THE SUBGROUP HOMOLOGY DECOMPOSITION FOR FUSION SYSTEMS IS SHARP

ANTONIO DÍAZ AND SEJONG PARK

Abstract. We extend Dwyer’s sharp subgroup homology decomposition of the classifying space of a finite group to arbitrary saturated fusion systems and arbitrary Mackey functors.

1. Introduction

Let G be a finite group and p a prime. In the nineties there was an intense interest among topologists in reconstructing the classifying space BG, up to mod-p cohomology, by gluing together classifying spaces of subgroups of G. These are so called homology decompositions and among the main contributors are Benson, Dwyer, Jackowski, McClure, Oliver and Wilkerson. See [8] for a unified treatment and references. Some homology decompositions turn out to be sharp, in the sense that the associated Bousfield–Kan mod-p cohomology spectral sequence collapses onto the vertical axis. As a consequence, a sharp homology decomposition describes the mod-p cohomology of BG as a limit of the cohomology of the classifying spaces of certain subgroups of G. The systematic study of this phenomenon was initiated by Dwyer in [9] and completed by Grodal and Smith in [10].

It is well-known that the properties of BG up to mod-p cohomology are closely related to the p-fusion pattern of G, namely the p-subgroups of G and conjugation maps between them. Two examples of this deep connection are the Cartan– Eilenberg stable elements theorem [4, XII.10.1] and the, now proven, Martino– Priddy conjecture [12, 13]. The notion of the p-fusion pattern of a finite group has been successfully extended to that of a saturated fusion system. A saturated fusion system is a category F whose objects are the subgroups of a fixed finite p-group S and whose morphisms are injective group homomorphisms between them. These morphisms must satisfy some additional conditions modelled on the p-fusion pattern of a finite group. By the combined work of group theorists, representation theorists and topologists, now saturated fusion systems are considered to be the right setup for studying p-local aspects of finite groups and related spaces. See [1] for an introduction to this subject and for precise definitions.

Saturated fusion systems have a rich homotopy theory similar to that of pcompleted classifying spaces of finite groups. After much effort, it has been proven recently [5, 14] that each saturated fusion system F has a unique classifying space BF. This space has homology decompositions [2, §2] [1, III.5.5, 5.6, 5.8] similar to those for BG. In particular, BF has a subgroup homology decomposition

Date: December 1, 2014.

The first author is supported by MICINN grant RYC-2010-05663 and partially supported by FEDER-MCI grant MTM2010-18089 and Junta de Andalucía grant FQM-213.

BF ≃ hocolim_{O(F^{c})} B̃.

Dwyer [9, Theorem 10.3] showed that the corresponding decomposition for BG

BG ≃ hocolim_{O_{p}^{c}(G)} B̃

is sharp, but it has not been known whether the decomposition for BF is sharp or not. This problem figures in Oliver’s list of open questions [1, III.7(9)]. In this paper, we answer this question in the affirmative.

Theorem 1.1. Let F be a saturated fusion system on a finite p-group S. Then \underleftarrow{lim}_{O(F^{c})}^{∗} H^{j}(−; 𝔽_{p}) = 0 for ∗ > 0 and for j ≥ 0.

This also gives a new proof of the following theorem.

Theorem 1.2 ([2, Theorem 5.8]). For any saturated fusion system F on a finite p-group S, the natural homomorphism

H^{∗}(BF; 𝔽_{p}) → \underleftarrow{lim}_{F} H^{∗}(−; 𝔽_{p})

is an isomorphism, and the ring H^{∗}(BF; 𝔽_{p}) is Noetherian.

In fact, we prove the following more general theorem, which we posed as a conjecture and proved in some special cases in our previous work [6].

Theorem 1.3 (Subgroup sharpness for fusion systems). Let F be a saturated fusion system on a finite p-group S and let k be a field of characteristic p. Let M = (M^{∗}, M_{∗}): O(F) → k-mod be a Mackey functor over the full orbit category of F with values finite dimesional k-vector spaces. Then \underleftarrow{lim}_{O(F^{c})}^{i} M^{∗}|_{O(F^{c})} = 0 for i > 0.

