EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT ODD PRIMES

BOB OLIVER

Abstract. We prove here the Martino-Priddy conjecture for an odd prime p: the pcompletions of the classifying spaces of two groups G and G′ are homotopy equivalent if and only if there is an isomorphism between their Sylow p-subgroups which preserves fusion. A second theorem is a description for odd p of the group of homotopy classes of self homotopy equivalences of the p-completion of BG, in terms of automorphisms of a Sylow p-subgroup of G which preserve fusion in G. These are both consequences of a technical algebraic result, which says that for an odd prime p and a finite group G, all higher derived functors of the inverse limit vanish for a certain functor Z_{G} on the p-subgroup orbit category of G.

In an earlier paper [BLO1] in collaboration with Carles Broto and Ran Levi, we reduced certain problems involving equivalences between p-completed classifying spaces of finite groups to a question of whether certain obstruction groups vanish. The main technical result of this paper is that these groups do always vanish when p is odd. The proof of this result 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. Let

Z_{G}: O_{p}(G) ⟶ Ab

denote the functor Z_{G}(P) = Z(P) if P is p-centric in G, and Z_{G}(P) = 0 otherwise. We refer to [BLO1, §6] for more details on how this is made into a functor.

This paper is centered around the proof of the following theorem.

Theorem A. For any odd prime p and any finite group G,

\underleftarrow{lim}^{i}_{O_{p}(G)}(Z_{G}) = 0 for all i ≥ 1.

1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 20D05.

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

Partially supported by UMR 7539 of the CNRS.

Theorem A is proven as Theorem 4.5 below. It was motivated by applications for studying equivalences between p-completed classifying spaces of finite groups. 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 (cf. [BLO1, Proposition 2.1]). Conversely, by [BLO1, Proposition 6.1], given a fusion preserving isomorphism between Sylow psubgroups 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 odd primes). For any odd prime p, and any pair G and G′ of finite groups with Sylow p-subgroups S ≤ G and S′ ≤ G′, BG_{p}^{∧} ≃ BG′_{p}^{∧} 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 [BLO1, Theorem 6.2], gives the following description of Out(BG_{p}^{∧}).

Theorem C. For any odd prime p and any finite group G, with Sylow p-subgroup S ≤ G,

Out(BG_{p}^{∧}) ≅ Out_{fus}(S).

Theorem A should be a special case of a more general vanishing result, formulated here as Conjecture 2.2, where orbit categories of groups are replaced by orbit categories of arbitrary “saturated fusion systems” in the sense of Puig [Pu]. We refer to [BLO2, §1], and to the summary in Section 2 below, for definitions of saturated fusion systems. Conjecture 2.2 would, in particular, imply the existence and uniqueness of linking systems, hence of classifying spaces, associated to an arbitrary saturated fusion system over a p-group. This has motivated us to state results here, as far as possible, in the context of abstract saturated fusion systems. It is only at the end that we translate our partial results to a condition on simple groups, which is then checked in the individual cases.

When p = 2, Theorem A is not true, since \underleftarrow{lim}^{1}(Z_{G}) can be nonzero. The simplest counterexamples occur for G = P SL_{2}(q), when q ≡ ±1 (mod 8). Recently, we have proved that \underleftarrow{lim}^{i}(Z_{G}) = 0 for all i ≥ 2, when p = 2 and G is an arbitrary finite group. This means that the Martino-Priddy conjecture does hold for p = 2, but that Theorem C is not true (as formulated above) in this case. The proof for p = 2 not only requires the classification theorem for finite simple groups, but also (in its current form) requires a long, detailed case-by-case check when handling the simple groups of Lie type in odd characteristic as well as the sporadic groups. For this reason, we have not tried to incorporate it into this paper, but will write it up separately.

Section 1 contains general material about higher limits over orbit categories of finite groups, and Section 2 some results about saturated fusion systems and higher limits over their orbit categories. Concrete criteria for proving the acyclicity of Z_{F} are then set up in Section 3, where the problem is reduced to a question about “simple” fusion systems (Proposition 3.8). Also, at the end of Section 3, there is a discussion of what further results would be necessary to prove Conjecture 2.2 for odd primes. Finally, in Section 4, we restrict attention to fusion systems of finite groups, and apply the classification theorem for finite simple groups to finish the proof of Theorem A.

I would like to thank George Glauberman for his encouragement, and his efforts to prove a result about p-groups (Conjecture 3.9 below) which would have led to a proof of Conjecture 2.2 for odd p, and in particular to a “classification free” proof of Theorem A. I also want to point out the importance to this work of Jesper Grodal’s techniques in [Gr] for computing higher limits of functors on orbit categories. His main theorem, while not used here directly, was used in many of the computations which led to this proof. Finally, I thank Carles Broto and Ran Levi, not only for their collaboration in the papers [BLO1] and [BLO2] which are closely connected to this one, but also for introducing me to this problem in the first place.

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

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

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

• Ω_{n}(P) = ⟨g ∈ P | g^{p^{n}} = 1⟩ (for a p-group P)

• 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) (for H, K ≤ G) is the set of homomorphisms from H to K induced by conjugation in G

• Rep(H, K) = Inn(K)\Hom(H, K) and Rep_{G}(H, K) = Inn(K)\Hom_{G}(H, K)

• Aut_{G}(H) = Hom_{G}(H, H), and Out_{G}(H) = Rep_{G}(H, 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.

1. Higher limits over orbit categories of groups

We first collect some tools for computing higher limits of functors over the orbit category of a finite group G. Very roughly, these reduce to two general techniques. One is to filter a functor by a sequence of subfunctors, such that each of the subquotients vanishes except on one conjugacy class of p-subgroups of G. Proposition 1.1 then gives some tools which are very effective when computing the higher limits of these subquotients. The other method is to reduce computations to a situation, described in Proposition 1.3, where the functor extends to a Mackey functor, and hence is acyclic by a theorem of Jackowski and McClure [JM].

Fix a prime p, a finite group G, and a ℤ_{(p)}[G]-module M. Let F_{M} be the functor on O_{p}(G) defined by setting F_{M}(P) = M^{P} (the fixed submodule), and define

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

These graded groups were shown in [JMO] to be very effective tools when computing higher limits over functors on orbit categories. We first summarize the properties of the Λ^{∗} which will be needed here.

Proposition 1.1. Fix a prime p. Then the following hold.

(a) For any finite group G and any functor F : O_{p}(G)^{op} ⟶ ℤ_{(p)}-mod which vanishes except on subgroups conjugate to some given p-subgroup P ≤ G,

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

(b) If G is a finite group, H ◁ G is a normal subgroup which acts trivially on the ℤ_{(p)}[G]-module M, and p||H|, then Λ^{∗}(G; M) = 0.

(c) If G is a finite group, and H ◁ G is a normal subgroup of order prime to p which acts trivially on the ℤ_{(p)}[G]-module M, then

Λ^{∗}(G; M) ≅ Λ^{∗}(G/H; M).

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

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

The idea now is to filter an arbitrary functor F : O_{p}(G) → ℤ_{(p)}-mod in such a way that all quotient functors vanish except on one conjugacy class, and hence are described via Proposition 1.1(a).

We next look for some conditions on a pair of finite groups H ≤ G, and a functor F on O_{p}(G), which reduce the computation of \underleftarrow{lim}^{∗}(F) to one of higher limits of a functor over O_{p}(N(H)/H). In general, for any small categories C and D and any functors

C ⟶^{Φ} D ⟶^{F} Ab,

there is an induced homomorphism

\underleftarrow{lim}_{D}^{∗}(F) ⟶^{Φ^{∗}} \underleftarrow{lim}_{C}^{∗}(F ∘ Φ)

defined as follows. Let I^{∗} be an injective resolution of F (in the category of functors D → Ab), and let Î^{∗} be an injective resolution of F ∘ Φ. Since I^{∗} ∘ Φ is a resolution of F ∘ Φ (though not injective), there is a chain homomorphism I^{∗} ∘ Φ ⟶^{ψ^{∗}} Î^{∗} which extends the identity on F ∘ Φ, and which is unique up to chain homotopy. Then Φ^{∗} is the homology of the composite homomorphism

\underleftarrow{lim}_{D}(I^{∗}) ⟶ \underleftarrow{lim}_{C}(I^{∗} ∘ Φ) ⟶^{lim(ψ^{∗})} \underleftarrow{lim}_{C}(Î^{∗}),

where the first map is induced by the universal property of inverse limits over C.

Lemma 1.2. Fix a finite group G and a p-subgroup Q ≤ G. Then there is a well defined functor

Φ = Φ_{Q}^{G}: O_{p}(N_{G}(Q)/Q) ⟶ O_{p}(G)

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

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

is an isomorphism.

Proof. Clearly, Φ 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 Φ is defined on morphism sets to be this inclusion.

Composition with Φ 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. Hence it 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 ∘ Φ vanishes except on the O_{p}(N_{G}(Q)/Q)-isomorphism class of P/Q. Let

Ψ = Φ_{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{^{Φ^{∗}} →} \underleftarrow{lim}^{∗}_{O_{p}(N(Q)/Q)}(F ∘ Φ)

Λ^{∗}(N_{G}(P)/P; F (P)) ↓^{(Φ∘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 Φ^{∗} is an isomorphism.

The next proposition describes a different condition which implies the acyclicity of a functor on the orbit category of a finite group.

Proposition 1.3. Fix a finite group G, a prime p, and a ℤ_{(p)}[G]-module M, and let

H^{0}M: O_{p}(G)^{op} ⟶ ℤ_{(p)}-mod,

be the functor defined by setting

H^{0}M(P) = H^{0}(P; M) = M^{P} .

Let

F: O_{p}(G)^{op} ⟶ ℤ_{(p)}-mod

be any subfunctor of H^{0}M (thus F (P) ≤ M^{P} for all P) which satisfies the following “relative norm property”: for each pair of p-subgroups P ≤ Q ≤ G,

𝔑_{P}^{Q}(F (P)) =^{def} { ∑_{gP ∈Q/P} gx | x ∈ F (P) ≤ M^{P} } ⊆ F (Q).  (∗)

Then F is acyclic: \underleftarrow{lim}^{i}(F) = 0 for all i > 0.

Proof. The relative norms 𝔑_{P}^{Q} make F into a proto-Mackey functor in the sense of [JM], and hence it is acyclic by [JM, Proposition 5.14].

