p}[G]-module. For 0 ≤ i ≤ m, set X_{i} = X∩(W_{1}⊕⋯⊕W_{i}). Thus 0 = X_{0} ≤ X_{1} ≤ ⋯ ≤ X_{m} = X are 𝔽_{p}[N_{G}(P_{0})/P_{0}]-submodules, and X_{i}/X_{i−1} is isomorphic to a submodule of C_{W_{i}}(P_{0}) for each i ≥ 1.

Since Λ^{k}(N_{G}(P_{0})/P_{0}; X) ≠ 0, the exact sequences for the pairs X_{i−1} ≤ X_{i} (Proposition 1.11(d)) imply that Λ^{k}(N_{G}(P_{0})/P_{0}; X_{i}/X_{i−1}) ≠ 0 for some 1 ≤ i ≤ m. Set K = C_{G}(W_{i}), G = G/K, and H = HK/K ≤ G for each H ≤ G. The action of N_{G}(P_{0})/P_{0} on C_{W_{i}}(P_{0}) factors through

N_{G}(P_{0})/P_{0} = N_{G/K}(P_{0}K/K)/(P_{0}K/K) ≅ N_{G}(P_{0}K)/P_{0}K ,

and N_{G}(P_{0})/P_{0} surjects onto N_{G}(P_{0})/P_{0} with kernel N_{P_{0}K}(P_{0})/P_{0}. By Proposition 1.11(b), p ∤ |N_{P_{0}K}(P_{0})/P_{0}| and Λ^{k}(N_{G}(P_{0})/P_{0}; X_{i}/X_{i−1}) ≠ 0. By Lemma 1.14(a), P_{0} ∈ Syl_{p}(P_{0}K).

Since G = ⟨U⟩ where U is the set of quadratic best offenders on W, G = ⟨U⟩ where U = {U | U ∈ U} is a set of quadratic best offenders on W_{i} by Lemma 2.2(a). By assumption, W_{i} is a faithful, simple 𝔽_{p}[G]-module. So by Case 1, each U ∈ U is G-conjugate to a subgroup of P_{0} = P_{0}K/K. Hence each U ∈ U is G-conjugate to a subgroup of P_{0} ∈ Syl_{p}(P_{0}K). □

The following lemma was needed to prove Proposition 4.5. This is where the explicit list in Theorem 4.2 was needed.

Lemma 4.6. Let G be a nontrivial finite group, let W be a faithful, simple 𝔽_{p}[G]- module, and assume G is generated by its quadratic best offenders on W. Let P_{0} < P_{1} < ⋯ < P_{k} be a radical p-chain in G with k ≥ k(p). Set

H_{0} = C_{N_{G}(P_{0})}(C_{W} (P_{0})) ,

and assume also that p ∤ |H_{0}/P_{0}| and that 1 < P_{1}H_{0}/H_{0} < ⋯ < P_{k}H_{0}/H_{0} is a radical p-chain in N_{G}(P_{0})/H_{0}. Then either

(a) C_{W} (P_{0}), with its induced action of N_{G}(P_{0})/P_{0}, does not contain a copy of the free module 𝔽_{p}[P_{k}/P_{0}]; or

(b) P_{k} ∈ Syl_{p}(G).

Proof. Quadratic offenders with faithful action are elementary abelian by Lemma 2.4. Also, O_{p}(G) = 1, since C_{W} (O_{p}(G)) is a nontrivial 𝔽_{p}[G]-submodule of W and G acts faithfully. So we are in the situation of Theorem 4.2(d). The cases listed there will be considered individually.

Assume C_{W} (P_{0}) does contain a copy of 𝔽_{2}[P_{k}/P_{0}]. We must show that P_{k} ∈ Syl_{p}(G). Note that if p = 2, then rk(C_{W} (P_{0})) ≥ |P_{k}/P_{0}| ≥ 4 since k ≥ k(2) = 2.

Case 1: Assume G ∈ 𝔏𝔦𝔢(p). The nontrivial radical p-subgroups of G are well known: by a theorem of Borel and Tits (see [GLS3, Corollary 3.1.5]), they are all conjugate to maximal normal unipotent subgroups in parabolic subgroups. Hence the normalizers N_{G}(P_{0}, …, P_{i}) all contain Sylow p-subgroups of G, and the quotients N_{G}(P_{0}, …, P_{i})/P_{i} (the Levi complements) are central products of groups in 𝔏𝔦𝔢(p) (see [GLS3, Theorem 2.6.5(f)]). Since P_{k} ∈ Syl_{p}(N_{G}(P_{0}, …, P_{k−1})), P_{k} ∈ Syl_{p}(G) in this case.