The proof of this theorem appears in Section 3. The cohomology functor H^{j}(−; 𝔽_{p}) restricts to a Mackey functor for F (see Section 2 for the relevant definitions) and so Theorem 1.1 is a special case of Theorem 1.3. The reason why we conjectured this more general version is the result [6, Theorem A] going back to Jackowski and McClure [11] that Mackey functors over certain categories have vanishing higher limits. Axioms for Mackey functors were first formulated by Dress [7], and they have been succesfully employed in different areas. There exists a general theory of Mackey functors, including a parametrization and an explicit description of simple Mackey functors. See the work of Webb [15] for example.

We have adapted the theory of Mackey functors to the fusion system setting [6], and they play an essential role in the proof of Theorem 1.3. The other key ingredient in the proof is an adaptation of Oliver’s homological algebraic version [14] of Chermak’s proof [5] of the existence and uniqueness of a classifying space BF. Theorem 1.3 is thus proven by combining two filtrations, one for the functor and one for the category on which the functor is defined: First, we filter the Mackey functor M to reduce the problem to the simple Mackey funtors S_{Q,V} (see Proposition 2.5 and Lemma 2.8) that appear as composition factors. Then, for each simple Mackey functor S_{Q,V}, we construct a filtration of the category O(F^{c}) based on Chermak’s ideas: Chermak’s filtration was built using the Thompson subgroups J(P) of P ≤ S. In terms of Oliver’s interpretation of Chermak’s proof, this filtration is a correct one for the center functor Z_{F}, the vanishing of whose second and third derived limits over O(F^{c}) implies the existence and uniqueness of BF by obstruction theory. We get a filtration of O(F^{c}) which is correct for the simple Mackey functor S_{Q,V} at work by using an analogue Q^{∗}(P) (Definition 3.1) of the Thompson subgroup.

2. Simple Mackey functors and local vanishing results

In this section we briefly review the relevant results on higher limits from Oliver’s work [14] and basic notions about Mackey functors for fusion systems [6]. Throughout this section we denote by k a commutative ring with identity (we will be mainly interested in the case where k is a field of characteristic p > 0), by k-Mod the category of k-modules and by k-mod the category of finitely generated k-modules.

Definition 2.1. Let F be a fusion system on the finite p-group S. An interval is a collection R of subgroups of S such that P < Q < R and P, R ∈ R imply Q ∈ R. An interval is F-invariant if it is closed under taking F-conjugacy.

For instance, the F-conjugacy class R = P^{F} of the subgroup P ≤ S is an Finvariant interval. Note that every F-invariant interval is of the form X \ X_{0} for F-invariant intervals X_{0} ⊆ X which are closed under overgroups. Now let X be an F-invariant interval which is closed under overgroups; for example, X = F^{c}, the collection of all F-centric subgroups of S. The orbit category O(X) has objects the subgroups in X and morphisms given by

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

Definition 2.2. Let F be a fusion system on the p-group S and let N: O(F^{c})^{op} → k-Mod be a functor. If R ⊆ F^{c} is an F-invariant interval define

N^{R}: O(F^{c})^{op} → k-Mod

as the subquotient functor of N with value N^{R}(P) = N(P) for P ∈ R and 0 otherwise. An F-partition is a collection of F-invariant intervals R_{0}, R_{1},…,R_{n} such that:

(1) F^{c} = R_{0} ∪ R_{1} ∪ … R_{n} is a partition of F^{c}.

(2) R_{i} ∈ R_{i}, R_{j} ∈ R_{j} and i < j imply R_{i} ≰ R_{j}.

The F-partitions defined above permit to reduce higher limits to simpler pieces.

Lemma 2.3 ([14, Lemma 1.7]). Let F be a fusion system on the finite p-group S and let N: O(F^{c})^{op} → k-Mod be a functor. If R_{0}, R_{1},…,R_{n} is an F-partition such that \underleftarrow{lim}_{O(F^{c})}^{∗} N^{R_{i}} = 0 for all i and ∗ > 0, then \underleftarrow{lim}_{O(F^{c})}^{∗} N = 0 for ∗ > 0.