The following application of Proposition 1.3 plays an important role in Section 3. If G is a finite group and S ∈ Syl_{p}(G), then O_{S}(G) ⊆ O_{p}(G) denotes the full subcategory whose objects are the subgroups of S (the inclusion is clearly an equivalence of categories). As usual, a subgroup T ≤ S is called strongly closed in S with respect to G if no element of T is G-conjugate to any element of S∖T.

Recall that for any p-group P and any n ≥ 1, Ω_{n}(P) denotes the subgroup generated by all x ∈ P such that x^{p^{n}} = 1.

Proposition 1.4. Fix a finite group G, a Sylow subgroup S ∈ Syl_{p}(G), and a subgroup T ≤ S which is strongly closed in S with respect to G. Let M be a finite ℤ_{(p)}[G]-module, and let

F: O_{S}(G)^{op} ⟶ Ab

be a subfunctor of H^{0}M (in particular, F (P) ≤ M^{P} for all P), which satisfies the relative norm property:

𝔑_{P}^{Q}(F (P)) ≤ F (Q)  (∗)

for each pair of subgroups P ≤ Q ≤ S. Let F_{1} ⊆ F be the subfunctor

F_{1}(P) = \begin{cases}F (P) & if P ∩ T = 1 \\ 0 & otherwise. \end{cases}

Assume 𝔑_{Z(T)}·Ω_{1}(M) = 0. Then \underleftarrow{lim}^{i}_{O_{S}(G)}(F_{1}) = 0 for all i ≥ 1.

Proof. Note first that F_{1} is a functor: if P, P′ ≤ S are G-conjugate, then P ∩ T = 1 if and only if P′ ∩ T = 1 since T is strongly closed.

Assume \underleftarrow{lim}^{n}(F_{1}) ≠ 0 for some n ≥ 1. We must prove that 𝔑_{Z(T)}·Ω_{1}(M) ≠ 0. This will be shown by induction on n.

Let Ω_{k}F ⊆ F be the p^{k}-torsion subfunctor in F; i.e., (Ω_{k}F)(P) = Ω_{k}(F (P)) for each P. Then for some k, \underleftarrow{lim}^{n}(Ω_{k}F/Ω_{k−1}F) ≠ 0. This functor Ω_{k}F/Ω_{k−1}F satisfies all of the hypotheses of the proposition with respect to the 𝔽_{p}[G]-module Ω_{k}(M)/Ω_{k−1}(M) ⊆ Ω_{1}(M). It thus suffices to prove the proposition when M = Ω_{1}(M); i.e., when pM = 0.

Without loss of generality, we can assume M = F (1). Define a functor F ⊆ F by setting

F (P) = F (P) ∩ (𝔑_{P ∩T} ·M)

for all P ≤ S. We claim that F still satisfies condition (∗) above. To see this, fix subgroups P ≤ Q ≤ S, set P′ = P ∩ T ◁ P and Q′ = Q ∩ T ◁ Q, and set Q′′ = P Q′. Then P′ = P ∩Q′, so coset representatives for Q′/P′ are also representatives for Q′′/P. Hence

𝔑_{P}^{Q}(F (P)) = 𝔑_{P}^{Q}(F (P) ∩ 𝔑_{P′}·M) ⊆ 𝔑_{P}^{Q}(F (P)) ∩ 𝔑_{P′}^{Q′}(𝔑_{P′}·M)

⊆ F (Q) ∩ (𝔑_{Q′}·M) = F (Q).

Thus, upon replacing F by F (without changing F_{1}), we can assume that F (P) = 𝔑_{P} ·M for all P ≤ T.

By Proposition 1.3, \underleftarrow{lim}^{i}(F) = 0 for all i > 0. Hence \underleftarrow{lim}^{n−1}(F/F_{1}) ≠ 0, since \underleftarrow{lim}^{n}(F_{1}) ≠ 0 by assumption. For each subgroup Q ≤ T, let F_{Q} be the functor on O_{S}(G) defined by

F_{Q}(P) = \begin{cases}F (P) & if P ∩ T is G-conjugate to Q \\ 0 & otherwise. \end{cases}

There is an obvious filtration of F/F_{1} whose quotients are all isomorphic to F_{Q} for various 1 ≠ Q ≤ T. Hence there is some Q ≤ T such that

\underleftarrow{lim}^{n−1}_{O_{S}(G)}(F_{Q}) ≠ 0.  (1)

Since we can replace Q by any other subgroup of S in its G-conjugacy class, we can assume that N_{S}(Q) ∈ Syl_{p}(N_{G}(Q)).

If n = 1, then (1) implies that F_{Q}(S) ≠ 0. Hence S ∩ T = Q, so Q = T, and 0 ≠ F_{T} (S) ⊆ F_{T} (T) = 𝔑_{T} ·M. In particular, 𝔑_{Z(T)}·M ≠ 0.

Now assume n > 1. Set G′ = N_{G}(Q)/Q, S′ = N_{S}(Q)/Q ∈ Syl_{p}(G′), and T′ = N_{T} (Q)/Q. Then T′ is strongly closed in S′ with respect to G′: no element of N_{T} (Q) is N_{G}(Q)-conjugate to any element of N_{S}(Q)∖N_{T} (Q) since no element of T is G-conjugate to any element of S∖T. Define functors

F′, F′_{1} : O_{S′}(G′) ⟶ Ab

by setting

F′(P/Q) = F (P) and F′_{1}(P/Q) = F_{Q}(P).

Thus F′_{1}(P/Q) = F′(P/Q) whenever P ∩T = Q, equivalently whenever (P/Q)∩T′ = 1; and F′_{1}(P/Q) = 0 otherwise.

Consider the set

T_{0} = {P ≤ S | P ∩ T = Q}.

If F_{Q}(P) ≠ 0 (and P ≤ S), then P ∩ T is G-conjugate to Q, P is G-conjugate to some P′ such that Q ≤ P′ ≤ N_{S}(Q) (since N_{S}(Q) ∈ Syl_{p}(N_{G}(Q))), P′ ∩ T is G-conjugate to P ∩ T since T is strongly closed, and thus P′ ∩ T = Q and P′ ∈ T_{0}. By a similar argument, if P ∈ T_{0} and Q ◁ xP x^{−1}, then (yx)P (yx)^{−1} ≤ N_{S}(Q) for some y ∈ N_{G}(Q), (yx)P (yx)^{−1} ∩ T = Q, and so x ∈ N_{G}(Q). This shows that T_{0} is contained in the set T defined in Lemma 1.2, and thus that each subgroup of G for which F_{Q}(P) ≠ 0 is G-conjugate to a subgroup in T . Hence by Lemma 1.2,

\underleftarrow{lim}^{∗}_{O_{S′}(G′)}(F′_{1}) ≅ \underleftarrow{lim}^{∗}_{O_{S}(G)}(F_{Q}).

In particular, \underleftarrow{lim}^{n−1}(F′_{1}) ≠ 0 by (1). All of the conditions of the proposition are satisfied (with G, S, T, F, and M replaced by G′, S′, T′, F′, and M′ =^{def} F (Q) = 𝔑_{Q}·M). So by the induction hypothesis, if we set Z′/Q = Z(T′), then

𝔑_{Z′}·M = 𝔑_{Z(T′)}·(𝔑_{Q}·M) = 𝔑_{Z(T′)}·M′ ≠ 0,

and hence 𝔑_{Z(T)}·M ≠ 0 since Z(T) ≤ Z′.

2. Higher limits over orbit categories of fusion systems

We first briefly recall some definitions. We refer to [BLO2, §1] or [Pu] for more details.

A fusion system over a finite p-group S is a category F whose objects are the subgroups of S, and whose morphisms satisfy the following conditions:

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

• each morphism in F is the composite of an F-isomorphism followed by an inclusion.

To emphasize that the morphisms in F are all homomorphisms of groups, we write Hom_{F}(P, Q) = Mor_{F}(P, Q) for the morphism sets. Two subgroups of F are called F-conjugate if they are isomorphic in F. A subgroup P ≤ S is called fully centralized in F (fully normalized in F) if |C_{S}(P)| ≥ |C_{S}(P′)| (|N_{S}(P)| ≥ |N_{S}(P′)|) for all P′ in the F-conjugacy class of P. The fusion system F is saturated if

(I) for each fully normalized subgroup P ≤ S, P is fully centralized and Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)); and

(II) for each φ ∈ Hom_{F}(P, S) whose image is fully centralized in F, if we set

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

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

If G is a finite group and S ∈ Syl_{p}(G), then we let F_{S}(G) denote the category whose objects are the subgroups of S, and where

Mor_{F_{S}(G)}(P, Q) = Hom_{G}(P, Q) ≅ N_{G}(P, Q)/C_{G}(P).

It is not hard to see [BLO2, Proposition 1.3] that F_{S}(G) is a saturated fusion system over S, and that a subgroup P ≤ S is fully centralized (fully normalized) if and only if C_{S}(P) ∈ Syl_{p}(C_{G}(P)) (N_{S}(P) ∈ Syl_{p}(N_{G}(P))).

By analogy with the orbit category of a finite group, when F is a saturated fusion system over a p-group S, we let O(F) (the orbit category of F) be the category with the same objects, and with morphism sets

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

If F = F_{S}(G) for some finite group G, then O(F) is a quotient category of O_{S}(G) (the full subcategory of O_{p}(G) whose objects are the subgroups of S), but its morphism sets are much smaller in general. More precisely, if P and Q are two p-subgroups of G, then

Mor_{O_{p}(G)}(P, Q) ≅ Q\N_{G}(P, Q),

while

Mor_{O(F_{p}(G))}(P, Q) ≅ Q\N_{G}(P, Q)/C_{G}(P).

Thus, there is a natural projection functor O_{p}(G) ⟶^{Φ} O(F_{p}(G)) which is the identity on objects and a surjection on all morphism sets, but these maps of morphism sets are very far in general from being bijections. However, the next lemma shows that if one restricts to p-centric subgroups of G, then p-local functors over these two categories have the same higher limits.

If F is a saturated fusion system over S, then a subgroup P ≤ S is F-centric if C_{S}(P′) = Z(P′) for all P′ F-conjugate to P. If F = F_{S}(G), then a subgroup is Fcentric if and only if it is p-centric in G (see [BLO1, Lemma A.5]). Let F^{c} ⊆ F and O(F^{c}) ⊆ O(F) be the full subcategories whose objects are the F-centric subgroups of S. Similarly, for any finite group G, and any S ∈ Syl_{p}(G), we let O_{p}^{c}(G) ⊆ O_{p}(G) and O_{S}^{c}(G) ⊆ O_{S}(G) be the full subcategories whose objects are the p-centric subgroups of G, and those contained in S, respectively.

