Stable splittings of classifying spaces of finite groups

D. J. Benson Mathematical Institute 24–29 St. Giles Oxford OX1 3LB Great Britain

M. Feshbach School of Mathematics University of Minnesota Minneapolis, MN55455 U.S.A.

1 Introduction

The object of this paper is to describe the stable splittings of classifying spaces of finite groups. The study of such splittings goes back at least to papers of Holzsager [13], Cooke and Smith [6] and Witten [22]. Homological splittings go back even further (cf. Dold [9]). There is now a long list of authors who have contributed to the subject. Stable splittings of spaces have had numerous applications in homotopy theory. The general theory developed here should make the calculation of many more stable splittings accessible. In particular we give a complete splitting of BG based only on the subgroup structure of G and the modular representation theory of the automorphism groups of subgroups of G. A different approach to the calculation of stable splittings is given in Martino and Priddy [17].

If G is a finite group and P is a Sylow p-subgroup of G then the restriction map

res_{G,P} : H^{∗}(G, Z_{p}^{∧}) → H^{∗}(P, Z_{p}^{∧})

is injective, where Z_{p}^{∧} denotes the p-adic integers. Indeed, a similar statement is true of any generalised cohomology theory, because there is a transfer map, and restriction followed by transfer is |G : P | times the identity map in ordinary cohomology (though not in other theories). The transfer map happens at the level of classifying spaces, after stabilising with respect to suspension, and the corresponding statement there is that BG is a wedge summand of BP after stabilising and completing at p. Thus a study of stable p-completed splittings of BP should shed light on the question of how P can be embedded as a Sylow p-subgroup in a finite group G.

Graeme Segal’s conjecture for the stable cohomotopy of the classifying space of a finite group was proved by Gunnar Carlsson [4], based in part on the proof for elementary abelian groups by Adams, Gunawardena and Miller [2]. As was predicted by Adams [1] and shown by Lewis, May and McClure [14], one consequence of Segal’s conjecture is an algebraic description of the abelian group of stable p-completed maps from one classifying space (with disjoint basepoint) to another {BG′; BG}_{p}^{∧} as the completion of a certain finitely generated free abelian group A(G′, G) with respect to a certain filtration. Composition of maps

{BG′; BG}_{p}^{∧} × {BG′′; BG′}_{p}^{∧} → {BG′′; BG}_{p}^{∧}

corresponds to an algebraically defined associative composition

A(G′, G) × A(G′′, G′) → A(G′′, G).

In particular A(G, G) is a ring, and A(G′, G) is an A(G, G)-A(G′, G′)-bimodule. One can view A(G, G) as a sort of non-commutative version of the Burnside ring (see Section 3). In fact, the Burnside ring is just A(G) = A(G, 1), and is hence an A(G, G)-module. Note, though, that the multiplication on A(G) is not easy to relate to the multiplication on A(G, G).

The filtration on A(G, G) is by powers of a certain ideal, and we write A(G, G)^{∧} for the completion with respect to this ideal, so that {BG; BG}_{p}^{∧} ≅ A(G, G)^{∧} as rings. If we denote by Ã(G, G) the quotient of A(G, G) by the two-sided ideal which corresponds to maps factoring through the trivial subgroup, then in the case where G = P is a p-group, the completion of Ã(P, P) is isomorphic to Z_{p}^{∧} ⊗_{Z} Ã(P, P) where Z_{p}^{∧} denotes the p-adic completion of the integers (see Nishida [20]).

It follows from this description of {BG; BG}_{p}^{∧} that stable splittings of BG (as a wedge sum of p-completed spectra) are in one-one correspondence with decompositions

1 = e_{1} + ⋯ + e_{s}

of the identity element of A(G, G)^{∧} as a sum of orthogonal idempotents. The summand e_{i}.BG is indecomposable if and only if e_{i} is a primitive idempotent; and e_{i}.BG ≅ e_{j}.BG if and only if e_{i} is conjugate to e_{j}.

In the case where G = P is a p-group, the idempotent refinement theorem implies that decompositions of the identity element of A(P, P)^{∧} as a sum of primitive orthogonal idempotents may be obtained by lifting such decompositions of the identity element of Ā(P, P) = F_{p} ⊗_{Z} A(P, P). It follows that there is a one-one correspondence between isomorphism types of summands e_{i}.BP and isomorphism types of simple Ā(P, P)-modules. The multiplicity of the summand e_{i}.BP in BP is equal to the multiplicity of the corresponding projective indecomposable Ā(P, P)-module in the regular representation; and this is equal to the dimension of the corresponding simple Ā(P, P)-module over its endomorphism ring, which is a finite field of order a power of p by Wedderburn’s theorem on finite division rings.

In this paper we introduce the coadjoint module M(P, P) for A(P, P). This is a certain submodule of finite index in the dual of the regular representation Hom_{Z}(A(P, P), Z). Most of the results in this paper are a consequence of our formula (Proposition 3.1) for the action of A(P, P) on M(P, P). It follows from this formula that the reduction modulo p, M̄(P, P) = F_{p} ⊗_{Z} M(P, P) has an Ā(P, P)-invariant filtration, whose subquotients we then proceed to study.

The final result of this analysis (Theorem 5.7) is a collection of Ā(P, P)-modules L̄(P, K, S), each labelled by a “type” of subgroup K and a simple OutK-module S, where OutK = AutK/InnK is the group of outer automorphisms of K. Each L̄(P, K, S) is either simple or zero, and the non-zero ones form a set of representatives for the isomorphism types of simple Ā(P, P)-modules. A necessary and sufficient condition is given for L̄(P, K, S) to be non-zero. In particular, given an OutK-module S, there can be at most one possible type of K from an isomorphism class for which L̄(P, K, S) ≠ 0. For this K, the subgroup Stab_{P} (S) of N_{P} (K) is a direct product of K and another subgroup; see Proposition 5.9. There is a discussion in Theorem 5.2 of the corresponding stable wedge summands of BP, and the form of the corresponding idempotents.

The module L̄(P, K, S) is a quotient of a module denoted S ⊗_{F_{p}OutK} L̄(P, K), where L̄(P, K) is an F_{p}OutK ⊗_{F_{p}} Ā(P, P)-module with an explicit basis described in Proposition 4.8. The quotient is by a submodule M described in Definition 4.9, which in case L̄(P, K, S) is non-zero, is the unique maximal submodule of S ⊗_{F_{p}OutK} L̄(P, K).

Returning to the case of a general finite group G having P as a Sylow p-subgroup, we show in Section 6 that the element [G] of Ā(P, P) has the property that

Ā(G, G) = [G].Ā(P, P).[G].

This means that the modules

L̄(G, K, S) = [G].L̄(P, K, S)

are simple or zero, and the non-zero ones form a set of representatives for the isomorphism types of simple Ā(G, G)-modules. The same necessary and sufficient condition as before picks out the non-zero ones. As in the case of a p-group, the number of copies of the corresponding indecomposable summand is equal to the dimension of L̄(G, K, S) as a module over its endomorphism ring, which is a finite field of order a power of p.

We describe here how to split BP for any particular p-group P. Proposition 5.9 allows one to rule out many subgroups K as possible vertices of summands. In particular KC_{P} (K) must split as a product of K and some other subgroup. Moreover, given any particular simple F_{p}OutK-module S, we have that Stab_{P} (S) is a direct product of K and some other subgroup. These conditions on K and S are in practice very close to being sufficient (Example 4 in Section 7 shows that they are not sufficient in general). One needs to calculate L̄(P, K, S) next. If there are few conjugacy classes of subgroups of the same type as K, then this is usually quite practical. Otherwise, it may be easier to rely on some ad hoc techniques to simplify the calculation (such as those involving cohomology). If one is only interested in whether a particular indecomposable summand occurs, Theorem 5.2 is sufficient. Corollary 6.4 and Proposition 6.6 are used to split BG for G a general finite group.

We give several examples in Section 7 of the application of this theory to some particular groups. The example of the extraspecial p-groups (Example 5) is a good illustration of the power of the theory.