Case 2: Now assume p = 2 and G ≅ SO_{2m}^{±}(q), where 2m ≥ 6, q = 2^{a} (a ≥ 1), and W is the natural 𝔽_{2}[G]-module of rank 2am. Set G_{0} = Ω_{2m}^{±}(q), so [G:G_{0}] = 2.

For any radical 2-subgroup P ≤ G, P ∩ G_{0} is a radical 2-subgroup of G_{0} by Lemma A.2, and hence is either trivial, or is a maximal normal unipotent subgroup in a parabolic subgroup. If P ∩ G_{0} = 1 and P ≠ 1, then P = ⟨t⟩ for some involution t ∈ SO_{2m}^{±}(q)∖Ω_{2m}^{±}(q). Set W_{1} = C_{W} (t) and W_{2} = [t, W] ≤ W_{1}. Then W_{1} ⊥ W_{2}, so the quadratic form 𝔮 on W is linear on W_{2} with W_{3} =^{def} Ker(𝔮|_{W_{2}}) ≤ W_{2} of index at most 2. If W_{3} ≠ 0, then by Witt’s theorem (cf. [Ta, Theorem 7.4]), each α ∈ Aut_{𝔽_{q}}(W_{1}) which induces the identity on W_{2} and on W_{1}/W_{3} extends to some α ∈ G, then α ∈ O_{2}(N_{G}(P)), so P is not radical. Thus W_{3} = 0, rk(W_{2}) = 1, and t is a transvection. By Witt’s theorem again, restriction to W_{1} induces an isomorphism N_{G}(P)/P ≅ SO_{2m−1}(q) ≅ Sp_{2m−2}(q).

Assume first that P_{0} = 1. If P_{1} ∩ G_{0} = 1, then P_{1} is generated by a transvection, so P_{1} is a quadratic best offender, which contradicts Lemma 2.7. Thus P_{1} ∩ G_{0} is a maximal normal unipotent subgroup of a parabolic subgroup. So |P_{1}| ≥ q^{2m−3} by Lemma A.5, |P_{2}| ≥ q^{2m−2}, and q^{2m−2} > rk_{𝔽_{2}}(W) = 2am since m ≥ 3. Hence this case is impossible.

Next assume P_{0} ≠ 1 and P_{0} ∩ G_{0} = 1. As noted above, P_{0} is generated by a transvection (hence rk_{𝔽_{q}}(C_{W} (P_{0})) = 2m − 1), and N_{G}(P_{0})/P_{0} ≅ Sp_{2m−2}(q). By Case 1 (applied with P_{0} = 1), P_{k}/P_{0} ∈ Syl_{2}(N_{G}(P_{0})/P_{0}). Hence |P_{k}/P_{0}| = q^{(m−1)^{2}} ≤ 2m−1 (cf. [Ta, p. 70]), which is impossible since m ≥ 3. (Alternatively, N_{G}(P_{0})∖P_{0} and hence P_{k}∖P_{0} contains a transvection, which contradicts Lemma 2.7.)

Finally, assume P_{0} ∩ G_{0} ≠ 1, and hence is a maximal normal unipotent subgroup of a parabolic subgroup. Set W_{0} = C_{W} (P_{0}) < W. Then W_{0} ≤ C_{W} (P_{0} ∩ G_{0}) is a totally isotropic subspace of W of rank at most m, and N_{G}(P_{0})/P_{0} ≅ GL(W_{0}) × N_{SO(W_{0}^{⊥}/W_{0})}(P_{0})/P_{0} acts on it via projection to the first factor. Thus N_{G}(P_{0})/H_{0} ≅ GL(W_{0}), and P_{k}/P_{0} ≅ P_{k}H_{0}/H_{0} ∈ Syl_{2}(N_{G}(P_{0})/H_{0}) by Case 1 (applied with G = SL(W_{0}) and P_{0} = 1). Set r = rk(W_{0}); then |P_{k}/P_{0}| = 2^{r(r−1)/2} ≤ r implies r ≤ 2, which contradicts the above observation that r ≥ 4.

Case 3: Assume p = 2, and G ≅ 3A_{6} or A_{7}. Then the Sylow 2-subgroups of G have order 8, the nontrivial radical 2-subgroups have order 4 or 8, hence are normal in Sylow 2-subgroups, and thus P_{k} ∈ Syl_{2}(G) (k ≥ 2).