Lemma 2.1. Fix a prime p and a finite group G. Let

F: O(F_{p}^{c}(G))^{op} ⟶ ℤ_{(p)}-mod

be any functor, and let

Φ: O_{p}^{c}(G) ⟶ O(F_{p}^{c}(G))

be the projection functor. Define

F̂: O_{p}(G)^{op} ⟶ ℤ_{(p)}-mod

by setting F̂|_{O_{p}^{c}(G)} = F ∘ Φ, and F̂(P) = 0 if P is not p-centric in G. Then

\underleftarrow{lim}^{∗}_{O(F_{p}^{c}(G))}(F) ≅ \underleftarrow{lim}^{∗}_{O_{p}^{c}(G)}(F ∘ Φ) ≅ \underleftarrow{lim}^{∗}_{O_{p}(G)}(F̂).  (1)

Proof. For any pair of p-centric subgroups P, Q ≤ G, write C_{G}(P) = Z(P) × C′_{G}(P) where C′_{G}(P) has order prime to p. For any x ∈ N_{G}(P, Q) and a ∈ Z(P), xa = (xax^{−1})x ∈ Qx, since xax^{−1} ∈ Q by definition of the transporter. Thus

Mor_{O(F_{p}^{c}(G))}(P, Q) = Q\N_{G}(P, Q)/C_{G}(P)

≅ Q\N_{G}(P, Q)/C′_{G}(P) ≅ Mor_{O_{p}^{c}(G)}(P, Q)/C′_{G}(P).

Since C′_{G}(P) has order prime to p, the first isomorphism in (1) now follows as an immediate consequence of [BLO1, Lemma 1.3]. The second isomorphism holds since F̂ vanishes on all p-subgroups which are not p-centric, and since every p-subgroup of G which contains a p-centric subgroup is also p-centric.

For any saturated fusion system F over a finite p-group S, let

Z_{F}: O(F^{c})^{op} ⟶ Ab

denote the functor Z_{F}(P) = Z(P) for all F-centric subgroups P ≤ S. If φ ∈ Hom_{F}(P, Q), then Z_{F}(φ) is the composite

Z(Q) ≤ Z(φ(P)) ⟶^{φ^{−1}} Z(P).

If F = F_{S}(G) for some finite group G with Sylow p-subgroup S, then Z_{G}|_{O_{S}(G)} is the composite of Z_{F} with the projection between orbit categories. So Lemma 2.1 implies as a special case that

\underleftarrow{lim}^{∗}_{O_{p}(G)}(Z_{G}) ≅ \underleftarrow{lim}^{∗}_{O(F^{c})}(Z_{F}).

What we would like to prove is the following conjecture, of which Theorem A is just the special case where F = F_{S}(G) and p is odd.

Conjecture 2.2. Fix a prime p, and let F be a saturated fusion system over a p-group S. Then

\underleftarrow{lim}^{i}_{O(F^{c})}(Z_{F}) = 0

if p is odd and i ≥ 1, or if p = 2 and i ≥ 2.

Conjecture 2.2 would imply that each saturated fusion system over a p-group S has a unique associated “linking system”, in the sense of [BLO2, §1], and hence a unique associated classifying space (see [BLO2, Proposition 3.1]). The vanishing of \underleftarrow{lim}^{1}(Z_{F}) would also imply (when p is odd) a description of the group of homotopy classes of self equivalences of the classifying space, similar to the description of Out(BG_{p}^{∧}) in Theorem C.

Throughout this section and the next, we will be developping tools for computing higher limits of functors on centric orbit categories of saturated fusion systems; in particular, those with connections to Conjecture 2.2. Only in the last section do we again return to the special case of fusion systems of finite groups, and finish the proof of Theorem A.

If F is any saturated fusion system over a p-group S, and Q ≤ S is fully normalized in F, then N_{F}(Q) is defined to be the fusion system over N_{S}(Q) whose morphisms are defined by the formula

Hom_{N_{F} (Q)}(P, P′) = {α|_{P} | α ∈ Hom_{F}(P Q, P′Q), α(P) ≤ P′, α(Q) = Q}.

By [BLO2, Proposition A.6], this is a saturated fusion system over N_{S}(Q). We also let O^{≥Q}(N_{F}(Q)) denote the full subcategory of the orbit category of N_{F}(Q) whose objects are the subgroups which contain Q.

Lemma 2.3. Fix a saturated fusion system F over a p-group S, and a fully normalized F-centric subgroup Q ≤ S. Consider the functor

Ψ = Ψ_{Q}^{F}: O^{≥Q}(N_{F}(Q)) ⟶ O_{Out_{S}(Q)}(Out_{F}(Q))

defined by setting

Ψ(P) = Out_{P} (Q) and Ψ(P ⟶^{α} P′) = [α|_{Q}].

Then Ψ is an isomorphism of categories. Hence there is a functor

Φ = Φ_{Q}^{F}: O_{p}(Out_{F}(Q)) ⟶ O(F^{c}),

unique up to natural isomorphism, whose restriction to O_{Out_{S}(Q)}(Out_{F}(Q)) is equal to Φ^{−1}.

Proof. Write Γ = Out_{F}(Q) and S′ = Out_{S}(Q) for short. Since Q is fully normalized in F, S′ is a Sylow p-subgroup of Γ (condition (I) in the definition of a saturated fusion system), and so the inclusion O_{S′}(Γ) ⊆ O_{p}(Γ) is an equivalence of categories.

Now, S′ ≅ N_{S}(Q)/Q since Q is F-centric in S. So Ψ defines a bijection between objects of O^{≥Q}(N_{F}(Q)) and objects of O_{S′}(Γ), sending P to Out_{P} (Q) ≅ P/Q.

Fix subgroups P, P′ ≤ N_{S}(Q) containing Q, and consider the function

Ψ_{P,P′}: Rep_{N_{F} (Q)}(P, P′) = Mor_{O(N_{F} (Q))}(P, P′) ⟶ Mor_{O_{p}(Γ)}(Out_{P} (Q), Out_{P′}(Q))

which sends the class [α], for α ∈ Hom_{N_{F} (Q)}(P, P′), to the class of [α|_{Q}] ∈ Out_{F}(Q) = Γ. For any such α, the following square commutes

\begin{matrix}P & \overline{^{α} →} & P′ \\ P & ↓^{c_{g}} & \overline{^{α} →} & P′ & ↓^{c_{α(g)}} \end{matrix}

for all g ∈ P, so α lies in the transporter N_{Γ}(Out_{P} (Q), Out_{P′}(Q)), and the map Ψ_{P,P′} is well defined. If β ∈ Aut_{F}(Q) is such that conjugation by [β] ∈ Γ = Out_{F}(Q) sends Out_{P} (Q) into Out_{P′}(Q), then βc_{g}β^{−1} ∈ Aut_{P′}(Q) for all g ∈ P, so β extends to some α ∈ Hom_{F}(P, P′) by condition (II) in the definition of a saturated fusion system, and Ψ_{P,P′} sends [α] to [β]. Thus Ψ_{P,P′} is onto. If α_{1}, α_{2} ∈ Hom_{N_{F} (Q)}(P, P′) are such that Ψ_{P,P′}([α_{1}]) = Ψ_{P,P′}([α_{2}]) in O_{p}(Γ), then α_{1}|_{Q} = c_{g} ∘ α_{2}|_{Q} for some g ∈ Q, hence α_{1} = c_{g} ∘ α_{2} ∘ c_{z} for some z ∈ Z(Q) by [BLO2, Proposition A.8], and so [α_{1}] = [α_{2}] in Rep_{F}(P, P′). Thus, Ψ_{P,P′} is a bijection for each pair of objects P, P′, and this finishes the proof that Ψ is an isomorphism of categories.

The last statement now follows by letting Φ be the composite of a retraction of O_{p}(Γ) onto O_{S′}(Γ), followed by Ψ^{−1}, followed by the inclusion of O^{≥Q}(N_{F}(Q)) into O(F^{c}).

The next proposition describes how higher limits over O(F^{c}) can be reduced in certain cases to higher limits over the orbit category of Out_{F}(Q) for some subgroup Q. Note its similarity with Lemma 1.2, in both the statement and the proof.

By analogy with the usual definition for subgroups of finite groups, for any saturated fusion system F over a p-group S, a subgroup P ≤ S is called weakly F-closed (or weakly F-closed in S) if P is not F-conjugate to any other subgroup of S.

Proposition 2.4. Fix a saturated fusion system F over a p-group S and a fully normalized F-centric subgroup Q ≤ S, and let

Φ = Φ_{Q}^{F}: O_{p}(Out_{F}(Q)) ⟶ O(F^{c})

be the functor of Lemma 2.3. Let T be the set of all subgroups P ≤ S such that

Q ◁ P, and Q ◁ α(P) for α ∈ Hom_{F}(P, S) implies α(Q) = Q.  (∗)

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

\underleftarrow{lim}^{∗}_{O(F^{c})}(F) ⟶^{Φ^{∗}} \underleftarrow{lim}^{∗}_{O_{p}(Out_{F} (Q))}(F ∘ Φ)  (1)

is an isomorphism. In particular, if Q is weakly F-closed in S, then (1) holds for any functor F which vanishes except on subgroups which contain Q.

Proof. Composition with Φ is natural in F and preserves short exact sequences of functors. If F′ ⊆ F is a pair of functors from O(F^{c}) to ℤ_{(p)}-mod, and the lemma holds for F′ and for F/F′, then it also holds for F by the 5-lemma. Hence it suffices to prove that (1) is an isomorphism when F vanishes except on the F-conjugacy class of one subgroup P ∈ T .

Fix P ∈ T , and set P = Out_{P} (Q) ≤ Out_{F}(Q). By condition (∗), Q ◁ P (so P ≅ P/Q), and F ∘Φ vanishes except on the O_{p}(Out_{F}(Q))-isomorphism class of Out_{P} (Q) ≅ P/Q. Also, by (∗) again,

Out_{F}(P) ≅ Out_{N_{F} (Q)}(P),

and

Out_{N_{F} (Q)}(P) = Aut_{O_{p}(N_{F} (Q))}(P) ≅ Aut_{O_{p}(Out_{F} (Q))}(P) = N_{Out_{F} (Q)}(P)/P

by Lemma 2.3. Let

Ψ = Φ_{Out_{P} (Q)}^{Out_{F} (Q)} : O_{p}(Out_{F}(P)) ⟶ O_{p}(Out_{F}(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(F)}(F)  \overline{^{Φ^{∗}} →} \underleftarrow{lim}^{∗}_{O_{p}(Out_{F} (Q))}(F ∘ Φ)

Λ^{∗}(Out_{F}(P); F (P)) ↓^{(Φ∘Ψ)^{∗} ≅}  →^{≅} Λ^{∗}(N_{Out_{F} (Q)}(P)/P; F (P)) , ↓^{Ψ^{∗} ≅}

and the vertical maps are isomorphisms by [BLO2, Proposition 3.2] (and its proof) and Proposition 1.1(a). It follows that Φ^{∗} is an isomorphism.