Throughout this paper, when we talk of the classifying space BG of a finite group G, we really mean the suspension spectrum of the space (BG_{+})_{p}^{∧} obtained by adjoining a disjoint basepoint, and then completing at a fixed prime p. If α, β : Σ^{n}BG′ → Σ^{n}BG are maps at the level of spaces, then α + β : Σ^{n+1}BG′ → Σ^{n+1}BG is defined using the suspension coordinate. This gives {BG′; BG}_{p}^{∧} the structure of an abelian group. Because of these conventions, the classifying space B1 of the trivial group is actually the sphere spectrum, and splits off BG for all finite groups G. The remaining piece, which we shall denote \overline{BG}, is the same as BG but without the disjoint basepoint.

If H is a subgroup of G then transfer is defined as a map BG → Ω^{∞}Σ^{∞}BH at the level of spaces, and hence gives a map BG → BH at the level of suspension spectra.

We write our maps on the left, which means in particular that we shall write ^{g}H to denote gHg^{−1}.

2 The ring A(G, G)

The set A(G′, G) is a free abelian group with one basis element ζ_{H′,ϕ} for each conjugacy class of pairs (H′, ϕ), where H′ is a subgroup of G′ and ϕ is a (not necessarily injective) homomorphism ϕ : H′ → G. This basis element corresponds to the map BG′ → BG given by composing the transfer tr = tr_{G′,H′} : BG′ → BH′ with the map B(ϕ) : BH′ → BG. If ζ_{H′,ϕ} ∈ A(G′, G) and ζ_{H′′,ϕ′} ∈ A(G′′, G′) then the composite ζ_{H′,ϕ} ∘ ζ_{H′′,ϕ′} ∈ A(G′′, G) is obtained formally as follows. Factor ϕ′ : H′′ → G′ as a surjection H′′ → K′ followed by an inclusion i′ : K′ ↪ G′. Then the Mackey formula enables us to express the composite of B(i′) : BK′ → BG′ and tr : BG′ → BH′ as a sum of composites

BK′ →^{tr} B(K′ ∩^{x}H′) ⟶^{c_{x^{−1}}} B(^{x^{−1}}K′ ∩ H′) ⟶^{B(i′)} BH′

where x runs over a set of double coset representatives K′\G′/H′ and c_{x^{−1}} denotes con-

jugation by x^{−1}. Then the composite BH′′ ⟶^{B(ϕ′)} BK′ →^{tr} B(K′ ∩ ^{x}H′) is equal to the

composite BH′′ →^{tr} B((ϕ′)^{−1}(K′ ∩^{x}H′)) ⟶^{B(ϕ′)} B(K′ ∩^{x}H′), and so we have expressed the composite as a sum of basis elements of the given kind

ζ_{H′,ϕ} ∘ ζ_{H′′,ϕ′} = ∑_{x∈K′\G′/H′} ζ_{(ϕ′)^{−1}(K′∩^{x}H′),ϕ∘i′∘c_{x^{−1}}∘ϕ′}.  (2.1)

This formula for the composite is illustrated in the following diagram.

B((ϕ′)^{−1}(K′ ∩^{x}H′))
↑ tr ↘B(ϕ′)
BH′′  B(K′ ∩^{x}H′) ⟶^{B(i′∘c_{x^{−1}})} BH′
↑ tr ↘B(ϕ′)  ↑ tr  ↑ tr ↘B(ϕ)
BG′′  BK′  ⟶^{B(i′)}  BG′  BG

An alternative description (Adams [1] and May [15]) of A(G′, G) is as follows. The set of G-free (G′ × G)-sets has an additive structure given by disjoint union, and satisfies cancellation, so we may adjoin formal negatives as in the construction of the Burnside ring to form the additive group A(G′, G). The basis element ζ_{H′,ϕ} corresponds to the transitive (G′ × G)-action on the cosets of the subgroup

Δ_{H′,ϕ} = {(x, ϕ(x)), x ∈ H′} ⊆ G′ × G.

Note that the subgroup H′ and the map ϕ are only well defined up to conjugacy in the domain and range. It is easily seen that every transitive G-free (G′ × G)-set is of this form. Composition is then defined as follows. If X is a G-free (G′ × G)-set and Y is a G′-free (G′′ × G′)-set then Y ×_{G′} X = (Y × X)/G′ (where G′ acts diagonally on Y × X) is a G-free (G′′ × G)-set. It is easy to check that if K′ = Im(ϕ′) as before then

((G′′ ×G′)/Δ_{H′′,ϕ′})×_{G′} ((G′ ×G)/Δ_{H′,ϕ}) ≅ ⋃_{x∈K′\G′/H′}^{.}(G′′ ×G)/Δ_{(ϕ′)^{−1}(K′∩^{x}H′),ϕ∘i′∘c_{x^{−1}}∘ϕ′}

so that this defines the same multiplication as before.

3 The coadjoint module

Suppose H is a collection of subgroups of a finite group G, with the following properties. (i) If H ∈ H then every conjugate ^{x}H is also in H. (ii) If H, K ∈ H then the intersection H ∩ K ∈ H.

Then there is a subring A(G, H) of the Burnside ring A(G) corresponding to G-sets where the point stabilisers are all in H. For each conjugacy class of H ∈ H, there is a ring homomorphism

f_{H} : A(G, H) → Z
X ↦ |X^{H}|

given by counting fixed points (see for example tom Dieck [tD], Section 1.2). It is well known that the resulting map

∑ f_{H} : A(G, H) → ⨁_{\begin{matrix}ccl’s of \\ H∈H \end{matrix}} Z

is an injective map with finite cokernel. It follows from linear independence of ring homomorphisms (see for example the discussion on pp. 34–35 of Artin [3]) that every ring homomorphism A(G, H) → Z is of the form f_{H} for some H ∈ H.

Taking as H the collection of subgroups of the form Δ_{H′,ϕ}, we have A(G′, G) = A(G′ × G, H) as abelian groups, and we obtain maps

f_{H′,ϕ} : A(G′, G) → Z
X ↦ |X^{Δ_{H′,ϕ}}|.

The sum of these maps

∑ f_{H′,ϕ} : A(G′, G) → ⨁_{ccl’s of ; pairs (H′,ϕ)} Z

is injective with finite cokernel.

Remark The commutative ring structure obtained by regarding A(G′, G) as contained in A(G′ × G) has nothing to do with the multiplication coming from composition of maps. All we need to know is that the above map ∑ f_{H′,ϕ} is injective with finite cokernel.

We now calculate the composite of right multiplication by ζ_{H,ϕ} with a map of the form f_{K,ψ}.

Proposition 3.1 If H, K ≤ G′′, ϕ : H → G′ and ψ : K → G then the composite map

A(G′, G) ⟶^{−∘ζ_{H,ϕ}} A(G′′, G) →^{f_{K,ψ}} Z

is given by the formula

ζ_{H,ϕ}.f_{K,ψ} = ∑_{\begin{matrix}x∈K\G′′/H \\ K≤^{x}H \\ K∩^{x}(Kerϕ)≤Kerψ \end{matrix}} f_{ϕ(^{x^{−1}}K),ψ∘c_{x}∘ϕ^{−1}}

Remark The sum here runs over those double coset representatives x of K and H in G which satisfy the two conditions K ≤ ^{x}H and K ∩^{x}(Kerϕ) ≤ Kerψ. Under these two conditions the composite map

ϕ(^{x^{−1}}K) ⟶^{ϕ^{−1}} ^{x^{−1}}K →^{c_{x}} K →^{ψ} G

is well defined since the ambiguity created by ϕ^{−1} is in ^{x^{−1}}K ∩ Kerϕ ≤ ^{x^{−1}}(Kerψ) so that conjugating by x takes this ambiguity to an element killed by ψ.

The proof of this proposition depends on two lemmas.