Case 4: Assume p = 2, G ≅ Σ_{m} or A_{m}, and W is a natural module for G. Set m = {1, 2, …, m}, with the canonical action of G. Set V = 𝔽_{2}(m): the 𝔽_{2}-vector space with basis m, and G-action induced by that on m. Set Δ = C_{V} (G): the subgroup generated by the sum of all elements in m. Identify W = V/Δ if m is odd (so rk(W) = m−1), and W ≤ V/Δ with index two if m is even (so rk(W) = m−2). Since rk(W) ≥ 4, m ≥ 5.

For any H ≤ G, let m/H be the set of orbits of H acting on m (with induced action of N_{G}(H)/H), and let 𝔽_{2}(m/H) be the permutation module with basis m/H. Since C_{V} (H) is the group of elements of V = 𝔽_{2}(m) whose coefficients are constant on each H-orbit, we can identify C_{V} (H) with 𝔽_{2}(m/H) as 𝔽_{2}[N_{G}(H)/H]-modules.

If P_{0} acts on m with more than one orbit, then C_{V/Δ}(P_{0}) = C_{V} (P_{0})/Δ by Lemma A.8(a). Thus C_{W} (P_{0}) ≤ C_{V} (P_{0})/Δ, so C_{V} (P_{0}) also contains a copy of the free module 𝔽_{2}[P_{k}/P_{0}], and its basis m/P_{0} contains a free (P_{k}/P_{0})-orbit by Lemma A.1. Since k ≥ 2, m/P_{0} contains |P_{k}/P_{1}| ≥ 2 free (P_{1}/P_{0})-orbits, which contradicts Lemma A.7(i).

Now assume P_{0} acts transitively on m. Let U ≤ V be such that U/Δ = C_{V/Δ}(P_{0}) ≥ C_{W} (P_{0}). Each g ∈ N_{G}(P_{0}) normalizes U/Δ, so N_{G}(P_{0}) ≤ N_{G}(U), and P_{0} < ⋯ < P_{k} is also a radical p-chain in N_{G}(U). Also, U/Δ contains a copy of 𝔽_{2}[P_{k}/P_{0}], and in particular, P_{k} ∩ C_{G}(U) = P_{0} ∩ C_{G}(U). By Lemma A.3, applied with C_{G}(U) ⊴ N_{G}(U) in the role of H ⊴ G, C_{P_{0}}(U) ∈ Syl_{p}(C_{G}(U)).

By Lemma A.8(b), N_{G}(P_{0})/H_{0} ≅ GL(U/Δ) with the canonical action on U/Δ = C_{V/Δ}(P_{0}). Hence C_{W} (P_{0}) = U/Δ. By Case 1 (applied with G = SL(C_{W} (P_{0})) and P_{0} = 1), P_{k}/P_{0} ≅ P_{k}H_{0}/H_{0} ∈ Syl_{2}(N_{G}(P_{0})/H_{0}). Set r = rk(C_{W} (P_{0})). Then |P_{k}/P_{0}| = 2^{r(r−1)/2} ≤ r since C_{W} (P_{0}) contains a copy of 𝔽_{2}[P_{k}/P_{0}]. Thus r = 2 and |P_{k}/P_{0}| = 2, which is impossible since k ≥ 2. □

Proof of Proposition 4.1. Fix a reduced setup (Γ, S, Y), set D = Z(Y), V = Ω_{1}(D), and G = Γ/C_{Γ}(D), and assume G = ⟨U⟩ where

U = {1 ≠ P ≤ G | P a quadratic best offender on D}.

Since O_{p}(G) = 1 by definition of a reduced setup, G acts faithfully on V by Lemma 1.15. Hence U is a set of quadratic best offenders on V by Lemma 2.2(a).

Recall that R = {P ∈ F^{c} | J(P, D) = Y }. By Timmesfeld’s replacement theorem (Theorem 2.5), R is the set of all P ∈ 𝒮 (S)_{≥Y} such that P/Y = P/C_{S}(D) contains no nontrivial quadratic best offender on D; i.e., no subgroups in U.

Set D_{0} = 1. For each i ≥ 1, set D_{i} = Ω_{i}(D) = {g ∈ D | g^{p^{i}} = 1} and V_{i} = D_{i}/D_{i−1}. Thus each V_{i} is an 𝔽_{p}[G]-module, and (x ↦ x^{p}) sends V_{i} injectively to V_{i−1} for each i > 0.