The last statement follows since if Q is weakly F-closed in S, then T = {P ≤ S | P ≥ Q}: every subgroup which contains Q satisfies (∗).

The following lemma describes how quotient fusion systems are obtained by dividing out by weakly F-closed subgroups.

Lemma 2.5. Let F be a saturated fusion system over a p-group S, and let Q ◁ S be a weakly F-closed subgroup. Let F/Q be the fusion system over S/Q defined by setting

Hom_{F/Q}(P/Q, P′/Q) = {φ/Q | φ ∈ Hom_{F}(P, P′)}

for all P, P′ ≤ S which contain Q. Then F/Q is saturated. Also, for any P/Q ≤ S/Q, P/Q is fully normalized in F/Q if and only if P is fully normalized in F, while P is fully centralized in F whenever P/Q is fully centralized in F/Q.

Proof. For each P ≤ S which contains Q, set

K_{P} = Ker[Aut_{F}(P) −↠ Aut_{F/Q}(P/Q)].

and

K_{P}^{0} = Ker[Aut_{S}(P) −↠ Aut_{S/Q}(P/Q)].

Then

|C_{S/Q}(P/Q)| = |C_{S}(P)|·|K_{P}^{0} |/|Q| and |N_{S/Q}(P/Q)| = |N_{S}(P)|/|Q|. (1)

By the second formula, P/Q is fully normalized in F/Q if and only if P is fully normalized in F.

Assume P/Q is fully normalized in F/Q. Then P is fully normalized in F, so by condition (I) in the definition of a saturated fusion system applied to F, P is fully centralized in F and Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)). This last condition implies that

K_{P}^{0} ∈ Syl_{p}(K_{P}) and Aut_{S/Q}(P/Q) ∈ Syl_{p}(Aut_{F/Q}(P/Q)).

Thus |C_{S}(P)| and |K_{P}^{0} | both take the largest possible values among subgroups in the F-conjugacy class of P, and hence P/Q is fully centralized by (1). This finishes the proof that condition (I) holds for F/Q. It also shows that if P/Q is fully centralized in F/Q, then |C_{S}(P)| and |K_{P}^{0} | must both take the largest possible values among subgroups in the F-conjugacy class of P, and in particular P is fully centralized in F.

To prove condition (II), fix a morphism φ/Q ∈ Hom_{F/Q}(P/Q, S/Q) such that φ(P)/Q is fully centralized in F/Q, and set

Ñ_{φ} = {g ∈ N_{S}(P) | φc_{g}φ^{−1} ∈ K_{φ(P)}· Aut_{S}(φ(P))}.

Then

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

and we must show that φ/Q extends to Ñ_{φ}/Q. Set P′ = φ(P) for short; P′ is fully centralized in F since P′/Q is fully centralized in F/Q. Since

φ· AutÑ_{φ}(P)·φ^{−1} ≤ K_{P′}· Aut_{S}(P′),

where K_{P′} ◁ Aut_{F}(P′), Aut_{S}(P′) ∈ Syl_{p}(Aut_{F}(P′)), and the left hand side is a pgroup, there is ψ ∈ K_{P′} such that

(ψφ)· AutÑ_{φ}(P)·(ψφ)^{−1} ≤ Aut_{S}(P′).

So by condition (II) for the saturated fusion system F, ψφ extends to a homomorphism φ ∈ Hom_{F}(Ñ_{φ}, S), and φ/Q is an extension of φ/Q to N_{φ/Q}.

3. Reduction to simple fusion systems

In this section, we establish a sufficient condition for proving the acyclicity of Z_{F}: a criterion which in the case F = F_{S}(G) will depend only on the simple components in the decomposition series of the finite group G.

Recall that for any p-group P and any n ≥ 1, Ω_{n}(P) denotes the subgroup of P generated by p^{n}-torsion elements.

If H and K are two subgroups of a group G (usually normal subgroups) and n ≥ 1, then we write [H, K; n] for the n-fold iterated commutator: [H, K; 1] = [H, K], [H, K; 2] = [[H, K], K], and [H, K; n+1] = [[H, K; n], K].

Definition 3.1. For any p-group S, 𝔛(S) denotes the largest subgroup of S for which there is a sequence

1 = Q_{0} ≤ Q_{1} ≤ ⋯ ≤ Q_{n} = 𝔛(S) ≤ S

of subgroups, all normal in S, such that

[Ω_{1}(C_{S}(Q_{i−1})), Q_{i}; p−1] = 1  (1)

for each i = 1, …, n.

It is easy to see that there always is such a largest subgroup. If

1 = Q_{0} ≤ Q_{1} ≤ ⋯ ≤ Q_{n} and 1 = Q′_{0} ≤ Q′_{1} ≤ ⋯ ≤ Q′_{m}

are two sequences of normal subgroups of S which satisfy condition (1) in Definition 3.1, then the sequence

1 = Q_{0} ≤ Q_{1} ≤ ⋯ ≤ Q_{n} = Q_{n}·Q′_{0} ≤ Q_{n}·Q′_{1} ≤ ⋯ ≤ Q_{n}·Q′_{n}

also satisfies the same condition.

When p = 2, 𝔛(S) = C_{S}(Ω_{1}(S)) for any finite 2-group S. In particular, 𝔛(S) = Z(S) if S is generated by elements of order 2. So these subgroups are not very interesting in that case.

We first note some elementary properties of these subgroups 𝔛(S):

Lemma 3.2. If p is odd and S is a p-group, then 𝔛(S) ≥ A for every normal abelian subgroup A ◁ S. In particular, 𝔛(S) is centric in S.

Proof. If A ◁ S is abelian, then [S, A; p−1] ≤ [[S, A], A] = 1, and so A ≤ 𝔛(S) by definition.

Now let A be maximal among the normal abelian subgroups of S. If C_{S}(A) ≰ A, then A·C_{S}(A)/A is a nontrivial normal subgroup of S/A, and hence contains an element xA ∈ Z(S/A) of order p. But then ⟨A, x⟩ is a larger normal abelian subgroup of S, which is a contradiction. Thus A is centric in S, and in particular 𝔛(S) ≥ A is centric in S.

The following lemma is useful when proving that certain subgroups of S are contained in 𝔛(S).

Lemma 3.3. Fix an odd prime p and a p-group S. Let Q ◁ S be any normal subgroup such that

[Ω_{1}(Z(𝔛(S))), Q; p−1] = 1.  (1)

Then 𝔛(S) ≥ Q.

Proof. Set 𝔛 = 𝔛(S) for short. By definition, there is a sequence

1 = Q_{0} ≤ Q_{1} ≤ ⋯ ≤ Q_{n} = 𝔛

of subgroups normal in S, such that [Ω_{1}(C_{S}(Q_{i−1})), Q_{i}; p−1] = 1 for each i. If Q ◁ S is normal and satisfies condition (1), then since Z(𝔛) = C_{S}(𝔛) by Lemma 3.2, we can set Q_{n+1} = Q·Q_{n}, and Q ≤ Q_{n+1} ≤ 𝔛 by definition.

The purpose of these subgroups 𝔛(S) is to provide a tool for applying Proposition 1.4, when trying to show that the functors Z_{F} are acyclic. These are most useful when applied to a filtration of these functors, described as follows.

For any saturated fusion system F over a p-group S, a subgroup P ≤ S is strongly F-closed in S if no element of P is F-conjugate to any element of S∖P. If T ≤ S is strongly F-closed subgroup in S, let

Z_{F}^{T}: O(F^{c})^{op} ⟶ ℤ_{(p)}-mod

be the subfunctor of Z_{F} defined by setting Z_{F}^{T}(P) = Z(P) ∩ T.

When F is a saturated fusion system over a p-group S, and T ◁ S is a strongly Fclosed subgroup, then a fully F-normalized subgroup P ≤ T will be called F|_{T} -radical if

O_{p}(Out_{F}(P)) ∩ Out_{T} (P) = 1.

Lemma 3.4. Fix an odd prime p, a saturated fusion system F over a p-group S, and a pair T_{0} ◁ T ◁ S of subgroups strongly F-closed in S. Write 𝔛(T/T_{0}) = 𝔛/T_{0} for short. For any fully F-normalized subgroup Q ≤ T, define

Z_{Q}: O(F^{c})^{op} ⟶ ℤ_{(p)}-mod

by setting, for F-centric P ≤ S,

Z_{Q}(P) = \begin{cases}(Z_{F}^{T}/Z_{F}^{T_{0}})(P) ≅ \frac{Z(P)∩T}{Z(P)∩T_{0}} & if P ∩ T is F-conjugate to Q \\ 0 & otherwise. \end{cases}

Assume that Q ≱ 𝔛, or that Q is not centric in T, or that Q is not F|_{T} -radical. Then

\underleftarrow{lim}^{∗}_{O(F^{c})}(Z_{Q}) = 0.

Proof. Since Q is fully F-normalized, for any Q′ ≤ S which is F-conjugate to Q, there is some φ ∈ Hom_{F}(N_{S}(Q′), N_{S}(Q)) such that φ(Q′) = Q [BLO2, Proposition A.2(c)]. Hence each subgroup P′ ≤ S for which Z_{Q}(P′) ≠ 0 is F-conjugate to a subgroup P such that P ∩ T = Q.

Assume first Q ≱ T_{0}. Then for each F-centric subgroup P such that P ∩ T = Q, N_{P T_{0}}(P)/P ≠ 1 and acts trivially on Z_{Q}(P), and so Λ^{∗}(Out_{F}(P); Z_{Q}(P)) = 0 by Proposition 1.1(b). Thus \underleftarrow{lim}^{∗}(Z_{Q}) = 0 in this case.

If Q is not centric in T, then for each F-centric subgroup P such that P ∩ T = Q, N_{P ·C_{T} (Q)}(P)/P ≠ 1 and acts trivially on Z_{Q}(P), and so Λ^{∗}(Out_{F}(P); Z_{Q}(P)) = 0 by Proposition 1.1(b). Again, \underleftarrow{lim}^{∗}(Z_{Q}) = 0 in this case.