Lemma 3.2 The restriction to K × G′ of the (G′′ × G′)-set (G′′ × G′)/Δ_{H,ϕ} is given by

((G′′ × G′)/Δ_{H,ϕ})↓_{K×G′}= ⋃_{x∈K\G′′/H}^{.}(K × G′)/Δ_{K∩^{x}H,ϕ∘c_{x^{−1}}}

Proof This follows immediately from the usual Mackey formula for restriction of permutation representations, since

(K × G) ∩^{x}(Δ_{H,ϕ}) = Δ_{K∩^{x}H,ϕ∘c_{x^{−1}}} . □

Lemma 3.3 If L is a subgroup of K, ρ : L → G′, and X is a G-free (G′ × G)-set then

|(((K × G′)/Δ_{L,ρ}) ×_{G′} X)^{Δ_{K,ψ}}| = \{ \begin{matrix}|X^{(ρ,ψ)(K)}| & if L = K and Kerρ ≤ Kerψ \\ 0 & otherwise. \end{matrix}

Proof We have

((K × G′)/Δ_{L,ρ}) ×_{G′} X ≅ K ×_{L} X

where L acts on X via the map ρ. Thus if L is a proper subgroup of K, Δ_{K,ψ} has no fixed points, while if L = K then K×_{L}X = X, and Δ_{K,ψ} ≅ K acts on X via (ρ, ψ) : K → G′×G.

Thus there are no fixed points unless Kerρ ≤ Kerψ, and then the number of fixed points is |X^{(ρ,ψ)(K)}|. □

Proof of Proposition 3.1 : If X is a G-free (G′ × G)-set then using Lemma 3.2 we have

(ζ_{H,ϕ}.f_{K,ψ})(X) = f_{K,ψ}(((G′′ × G′)/Δ_{H,ϕ}) ×_{G′} X)
= f_{K,ψ}(((G′′ × G′)/Δ_{H,ϕ})↓_{K×G′} ×_{G′}X)
= f_{K,ψ}( ⋃_{x∈K\G′′/H}^{.}((K × G′)/Δ_{K∩^{x}H,ϕ∘c_{x^{−1}}} ) ×_{G′} X)
= ∑_{x∈K\G′′/H} |(((K × G′)/Δ_{K∩^{x}H,ϕ∘c_{x^{−1}}} ) ×_{G′} X)^{Δ_{K,ψ}}|.

By Lemma 3.3, a double coset representative x contributes nothing to this sum unless K ∩^{x}H = K and the kernel of (ϕ ∘ c_{x^{−1}}) on K ∩^{x}H is contained in Kerψ; in other words unless K ≤ ^{x}H and K ∩^{x}(Kerϕ) ≤ Kerψ. In that case the contribution is

|X^{(ϕ∘c_{x^{−1}},ψ)(K)}| = |X^{(Δ_{ϕ(^{x^{−1}}K),ψ∘cx∘ϕ^{−1}})}|
= f_{ϕ(^{x^{−1}}K),ψ∘c_{x}∘ϕ^{−1}}(X). □

The effect of this proposition is the following. The free abelian group M(G′′, G) whose basis consists of the f_{K,ψ} admits a left action

A(G′′, G′) × M(G′′, G) → M(G′, G)

satisfying the obvious associativity formula.

In particular, setting G = G′ = G′′ (this would have made the proof of the Proposition impossible to follow if we had done so there!), we see that M(G, G) is a left A(G, G)- module, called the coadjoint module. We give it this name because we have

Q ⊗_{Z} M(G, G) ≅ Hom_{Z}(A(G, G), Q).

Note, however, that

M(G, G) ≅ H̸om_{Z}(A(G, G), Z)

in general, since not every homomorphism of abelian groups is necessarily an integer linear combination of the f_{K,ψ}. Thus the proposition is necessary in order to show that the obvious left action of Q ⊗_{Z} A(G, G) on Q ⊗_{Z} M(G, G) comes from a left action of A(G, G) on M(G, G).

Next, we assume that G = P is a p-group, and reduce modulo p. Set

Ā(P, P) = F_{p} ⊗_{Z} A(P, P)
M̄(P, P′) = F_{p} ⊗_{Z} M(P, P′).

Lemma 3.4 Every simple Ā(P, P)-module is a composition factor of M̄(P, P).

Proof This follows from the fact that M(P, P) is a Z-form in the dual of the regular representation of Q ⊗_{Z} A(P, P). See for example Curtis and Reiner [7], Proposition 16.16, whose proof does not depend on the algebra in question being a group algebra. □

4 Breaking up the coadjoint module

We now observe that there is a certain filtration on M̄(P, P) preserved by the action of Ā(P, P). At the coarsest level we could just observe that the linear span of those f_{K,ψ} with |K| less than or equal to a given bound, form an A(P, P)-submodule of M(P, P). We refine this notion as follows, and at this stage we retain the extra generality of considering M̄(P, P′) as a left Ā(P, P)-module, since we shall need this later. For the remainder of this section, P and P′ are p-groups.

Definition 4.1 If K ≤ P and ψ : K → P′, we define

Stab_{P} (K, ψ) = {g ∈ N_{P} (K) | ∃ g′ ∈ P′ with ψ ∘ c_{g} = c_{g′} ∘ ψ}.

Note that N_{P} (K) ∩ N_{P} (Kerψ) ≥ Stab_{P} (K, ψ) ≥ KC_{P} (K).

Lemma 4.2 In the reduction modulo p of the formula given in Proposition 3.1, as a formula for the action of Ā(P, P) on M̄(P, P′), we may replace the condition K ≤ ^{x}H by the condition Stab_{P} (K, ψ) ≤ ^{x}H.

Proof Suppose g ∈ Stab_{P} (K, ψ) and x is a double coset representative of K and H in P satisfying K ≤ ^{x}H and K ∩^{x}(Kerϕ) ≤ Kerψ. Since g ∈ N_{P} (K) ∩ N_{P} (Kerψ), we have K ≤ ^{gx}H, K ∩^{gx}(Kerϕ) ≤ Kerψ, and ^{x^{−1}}K = ^{x^{−1}g^{−1}}K. Since there is an element g′ ∈ P′ with ψ ∘ c_{gx} = c_{g′} ∘ ψ ∘ c_{x}, we have

f_{ϕ(^{x^{−1}}K),ψ∘c_{x}∘ϕ^{−1}} = f_{ϕ(^{x^{−1}g^{−1}}K),ψ∘c_{gx}∘ϕ^{−1}}.

This shows that the contribution corresponding to a given double coset representative x is equal to the contribution from all the gx with g ∈ Stab_{P} (K, ψ). If Stab_{P} (K, ψ) ≰ ^{x}H, then |Stab_{P} (K, ψ) : Stab_{P} (K, ψ) ∩^{x}H| is divisible by p and so the sum of these contributions is zero modulo p. □

There is a natural filtration on M̄(P, P′) which is preserved by the action of Ā(P, P). This gives rise to the notion of types, which we now define. The main motivation for the definition is Proposition 4.4, whose proof is immediate.

Definition 4.3 We write (K_{1}, ψ_{1}) ⪰ (K_{2}, ψ_{2}) if there is a surjective homomorphism α : K_{1} → → K_{2} which extends to a (not necessarily surjective) homomorphism K_{1}C_{P} (K_{1}) → P, and there is an element g′ ∈ P′ such that ψ_{1} = c_{g′} ∘ ψ_{2} ∘ α. In particular Im(ψ_{1}) is P′- conjugate to Im(ψ_{2}).

If (K_{1}, ψ_{1}) ⪰ (K_{2}, ψ_{2}) and (K_{2}, ψ_{2}) ⪰ (K_{1}, ψ_{1}), we write (K_{1}, ψ_{1}) ∼ (K_{2}, ψ_{2}), and say that (K_{1}, ψ_{1}) has the same type as (K_{2}, ψ_{2}). Thus a type is an equivalence class of such pairs. In particular, if (K_{1}, ψ_{1}) ∼ (K_{2}, ψ_{2}) then there is an isomorphism α : K_{1} → K_{2} taking Kerψ_{1} to Kerψ_{2}. If (K_{1}, ψ_{1}) ⪰ (K_{2}, ψ_{2}) and (K_{1}, ψ_{1}) ≁ (K_{2}, ψ_{2}) we write (K_{1}, ψ_{1}) ≻ (K_{2}, ψ_{2}).

