Two Classifications of Simple Mackey Functors with Applications to Group Cohomology and the Decomposition of Classifying Spaces

Peter Webb

With best wishes to my teacher, Karl Gruenberg, on his 65th birthday

In previous joint work with Thévenaz we have classified simple Mackey functors [15] and developed a structural theory of Mackey functors which takes its standpoint from representation theory [16]. In this paper we show that this type of theory may be used to perform specific calculations of Mackey functor values, and also to obtain structural results about the decomposition of classifying spaces. It turns out in doing this that it is profitable to allow the precise nature of the Mackey functors we consider to vary, so as best to suit the problem in hand.

Our first application is to group cohomology H^{n}(G, k) with a trivial coefficient module. This gives a Mackey functor H^{n}( , k) defined on every finite group G, and provides an example of what we will call a ‘global Mackey functor’, namely a Mackey functor which has a consistent definition on all finite groups. Our approach is to consider a composition series as a global Mackey functor of H^{n}( , k), by which we mean a series of global Mackey functors

⋯ ⊂ M_{i−1} ⊂ M_{i} ⊂ M_{i+1} ⊂ ⋯ ⊂ H^{n}( , k)

such that ⋂ M_{i} = 0, ⋃ M_{i} = H^{n}( , k) and M_{i+1}/M_{i} is always a simple global Mackey functor. At this point the question of the existence of such series arises, and we show that group cohomology does indeed have such a series. We show also that the multiplicities of the composition factors are determined independently of the choice of series. This sequence of multiplicities of simple global Mackey functors as composition factors is now an invariant of group cohomology which it is of interest to compute.

For our application to the computation of group cohomology we rely on two more things. The first is the fact that the composition factor multiplicities can be computed from a knowledge only of p-groups. The second is a classification of the simple global Mackey functors, which is the first result we present in this paper. The arguments we use parallel those in [15]. By virtue of an explicit formula for these simple objects, knowing their multiplicities as composition factors we are able to give the dimensions of cohomology groups. A useful feature of this approach is that since the computation is done using a knowledge only of p-groups, we obtain simultaneously a formula for the mod p cohomology of all groups with a given Sylow p-subgroup, in terms of the cohomology of the Sylow psubgroup and the fusion of certain subgroups.

Turning to our second application of this type of theory, we show how a study of Mackey functors yields a proof of the theorem of Benson-Feshbach [1] and Martino-Priddy [8] on the stable decomposition of classifying spaces. For this we need to consider global Mackey functors with extra structure analogous to the that of the inflation map in group cohomology. To avoid a potentially longer name for these objects we will call them simply ‘inflation functors’. They were used by Symonds [13] where they are called ‘functors with Mackey structure’, and they are defined in [4, p.278] where they are called ‘global Mackey functors’ (thus differing from our use of the term). The term ’Burnside functor’ is also used by some to refer to a dual notion. Inflation functors are Mackey functors defined on all groups with the extra specification that for each group homomorphism θ : H → G (not just monomorphism) there is a contravariant map M^{∗}(θ) : M(G) → M(H). The classification of the simple inflation functors proceeds in a similar way to that for simple global Mackey functors, and it so happens that these two types of simple object are in bijection, although their detailed structure is different.