Now assume Q is centric in T, but not F|_{T} -radical. Set

Q̂ = {x ∈ N_{T} (Q) | c_{x} ∈ O_{p}(Out_{F}(Q))};

Q̂/Q ≠ 1 by assumption. Let P ≤ S be a F-centric subgroup such that P ∩ T = Q. Each element of Aut_{F}(P) leaves Q invariant (since T ◁ S), so we have a restriction map

ρ: Aut_{F}(P) ⟶ Aut_{F}(Q) × Aut(P/Q),

and Ker(ρ) is a p-group by [Go, Corollary 5.3.3]. Hence ρ^{−1}(O_{p}(Aut_{F}(Q)) × 1) is a normal p-subgroup of Aut_{F}(P). Also, P normalizes Q̂, since it normalizes Q, so

1 ≠ Q_{0}/Q =^{def} (Q̂/Q)^{P}

since Q̂/Q ≠ 1, and Q_{0} ≤ N_{S}(P) since [Q_{0}, P] ≤ Q by definition. For any x ∈ Q_{0}∖Q ⊆ N(P)∖P, c_{x} ∈ ρ^{−1}(O_{p}(Aut_{F}(Q)) × 1), its class in Out_{F}(P) is nontrivial since P is Fcentric and x ∉ P, and hence O_{p}(Out_{F}(P)) ≠ 1. Thus Λ^{∗}(Out_{F}(P); Z_{Q}(P)) = 0 by Proposition 1.1(b). Since this holds for all F-centric P with P ∩ T = Q, \underleftarrow{lim}^{∗}(Z_{Q}) = 0 in this case.

It remains to consider the case where Q ≥ T_{0}, Q is centric in T, and Q ≱ 𝔛. This will be done in three steps. In the first two steps, we show that \underleftarrow{lim}^{∗}(Z_{𝔛}) is isomorphic to the higher limits of a certain functor over an orbit category of a group. Only in Step 3 do we apply the assumption that Q ≱ 𝔛.

Step 1: In this case, we set

Q̂ = Q·C_{S}(Q).

Then Q̂ is F-centric, and Q̂ ∩ T = Q does not contain 𝔛. Also, Q̂ is fully normalized in F, since if Q′ is F-conjugate to Q̂ and fully normalized in F, then there is some α ∈ Hom_{F}(N_{S}(Q̂), N_{S}(Q′)) with α(Q̂) = Q′ (see [BLO2, Proposition A.2(c)]). Hence α(Q) = Q′ ∩ T ◁ N_{S}(Q′), and

|N_{S}(Q̂)| = |N_{S}(Q)| ≥ |N_{S}(α(Q))| ≥ |N_{S}(Q′)|

since Q is fully normalized.

Let Ẑ_{Q} be the quotient functor of Z_{Q} where

Ẑ_{Q}(P) = \begin{cases}Z_{Q}(P) & if P contains a subgroup F-conjugate to Q̂ \\ 0 & otherwise. \end{cases}

If Ẑ_{Q}(P) ≠ Z_{Q}(P) (i.e., if Ẑ_{Q}(P) = 0 and Z_{Q}(P) ≠ 0), then up to conjugacy, P is F-centric and P ∩ T = Q, but P ≱ Q̂. Then N_{P Q̂}(P)/P is a nontrivial p-subgroup of Out_{F}(P) which acts trivially on Z_{Q}(P), so Λ^{∗}(Out_{F}(P); Z_{Q}(P)) = 0 in this case. Thus

\underleftarrow{lim}^{∗}_{O(F^{c})}(Ẑ_{Q}) ≅ \underleftarrow{lim}^{∗}_{O(F^{c})}(Z_{Q}).  (1)

Step 2: Set

Γ = Out_{F}(Q̂),  S′ = Out_{S}(Q̂) ∈ Syl_{p}(Γ),  and T′ = Out_{T} (Q̂)

for short. Using the isomorphism

O^{≥Q̂}(N_{F}(Q̂)) ⟶_{≅}^{Ψ} O_{S′}(Γ)

of Lemma 2.3, we see that T′ = Ψ(T Q̂) is strongly closed in S′ = Ψ(S) with respect to Γ, since no element of T Q̂ can be N_{F}(Q)-conjugate to any element of S∖T Q̂.

By definition, each subgroup on which Ẑ_{Q} is nonvanishing is F-conjugate to some P ≥ Q̂ such that P ∩T = Q. In particular, Q ◁ P since T ◁ S, and so Q̂ = Q·C_{S}(Q) ◁ P. If P′ is any subgroup F-conjugate to P which contains Q̂, and α ∈ Iso_{F}(P, P′) is any isomorphism, then

α(Q) = α(P ∩ T) = P′ ∩ T ≥ Q̂ ∩ T = Q,

and this is an equality since |α(Q)| = |Q|. Hence α(Q̂) = Q̂. Hypothesis (∗) of Proposition 2.4 is thus satisfied, and hence

\underleftarrow{lim}^{∗}_{O_{S′}(Γ)}(Ẑ_{Q} ∘ Ψ^{−1}) ≅ \underleftarrow{lim}^{∗}_{O(F^{c})}(Ẑ_{Q}).  (2)

Set

M_{1} = Z(Q) = Z(Q̂) ∩ T and M_{0} = Z(Q) ∩ T_{0} = Z(Q̂) ∩ T_{0};

and set M = M_{1}/M_{0}. We regard these as ℤ_{(p)}[Γ]-modules. Let

F: O_{S′}(Γ)^{op} ⟶ ℤ_{(p)}-mod

be the functor F (P) = M_{1}^{P} /M_{0}^{P} for all P ≤ S′. This is clearly a subfunctor of H^{0}M which satisfies the relative norm condition (∗) in Propositions 1.3 and 1.4. Also, for P′ = Out_{P} (Q̂) ≤ S′ (i.e., Q̂ ◁ P and P′ ≅ P/Q̂),

(Ẑ_{Q} ∘ Ψ^{−1})(P′) = Ẑ_{Q}(P) = \begin{cases}\frac{Z(P)∩T}{Z(P)∩T_{0}} ≅ F (P′) & if P ∩ T = Q \\ 0 & otherwise; \end{cases}

and P ∩ T = Q if and only if P ∩ T Q̂ = Q̂, if and only if P′ ∩ T′ = 1. Hence by Proposition 1.4, together with (1) and (2),

𝔑_{Z(T′)}·Ω_{1}(M) = 0 implies \underleftarrow{lim}^{∗}_{O(F^{c})}(Z_{Q}) ≅ \underleftarrow{lim}^{∗}_{O_{S′}(Γ)}(Ẑ_{Q} ∘ Φ) = 0.  (3)

(More precisely, Proposition 1.4 only tells us that Z_{Q} is acyclic. But Q ≨ T since it does not contain 𝔛, so Z_{Q}(S) = 0, and this implies \underleftarrow{lim}^{0}(Z_{Q}) = 0.)

Step 3: By definition of 𝔛(T/T_{0}), there are subgroups

1 = Q_{0}/T_{0} ≤ Q_{1}/T_{0} ≤ ⋯ ≤ Q_{n}/T_{0} = 𝔛(T/T_{0}),

all normal in T/T_{0}, such that

[Ω_{1}(C_{T/T_{0}}(Q_{i−1}/T_{0})), Q_{i}/T_{0}; p−1] = 1  (4)

for all i = 1, …, n. Let i ≤ n (i ≥ 1) be the smallest integer such that Q ≱ Q_{i}. Then QQ_{i} ≩ Q, so N_{QQ_{i}}(Q)/Q is nontrivial, and is normal in N_{T} (Q)/Q since Q_{i} ◁ T. Hence the fixed subgroup

(N_{QQ_{i}}(Q)/Q)^{N_{T} (Q)/Q} = {xQ | x ∈ N_{QQ_{i}}(Q), [x, N_{T} (Q)] ≤ Q}

is also nontrivial. Fix some x ∈ N_{QQ_{i}}(Q)∖Q such that x ∈ Q_{i} and [x, N_{T} (Q)] ≤ Q. In particular,

xQ ∈ Z(N_{T} (Q)/Q).  (5)

Since Q ≥ Q_{i−1} by assumption,

[Ω_{1}(M), x; p−1] ≤ [Ω_{1}(Z(Q/T_{0})), x; p−1] ≤ [Ω_{1}(C_{T/T_{0}}(Q_{i−1}/T_{0})), Q_{i}; p−1] = 1, (6)

where the last equality holds by (4).

Now regard M additively as a ℤ_{(p)}[Γ]-module. Then (6) translates to the statement that

𝔑_{⟨x⟩}·Ω_{1}(M) = (1 − x)^{p−1}·Ω_{1}(M) = 0.

Also, x ∈ Z(Out_{T} (Q)) by (5), so x ∈ Z(T′) by (3). Hence 𝔑_{Z(T′)}·Ω_{1}(M) = 0, so \underleftarrow{lim}^{∗}(Z_{Q}) = 0 by (3), and this finishes the proof.

Using Proposition 2.4 and Lemma 3.4 (with T_{0} = 1 and T = S), it is not hard to show that for any saturated fusion system F over a p-group S, Z_{F} is acyclic if 𝔛(S) contains a subgroup which is both centric and weakly F-closed in S. Since we are unable to prove directly that this holds for all F, we instead filter Z_{F} via a maximal series of strongly F-closed subgroups of S, and use the following more general result.

Proposition 3.5. Fix a saturated fusion system F over a p-group S, and let T_{0} ◁ T ◁ S be a pair of subgroups strongly F-closed in S. Assume there is a subgroup 𝔛/T_{0} ≤ 𝔛(T/T_{0}) which is centric in T/T_{0} and weakly F/T_{0}-closed. Then the quotient functor Z_{F}^{T}/Z_{F}^{T_{0}} is acyclic.

More generally, let 𝔛_{F}(T/T_{0}) be the intersection of all subgroups Q/T_{0} ≤ T/T_{0} containing 𝔛(T/T_{0}) such that Q is fully F-normalized and F|_{T} -radical. Assume there is a subgroup 𝔛/T_{0} ≤ 𝔛_{F}(T/T_{0}) which is centric in T/T_{0} and weakly F/T_{0}-closed. Then the quotient functor Z_{F}^{T}/Z_{F}^{T_{0}} is acyclic.