If ψ_{1} and ψ_{2} are isomorphisms then by abuse of notation (since the notion does depend on the chosen isomorphism), we talk of types of subgroups of P, and we write K_{1} ⪰ K_{2}, etc. In this case we have Stab_{P} (K, ψ) = KC_{P} (K).

Proposition 4.4 If f_{K_{2},ψ_{2}} appears with non-zero multiplicity modulo p in the expression for ζ_{H,ϕ}.f_{K_{1},ψ_{1}} given in Proposition 3.1, then (K_{1}, ψ_{1}) ⪰ (K_{2}, ψ_{2}).

Proof This follows from the lemma, by taking for α the restriction of ϕ ∘ c_{x^{−1}} to K_{1}. □

It follows that the partial order ⪰ on the set of pairs (K, ψ) defines an Ā(P, P)-invariant filtration of M̄(P, P′). We now examine the subquotients.

Definition 4.5 Given a pair (K, ψ) with K ≤ P, ψ : K → P′, we define M̄(P, P′)_{K,ψ} to be the linear span of those f_{K′,ψ′} with (K, ψ) ⪰ (K′, ψ′). According to Proposition 4.4, M̄(P, P′)_{K,ψ} is an Ā(P, P′)-invariant submodule of M̄(P, P′). We define L̄(P, P′)_{K,ψ} to be the quotient of M̄(P, P′)_{K,ψ} by the submodule given by the linear span of the M̄(P, P′)_{K′,ψ′} with (K, ψ) ≻ (K′, ψ′). We define L̄(P, K) = ⨁ L̄(P, K)_{K,ϕ}, where the sum is over all types with ϕ an isomorphism.

Remark In case KC_{P} (K) is a direct product of K and some other group, we have L̄(P, K) = L̄(P, K)_{K,id_{K}}. It will follow from Proposition 5.9 that every simple Ā(P, P)- module is a subquotient of such an L̄(P, K).

Lemma 4.6 Every simple Ā(P, P)-module is isomorphic to a subquotient of some module of the form L̄(P, P)_{K,ψ}.

Proof This follows from Lemma 3.4 and the fact that M̄(P, P) has a filtration in which the quotients are of the form L̄(P, P)_{K,ψ}. □

Next we discuss homomorphisms between modules of the form M̄(P, P′). Suppose π : P′ → P′′ is a group homomorphism. Then π induces a homomorphism of Ā(P, P)- modules

π_{∗} : M̄(P, P′) → M̄(P, P′′)
π_{∗}(f_{K,ψ}) = f_{K,π∘ψ}.

This is a well defined map since if ψ′ : K′ → P′ is another map with (K, ψ) conjugate to (K′, ψ′) then (K, π ∘ ψ) is conjugate to (K′, π ∘ ψ′). It is a homomorphism of Ā(P, P)- modules by Proposition 3.1.

For example, if π : P′ → P′′ is injective then the image of π_{∗} is the sum of the M̄(P, P′′)_{K,ψ} with K ≤ P, ψ : K → P′′ and Imψ ≤ Imπ. Thus by Lemma 3.4 every simple Ā(P, P)-module is isomorphic to a subquotient of some module of the form L̄(P, P′)_{K,ψ} with K ≤ P and ψ : K → P′ surjective.

Next, if K ≤ P and ψ : K → P′ is surjective it follows that if (K, id_{K}) ∼ (K′, ψ′) then (K, ψ) ∼ (K′, ψ∘ψ′). Moreover, ψ induces a surjective homomorphism of Ā(P, P)-modules

ψ_{∗} : L̄(P, K)_{K,id_{K}} → → L̄(P, P′)_{K,ψ}.

So every subquotient of L̄(P, P′)_{K,ψ} is also a subquotient of L̄(P, K).

Finally if π is an element of the outer automorphism group OutK then π_{∗} is a module automorphism of L̄(P, K) and so F_{p}OutK acts as module endomorphisms. Thus L̄(P, K) is a left F_{p}OutK ⊗_{F_{p}} Ā(P, P)-module. In particular L̄(P, K) has a filtration in which the filtered quotients are Ā(P, P)-modules of the form S ⊗_{F_{p}OutK} L̄(P, K), where S is a simple right F_{p}OutK-module. In this tensor product, we have s ⊗ f̄_{K,π∘ψ} = sπ ⊗ f̄_{K,ψ}.

We have now proved the following Proposition.

Proposition 4.7 Every simple Ā(P, P)-module occurs as a subquotient of a module of the form

S ⊗_{F_{p}OutK} L̄(P, K)

where K is a subgroup of P and S is a simple right F_{p}OutK-module. The module L̄(P, K) depends only on the type of K. □

Remark Denote by Φ(K) the Frattini subgroup of K; namely the intersection of the maximal subgroups of K. Then K/Φ(K) ≅ (Z/p)^{r} for some r, and the kernel of the map

OutK → Out(K/Φ(K)) ≅ GL_{r}(F_{p})

is contained in O_{p}OutK (the largest normal p-subgroup of OutK), and hence in the kernel of every simple F_{p}OutK-module (see for example Gorenstein [11], Theorem 5.1.4).

It is worthwhile writing down explicitly at this stage the formula for the action of Ā(P, P) on S ⊗_{F_{p}OutK} L̄(P, K).

Proposition 4.8 The module L̄(P, K) has a basis consisting of the f̄_{K′,ψ} where K′ has the same type as K and ψ is an isomorphism from K′ to K. If π ∈ OutK then π_{∗}(f̄_{K′,ψ}) = f̄_{K′,π∘ψ}. An element ζ_{H,ϕ} ∈ Ā(P, P) acts on S ⊗_{F_{p}OutK} L̄(P, K) via the formula

ζ_{H,ϕ}.(s ⊗ f̄_{K′,ψ}) = ∑_{\begin{matrix}x∈K′\P/H \\ K′≤Stab_{P} (S)≤^{x}H \\ K′∩^{x}(Kerϕ)=1 \end{matrix}} s ⊗ f̄_{ϕ(^{x^{−1}}K′),ψ∘c_{x}∘ϕ^{−1}}. □

Here, the stabiliser of S, Stab_{P} (S) is defined to be the kernel of the action of N_{P} (K′) on S (via the natural map N_{P} (K′) → OutK′ ≅ OutK; note that this isomorphism is well defined up to conjugation, and so the correspondence between the simple modules is well defined). Note that unless Stab_{P} (S) ≤ ^{x}H, the same argument as in Lemma 4.2 shows that the corresponding contribution is zero modulo p. So the formula is true with or without this restriction in the sum.

We now define the modules L̄(P, K, S). In the next section, we shall show that these modules are always either simple or zero, and that the non-zero ones give a set of representatives for the isomorphism classes of simple Ā(P, P)-modules. Proposition 5.9 below shows that these modules are zero under many circumstances. We shall see in Theorem 5.7 that for the non-zero ones, the module M given in the following definition is a maximal submodule whose simple composition factors are Ā(P, P)-modules which have been accounted for by subgroups which are larger than K. In case K = P, L̄(P, P, S) is isomorphic to S^{∗}. These are the simple Ā(P, P)-modules considered by Nishida in his definition of dominant summands.

Definition 4.9 If S is a simple right F_{p}OutK-module, we define

L̄(P, K, S) = (S ⊗_{F_{p}OutK} L̄(P, K))/M

where M is defined as follows. Let M_{0} = ⋂_{Imϕ≅K} Ker(ζ_{L,ϕ}) ⊆ S ⊗_{F_{p}OutK} L̄(P, K). We note that M_{0} is a submodule by Formula 2.1. We now inductively define M_{i} to be the preimage of the submodule ⋂_{Imϕ≅K} Ker(ζ_{L,ϕ}) of (S ⊗_{F_{p}OutK} L̄(P, K))/M_{i−1} back in

S ⊗_{F_{p}OutK} L̄(P, K). Then for n large enough M_{n} = M_{n+1}, and we denote this by M.