The crucial key in applying this theory to the stable decomposition of classifying spaces is Carlsson’s verification of the Segal conjecture, which by work of Lewis, May and McClure [6] implies for a finite group G that the stable wedge decompositions of the suspension spectrum BG_{+} biject with decompositions of the identity in A(G, G)^{∧} as a sum of orthogonal idempotents (see also [1, 8, 10]). Here A(G, G) is the double Burnside ring and A(G, G)^{∧} is its completion at the augmentation ideal of the Burnside ring. We will see that when A( , G) is regarded as a functor in the first variable, it is projective in the category of inflation functors. In fact it is a representable functor, and its endomorphism ring is A(G, G). We will fix a prime and work over the p-adic integers ℤ_{p}, in which case when G is a p-group the stable summands of the p-completed spectrum (BG_{+})_{p}^{∧} biject with the summands of the functor A( , G) ⊗ ℤ_{p}. We will show (working over ℤ_{p}) that the indecomposable summands of these representable functors are the projective covers of the simple inflation functors, and so their isomorphism types are in bijection with the isomorphism types of the simple functors. Their multiplicities as summands of A( , G)⊗ℤ_{p} are immediately computable from the simple functors, and for arbitrary finite groups G these numbers are also the multiplicities as summands of the indecomposable stable pcompleted summands of (BG_{+})_{p}^{∧}. In this way using an explicit formula for the simple inflation functors we recover the computational approach to these multiplicities given in [1]. What our approach also gives is a structural interpretation of these numbers, since a single simple inflation functor contains the multiplicity information for the corresponding classifying space summand, as a summand of (BG_{+})_{p}^{∧} for every group G. We obtain also an equivalence of categories in which stable p-completed classifying space summands correspond to indecomposable projective inflation functors. The philosophy behind this approach is that the summands of (BG_{+})_{p}^{∧} are replaced by projective objects in a category which have very nice properties making them easy to deal with.

Thanks are due to many people for their technical assistance in connection with this work. I thank Dave Benson and Mark Feshbach; and especially Nick Kuhn, for an extremely valuable conversation concerning the application of Mackey functors to the decomposition of classifying spaces. At a time when the connection was still mysterious to me he was able to sketch in outline the way the theory would probably go, and with some technical modifications that is the approach presented here.

1. Global Mackey functors and inflation functors

In this section we will make the definitions of two very similar kinds of Mackey functor. The differences are apparently slight and much of the theory for one kind works very much the same way as for the other. These differences will become more apparent when we obtain explicit formulae for the functors, and it will turn out that one type of Mackey functor is harder to compute with, but is nevertheless necessary for the application we have in mind.

A global Mackey functor over a commutative ring R is given by the following specification. For each finite group G there is an R-module M(G) and for every monomorphism of groups α : H → G there are R-module homomorphisms

\begin{aligned}M_{∗}(α) & : M(H) → M(G) \\ M^{∗}(α) & : M(G) → M(H) \end{aligned}

such that

(1) M_{∗}(αβ) = M_{∗}(α)M_{∗}(β) and M^{∗}(αβ) = M^{∗}(β)M^{∗}(α) always,

(2) whenever α : H → H is an inner automorphism then M_{∗}(α) = M^{∗}(α) = 1,

(3) whenever α : H → H is an automorphism then M_{∗}(α^{−1}) = M^{∗}(α), and

(4) for every pair of monomorphisms H⟶^{α}G⟵^{β}K we have

M^{∗}(β)M_{∗}(α) = ∑_{x∈[α(H)\G/β(K)]} M_{∗}(ϕ_{x})M^{∗}(ψ_{x})

where

ϕ_{x} : α(H) ∩ ^{x}β(K)⟶^{c_{x}}α(H)^{x} ∩ β(K) ↪ β(K)⟶^{β^{−1}}K

and

ψ_{x} : α(H) ∩ ^{x}β(K) ↪ α(H)⟶^{α^{−1}}H.

In condition (4) we are using the notation that [X] denotes a set of representatives for the elements in a quotient object X: thus α(H)\G/β(K) is a set of double cosets and [α(H)\G/β(K)] a set of representatives for these double cosets, consisting of elements of G.

A morphism θ : M_{1} → M_{2} of global Mackey functors is a collection of R-module homomorphisms θ : M_{1}(G) → M_{2}(G) for every finite group G which is natural with respect to both M_{∗} and M^{∗}. Thus whenever α : H → G is a group monomorphism, the squares

M_{1}(H) ⟶^{M_{∗}(α)} M_{1}(G)
↓^{θ}  ↓^{θ}
M_{2}(H) ⟶^{M_{∗}(α)} M_{2}(G)

and

M_{1}(G) ⟶^{M^{∗}(α)} M_{1}(H)
↓^{θ}  ↓^{θ}
M_{2}(G) ⟶^{M^{∗}(α)} M_{2}(H)

must commute. Global Mackey functors over R form an abelian category, which will be denoted Mack_{R}. In cases where it is clear what R is without further specification, we may simply write Mack. We will speak of subfunctors, quotient functors etc., and in particular we have the notion of simple global Mackey functors, namely those which have no non-trivial subfunctors.