Proof. Write Z = Z_{F}^{T}/Z_{F}^{T_{0}} for short. Assume 𝔛/T_{0} ≤ 𝔛_{F}(T/T_{0}) is centric in T/T_{0} and weakly F/T_{0}-closed. In particular, 𝔛 is weakly F-closed. Let Z_{𝔛} be the functor on O(F^{c}) defined by setting, for all P ≤ S,

Z_{𝔛}(P) = \begin{cases}Z(P) & if P ≱ 𝔛 \\ 0 & otherwise. \end{cases}

(Note that since 𝔛 is weakly F-closed, if P ≱ 𝔛, then the same holds for all subgroups in its F-conjugacy class.) We regard Z_{𝔛} as a subfunctor of Z. Define Z_{Q} as in Lemma 3.4; then \underleftarrow{lim}^{∗}(Z_{Q}) = 0 for all fully F-normalized Q ≤ T such that Q/T_{0} ≱ 𝔛(T/T_{0}), or such that Q is not F|_{T} -radical. In particular, this applies to all Q ≱ 𝔛. Thus via the obvious filtration of Z_{𝔛}, we get that \underleftarrow{lim}^{∗}(Z_{𝔛}) = 0, and hence that

\underleftarrow{lim}^{∗}_{O(F^{c})}(Z/Z_{𝔛}) ≅ \underleftarrow{lim}^{∗}_{O(F^{c})}(Z) .  (1)

Set 𝔛^{∗} = 𝔛·C_{S}(𝔛). Then 𝔛^{∗} is F-centric; and 𝔛^{∗} ∩ T = 𝔛 since 𝔛 is centric in T (since 𝔛/T_{0} is centric in T/T_{0}). If 𝔛 ≤ P ≤ S and P ≱ 𝔛^{∗}, and P is F-centric, then N_{𝔛^{∗}P}(P)/P ≅ Out_{𝔛^{∗}}(P) is a nontrivial p-subgroup of Out_{F}(P) which acts trivially on (Z(P) ∩ T)/(Z(P) ∩ T_{0}), and so

Λ^{∗}(Out_{F}(P); Z(P)) = 0

for such P. Hence if we let F denote the functor

F (P) = \begin{cases}Z(P) = \frac{Z(P)∩T}{Z(P)∩T_{0}} & if P′ ≥ 𝔛^{∗} for some P′ F-conjugate to P \\ 0 & otherwise; \end{cases}

then

\underleftarrow{lim}^{∗}_{O(F^{c})}(F) ≅ \underleftarrow{lim}^{∗}_{O(F^{c})}(Z/Z_{𝔛}).  (2)

Set

M_{1} = Z(𝔛^{∗}) ∩ T = Z(𝔛) and M_{0} = Z(𝔛^{∗}) ∩ T_{0}.

Since 𝔛 is weakly F-closed, 𝔛^{∗} =^{def} 𝔛·C_{S}(𝔛) is both centric in S and weakly F-closed. So by Proposition 2.4, there is a functor

F: O_{p}(Out_{F}(𝔛^{∗})) ⟶ Ab,

where

F (P/𝔛^{∗}) = \frac{Z(P) ∩ T}{Z(P) ∩ T_{0}} ≅ M_{1}^{P} /M_{0}^{P}

for all

P/𝔛^{∗} ≅ Out_{P} (𝔛^{∗}) ≤ Out_{S}(𝔛^{∗}) ∈ Syl_{p}(Out_{F}(𝔛^{∗}));

and such that

\underleftarrow{lim}^{∗}_{O_{p}(Out_{F} (𝔛^{∗}))}(F) ≅ \underleftarrow{lim}^{∗}_{O(F^{∗})}(F).  (3)

Finally, \underleftarrow{lim}^{i}(F) = 0 for i > 0 by Proposition 1.3 (F ≅ H^{0}M_{1}/H^{0}M_{0}). Together with (1), (2), and (3), this finishes the proof of the proposition.

It now remains to determine, for each saturated fusion system F over a p-group S (p odd), whether there always exists a sequence of strongly F-closed subgroups for which Proposition 3.5 applies to each successive pair. For convenience, we define a subgroup Q ≤ S to be universally weakly closed in S if for every saturated fusion system F over a p-group S′ ≥ S such that S is strongly F-closed, Q is weakly F-closed in S′.

Lemma 3.6. Fix an odd prime p and a p-group S. Then a subgroup Q ≤ S is universally weakly closed if for all P ≤ S containing Q, Q is a characteristic subgroup of P.

Proof. Assume that Q ≤ S is not universally weakly closed. Then there exist a saturated fusion system F over a p-group S′ ≥ S such that S is strongly F-closed, and such that Q is not weakly F-closed in S′. By Alperin’s fusion theorem for saturated fusion systems [BLO2, Theorem A.10], there is a subgroup P′ ≤ S′ containing Q, and an automorphism α ∈ Aut_{F}(P′) such that α(Q) ≠ Q. Set P = T ∩P′. Then α(P) = P since T is strongly F-closed, and hence α induces an automorphism of P ≤ S which does not send Q to itself. Thus Q is not a characteristic subgroup of P.

The next lemma gives some simple conditions on the p-group for being able to apply Proposition 3.5. For any p-group S, let J(S) denote Thompson’s subgroup: the subgroup generated by all elementary abelian subgroups of S of maximal rank.

Proposition 3.7. Fix an odd prime p, and a p-group S which satisfies any of the following conditions.

(a) 𝔛(S) ≥ J(S).

(b) S contains a unique elementary abelian p-subgroup E of maximal rank.

(c) S/𝔛(S) is abelian.

Then there is a subgroup P ≤ 𝔛(S) which is centric and universally weakly closed in S.

Proof. Write 𝔛 = 𝔛(S) for short.

(a) Assume 𝔛 ≥ J(S). Clearly, J(S) is universally weakly closed in S; however, it need not be centric. So instead, consider the subgroup Q = J(S)·C_{S}(J(S)) ≤ S. This is clearly normal and centric in S, and is characteristic in any subgroup of S which contains it since J(S) is. Thus Q is universally weakly closed in S by Lemma 3.6.

It remains to check that Q ≤ 𝔛. Since J(S) ≤ 𝔛, every elementary abelian subgroup of S of maximal rank commutes with Z(𝔛), and thus contains Ω_{1}(Z(𝔛)) since otherwise it would not be maximal. Thus Ω_{1}(Z(𝔛)) ≤ Z(J(S)), so

[Ω_{1}(Z(𝔛)), Q] ≤ [Z(J(S)), J(S)·C_{S}(J(S))] = 1.

Hence Q ≤ 𝔛 by Lemma 3.3.

(b) If E ≤ S is the unique elementary abelian subgroup of maximal rank, then J(S) = E, and E ≤ 𝔛 by Lemma 3.2. The result thus follows from (a).

(c) Assume that S/𝔛 is abelian, and that 𝔛 is not universally weakly closed in S. By Lemma 3.6, there is a subgroup P ≤ S containing 𝔛, and an automorphism α ∈ Aut(P) such that α(𝔛) ≠ 𝔛. We claim that this is impossible.

Assume first that α(Z(𝔛)) ≰ 𝔛, and fix an element g ∈ α(Z(𝔛))∖𝔛. Then [Ω_{1}(Z(𝔛)), g] ≤ α(𝔛), since α(𝔛) ◁ P, and hence

[[Ω_{1}(Z(𝔛)), g], g] ≤ [α(𝔛), g] = 1

since g ∈ Z(α(𝔛)). Set Q = ⟨g, 𝔛⟩; then [Ω_{1}(Z(𝔛)), Q; 2] = 1, and Q ◁ S since S/𝔛 is abelian. Then Q ∈ 𝔛 by Lemma 3.3, and this contradicts the original assumption on g.

Now assume that α(Z(𝔛)) ≤ 𝔛, and thus that Z(𝔛) ≤ α^{−1}(𝔛) (and α^{−1}(𝔛) ≠ 𝔛). Fix a chain of subgroups

1 = Q_{0} ≤ Q_{1} ≤ ⋯ ≤ Q_{n} = 𝔛,

all normal in S (hence in P), which satisfy condition (1) in Definition 3.1. Let i ≤ n be such that Q_{i} ≰ α(𝔛) but Q_{i−1} ≤ α(𝔛). Then

α^{−1}(Ω_{1}(C_{P} (Q_{i−1}))) = Ω_{1}(C_{P} (α^{−1}Q_{i−1})) ≥ Ω_{1}(C_{P} (𝔛)) = Ω_{1}(Z(𝔛)),

and hence

[Ω_{1}(Z(𝔛)), α^{−1}Q_{i}; p−1] ≤ α^{−1}([Ω_{1}(C_{P} (Q_{i−1})), Q_{i}; p−1]) = 1

by the assumption on the Q_{i}. Hence by Lemma 3.3 again, ⟨𝔛, Q_{i}⟩ ≤ 𝔛, which contradicts the original assumption on Q_{i}.

We note the following immediate corollary to Propositions 3.7(a) and 3.5.

Corollary 3.8. Let F be a saturated fusion system over a p-group S, and let 1 = T_{0} ≤ T_{1} ≤ ⋯ ≤ T_{k} = S be any sequence of subgroups which are all strongly F-closed in S. Assume, for all 1 ≤ i ≤ k, that 𝔛(T_{i}/T_{i−1}) ≥ J(T_{i}/T_{i−1}). Then \underleftarrow{lim}^{i}(Z_{F}) = 0 for all i > 0.

Corollary 3.8 motivates the following

Conjecture 3.9. For any odd prime p and any p-group P, 𝔛(P) ≥ J(P).

By Corollary 3.8 (together with Lemma 2.5), in order to prove that Z_{F} is acyclic for all saturated fusion systems F, it suffices to prove Conjecture 3.9 for all p-groups P which can occur as minimal strongly closed subgroups in saturated fusion systems. However, it seems to be very difficult to prove or find a counterexample to this conjecture, even in this restricted form. This also indicates that it will be very difficult to find an example of a saturated fusion system F for which Z_{F} is not acyclic, if there are any.

We finish this section with one other elementary result about the groups 𝔛(S), a result which will be useful in the next section.

Proposition 3.10. Fix an odd prime p and a p-group S. Then either rk(Z(𝔛(S))) ≥ p, or 𝔛(S) = S. In particular, 𝔛(S) = S if rk(S) ≤ p − 1.