5 Dominant summands

In this section we discuss some topological aspects of the theory. We first give a brief review of Nishida theory.

There is a surjective ring homomorphism ρ : Ā(P, P) → F_{p}OutP which sends every basis element ζ_{H,ϕ} to zero unless H = P and ϕ is an automorphism, in which case it is sent to the corresponding outer automorphism. This is clearly a split surjection. A primitive idempotent in the kernel of ρ corresponds to an indecomposable stable summand of BP which factors through a wedge of BK’s where the K’s are proper subgroups. Since the endomorphism rings of indecomposable summands of these BK’s are local rings, it follows that such a summand of BP factors through a single BK.

Definition 5.1 An indecomposable stable wedge summand of BP is dominant if it is not homotopy equivalent to a summand of BK for any proper subgroup K < P.

Isomorphism classes of dominant summands are in one-one correspondence with conjugacy classes of primitive idempotents of Ā(P, P) which are not in the kernel of ρ. Each such idempotent ē is labelled by ρ(ē) (or a lift of this to a primitive idempotent in Z_{p}^{∧}OutP), and hence by a simple left F_{p}OutP -module. The multiplicity as a dominant summand is equal to the dimension of the simple module over its endomorphism ring, which is a finite field. If the simple F_{p}Out(P)-module is trivial, the dominant summand is called the principal dominant summand. For further information see Nishida [20] or Priddy [21].

Let us now consider a non-dominant stable summand X of BP. X is a dominant summand of BK for some proper subgroup K of P. Let S^{∗} be the simple left F_{p}OutKmodule that labels X (we use ∗ to denote duality between left and right F_{p}OutK-modules, and also to denote the antiautomorphism of the group algebra given by ∑ a_{g}g ↦ ∑ a_{g}g^{−1}). Since we are only concerned with the dominant summand X, we work throughout this section modulo the two-sided ideal generated by all ζ_{L,ψ} where the image of ψ has order less than that of K, except where explicitly stated otherwise. The map e : BP → X → BP consisting of projection followed by inclusion is an idempotent. This map factors through BK, and so ē is a composition of an element in Ā(P, K) and one in Ā(K, P). By Formula 2.1, ē is a sum of basis elements ζ_{H,ϕ} with the image of ϕ isomorphic to K. One cannot assume that this image is constant over the sum. However, one can write each term as a map of the form ζ_{H,i∘π}, where π is a surjection of some subgroup H onto K with kernel N, say, and i is an isomorphism of K with some subgroup K′ of P.

The following theorem, giving a necessary and sufficient condition for the summand X to occur as a summand of BP, is essentially the same as a theorem of Priddy and Martino ([21] II, Theorem 0.1), although our proof is different.

Theorem 5.2 The following are equivalent.

(i) The dominant summand X of BK labelled by a primitive idempotent e_{0} ∈ Z_{p}^{∧}OutK is isomorphic to a summand e.BP of BP.

(ii) K is isomorphic to a subgroup K′ of P with the following properties. There is a split surjection π : H → K, where H is a subgroup of P. Letting

γ = ∑_{\begin{matrix}x∈K′\P/H \\ K′≤^{x}H \end{matrix}} K →^{≅} K′ ⟶^{c_{x^{−1}}} ^{x^{−1}}K′ ↪ H →^{π} K

considered as an element of F_{p}OutK, the element ē_{0}γ has non-zero image in F_{p}OutK/J, where J denotes the radical. In particular, γ induces a non-zero action on the left F_{p}OutKmodule S^{∗} and γ^{∗} induces a non-zero action on the right F_{p}OutK-module S.

Replacing K by the K′ given in (ii), the primitive idempotent splitting off X is given by transfer to H followed by the split surjection π, and then by an element of F_{p}OutK and finally the inclusion of K in P (modulo the ideal in Ā(P, P) generated by all ζ_{L,ψ} where the image of ψ has order less than that of K). Even without working modulo this ideal, the idempotent e splitting off X from BP may be taken to factor through Bi : BK → BP.

Proof We begin with a sequence of lemmas.

Lemma 5.3 Write \overline{N_{P} (K)} for the image of N_{P} (K) in OutK. Inside S ⊗_{F_{p}OutK} L̄(P, K) we have

S ⊗ f̄_{K,id} ≅ S ⊗_{F_{p}\overline{N_{P} (K)}} F_{p} ≠ 0.

Proof The sub-F_{p}OutK-module of L̄(P, K) generated by the element f̄_{K,id} is a copy of F_{p}(OutK/\overline{N_{P} (K)}). So we have

S ⊗ f̄_{K,id} ≅ S ⊗_{F_{p}OutK} (F_{p}(OutK/\overline{N_{P} (K)}))
≅ S ⊗_{F_{p}OutK} (F_{p}OutK ⊗_{F_{p}\overline{N_{P} (K)}} F_{p})
≅ S ⊗_{F_{p}\overline{N_{P} (K)}} F_{p}
= H_{0}(\overline{N_{P} (K)}, S).

This is non-zero, because \overline{N_{P} (K)} is a p-group. □

Notation Let e_{0} be the primitive idempotent in Z_{p}^{∧}OutK labelling a dominant summand X of BK, and with image ē_{0} in F_{p}OutK.

If u = ∑_{j} a_{j}g_{j} in F_{p}OutK, we write ζ_{H,i∘u∘π} for the element ∑_{j} a_{j}ζ_{H,i∘g_{j}∘π} of Ā(P, P).

Lemma 5.4 Suppose s, t ∈ S and γ^{∗} is the image of the element γ of Theorem 5.2 under the duality automorphism ∗ of F_{p}OutK (given by inverting group elements). If sγ^{∗}ē_{0}^{∗} ≠ 0 in S and t ⊗ f̄_{K,id} ≠ 0 in S ⊗_{F_{p}OutK} L̄(P, K), then there exists u ∈ F_{p}OutK such that

ζ_{H,i∘u∘ē_{0}∘π}(s ⊗ f̄_{K,id}) = t ⊗ f̄_{K,id},

and in particular s ⊗ f̄_{K,id} ≠ 0 in S ⊗_{F_{p}OutK} L̄(P, K).

Proof Since S is simple and sγ^{∗}ē_{0} ≠ 0, we may choose u ∈ F_{p}OutK so that sγ^{∗}ē_{0}^{∗}u^{∗} = t. By Proposition 4.8 (ignoring the restriction on Stab_{P} (S)), we have

ζ_{H,i∘u∘ē_{0}∘π}(s ⊗ f̄_{K,id}) = s ⊗ f̄_{K,id∘γ^{∗}∘ē_{0}^{∗}∘u^{∗}}
= sγ^{∗}ē_{0}^{∗}u^{∗} ⊗ f̄_{K,id}
= t ⊗ f̄_{K,id} ≠ 0. □

Remark This lemma holds just as well without the ē_{0}’s, and hence shows that the kernel of γ^{∗} on S is contained in S.J(F_{p}\overline{N_{P} (K)}), where J denotes the radical.

Lemma 5.5 Choose s = t with t⊗f̄_{K,id} ≠ 0 as in Lemma 5.4, and let u be as given there. Then some power of ζ = ζ_{H,i∘u∘ē_{0}∘π} is an idempotent ε ∈ Ā(P, P) with the property that ε.(S ⊗_{F_{p}OutK} L̄(P, K)) is the one dimensional subspace spanned by t ⊗ f̄_{K,id}. Moreover, ε.(S′ ⊗_{F_{p}OutK′} L̄(P, K′)) = 0 for any K′ with |K′| ≥ |K| and any simple F_{p}OutK′-module S′ not isomorphic to S in case K′ has the same type as K. In particular, ε is a primitive idempotent modulo the ideal in Ā(P, P) generated by all ζ_{L,ψ} where the image of ψ has order less than that of K.

Note that ε is given by transfer to H followed by the split surjection π, and then by an element of F_{p}OutK and finally the inclusion of K in P.