Set k = k(p). We will show that Λ^{k}(Out_{Γ}(R); Z(R)) = 0 for each R ∈ R; the proposition then follows from Corollary 1.10. Here, Z(R) = C_{D}(R) and Out_{Γ}(R) ≅ N_{Γ}(R)/R since R ≥ Y and C_{S}(Y) = Z(Y) = D. So by Proposition 1.11(d), it suffices to show, for each R and i, that Λ^{k}(N_{Γ}(R)/R; C_{D_{i}}(R)/C_{D_{i−1}}(R)) = 0. Also, for each i, C_{D_{i}}(R)/C_{D_{i−1}}(R) can be identified with an N_{Γ}(R)-invariant subgroup of C_{V_{i}}(R) ≤ C_{V} (R). It thus suffices to show that

Λ^{k}(N_{Γ}(R)/R; X) = 0 ∀ R ∈ R, ∀ N_{Γ}(R)-invariant X ≤ C_{V} (R).  (1)

Set W_{1} = C_{V} (O^{p}(G)), W_{2} = W_{1}[O^{p}(G), V], and W = W_{2}/W_{1}. Thus the Gactions on W_{1} and on V/W_{2} factor through the quotient p-group G/O^{p}(G). So by Proposition 1.11(a,b,c), for each R ∈ R and each X ≤ C_{V} (R) as in (1),

Λ^{k}(N_{Γ}(R)/R; X ∩ W_{1}) = 0 and Λ^{k}(N_{Γ}(R)/R; X/(X ∩ W_{2})) = 0.

By the exact sequences of Proposition 1.11(d), we are now reduced to showing that Λ^{k}(N_{Γ}(R)/R; (X ∩ W_{2})/(X ∩ W_{1})) = 0 for all such X; or more generally that

Λ^{k}(N_{Γ}(R)/R; X) = 0 ∀ R ∈ R, ∀ N_{Γ}(R)-invariant X ≤ C_{W} (R).  (2)

Since R ∈ R, no U ∈ U is contained in R, and hence (2) follows from Proposition 4.5. □

Appendix A. Radical p-chains and free submodules

We collect here some lemmas needed in the proofs in Section 4.

Lemma A.1. Let P be a p-group, and let V be an 𝔽_{p}[P]-module. Assume V has an 𝔽_{p}-basis B which is permuted by P, and also contains a copy of the free module 𝔽_{p}[P]. Then B contains a free P -orbit.

Proof. Write V = V_{1} ⊕ ⋯ ⊕ V_{n}, where each V_{i} has as basis one P -orbit B_{i} ⊆ B. For each i, let pr_{i}: V → V_{i} be the projection.

Let F ≤ V be a submodule isomorphic to 𝔽_{p}[P]. Then C_{F} (P) ≅ 𝔽_{p}. Choose i such that C_{F} (P) ≰ Ker(pr_{i}). Then Ker(pr_{i}|_{F}) = 0 (otherwise it contains nontrivial elements fixed by P), so pr_{i} sends F injectively into V_{i}. Thus |P | ≤ rk(V_{i}) = |B_{i}|, so B_{i} is a free orbit. □

Lemma A.2. Assume H ⊴ G are finite groups, and let P ≤ G be a radical psubgroup. Then P ∩ H is a radical p-subgroup of H.

Proof. Set Q = O_{p}(N_{H}(P ∩ H)). Then N_{G}(P) normalizes Q, so N_{QP} (P) ⊴ N_{G}(P). Hence N_{QP} (P) ≤ O_{p}(N_{G}(P)) = P, so Q ≤ P by Lemma 1.14(a), and P ∩ H is radical in H. □

The next two lemmas are useful when manipulating radical p-chains. The first was suggested by one of the referees.

Lemma A.3. Fix a finite group G and a normal subgroup H ⊴ G. Let P_{0} < P_{1} < ⋯ < P_{k} be a radical p-chain in G such that P_{0}∩H = P_{k}∩H. Then P_{0}∩H ∈ Syl_{p}(H).

Proof. Choose S ∈ Syl_{p}(G) such that S ≥ P_{k}, and set Q = S ∩ H ∈ Syl_{p}(H). Then Q ≥ P_{k} ∩ H = P_{0} ∩ H. We must show that Q = P_{0} ∩ H.