Now recall that, roughly speaking, a Mackey functor consists of a contravariant functor and a covariant functor that satisfy various compatibility conditions, including a Mackey decomposition formula. This notion extends naturally to the setup of fusion systems.

Definition 2.4. Let F be a fusion system on a finite p-group S and let X be an F-invariant interval which is closed under overgroups. An X -restricted Mackey functor for F over k is a pair of functors

M = (M^{∗}, M_{∗}): O(X) → k-Mod

satisfying the following conditions.

(1) (Bivariance) M^{∗}: O(X) → k-Mod is a contravariant functor, M_{∗}: O(X) → k-Mod is a covariant functor, and M^{∗}(P) = M_{∗}(P) =: M(P) for all objects P of X.

(2) (Isomorphism) M^{∗}([α]) = M_{∗}([α]^{−1}) if [α] is an isomorphism in O(X).

(3) (Mackey decomposition) If Q, R ≤ P ≤ S are in X, then

M^{∗}([ι_{Q}^{P}]) ∘ M_{∗}([ι_{R}^{P}]) = ∑_{x∈[Q\P/R]_{X}} M_{∗}([ι_{Q∩^{x}R}^{Q}]) ∘ M^{∗}([ι_{Q∩^{x}R}^{^{x}R}]) ∘ M_{∗}([c_{x}|_{R}]),

where [Q\P/R]_{X} denotes a set of representatives x of the double cosets QxR in P such that Q ∩^{x}R ∈ X.

When X consists of all subgroups of S, we simply say that M is a Mackey functor for F.

We denote by Mack_{k}^{X} (F) the category of X -restricted Mackey functors for F over k, and we simply write Mack_{k}(F) for the category of Mackey functors for F over k. The category Mack_{k}^{X} (F) is an abelian category in which kernels and cokernels are constructed “objectwise”. The simple objects in the abelian category Mack_{k}(F) can be parametrized and precisely described.

Proposition 2.5 ([6, Proposition 3.2]). Let F be a fusion system on a finite pgroup S. The simple Mackey functors in Mack_{k}(F) are of the form S_{Q,V}, where Q runs over the subgroups of S and V runs over the simple k Out_{F}(Q)-modules. Moreover, S_{Q,V} and S_{R,W} are isomorphic in Mack_{k}(F) if and only if there is an F-isomorphism α: Q → R such that ^{α}V ≅ W as k Out_{F}(R)-modules.

Here ^{α}V = V as a set and its k Out_{F}(R)-module structure is obtained from the k Out_{F}(Q)-module structure of V by transporting the action along the Fisomorphism α: Q → L. From the above proposition, it is easy to deduce [6, Proposition 3.3] that the simple objects in Mack_{k}^{X} (F) are the restrictions to O(X) of the functors S_{Q,V} for Q ∈ X, although we do not need this fact in this work.

The full Mackey functor structure of S_{Q,V} is described in [6, §3]. Here we only describe the values S_{Q,V} (P) for P ≤ S and its structure as an Out_{F}(P)-module as these are the only parts relevant to our main theorem. We start by setting S_{Q,V} (L) = V as a set for each F-conjugate L of Q. For each such subgroup L we also fix an F-isomorphism α: Q → L. If α_{i}: Q → L_{i} (i = 1, 2) are the chosen isomorphisms for subgroups L_{1} and L_{2} that are F-conjugate to Q and [γ]: L_{1} → L_{2} is an isomorphism in O(F) we define

(1)  S_{Q,V} _{∗}([γ])(v) = [α_{2}^{−1}γα_{1}] · v for v ∈ V

and S_{Q,V} ^{∗}([γ]) = S_{Q,V} _{∗}([γ^{−1}]). Consequently we have S_{Q,V} (L) ≅ ^{α}V as k Out_{F}(L)- modules.