Proof. Set 𝔛 = 𝔛(S) for short. Assume rk(Z(𝔛)) ≤ p−1, and set E =^{def} Ω_{1}(Z(𝔛)) ◁ S. For each i ≥ 0, either

[E, S; i+1] = [[E, S; i], S] ≨ [E, S; i],

or [E, S; i] = 1. Since |E| ≅ (C_{p})^{k} for k ≤ p − 1, this shows that [E, S; p−1] = 1, and hence that 𝔛(S) = S by Lemma 3.3.

4. The acyclicity of Z_{G} at odd primes

We are now ready to show, for any finite group G and any odd prime p, that all higher limits of Z_{G} vanish when p is odd. This will be based on the following proposition, which gives for any finite group G a sufficient condition for the acyclicity of Z_{G} in terms of its simple composition factors. When G is a finite group and S ∈ Syl_{p}(G), set

𝔛_{G}(S) = ⋂{P ≤ S | P ≥ 𝔛(S), O_{p}(Out_{G}(P)) = 1, N_{S}(P) ∈ Syl_{p}(N_{G}(P))} :

the intersection of all subgroups of S which contain 𝔛(S), and are fully normalized and F_{p}(G)-radical.

Proposition 4.1. For any prime p and any finite group G, Z_{G} is acyclic if for each nonabelian simple group L which occurs in the decomposition series for G, and any S ∈ Syl_{p}(L), there is a subgroup Q ≤ 𝔛_{L}(S) which is centric and weakly Aut(L)-closed in S. In particular, Z_{G} is acyclic for each finite solvable group G.

Proof. Fix a sequence of normal subgroups

1 = K_{0} ≨ K_{1} ≨ ⋯ ≨ K_{n} = G

such that each subquotient K_{i+1}/K_{i} is a minimal normal subgroup of G/K_{i}. We show that Z_{G}^{K_{i+1}}/Z_{G}^{K_{i}} is acyclic for each i. Choose S ∈ Syl_{p}(G), and set S_{i} = S ∩ K_{i} ∈ Syl_{p}(K_{i}) and F = F_{S}(G).

Assume that Q = Q̃/S_{i} ≤ S_{i+1}/S_{i} is centric in S_{i+1}/S_{i} and that Q̃ is fully Fnormalized (i.e., N_{S}(Q̃) ∈ Syl_{p}(N_{G}(Q̃))). If Q̃ is F|_{S_{i+1}}-radical, then

1 = O_{p}(Out_{G}(Q̃)) ∩ Out_{K_{i+1}}(Q̃) = O_{p}(Out_{K_{i+1}}(Q̃)) −↠^{proj} O_{p}(Out_{K_{i+1}/K_{i}}(Q)),

and hence Q is a radical p-subgroup of K_{i+1}/K_{i}. This proves that

𝔛_{F}(S_{i+1}/S_{i}) ≤ 𝔛_{K_{i+1}/K_{i}}(S_{i+1}/S_{i}).

So by Proposition 3.5 (and Lemma 2.1), to prove that Z_{G}^{K_{i+1}}/Z_{G}^{K_{i}} is acyclic, it suffices to show

𝔛_{K_{i+1}/K_{i}}(S_{i+1}/S_{i}) contains a subgroup Q which is centric and weakly G/K_{i}-

closed in S_{i+1}/S_{i}.

(1)

To simplify notation, we replace G by G/K_{i} (so K_{i} = 1), and set K = K_{i+1} and P = S_{i+1} ∈ Syl_{p}(K). Thus, K is a minimal normal subgroup of G, and we must find Q ≤ 𝔛_{K}(P) which is centric and weakly G-closed in P. This is clear if K has order prime to p (i.e., Q = P = 1).

Since K is a minimal normal subgroup, it is a product of finite simple groups isomorphic to each other (cf. [Go, Theorem 2.1.5]). If K is an elementary abelian p-group, then 𝔛(K) = K, and is centric and weakly closed in K. So assume K ≅ L^{n} where L is simple and nonabelian and n ≥ 1. We can choose this identification in a way such that P = (P′)^{n} for some fixed P′ ∈ Syl_{p}(L). Then 𝔛(P) = 𝔛(P′)^{n} (see Definition 3.1), and 𝔛_{K}(P) = 𝔛_{L}(P′)^{n} since each radical p-subgroup of K splits as a product of n radical p-subgroups of L [JMO, Proposition 1.6(ii)]. By assumption, there is a subgroup Q′ ≤ 𝔛_{L}(P′) which is centric and weakly Aut(L)-closed in P′. Then Q =^{def} (Q′)^{n} is centric in P, and Q ≤ 𝔛_{K}(P). It remains to show that Q is weakly Aut(K)-closed in P, and hence weakly G-closed in P.

Assume otherwise: assume there is α ∈ Aut(L^{n}) such that Q ≠ α(Q) ≤ P. The n factors L are the unique minimal normal subgroups of L^{n}, so each automorphism of L^{n} permutes these factors, and hence Aut(L^{n}) ≅ Aut(L) ≀ Σ_{n}. Thus α = σ ∘ (α_{1}, …, α_{n}) for some α_{i} ∈ Aut(L) and some σ ∈ Σ_{n} (regarded as an automorphism of L^{n}); and Q′ ≠ α_{i}(Q′) ≤ P′ for some i. Which contradicts the assumption that Q′ is weakly Aut(L)-closed in P′.

We now prove that all finite nonabelian simple groups L satisfy the condition in Proposition 4.1: for any odd prime p||L| and any S ∈ Syl_{p}(L), there is a subgroup Q ≤ 𝔛_{L}(S) (or ℚ ≤ 𝔛(S)) which is centric and weakly Aut(L)-closed in S. We first consider some cases where this can be shown using Proposition 3.7(b).

Proposition 4.2. Assume p is odd, and let L be a simple group which is either an alternating group, or a group of Lie type in characteristic different from p. Then for S ∈ Syl_{p}(L), J(S) ≤ 𝔛(S), and hence there is a subgroup Q ≤ 𝔛(S) which is centric and weakly Aut(L)-closed in S.

Proof. If L ≅ A_{n}, then S contains a unique elementary abelian p-subgroup E of maximal rank, generated by a product of [n/p] disjoint p-cycles (cf. [GL, 10-5]). Hence J(S) = E ≤ 𝔛(S) by Lemma 3.2, and the result follows from Proposition 3.7(a) or (b).

Now assume that L is a simple group of Lie type in characteristic ≠p. If rk_{p}(L) ≤ 2, then 𝔛(S) = S by Proposition 3.10. So assume rk_{p}(L) > 2. Then by [GL, 10-2(1)], each Sylow p-subgroup of L contains a unique elementary abelian p-subgroup of maximal rank; and the result follows from Lemma 3.2 and Proposition 3.7(a,b) again. (Note that all of the exceptional cases listed in [GL] — the simple groups A_{2}(q), ^{2}A_{2}(q), G_{2}(q), 3D_{4}(q), and ^{2}F_{4}(q) when p = 3 — have 3-rank at most 2 by [GL, 10-2(2)] and Tables 10:1 and 10:2.)

We next consider simple groups of Lie type in characteristic p. We first summarize the structures in these groups which will be needed, referring to [Ca] as a general reference.

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. 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.

To each root r ∈ Φ corresponds a root subgroup X_{r} ≅ 𝔽_{q} in L = 𝔾(q). Then 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 is the subgroup of diagonal elements (and has order prime to p). 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 = {e_{i} − e_{j} | i ≠ j}

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

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

Then X_{e_{i}−e_{j}} is the subgroup of matrices which have 1’s along the diagonal and are zero elsewhere except at entry (i, j), 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 represented by diagonal matrices, N is the image of the subgroup of monomial matrices, and W ≅ Σ_{n+1}.

We next fix the notation for the twisted groups ^{t}𝔾(q). Let τ ∈ Aut(𝕍, Φ, Φ_{+}) be an automorphism of the root system of 𝔾 of order t. Set σ = τ̂∘φ̂ ∈ Aut(𝔾(q′)), where τ̂ is induced by τ and φ̂ is induced by φ ∈ Aut(𝔽_{q′}) of order t. (In most cases, q′ = q^{t}, and so 𝔽_{q} is the fixed subfield of the automorphism φ.) 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 commutator subgroup of 𝔾(q′)^{σ}, or alternatively as the subgroup of 𝔾(q′) generated by U^{σ} and the analogous subgroup for the root groups of negative roots. Its Borel subgroup is defined to be B = N_{L}(U^{σ}) = U^{σ}⋊(H^{σ} ∩ L).

Proposition 4.3. Assume p is odd, let L be a simple group of Lie type in characteristic p, and fix S ∈ Syl_{p}(L). Then 𝔛_{L}(S) is weakly Aut(L)-closed in S.

Proof. Write L = ^{t}𝔾(q), where q = p^{a} (possibly t = 1). We use the above notation. In particular, Φ_{+} denotes the set of positive roots, and U ∈ Syl_{p}(L) is the product of the root subgroups X_{r} for r ∈ Φ_{+}.

For each τ-invariant subset J ⊆ I, consider the subgroups

U_{J} = ∏_{r∈Φ_{+}∖⟨J⟩} X_{r},

In particular, U_{∅} = U, and U_{I} = 1. We claim that the following statement holds:

The subgroups U_{J}^{σ}, for τ-invariant subsets J ⊆ I, are the only subgroups of S = U^{σ} which are radical p-subgroups of L; and they are all weakly L-closed in U^{σ}.

(1)

By a theorem of Borel and Tits (see the corollary in [BW]), every radical p-subgroup of L is conjugate to one of the subgroups U_{J}^{σ}, and by [Gr, Lemma 4.2], if U_{J}^{σ} is Lconjugate to U_{J′}^{σ} for J, J′ ≤ I then J = J′. But since we need to know that each radical p-subgroup of L contained in S is actually equal to one of the U_{J}^{σ}, we modify Grodal’s proof to show this.

Assume that P ≤ S = U^{σ} is L-conjugate to U_{J}^{σ}. We show that P = U_{J}^{σ}; this proves that U_{J}^{σ} is weakly L-closed in S, and hence (using [BW]) proves (1). Since L ≤ U^{σ}·N^{σ}·U^{σ} [Ca, Proposition 8.2.2], and since U_{J} ◁ U, we have P = ux(U_{J}^{σ})x^{−1}u^{−1} for some u ∈ U^{σ} and x ∈ N^{σ}, and P = U_{J}^{σ} if and only if x(U_{J}^{σ})x^{−1} = U_{J}^{σ}. So we can assume u = 1 and P = x(U_{J}^{σ})x^{−1}. Now, x permutes the root subgroups via the action of w = xH ∈ 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) has the same property. This implies that all primitive roots for the system w(±⟨J⟩)∩Φ_{+} are primitive roots in Φ_{+}, and thus that w(Δ) = Φ_{+}∖⟨J′⟩ for some J′ ⊆ I. 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. So J′ = J, and P = U_{J}^{σ}.