Assume otherwise: assume Q > P_{0}∩H. By construction, Q is normalized by S and hence by each of the P_{i}. Set Q = Q_{−1}, and define recursively Q_{i} = Q_{i−1}∩N_{P_{i}Q_{i−1}}(P_{i}) for each 0 ≤ i ≤ k. Since P_{k} normalizes Q and each of the P_{i}, it normalizes each Q_{i} (so the P_{i}Q_{i−1} are subgroups of G). By Lemma 1.14(a), if Q_{i−1} > P_{0} ∩ H = P_{i} ∩ H, then N_{P_{i}Q_{i−1}}(P_{i}) > P_{i} so that Q_{i} > P_{i} ∩ H = P_{0} ∩ H. Thus P_{k}Q_{k} > P_{k}, Q_{k} ≤ N_{G}(P_{0}, P_{1}, …, P_{k−1}), and this is impossible since P_{k} ∈ Syl_{p}(N_{G}(P_{0}, P_{1}, …, P_{k−1})) by definition of a radical p-chain. □

Lemma A.4. Let G be a finite group, and let H ⊴ G be a normal subgroup of order prime to p. Set G = G/H, and set X = XH/H for each X ≤ G.

(a) If P ≤ G is a p-subgroup such that P is radical in G, then P is radical in G.

(b) If 1 < R_{1}/H < ⋯ < R_{k}/H is a radical p-chain in G, then there is a radical p-chain 1 < P_{1} < ⋯ < P_{k} in G such that P_{i} = R_{i}/H for each i ≤ k.

Proof. For each p-subgroup P ≤ G, the Frattini argument (Lemma 1.14(b)), applied with P ≤ P H ⊴ N_{G}(P H) in the role of T ≤ H ⊴ G, implies that N_{G}(P H) = P H·(N_{N_{G}(P H)}(P)) = HN_{G}(P).

(a) By assumption, P H/H = O_{p}(N_{G}(P H)/H) = O_{p}(N_{G}(P)H/H). Set Q = O_{p}(N_{G}(P)); then QH/H ≤ P H/H, and Q = P since p ∤ |H|. Thus P is radical in G.

(b) Choose P_{k} ∈ Syl_{p}(R_{k}), and set P_{i} = R_{i} ∩ P_{k} ∈ Syl_{p}(R_{i}) for each i < k (recall R_{i} ⊴ R_{k}). Thus P_{i}H = R_{i} for each i ≤ k (hence P_{i} = R_{i}/H), since R_{i}/H is a p-group and p ∤ |H|.

Since N_{G}(P_{i}H) = N_{G}(P_{i})H for each i, N_{G}(R_{1}, …, R_{i}) = N_{G}(P_{1}, …, P_{i})H. Since P_{i} is radical in N_{G}(P_{1}, …, P_{i−1}) for each i, P_{i} is radical in N_{G}(P_{1}, …, P_{i−1}) by (a). Also, P_{k} ∈ Syl_{p}(N_{G}(R_{1}, …, R_{k−1})/H), so P_{k} ∈ Syl_{p}(N_{G}(P_{1}, …, P_{k−1})). Thus 1 < P_{1} < ⋯ < P_{k} is a radical p-chain in G. □

We need the following lower bounds for orders of radical subgroups of Ω_{2m}^{±}(q).

Lemma A.5. Let P be a radical 2-subgroup of G = Ω_{2m}^{±}(q), where m ≥ 2 and q = 2^{a}. Then |P | ≥ q^{2m−3}, and |P | ≥ q^{2m−2} if m ≥ 4.

Proof. By a theorem of Borel and Tits (see [GLS3, Corollary 3.1.5]), P is conjugate to the maximal normal unipotent subgroup in a parabolic subgroup N < G. We can assume that P is minimal, and hence that N is a maximal parabolic subgroup. Thus P = O_{2}(N), N contains a Borel subgroup and hence a Sylow 2-subgroup of G, and N is the stabilizer of a totally isotropic subspace of dimension ℓ for some 1 ≤ ℓ ≤ m (see [GL, pp. 100–101]). Let L ≅ N/O_{2}(N) be a Levi factor for N; then O^{2′}(L) ≅ SL_{ℓ}(q) × Ω_{2m−2ℓ}^{±}(q) ([GLS3, Example 3.2.3]).