For an arbitrary subgroup P ≤ S we set

(2)  S_{Q,V} (P) ≅ ⨁_{Q≅^{α}L≤_{P} P} tr_{L}^{N_{P} (L)}(^{α}V),

where the direct sum is taken over the P -conjugacy classes of the subgroups L of P that are F-conjugate to Q. For each of these classes a representative L is chosen and an F-isomorphism α: Q → L is also fixed. The map tr_{L}^{N_{P} (L)} : ^{α}V → ^{α}V is the relative trace map, where N_{P} (L) acts on the k Out_{F}(L)-module ^{α}V via the map N_{P} (L) → Out_{F}(L) given by conjugation. It is easy to check that the above description of S_{Q,V} (P) does not depend on the choices of L and α up to isomorphism.

From this description, we obtain some vanishing properties of S_{Q,V} below. Combined with a correct F-partition they produce the desired sharpness result. First we recall a group theoretic definition.

Definition 2.6. Let Q ≤ P be groups. We say that Q is centric in P if C_{P} (Q) ≤ Q.

Thus, if F is a fusion system on a finite p-group S, the subgroup P ≤ S is F-centric if and only if all its F-conjugates are centric in S.

Lemma 2.7. Let F be a fusion system on a finite p-group S. Consider the simple Mackey functor S_{Q,V} for F where Q ≤ S and V is a simple k Out_{F}(Q)-module. Let P ≤ S and let tr_{L}^{N_{P} (L)}(^{α}V) be a direct summand of S_{Q,V} (P) as in (2) with a chosen F-isomorphism α: Q → L ≤ P.

(a) If k has characteristic p and C_{P} (L) ≰ L, then tr_{L}^{N_{P} (L)}(^{α}V) = 0.

(b) If g ∈ (N_{S}(P) ∩ C_{S}(L)) \ P and P is centric in S, then [c_{g}] ∈ Out_{F}(P) is a nontrivial p-element that acts trivially on tr_{L}^{N_{P} (L)}(^{α}V).

Proof. The first item follows from the transitivity of the relative trace map

tr_{L}^{N_{P} (L)}(^{α}V) = (tr_{C_{P} (L)L}^{N_{P} (L)} ∘ tr_{L}^{C_{P} (L)L})(^{α}V)

and the fact that tr_{L}^{C_{P} (L)L}(^{α}V) = 0. For the second item, as P is centric in S, it is clear that [c_{g}] ∈ Out_{F}(P) is a nontrivial p-element. Moreover, conjugation by g fixes L elementwise and hence, by (1), [c_{g}] acts on tr_{L}^{N_{P} (L)}(^{α}V) as

S_{Q,V} _{∗}([c_{g}])(v) = [α^{−1}c_{g}α] · v = v for v ∈ ^{α}V .  □

If k is a field and M ∈ Mack_{k}(F) takes as values finite dimensional k-vector spaces, then M has a composition series of finite length, which can be used to reduce the acyclicity of M over O(F^{c}) to that of simple Mackey functors.

Lemma 2.8 ([6, Proposition 4.3]). Let F be a fusion system on a finite p-group S and let k be a field. Consider a Mackey functor M: O(F) → k-mod. If for each composition factor S_{Q,V} of M, where Q ≤ S and V is a simple k Out_{F}(Q)-module, we have \underleftarrow{lim}_{O(F^{c})}^{i} S_{Q,V}^{∗} |_{O(F^{c})} = 0 for i > 0, then \underleftarrow{lim}_{O(F^{c})}^{i} M^{∗}|_{O(F^{c})} = 0 for i > 0.

The next lemma describes local vanishing conditions for higher limits on an F-invariant interval.

Definition 2.9. Let F be a fusion system on the finite p-group S, let R ⊆ F^{c} be an F-invariant interval and let Q ≤ S be an F-centric subgroup.

(a) We say that R is Q-atomic if R = Q^{F}.