Proof The image of ζ on S ⊗_{F_{p}OutK} L̄(P, K) is contained in S ⊗ f̄_{K,id}. Applying ζ a second time, the image is Sγ^{∗}ē_{0}^{∗}u^{∗} ⊗ f̄_{K,id} = ⟨t ⊗ f̄_{K,id}⟩ and so some power of ζ^{2} is an idempotent projection onto ⟨t ⊗ f̄_{K,id}⟩.

If S′ ≅ S̸ is another F_{p}OutK-module then ē_{0}^{∗} is zero on S′, and hence ζ^{2} is zero on S′ ⊗_{F_{p}OutK} L̄(P, K).

If |K′| ≥ |K| and K′ is not of the same type as K, then ζ is zero on S′⊗_{F_{p}OutK′} L̄(P, K′). □

Proof of Theorem 5.2 : We start with the implication (i) ⇒ (ii). The map

BK → X → BP →^{e} BP → X → BK

is equal to e_{0}. Hence its image in F_{p}OutK/J is non-zero.

The idempotent e is a linear combination of maps of the following form

BH
↑ tr ↘Bπ
BP  BK →^{e_{0}} BK →^{Bi} BK′ → BP

(recall that we are working modulo the ideal generated by all ζ_{L,ψ} where the image of ψ has order less than that of K).

Consider the following diagram for composing two such maps.

BH′  BH
↑ tr ↘  ↑ tr ↘Bπ
BP BK →^{e_{0}}BK →^{≃} BK′ →BP  BK→^{e_{0}} BK → BK′′ → BP

This diagram shows that the product of two such maps factors through a map from BK to BK.

Since the radical is a two-sided ideal, at least one such product must have non-zero image modulo the radical. Now apply the double coset formula to the map from BK′ to BH, to express the map from BK to BK as γ.

We now remark that we may as well assume K is equal to K′, since we have not specified a particular subgroup isomorphic to K yet. In particular, we can assume the first map in the definition of γ is the identity. Moreover, one can alter the split surjection π by following it with an automorphism, say α, of K. Call the resulting split surjection π′. The sum is non-zero in the composition involving π′ if and only if it is non-zero with π. Indeed, the formulae are related by postcomposing with α.

This completes the proof that (i) ⇒ (ii). Lemma 5.5 now shows that (ii) ⇒ (i), and also gives the form of the idempotent, modulo the ideal.

So we now have maps of the form

X → BK ⟶^{Bi∨?} BP ∨ Y ⟶^{id∨∗} BP → X

whose composite is the identity, where Y is a wedge sum of BQ’s with |Q| < |K|, and in particular has no summand isomorphic to X. This composite takes X to a summand of BP isomorphic to X. Decomposing Y as a wedge sum of indecomposable summands, we express this composite map as a sum of maps factoring through BP and indecomposable summands of Y. Now the endomorphism ring of each indecomposable is a local ring, and so any such composite which factors through a summand of Y is in the radical of the endomorphism ring of X. Thus the remaining component X → BK →^{Bi} BP → → X splits off X as a summand of BP. This completes the proof of the last part of Theorem 5.2. □

Lemma 5.6 If the equivalent conditions of Theorem 5.2 hold, then S ⊗_{F_{p}OutK} L̄(P, K) is a cyclic Ā(P, P)-module generated by the eigenvector t ⊗ f̄_{K,id} of the primitive idempotent ε as described in Lemma 5.5. In particular it has a unique maximal submodule, and the quotient module is the simple Ā(P, P)-module on which ε acts non-trivially.

Proof It follows from Lemma 5.4 that S ⊗ f̄_{K,id} ⊆ Ā(P, P).(t ⊗ f̄_{K,id}). If α : K → K′′ is an isomorphism and i′′ : K′′ → P is the inclusion, then ζ_{H,i′′∘α∘u∘ē_{0}∘π} takes t ⊗ f̄_{K,id} to tγ^{∗}ē_{0}^{∗}u^{∗} ⊗ f̄_{K′′,α^{−1}}. By choosing u suitably, this runs through all elements of S ⊗ f̄_{K′′,α^{−1}}. □

Theorem 5.7 The modules L̄(P, K, S) = (S ⊗_{F_{p}OutK} L̄(P, K))/M described in Definition 4.9 are either simple or zero. As K runs through the types of subgroups of P and S runs through the simple F_{p}OutK-moduls, the non-zero modules of the form L̄(P, K, S) run through a complete set of representatives for the isomorphism types of simple Ā(P, P)- modules. The module L̄(P, K, S) is non-zero if and only if the equivalent conditions given in Theorem 5.2 hold. If L̄(P, K, S) is non-zero then M is the unique maximal submodule of S ⊗_{F_{p}OutK} L̄(P, K).

Every composition factor of M is of the form L̄(P, K′, S′) with K′ ≻ K and S′ a simple F_{p}OutK′-module.

Proof We prove this by induction on |P : K|. If P = K, then S ⊗_{F_{p}OutP} L̄(P, P) = S ⊗_{F_{p}OutP} F_{p}OutP = S and M = 0, so there is nothing to prove.

By Lemma 5.6, the statement of the theorem is equivalent to the statement that under the equivalent conditions of Theorem 5.2, the radical of the module S ⊗_{F_{p}OutK} L̄(P, K) is equal to M, and that if these conditions do not hold then M = S ⊗_{F_{p}OutK} L̄(P, K). Since each ζ_{L,ϕ} with Im(ϕ) ≅ K acts as zero on composition factors of the form L̄(P, K′, S′) with |K′| > |K|, it suffices to show that every composition factor of S ⊗_{F_{p}OutK} L̄(P, K) (other than the top one in case the conditions of Theorem 5.2 hold) is of the form L̄(P, K′, S′) with |K′| > |K|. By Lemma 5.5, the simple top composition factor of S′⊗_{F_{p}OutK′} L̄(P, K′) with |K| ≥ |K′| and S not equal to S′ can never be a composition factor of S ⊗_{F_{p}OutK} L̄(P, K), so we are done. □

Definition 5.8 Let K be a subgroup of P with inclusion i : K → P. Suppose Bi : BK → BP induces a homotopy equivalence between a dominant summand of BK and an indecomposable summand of BP isomorphic to X = e.BP. Then we say that K is a vertex of X, and also of e. Note that by Theorem 5.2, every indecomposable summand of BP has a vertex. Furthermore, by Theorem 5.7, the idempotent e corresponds to a simple Ā(P, P)-module of the form L̄(P, K, S), where S is a simple right F_{p}OutK-module well defined up to isomorphism. We call S a source of X and of e. Note that Theorem 5.7 also shows that K is well defined, not up to conjugacy, but at least up to type.

Proposition 5.9 If e.BP has vertex K and source S, then Stab_{P} (S) is a direct product of K and some other subgroup of N_{P} (K).

Proof As in Theorem 5.2, ē is conjugate to an idempotent given by transfer to H followed by a split surjection π, with kernel N, say, and then by an element of F_{p}OutK and finally the inclusion of K in P, modulo a certain ideal. So we have H = N.K and N ∩ K = 1. Now by Proposition 4.8, such an element can only act in a non-nilpotent fashion if Stab_{P} (S) ≤ ^{x}H for some x. Thus since K is normal in Stab_{P} (S) we have

Stab_{P} (S) = K × (N ∩ Stab_{P} (S)). □

Corollary 5.10 If e.BP has vertex K and OutK is a p-group then N_{P} (K) is a direct product of K and some other subgroup of P.

Proof In this case OutK has only one simple module, namely the trivial module, and the stabiliser is N_{P} (K). □

Note that the condition on Stab_{P} (S) given in the above proposition is not sufficient to guarantee that there is a corresponding summand with vertex K. See Example 4 in Section 7 for an example of this.

Corollary 5.11 Suppose an indecomposable stable summand X of BP has vertex K. Then X is a stable summand of BH for every subgroup H of P containing K.

Proof One can factor the idempotent ζ_{H,i∘ē_{0}∘π} through any subgroup containing K, since i an inclusion. □

The following corollary generalises a result of Priddy, which deals with the case where K is elementary abelian and S is the Steinberg module for GL_{n}(F_{p}).