Thus G has Sylow 2-subgroups of order q^{m(m−1)}, while N/O_{2}(N) has Sylow 2subgroups of order q^{ℓ(ℓ−1)/2}·q^{(m−ℓ)(m−ℓ−1)}. So |P | = q^{ℓ(4m−3ℓ−1)/2}. Since m ≥ ℓ ≥ 1,

ℓ(4m − 3ℓ − 1)/2 = (2m − 3) + (2m(ℓ − 1) − \frac{3}{2}ℓ^{2} − \frac{1}{2}ℓ + 3)
≥ (2m − 3) + (\frac{1}{2}ℓ(ℓ − 5) + 3) ≥ 2m − 3,

with equality only when ℓ = m ∈ {2, 3}. □

The remaining lemmas involve symmetric and alternating groups. For any m > 0, we set m = {1, …, m}, and regard A_{m} < Σ_{m} as the alternating and symmetric groups on the set m.

Lemma A.6. Assume 4|m, let {X_{1}, X_{2}} be a partition of m, and let σ ∈ Σ_{m} be a permutation which exchanges X_{1} and X_{2}. Set σ^{2} = τ_{i}τ_{2}, where τ_{i} is a permutation of X_{i} for i = 1, 2. Then σ and the τ_{i} have the same partity.

Proof. Assume σ is a product of disjoint cycles of length 2k_{1}, 2k_{2}, …, 2k_{r} (thus m = ∑_{i=1}^{r} 2k_{i}). Then τ_{1} and τ_{2} are each products of cycles of length k_{1}, …, k_{r}. Hence sgn(σ) = ∏_{i=1}^{r}(−1)^{2k_{i}−1} = (−1)^{m−r}, while sgn(τ_{i}) = ∏_{i=1}^{r}(−1)^{k_{i}−1} = (−1)^{(m/2)−r}. Since m and m/2 are both even, sgn(σ) = (−1)^{r} = sgn(τ_{i}). □

In the next two lemmas, when a group G acts on a set X, X/G denotes the set of G-orbits in X.

Lemma A.7. Assume G = Σ_{m} or A_{m} for some m ≥ 2. Let Q ⊴ P ≤ G be 2-subgroups such that Q < P, and either

(i) Q is radical in G and P is radical in N_{G}(Q); or

(ii) P is radical in G.

Then the action of P/Q on m/Q contains at most one free orbit.

Proof. Assume otherwise. Let X_{1}, …, X_{r} ⊆ m be the orbits under the action of P, arranged so that P/Q acts freely on X_{1}/Q and X_{2}/Q. For each X ⊆ m, let A_{X} ⊴ Σ_{X} be the alternating and symmetric groups on X, regarded as subgroups of Σ_{m} (groups of permutations of m which leave m∖X pointwise fixed). Set H = Σ_{X_{1}} × ⋯ × Σ_{X_{r}} ≤ Σ_{m}. Thus P ≤ H.

For each i = 1, …, r, let Q_{i} ≤ P_{i} ≤ Σ_{X_{i}} be the images of Q < P under the i-th projection. Set P = P_{1} ⋯ P_{r}, Q = Q_{1} ⋯ Q_{r}, P^{∗} = P ∩ G, and Q^{∗} = Q ∩ G. Thus Q ≤ Q^{∗} ≤ Q, P ≤ P^{∗} ≤ P, and Q ⊴ P ≤ H.

In case (ii), N_{P^{∗}}(P) ⊴ N_{G}(P), so N_{P^{∗}}(P) ≤ O_{2}(N_{G}(P)) = P, and P^{∗} = P by Lemma 1.14(a). In case (i), Q = Q^{∗} ⊴ P^{∗} by a similar argument,

N_{P^{∗}}(Q, P) = N_{P^{∗}}(P) ⊴ N_{G}(Q, P) ⟹ N_{P^{∗}}(P) ≤ O_{2}(N_{G}(Q, P)) = P ,

and again P^{∗} = P by Lemma 1.14(a).

Let I ⊆ {1, …, r} be the set of all i such that P_{i} ≰ Q, and choose σ_{i} ∈ P_{i}∖Q for i ∈ I. For each i ∈ I and σ ∈ P_{i}∖Q, σ ∉ P since it acts trivially on at least one of the sets X_{1} or X_{2} (and P/Q acts freely on (X_{1} ∪ X_{2})/Q), and hence σ is an odd permutation. Since P/Q acts nontrivially on X_{i}/Q for i = 1, 2, 1, 2 ∈ I. If i ∈ I for i ≥ 3, then σ_{1}σ_{i} ∈ P^{∗} = P since it is an even permutation, but it acts trivially on X_{2}/Q, which is a contradiction. Thus I = {1, 2}. Also, G = A_{m}.