(b) We say that R is Q-normal if Q^{F} ⊆ R ⊆ T^{F} = {P^{F}|P ∈ T } with

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

Lemma 2.10. Let F be a saturated fusion system on the finite p-group S, let R ⊆ F^{c} be an F-invariant interval and let Q ≤ S be an F-centric subgroup which is fully F-normalized.

(a) If R is Q-atomic and N: O(F^{c})^{op} → k-Mod is a functor such that p divides the order of the kernel of the Out_{F}(Q)-action on N(Q), then \underleftarrow{lim}_{O(F^{c})}^{∗} N^{R} = 0.

(b) If R is Q-normal and N = (N^{∗}, N_{∗}): O(F) → k-Mod is a Mackey functor for F, then \underleftarrow{lim}_{O(F^{c})}^{i}(N^{∗}|_{O(F^{c})})^{R} = 0 for i > 0.

Proof. For part (a), we have

\underleftarrow{lim}_{O(F^{c})}^{∗}N^{R} ≅ Λ^{∗}(Out_{F}(Q), N(Q))

by [14, Proposition 1.9] and this vanishes by [14, Proposition 1.11(b)]. For part (b), let E = N_{F}(Q) be the normalizer fusion subsystem of F on N_{S}(Q) ([1, I.5]). Then, by [14, Lemma 1.12] and as R ∩ E^{c} ⊆ F^{c} ∩ E^{c}, we have

\underleftarrow{lim}_{O(F^{c})}^{i}(N^{∗}|_{O(F^{c})})^{R} ≅ \underleftarrow{lim}_{O(E^{c})}^{i}N_{1},

where N_{1} is the restriction of (N^{∗}|_{O(F^{c})})^{R} to O(E^{c}). Set E_{Q} = {P ≤ S | Q ≤ P ≤ N_{S}(Q)}. Then it is a standard argument using Λ^{∗}-functors (see for instance [3, Proposition 4.2]) that we can get rid of the subgroups in E^{c} \ E_{Q}. We obtain this way that

\underleftarrow{lim}_{O(E^{c})}^{i}N_{1} ≅ \underleftarrow{lim}_{O(E^{c})}^{i}(N^{∗}|_{O(E^{c})})^{R∩E_{Q}}.

We will be done by [6, Theorem A] if we prove that (N|_{O(E^{c})})^{R∩E_{Q}} is an E^{c}- restricted Mackey functor for E. This is straightforward. Note that (N^{∗}|_{O(E^{c})})^{R∩E_{Q}} vanishes outside of R ∩ E_{Q} and that Q ∈ R ∩ E_{Q}. Then, in view of Definition 2.4, it suffices to show that no term in the Mackey decomposition formula for P, R ≤ U becomes non-E-centric, where the three subgroups contain Q and are contained in N_{S}(Q). But then the terms P ∩ ^{x}R with x ∈ U also contain Q and hence are E-centric as P and R do and x normalizes Q. □

3. Proof of Theorem 1.3

By Lemma 2.8, we only need to prove Theorem 1.3 when M = S_{Q,V}, where Q ≤ S and V is a simple k Out_{F}(Q)-module. One of the common key ingredients of the proofs in [5] and [14] was an F-partition of F^{c} defined in terms of the Thompson subgroups J(P) of the subgroups P ≤ S. We define below certain subgroups Q^{∗}(P) of the subgroups P ≤ S similarly, which will be used to construct an F-partition adapted to S_{Q,V}.

Definition 3.1. Let F be a fusion system on a finite p-group S and let Q ≤ S. For P ≤ S, we define Q^{∗}(P) to be the subgroup of P generated by all subgroups of P which are F-conjugate to Q.

The subgroup Q^{∗}(P) satisfies properties similar to those of the Thompson subgroup J(P). The proof is immediate from the definition of Q^{∗}(P) and hence omitted.

Lemma 3.2. Let F be a fusion system on a finite p-group S and let Q ≤ S. (1) If P ≤ R ≤ S, then Q^{∗}(P) ≤ Q^{∗}(R).