Corollary 5.12 Suppose C_{P} (K) ≤ K. If S is a simple F_{p}OutK-module whose restriction to \overline{N_{P} (K)} = N_{P} (K)/K ≤ OutK has a non-trivial projective summand (for example, this is true if S is a simple F_{p}OutK-module in a block of defect zero), then L̄(P, K, S) ≠ 0, and so the corresponding dominant summand of BK is a wedge summand of BP.

Proof Let H = K and π = id_{K} in part (ii) of Theorem 5.2. Then γ = ∑_{g∈\overline{N_{P} (K)}} g ∈ F_{p}OutK. Thus γ acts in a non-zero fashion on S if and only if S contains a non-zero projective summand on restriction to \overline{N_{P} (K)}. The result now follows from Theorem 5.2. □

6 The general finite group

Now that we have completed our discussion of stable splittings of classifying spaces of p-groups, we can attack the general finite group. We shall apply the following lemma, with H = K = P a Sylow p-subgroup of G.

Lemma 6.1 Suppose H, K are subgroups of G, and i : H → G is the inclusion. The composite of Bi : BH → BG and transfer BG → BK is represented by the H × K-set [G] (via left and right multiplication) as an element of A(H, K)^{∧}.

Proof Bi is represented by the H × G-set G in A(H, G)^{∧}, and transfer BG → BK is represented by the G × K-set G in A(G, K)^{∧}. The lemma follows from the fact that G ×_{G} G = G as an element of A(H, K)^{∧}. □

The following lemma is well known.

Lemma 6.2 Suppose i : P → G is the inclusion of a Sylow p-subgroup. The composite of transfer BG → BP and Bi : BP → BG is a self homotopy equivalence of BG.

Proof In cohomology, this composite is multiplication by |G : P |, which is invertible in Z_{p}^{∧}. The lemma now follows from the stable Whitehead lemma. □

Proposition 6.3 Regarding [G] as an element of A(P, P)^{∧} as above, we have BG = [G].BP. In particular

A(G, G)^{∧} = [G].A(P, P)^{∧}.[G].

Proof By Lemma 6.2, BG is homotopy equivalent to the image of the composite BP →^{Bi} BG →^{tr} BP. By Lemma 6.1, this composite is given by multiplication by G as an element of A(P, P)^{∧}. It follows that [G].A(P, P)^{∧} is the projective A(P, P)^{∧}-module corresponding to the summand BG of BP, and so its endomorphism ring [G].A(P, P)^{∧}.[G] gives the stable maps from BG to itself. □

It follows from this proposition that primitive idempotents in A(G, G)^{∧} may be obtained by lifting primitive idempotents in Ā(G, G) = F_{p} ⊗_{Z} A(G, G)^{∧}. See for example Curtis and Reiner [7], Theorem 6.7. Thus isomorphism types of simple Ā(G, G)-modules are in one-one correspondence with homotopy types of indecomposable stable summands of BG. The number of copies of such a summand in any decomposition of BG as a wedge sum of indecomposables is equal to the dimension of the corresponding simple Ā(G, G)- module as a module over its endomorphism ring, which is a finite field.

Corollary 6.4 The simple Ā(G, G)-modules are precisely the non-zero modules of the form [G].L̄(P, K, S). We denote this module by L̄(G, K, S). □

Proposition 6.5 The action of the element [G] of Ā(P, P) on S ⊗_{F_{p}OutK} L̄(P, K) is given by the formula

[G].(s ⊗ f̄_{K,ψ}) = ∑_{\begin{matrix}K\G/P \\ K≤^{x}P \end{matrix}} s ⊗ f̄_{^{x^{−1}}K,ψ∘c_{x}}.

Proof By Proposition 4.8 we have

[G].(s ⊗ f̄_{K,ψ}) = ∑_{g∈P \G/P}(P gP)(s ⊗ f̄_{K,ψ})
= ∑_{g∈P \G/P} ζ_{P ∩^{g}P,c_{g^{−1}}} (s ⊗ f̄_{K,ψ})
= ∑_{g∈P \G/P} ∑_{\begin{matrix}x∈K\P/P ∩^{g}P \\ K≤P ∩^{xg}P \end{matrix}} s ⊗ f̄_{^{g^{−1}x^{−1}}K,ψ∘c_{xg}}

= ∑_{\begin{matrix}y∈K\G/P \\ K≤P ∩^{y}P \end{matrix}} s ⊗ f̄_{^{y^{−1}}K,ψ∘c_{y}}. □

We have the following criterion for when L̄(G, K, S) ≠ 0.

Proposition 6.6 Theorem 5.2 holds, with P replaced by G.

Proof This follows by an argument similar in nature to the proof of Theorem 5.2. Note that since Proposition 3.1 holds for a general finite group G, the analogues of Proposition 4.8 and Lemma 5.4 also hold for a general finite group. □

7 Examples

We shall first repeat some of the stable splittings which have already been obtained by other authors, to demonstrate how easy they are with the machinery developed here. We make extensive use of Proposition 5.9. Identifying the stable summands is not a simple matter in general. See the comment after Example 1.

Example 1 Dihedral groups Let P be the dihedral group

D_{2^{n}} = ⟨x, y | x^{2^{n−1}} = 1, y^{2} = 1, yxy^{−1} = x^{−1}⟩,

of order 2^{n} for n ≥ 3. Then the only subgroups K for which OutK is not a 2-group are K ≅ (Z/2)^{2}. Thus by Corollary 5.10, the only possible vertices of summands of BP are 1, non-central involutions, (Z/2)^{2} and D_{2^{n}}.

There are two conjugacy classes of non-central involutions, and they have the same type and have a normal complement. Thus the corresponding L̄(P, K) is a two dimensional simple Ā(P, P)-module and so the corresponding summand \overline{BZ/2} is a summand with multiplicity two.

There are two conjugacy classes of subgroups K ≅ (Z/2)^{2}, and these have the same type. Since N_{P} (K) is not a direct product of K with another subgroup of P, the principal dominant summand of BK is not a summand of BP. There is only one non-principal dominant summand of BK, since F_{2}OutK = F_{2}Σ_{3} has only two simple modules, k and S. This non-dominant summand is usually denoted L(2). It is easy to calculate that S ⊗_{F_{2}Σ_{3}} L̄(P, K) is a two-dimensional simple Ā(P, P)-module on which the element ζ_{K,ϕ} acts non-nilpotently, where ϕ : K → P is the map which applies an automorphism of order three to K followed by the embedding of K as a subgroup of P. Alternatively, setting H = K in Theorem 5.2, it is easy to see that γ^{∗} is non-zero on S. Thus L(2) is a summand of BD_{2^{n}} with multiplicity two.

Finally since OutP is a 2-group, there is only one dominant summand. It is not hard to see that this is in fact \overline{BP SL_{2}(q)} for q any odd prime power such that D_{2^{n}} is a Sylow 2-subgroup of P SL_{2}(q).

Thus we have the Mitchell-Priddy splitting [19]

BD_{2^{n}} = B1 ∨ 2.\overline{BZ/2} ∨ 2.L(2) ∨ \overline{BP SL_{2}(q)}.

Note that in this example the modules of the form S ⊗_{F_{2}OutK} L̄(P, K) for which Stab_{P} (S) is a product of K and another subgroup of P, are precisely the simple Ā(P, P)- modules once each. The following table gives the modules S ⊗_{F_{2}OutK} L̄(P, K) for all types K, in the case of the dihedral group D_{8}. The simple modules are denoted 1_{a}, 2_{a}, 2_{b} and 1_{b}. The arrows point to simple modules associated with their vertices.

⟨xy⟩, ⟨y⟩ ⟨x^{2}⟩
1  Z/2  Z/2 Z/4 (Z/2)^{2} D_{8}
1_{a}  2_{a}  1_{b} 1_{b}  \begin{matrix}1_{b} \\ 1_{b} \end{matrix} 2_{b} 1_{b}
↑  ↑  ↑ ↑

We remark that we have made no attempt to identify the spectrum L(2) appearing in the above spectrum as that obtained from the symmetric products of the sphere spectrum; this identification is an essential part of the Mitchell-Priddy theorem [19].