For i ∈ I, P_{i} ∩ Q = P_{i} ∩ A_{X_{i}} has index two in P_{i}, while P_{i} ≤ Q ≤ A_{m} and hence P_{i} = Q_{i} ≤ A_{X_{i}} for i ∉ I. Hence for i ∈ I, Q_{i} = P_{i} or Q_{i} = P_{i} ∩ Q. Since Q ≤ G = A_{m}, either Q_{i} ≤ A_{X_{i}} for each i or Q_{i} ≰ A_{X_{i}} for at least two indices i. In the latter case, Q_{i} = P_{i} for i = 1, 2, so Q^{∗} = P^{∗} = P > Q, which would imply that the projections of Q into Σ_{X_{i}} for i = 1, 2 are Q ∩ P_{i} < Q_{i}, a contradiction. Thus Q_{i} ≤ Q for each i, so Q = Q = Q^{∗}. Then [P:Q] = ∏_{i=1}^{r}[P_{i}:Q_{i}] = 4, so [P:Q] = 2, and P = Q⟨σ⟩ where σ = σ_{1}σ_{2}.

For i = 1, 2, let Y_{i1}, Y_{i2} be the two orbits under the action of Q_{i} on X_{i}. Set σ_{i}^{2} = τ_{i1}τ_{i2}, where τ_{ij} ∈ Σ_{Y_{ij}}. By Lemma A.6 and since σ_{i} is an odd permutation, either |X_{i}| = 2, or τ_{i1}, τ_{i2} are also odd. Since Q_{i}⟨τ_{i1}⟩ ⊴ N_{Σ_{X_{i}}}(Q_{i}), and Q_{i} = 1 if |X_{i}| = 2, we get [O_{2}(N_{Σ_{X_{i}}}(Q_{i})):Q_{i}] ≥ 2 in either case. Hence

[O_{2}(N_{Σ_{m}}(Q)):Q] ≥ [O_{2}(N_{Σ_{X_{1}}}(Q_{1})):Q_{1}][O_{2}(N_{Σ_{X_{2}}}(Q_{2})):Q_{2}] ≥ 4 ,

Q is not radical in A_{m}, and (i) does not hold. So P is radical in G = A_{m}. Now, O_{2}(N_{Σ_{X_{1}}}(P_{1})N_{Σ_{X_{2}}}(P_{2})) ≤ O_{2}(N_{Σ_{m}}(P)). Since P is radical in A_{m},

[O_{2}(N_{Σ_{m}}(P)):P] ≤ 2 and [P:P] = 2 ⟹ O_{2}(N_{Σ_{X_{i}}}(P_{i})) = P_{i} for i = 1, 2.

So P_{i} is radical in Σ_{X_{i}} for i = 1, 2. Let R_{i} ⊴ P_{i} be the subgroup of elements of Q_{i} which act via even permutations on Y_{i1} and on Y_{i2}. If |X_{i}| > 2, then τ_{i1} and τ_{i2} are odd as just shown, so σ_{i}^{2} = τ_{i1}τ_{i2} ∉ R_{i}, and P_{i}/R_{i} ≅ C_{4}. But by [AF, Proposition (2A)], each radical 2-subgroup of Σ_{X_{i}} is an iterated wreath product of elementary abelian 2-groups, and hence P_{i}/[P_{i}, P_{i}] is elementary abelian. This is a contradiction, and we conclude that |X_{1}| = |X_{2}| = 2.

Thus P = Q⟨σ⟩, where Q acts trivially on X_{1} ∪ X_{2}, and σ acts on it as a product of two transpositions. Also, for each i ≥ 3, P_{i} contains only even permutations of X_{i} since P_{i} ≤ Q (i ∉ I). Hence each element of N_{G}(P) sends X_{1} ∪ X_{2} to itself, O_{2}(N_{G}(P)) ≥ O_{2}(A_{X_{1}∪X_{2}}) > ⟨σ⟩, and P is not radical in G. □

We thank two of the referees for suggesting the following lemma and proof, both simpler than those in the original version. Whenever X is a set with G-action, 𝔽_{2}(X) denotes the permutation module over the group ring 𝔽_{2}[G] with 𝔽_{2}-basis X.

Lemma A.8. Let G = Σ_{m} or A_{m}, and set V = 𝔽_{2}(m). Let Δ = C_{V} (Σ_{m}) ≤ V be the 1-dimensional submodule generated by the sum of the elements in m. Let P ≤ G be a radical 2-subgroup, and let U ≤ V be such that U/Δ = C_{V/Δ}(P).

(a) If P is not transitive on m, then U = C_{V} (P).