(2) If P, P′ ≤ S and φ: P → P′ is an F-isomorphism, then φ(Q^{∗}(P)) =

Q^{∗}(P′). In particular, N_{S}(P) ≤ N_{S}(Q^{∗}(P)).
(3) If P ≤ S, then Q^{∗}(Q^{∗}(P)) = Q^{∗}(P).

(4) If P ≤ S, then Q^{∗}(P) is weakly F-closed in P, i.e., if φ: Q^{∗}(P) → P is

an F-morphism, then φ(Q^{∗}(P)) = Q^{∗}(P).

Proof of Theorem 1.3. Consider the simple Mackey functor M = S_{Q,V}, where Q ≤ S and V is a simple k Out_{F}(Q)-module. We shall define an F-partition using the subgroups Q^{∗}(P): Choose inductively subgroups X_{0}, X_{1}, ⋯, X_{N} ∈ F^{c} and Finvariant intervals ∅ = Q_{−1} ⊆ Q_{0} ⊆ ⋯ ⊆ Q_{N} = F^{c} as follows. Assume that Q_{n−1} has been defined (n ≥ 0), and that Q_{n−1} ≠ F^{c}. Consider the following collections of subgroups of S:

U_{1} = {P ∈ F^{c} \ Q_{n−1} | |Q^{∗}(P)| maximal},
U_{2} = {P ∈ U_{1} | Q^{∗}(P) ∈ F^{c}},
U_{3} = \begin{cases}\{P ∈ U_{2} | |P | minimal\}, & if U_{2} ≠ ∅, \\ \{P ∈ U_{1} | |P | maximal\}, & if U_{2} = ∅. \end{cases}

Let X_{n} be any subgroup in U_{3} such that both X_{n} and Q^{∗}(X_{n}) are fully Fnormalized. We first check that there is such an X_{n}. By Lemma 3.2, the collections U_{1}, U_{2} and U_{3} are all F-invariant intervals. So we may take X′_{n} ∈ U_{3} which is fully F-normalized. By Lemma [1, I.2.6(c)], for each F-conjugate Y of Q^{∗}(X′_{n}) which is fully F-normalized, there is an F-morphism φ: N_{S}(Q^{∗}(X′_{n})) → N_{S}(Y) such that φ(Q^{∗}(X′_{n})) = Y. By Lemma 3.2 again, N_{S}(X′_{n}) ≤ N_{S}(Q^{∗}(X′_{n})) and so φ(N_{S}(X′_{n})) ≤ N_{S}(φ(X′_{n})) and Y = Q^{∗}(φ(X′_{n})). Since X′_{n} is fully F-normalized, it follows that φ(N_{S}(X′_{n})) = N_{S}(φ(X′_{n})) and hence φ(X′_{n}) is also fully F-normalized. Thus X_{n} := φ(X′_{n}) ∈ U_{3} has the desired property that both X_{n} and Q^{∗}(X_{n}) are fully F-normalized.

Let Q_{n} be the union of Q_{n−1} with the collection of all subgroups P ≤ S which contain some F-conjugate of X_{n}. Set R_{n} = Q_{n} \ Q_{n−1}. Thus the collections Q_{n} and R_{n} are F-invariant intervals and Q_{n} is closed under overgroups too. By the definition of U_{3} and Lemma 3.2, X_{n} = Q^{∗}(X_{n}) if Q^{∗}(X_{n}) ∈ F^{c}, while R_{n} = X_{n}^{F} if Q^{∗}(X_{n}) ∉ F^{c}. For example, X_{0} = Q^{∗}(S) and R_{0} = Q_{0} = {P ≤ S | P ≥ Q^{∗}(S)} if Q^{∗}(S) ∈ F^{c}, while X_{0} = S and R_{0} = Q_{0} = {S} if Q^{∗}(S) ∉ F^{c}. By Lemma 2.3, it suffices to prove, for each n, that

\underleftarrow{lim}_{O(F^{c})}^{i}(S_{Q,V}^{∗} |_{O(F^{c})})^{R_{n}} = 0 for all i > 0.