Thus, by (1), if Q ≤ S is a radical p-subgroup of L, then Q = U_{J}^{σ} for some J. Hence 𝔛_{L}(S) is the intersection of the subgroups U_{J}^{σ} which contain 𝔛(S). Also, U_{J} ∩ U_{J′} = U_{J∪J′}, since each element of U has a unique decomposition as product of elements of the root groups taken in an appropriate order [Ca, Theorem 5.3.3(ii)]. So any intersection of subgroups U_{J}^{σ} is again of the same form, and thus 𝔛_{L}(S) = U_{J}^{σ} for some τ-invariant subset J ⊆ I. Hence 𝔛_{L}(S) is weakly L-closed in S by (1) again.

Each automorphism of L is congruent mod Inn(L) to some α ∈ Aut(L) which sends S to itself, and α permutes the radical p-subgroups of L contained in S and sends 𝔛(L) to itself. Thus each coset in Out(L) contains an automorphism which sends 𝔛_{L}(S) to itself, and 𝔛_{L}(S) is weakly Aut(L)-closed in S since it is weakly L-closed.

In fact, when L is simple of Lie type in characteristic p (and p is odd, as usual), then for S ∈ Syl_{p}(L), 𝔛(S) = S except when p = 3 and and L ≅ C_{n}(q) ≅ P Sp_{2n}(q) (n ≥ 2) or L ≅ ^{2}A_{n}(q) ≅ P SU_{n+1}(q^{2}) (n ≥ 3).

We are now ready to consider the sporadic groups.

Proposition 4.4. Assume p is odd, let L be a sporadic simple group, and fix S ∈ Syl_{p}(L). Then there is a subgroup Q ≤ 𝔛(S) which is centric and weakly Aut(L)- closed in S.

Proof. If p ≥ 5, then rk_{p}(L) < p by [GLS, §5.6], and so 𝔛(S) = S by Proposition 3.10. So assume p = 3. We consider several different cases.

(a) If L is one of the groups M_{11}, M_{12}, M_{22}, M_{23}, M_{24}, J_{1}, J_{2}, J_{4}, HS, He, or Ru, then rk_{3}(L) ≤ 2 by [GL, p.123], and so 𝔛(S) = S by Proposition 3.10.

(b) Assume L is one of the groups J_{3}, Co_{3}, Co_{2}, McL, Suz, Ly, O’N, or F_{5}. In all of these cases, S contains a normal elementary abelian 3-subgroup E of index ≤ 9, 𝔛(S) ≥ E by Lemma 3.2, so S/𝔛(S) is abelian, and 𝔛(S) is weakly Aut(L)-closed by Proposition 3.7(c). More precisely, there are the following inclusions of index prime to

L  J_{3}  Co_{3} Co_{2} McL Suz  Ly  O’N  F_{5}
E  C_{3}^{3}  C_{3}^{5}  C_{3}^{4} C_{3}^{4} C_{3}^{5}  C_{3}^{5}  C_{3}^{4}  C_{3}^{4}
N_{L}(E)/E C_{3}^{2}⋊C_{8} 2 × M_{11} M_{10} M_{10} M_{11} 2 × M_{11} order 320 order 1152

See [GL, §5] for references. (In fact, in all of the above cases, E is the unique elementary abelian subgroup of S of maximal rank.)

(c) Assume L ≅ Co_{1}. By [Cu, p.424], S is contained in a semidirect product C_{3}^{6}⋊2M_{12}, and the elementary abelian subgroup C_{3}^{6} is generated by all elements of order 3 in S which lie in the conjugacy class (3A). Thus S contains a unique elementary abelian 3-subgroup E of maximal rank, and hence C_{S}(E) ≤ 𝔛(S) is centric and weakly Aut(L)- closed in S by Proposition 3.7(b).

(e) Assume L = F_{3}. By [Ho] or [Pa] (see also [As, 14.2]), there are subgroups

D ≤ K ≤ M ≤ S,

all normal in S, such that K ≅ C_{3}^{5} is abelian, C_{S}(K) = K, and [M, K] = D = Z(M) ≅ (C_{3})^{2}. (Also, M/K ≅ C_{3}^{4} and N_{L}(D)/M ≅ GL_{2}(3).) Thus M ≤ 𝔛(S), so rk(Z(𝔛(S))) ≤ rk(Z(M)) = 2, and hence 𝔛(S) = S by Proposition 3.10.

In the remaining cases, for a p-group R, we use the notation Z_{n}(R) ◁ R: Z_{1}(R) = Z(R), and Z_{n}(R)/Z_{n−1}(R) = Z(R/Z_{n−1}(R)). The group R is of class n if R = Z_{n}(R) ≩ Z_{n−1}(R). Also, following the notation of [As], we say that a subgroup H ≤ L is of type H′/m_{t}/m_{t−1}/ ⋯ /m_{1} if upon setting R = O_{p}(H), then H/R ≅ H′, Z(R) ≅ C_{p}^{m_{1}}, and Z_{i}(R)/Z_{i−1}(R) ≅ C_{p}^{m_{i}} for all i. (We restrict, for simplicity, to the case where Z_{i}(R)/Z_{i−1}(R) is elementary abelian for all i.)

(d) Assume L ≅ Fi_{22}. Then L contains a subgroup L_{0} ≅ Ω_{7}(3) with index prime to 3 (cf. [As, p.26]). Regard L_{0} as acting on V = 𝔽_{3}^{7}, let W ⊆ V be a maximal isotropic subspace (dim(W) = 3), and let H ≤ L_{0} be the subgroup of elements which leave W invariant. Then [L_{0}:H] is prime to 3, and hence we can assume that S ≤ H. One easily checks that H is of type SL_{3}(3)/3/3, where R =^{def} O_{3}(H) is the subgroup of elements whose restriction to W (and to V/W^{⊥}) is the identity, and Z(R) is the subgroup of elements whose restriction to W^{⊥} and to V/W are the identity. Also, Z(R) = [R, R], and H/R ≅ SL_{3}(3) acts on Z(R) as the group of 3 × 3 antisymmetric matrices. From this, one quickly sees that R ≤ 𝔛(S). If we set S_{0}/R = Z(S/R) ≅ C_{3}, then [[Z(R), S_{0}], S_{0}] = 1, so S_{0} ≤ 𝔛(S). Hence rk(Z(𝔛(S))) ≤ rk(Z(S_{0}) = 2, so 𝔛(S) = S by Proposition 3.10. (Alternatively, one can show that S contains a unique elementary abelian subgroup of maximal rank 5, and then apply Proposition 3.7(b).)

(f) Assume L = Fi_{23} or F_{2}. By [As, p. 33], there is an inclusion Fi_{23} ≤ F_{2} with index prime to 3, so these groups have isomorphic Sylow 3-subgroups. By [As, p. 27 & 208– 209], there is a subgroup H ≤ Fi_{23} of index prime to 3 and of type SL_{3}(3)/3/3/1/3. We can thus assume R =^{def} O_{3}(H) ≤ S ≤ H. Also, Z_{2}(R) ≅ C_{3}^{4}, Z_{3}(R) = C_{H}(Z_{2}(R)) ≅ Q×C_{3} where Z(Q) = [Q, Q] = Z(R) and Q/Z(Q) ≅ Z_{3}(R)/Z_{2}(R) ≅ C_{3}^{3}, and R/Z_{3}(R) acts on Z_{2}(R) as the group of all automorphisms which are the identity on Z(R) and on Z_{2}(R)/Z(R). Since [[Z_{3}(R), Z_{3}(R)], Z_{3}(R)] = 1 and [[Z_{2}(R), R], R] = 1, we see that R ≤ 𝔛(S); and hence 𝔛(S) = S by the same argument as was used for L ≅ Fi_{22}.

(g) Assume L = Fi′_{24}. By [As, pp. 29 & 210–211], there is a subgroup H ≤ Fi′_{24} of index prime to 3 and of type (A_{5} ×SL_{2}(3))/8/4/2, and we can assume R =^{def} O_{3}(H) ≤ S ≤ H. Also, R/Z_{2}(R) acts on Z_{2}(R) ≅ C_{3}^{8} as the group of all automorphisms which are the identity on Z(R) and on Z_{2}(R)/Z(R); and the actions of SL_{2}(3) ≤ H/R on Z(R) and of A_{5} ≤ H/R on Z_{2}(R)/Z(R) ≅ C_{3}^{4} are faithful. Thus Z_{2}(R) is centric in H and [[Z_{2}(R), R], R] = 1. It follows that R ≤ 𝔛(S), and hence (since rk(Z(R)) = 2) that 𝔛(S) = S by Proposition 3.10.

(h) Assume L = F_{1}. By [As, pp. 35 & 211–212], there is a subgroup H ≤ F_{1} of index prime to 3 of type (GL_{2}(3) × M_{11})/10/5/2. We can thus assume that R =^{def} O_{3}(H) ≤ S ≤ H. Also, R/Z_{2}(R) acts on Z_{2}(R) ≅ C_{3}^{7} as the group of automorphisms which are the identity on Z(R) and on Z_{2}(R)/Z(R), and the actions of GL_{2}(3) ≤ H/R on Z(R) and of M_{11} ≤ H/R on Z_{2}(R)/Z(R) are faithful. Thus Z_{2}(R) is centric in H and [[Z_{2}(R), R], R] = 1. It follows that R ≤ 𝔛(S), and hence by Proposition 3.10 (since rk(Z(R)) = 2) that 𝔛(S) = S.

We are now ready to prove Theorem A.

Theorem 4.5. For any odd prime p and any finite group G, Z_{G} is acyclic.

Proof. Let L be a finite simple group, and fix S ∈ Syl_{p}(L). If L is an alternating group, or of Lie type in characteristic ≠ p, then by Proposition 4.2, there is a subgroup Q ≤ 𝔛(S) which is centric and weakly Aut(L)-closed in S. If L is of Lie type in characteristic p, then 𝔛_{L}(S) itself is centric and weakly Aut(L)-closed in S by Proposition 4.3. If L is a sporadic group, then there is a subgroup Q ≤ 𝔛(S) which is centric and weakly Aut(L)-closed in S by Proposition 4.4. The theorem now follows from Proposition 4.1, together with the classification theorem for finite simple groups.

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

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