(b) If P is transitive on m and P_{0} is the stabilizer of some point in m, then U/Δ ≅ (P/Fr(P)P_{0})^{∗}. If, in addition, C_{P} (U) ∈ Syl_{p}(C_{G}(U)) and m ≥ 8, then N_{G}(P) acts on U/Δ via its full general linear group GL(U/Δ).

Proof. We identify V with the power set of m, with addition given by symmetric difference. Thus Δ = ⟨m⟩. If X ∈ U∖C_{V} (P), then |X| = m/2 and P acts transitively on the partition {X, X + m}.

(b) If m ≥ 8 and P is transitive on m, then C_{V} (P) = Δ, so all elements of U∖Δ are partitions as just described. The map R ↦ m/R defines a bijection from the set of subgroups of index two in P containing P_{0} to the set of partitions of order two on which P acts transitively, and thus a natural bijection

Ψ: (P/Fr(P)P_{0})^{∗} ⟶^{≅} C_{V/Δ}(P) = U/Δ .

If φ_{1}, φ_{2}, φ_{3} ∈ (P/Fr(P)P_{0})^{∗} are nonzero elements such that φ_{3} = φ_{1} + φ_{2}, then the Ker(φ_{i}) are the three subgroups of index two which contain some fixed subgroup R_{0} of index four, so the Ψ(φ_{i}) are the three partitions into sets of order m/2 refined by m/R_{0}, and Ψ(φ_{3}) = Ψ(φ_{1}) + Ψ(φ_{2}). Thus Ψ is an isomorphism.

We claim that

Aut_{P} (U) = {α ∈ Aut(U) | [α, U] ≤ Δ}  (1)
Aut_{G}(U/Δ) = GL(U/Δ) .  (2)

By definition (U/Δ = C_{V/Δ}(P)), Aut_{P} (U) is contained in the right hand side of (1). For each g ∈ P ∖Fr(P)P_{0}, there is R < P of index two which contains P_{0} but not g, and g exchanges the two orbits of R on m. Thus C_{P} (U) ≤ Fr(P)P_{0} so |Aut_{P} (U)| ≥ |P/Fr(P)P_{0}| = |U/Δ|. Since the right side of (1) has order |U/Δ|, this proves (1).

To see (2), fix α ∈ GL(U/Δ), and let β ∈ Aut(P/Fr(P)P_{0}) be such that β^{∗} = Ψ^{−1}αΨ. Choose an orbit X ∈ m/Fr(P)P_{0}, and choose any σ ∈ Σ_{m} such that σ(g(X)) = β(g)(X) for each g ∈ P/Fr(P)P_{0}. For each φ ∈ (P/Fr(P)P_{0})^{∗}, β sends Ker(β^{∗}(φ)) to Ker(φ), so σ sends Ψ(β^{∗}(φ)) = α(Ψ(φ)) to Ψ(φ). Thus σ normalizes U and induces the automorphism α^{−1} on U/Δ. So Aut_{Σ_{m}}(U/Δ) = GL(U/Δ). If |X| ≥ 2, then we can always arrange that σ ∈ A_{m} ≤ G. If |X| = 1, then m = |U/Δ| ≥ 8, so rk(U/Δ) ≥ 3, and GL(U/Δ) has no subgroup of index two. Thus (2) holds in either case.

Now assume C_{P} (U) ∈ Syl_{p}(C_{G}(U)). By definition of U, N_{G}(P) ≤ N_{G}(U), and in particular, P normalizes C_{G}(U). So P ∈ Syl_{p}(P C_{G}(U)). By (1) (and since each element of Aut_{G}(U) fixes Δ)), Aut_{P} (U) is normal in Aut_{G}(U). Hence P C_{G}(U) ⊴ N_{G}(U). By the Frattini argument (Lemma 1.14(b)), N_{G}(U) = P C_{G}(U)·N_{N_{G}(U)}(P) = C_{G}(U)·N_{G}(P). So by (2), Aut_{N_{G}(P)}(U/Δ) = Aut_{G}(U/Δ) = GL(U/Δ).

(a) Now assume U > C_{V} (P); i.e., C_{V/Δ}(P) > C_{V} (P)/Δ. Then there is a partition {X, X′} of m upon which P acts transitively. Let Q < P be the subgroup of index two which stabilizes X and X′. Then P/Q acts freely on m/Q, so it acts transitively on m/Q by Lemma A.7(ii), and P acts transitively on m. □

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LAGA, Institut Galilée, Av. J-B Clément, 93430 Villetaneuse, France

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