Case 1: Assume that Q^{∗}(X_{n}) ∉ F^{c}. Hence R_{n} = X_{n}^{F} is X_{n}-atomic. Since Q^{∗}(X_{n}) is fully F-normalized and not F-centric, C_{S}(Q^{∗}(X_{n})) ≰ Q^{∗}(X_{n}).

On the one hand, if Q^{∗}(X_{n}) is not centric in X_{n}, then there is an element x ∈ C_{X_{n}}(Q^{∗}(X_{n})) \ Q^{∗}(X_{n}) and we have x ∈ C_{X_{n}}(L) \ L for any F-isomorphism α: Q → L with L ≤ X_{n}. Thus S_{Q,V} (X_{n}) = 0 by Lemma 2.7(a), so (S_{Q,V}^{∗} |_{O(F^{c})})^{R_{n}} is the zero functor and we are done.

On the other hand, if Q^{∗}(X_{n}) is centric in X_{n}, then C_{S}(Q^{∗}(X_{n})) ≰ Q^{∗}(X_{n}) implies that C_{S}(Q^{∗}(X_{n})) ≰ X_{n}. Thus X_{n}C_{S}(Q^{∗}(X_{n})) > X_{n} and so the normalizer N_{X_{n}C_{S}(Q^{∗}(X_{n}))}(X_{n}) properly contains X_{n}. Hence there exists x ∈ N_{S}(X_{n}) \ X_{n} which centralizes Q^{∗}(X_{n}). Then, by Lemma 2.7(b), [c_{x}] ∈ Out_{F}(X_{n}) is a nontrivial p-element that lies in the kernel of the action of Out_{F}(X_{n}) on S_{Q,V} (X_{n}) . Thus we are done by Lemma 2.10(a).

Case 2: Assume that Q^{∗}(X_{n}) ∈ F^{c}, and hence X_{n} = Q^{∗}(X_{n}). We show that R_{n} is X_{n}-normal. Note that X_{n}^{F} ⊆ R_{n} and set

T = {P ≤ S | X_{n} ⊴ P, and R ∈ X_{n}^{F} and R ⊴ P imply R = X_{n}}.

Then we are left with proving that R_{n} ⊆ T^{F}. Observe now that

(3) P ∈ R_{n} ⟹ Q^{∗}(P) is the only subgroup of P which is F-conjugate to X_{n}.

Indeed, suppose P ∈ R_{n}. This means that P contains an F-conjugate X′_{n} of X_{n} and P ∉ Q_{n−1}. By Lemma 3.2, Q^{∗}(P) ≥ Q^{∗}(X′_{n}) = X′_{n} and |Q^{∗}(X′_{n})| = |Q^{∗}(X_{n})|. By the definition of U_{1}, it follows that Q^{∗}(P) = Q^{∗}(X′_{n}) = X′_{n}, as claimed.

Let R be the collection of all P ∈ R_{n} such that P ≥ X_{n}. By (3), it is immediate that R ⊆ T and hence it is enough to show that R_{n} ⊆ R^{F}. Let P ∈ R_{n}. Then P contains an F-conjugate X′_{n} of X_{n}. By (3), X′_{n} = Q^{∗}(P) ⊴ P, so P ≤ N_{S}(X′_{n}). Since X_{n} is fully F-normalized, there is an F-morphism φ: N_{S}(X′_{n}) → N_{S}(X_{n}) such that φ(X′_{n}) = X_{n}. Thus φ(P) ≥ φ(X′_{n}) = X_{n} and so φ(P) ∈ R, as claimed. It follows that R_{n} is X_{n}-normal and hence we are done by Lemma 2.10(b). □

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[15] P. Webb, Two classifications of simple Mackey functors with applications to group cohomology and the decomposition of classifying spaces, J. Pure Appl. Algebra 88 (1993), no. 1–3, 265–304.

   E-mail address: adiaz@agt.cie.uma.es

Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, Apdo correos 59, 29080 Málaga, Spain

   E-mail address: sejong.park@nuigalway.ie

School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, University Road, Galway, Ireland