Example 2 Quaternion groups Let P be the generalised quaternion group

Q_{2^{n}} = ⟨x, y | x^{2^{n−1}} = 1, y^{2} = x^{2^{n−2}}, yxy^{−1} = x^{−1}⟩

of order 2^{n}. Then for every subgroup K apart from K ≅ Q_{8}, OutK is a 2-group and unless K = 1 or P, N_{P} (K) is not a product of K and another subgroup of P. Thus the only possible vertices of summands of BP are 1, Q_{8} and P. If S is the non-trivial simple F_{2}OutQ_{8}-module (note that OutQ_{8} ≅ Σ_{3}) then S ⊗_{F_{2}OutQ_{8}} L̄(P, Q_{8}) is a two-dimensional simple Ā(P, P)-module.

Thus BP decomposes as a sum of one copy of B1, two copies of the non-principal dominant summand of BQ_{8}, and one copy of the principal dominant summand of BP (which is isomorphic to \overline{BSL_{2}(q)} for those odd prime powers q for which Q_{2^{n}} is a Sylow 2-subgroup of SL_{2}(q)).

Here is a table of S ⊗_{F_{2}OutK} L̄(P, K) for all types K, in case P = Q_{8}.

1 Z/2 Z/4  Q_{8}
1_{a} 1_{b} 1_{b} ⊕ 2_{a} 1_{b} 2_{a}
↑  ↑ ↑

Example 3 Semidihedral groups Let P be the semidihedral group

SD_{2^{n}} = ⟨x, y | x^{2^{n−1}} = 1, y^{2} = 1, yxy^{−1} = x^{2^{n−2}−1}⟩

of order 2^{n}. Then the subgroups whose automorphism group is not a 2-group are isomorphic to (Z/2)^{2} and Q_{8}. The only types which have a normal complement in their normaliser are the conjugacy class of non-central subgroups of order two, represented by ⟨y⟩, the trivial subgroup 1, and SD_{2^{n}}. In these cases, it is easy to check that the corresponding one dimensional Ā(P, P)-modules are distinct one dimensional simple modules (Since (Z/2)^{2} and Q_{8} do not split off their normalisers, the corresponding summands are the non-principal ones). In fact all the S ⊗_{F_{2}OutK} L̄(P, K) are one dimensional, and here is a table of these modules in the case n = 4.

⟨y⟩ ⟨x^{4}⟩ ⟨x^{2}⟩ ⟨xy⟩ ⟨x^{4}, y⟩ ⟨x⟩ ⟨x^{2}, y⟩ ⟨x^{2}, xy⟩
1 Z/2 Z/2 Z/4 Z/4 (Z/2)^{2} Z/8 D_{8} Q_{8} SD_{16}
1_{a} 1_{b} 1_{c} 1_{c} 1_{e} 1_{c} 1_{d} 1_{c}  1_{c} 1_{c} 1_{e} 1_{c}
↑ ↑  ↑  ↑  ↑

It follows that \overline{BSD_{2^{n}}} = L(1) ∨ L(2) ∨ X ∨ Y, where X is the non-principal dominant summand of BQ_{8} and Y is the dominant summand of BSD_{2^{n}}, in accordance with the results of Martino [16].

Example 4 Let P be a split extension with normal subgroup Z/4 × Z/2 = ⟨x, y | x^{4} = y^{2} = [x, y] = 1⟩ and complement Z/4 = ⟨w | w^{4} = 1⟩ with action wxw^{−1} = x, wyw^{−1} = x^{2}y. Then the subgroup H = ⟨y, w^{2}⟩ ≅ (Z/2)^{2} has centraliser C_{P} (H) = ⟨x, y, w^{2}⟩ ≅ Z/4 × H. This is the stabiliser of the non-trivial two dimensional simple Out(H)-module. However, the corresponding non-principal dominant summand of BH is not a wedge summand of BP.

Example 5 Let P be an extraspecial p-group of order p^{1+2n}. If K is a subgroup which is not elementary abelian, then it includes the centre and is normal. The stabiliser of any representation of OutK is thus the whole group, since the centre is also the Frattini subgroup (see the remark after Proposition 4.7). Since the group is not a direct product in a non-trivial way, it follows from Proposition 5.9 that every summand has either the whole group or an elementary abelian subgroup as vertex. It is easy to see that an elementary abelian subgroup of rank n (resp. n − 1 in case p = 2 and the quadratic form has Arf invariant one) not containing the centre has a normal complement in P. Hence all summands of such subgroups occur in BP.

It remains to determine which dominant summands of the maximal elementary abelian subgroups occur. If p = 2 and the quadratic form has Arf invariant one there are no additional summands since no maximal elementary abelian subgroup is a direct factor of its centraliser. In the other cases we need to consider the criterion in Theorem 5.2. Since no maximal elementary abelian subgroup K has a normal complement in any larger subgroup, we must have K = H and hence γ is the sum over σ ∈ P/K of the maps described in the theorem. So the criterion is that the simple F_{p}GL_{n+1}(F_{p})-module S is a source if and only if γ^{∗} is non-zero on S. This happens at least for the Steinberg module by Corollary 5.12.

The number of homotopy types of indecomposable dominant summands of BP is the number of p-regular conjugacy classes in OutP, which is a symplectic or orthogonal group. The number of p-regular conjugacy classes is well known (see for example Carter [5]). The number of homotopy types of indecomposable summands of B(Z/p)^{k} was determined by Harris and Kuhn [12].

In the case of the extraspecial 2-groups of Arf invariant one, it is also easy to see using cohomology that the Steinberg summands (corresponding to the Steinberg module for the orthogonal group OutP) described in Feshbach and Priddy [10] are indecomposable after splitting off the obvious copies of the indecomposable Steinberg summands for elementary abelian groups (corresponding to the Steinberg module for the general linear groups). The key fact is that the Stiefel-Whitney class w_{2^{n+1}}(Δ_{2n}) restricts to zero on any elementary abelian subgroup not containing the centre.

Similar results hold for the 2-group of complex type considered in Feshbach and Priddy [10].

As a particular example consider P = E(4)^{−}, the extraspecial 2-group of order 32 with quadratic form x_{1}x_{−1} + x_{2}x_{−2} + x_{2}^{2} + x_{−2}^{2} of Arf invariant one. This group was considered in Section 3 of [10]. We use the notation used there as well as certain results. The outer automorphism group of P is isomorphic to Σ_{5}. Now F_{2}Σ_{5} has three simple modules of dimensions 1, 4 and 4. By the above discussion the only non-dominant summands of BP are S^{0} and a number of copies of L(1), the dominant summand of B(Z/2). We need only determine this number. The number of conjugacy classes of involutions which are a factor of their centralisers is 5, corresponding to x_{1}, x_{−1}, x_{1} + x_{−1} + x_{2}, x_{1} + x_{−1} + x_{−2}, and x_{1} + x_{−1} + x_{2} + x_{−2}. Thus L̄(P, K) is 5 dimensional, where K is any of these noncentral involutions. Moreover, as an F_{2}Σ_{5}-module, L̄(P, K) is isomorphic to the natural permutation representation (see [10], Section 2). The sum of all the basis elements is in M. The resulting quotient module is a simple Ā(P, P)-module, since it is a simple F_{2}Σ_{5}- module, and it is easy to see that L̄(P, K, S) is not zero for S the trivial OutK-module. Hence the number of copies of L(1) is 4. (One can also see this cohomologically, but we wish to avoid cohomology since it played no rôle in our theory.)

We thus have a complete splitting

\overline{BP} = 4L(1) ∨ 4Y ∨ 4X ∨ Z,

where Y is the dominant summand corresponding to the Steinberg module (for the group Σ_{5} considered as an orthogonal group), X is the other summand of multiplicity four, and Z is the principal dominant summand. It is shown in [10] that Z = \overline{BG}, where G is a semidirect product of P and a group of order five. The Poincaré series of these summands is given in [10]. It was unknown previously that this splitting was complete.

References

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