We now define the notion of an inflation functor. We say that M is an inflation functor over R provided that for each group G there is an R-module M(G) together with R-module homomorphisms

\begin{aligned}M_{∗}(α) & : M(H) → M(G) \\ M^{∗}(γ) & : M(G) → M(H) \end{aligned}

whenever α : H → G is a monomorphism of groups and γ : H → G is an arbitrary group homomoprhism. These must satisfy conditions (2), (3) and (4) previously given for global Mackey functors, together with (1′) M_{∗}(αβ) = M_{∗}(α)M_{∗}(β) for all monomorphisms α, β, and

M^{∗}(γδ) = M^{∗}(δ)M^{∗}(γ) for all homomorphisms γ, δ.

(5) Whenever

G′ ⟶^{γ} G
i′↑  ↑^{i}
H′ ⟶^{δ} H

is a commutative diagram of groups in which γ and δ are surjective and i, i′ are inclusions with H′ = γ^{−1}(H), then M^{∗}(γ)M_{∗}(i) = M_{∗}(i′)M^{∗}(δ). Morphisms of inflation functors are families of R-module homomorphisms natural with

respect to both M_{∗} and M^{∗}. Inflation functors form an abelian category which we will denote Mack_{R}^{∗}. In cases where it is clear what R is without further specification we will simply write Mack^{∗}.

We should point out at this stage that there is a dual notion in which the covariant part of a global Mackey functor is extended to all group homomorphisms but the contravariant part is not. We will not explicitly consider these covariant versions of inflation functors in this paper, but they also form an abelian category which one might denote Mack_{∗}. So, for example, in any fixed dimension n the cohomology H^{n}(G, R) is a functor in Mack^{∗}, while homology H_{n}(G, R) is a functor in Mack_{∗}. In general, the dual of an object in Mack^{∗} is an object in Mack_{∗} (using ‘dual’ in the manner of [16]) and vice versa, so that properties of one category may be deduced from those of the other. By this means the classification of the simple objects in Mack^{∗} which we are about to give also yields a classification of the simple objects in Mack_{∗}.

It is clear that every inflation functor yields a global Mackey functor by restricting the definition of the contravariant part M^{∗} to group monomorphisms, instead of arbitrary homomorphisms. It is also evident that if we restrict a global Mackey functor to a particular group G and its subgroups, together with the monomorphisms formed by inclusion of subgroups and conjugations by elements of G, we obtain a Mackey functor on G. Because of this connection with Mackey functors, if α : H → G happens to be an inclusion of subgroups, we may write I_{H}^{G} for M_{∗}(α) and R_{H}^{G} for M^{∗}(α), and if α : H → ^{g}H is conjugation by g ∈ G we write c_{g} for M_{∗}(α) = M^{∗}(α^{−1}). Since every monomorphism of groups may be expressed as an isomorphism followed by an inclusion of subgroups, it would be equivalent (in the presence of the other axioms) to replace axiom (4) by (4′) for every pair of subgroups H, K of G,

R_{K}^{G}I_{H}^{G} = ∑_{g∈[K\G/H]} I_{K∩^{g}H}^{K}c_{g}R_{K^{g}∩H}^{H}

which is just the usual Mackey decomposition formula. Another useful simplification of notation is that we may write α_{∗} instead of M_{∗}(α) and α^{∗} instead of M^{∗}(α).

Notice that for any group H the evaluation M(H) of a global Mackey functor or inflation functor has the structure of an R[Out(H)]-module. For the mapping

M_{∗} : Aut(H) → Aut(M(H))

specified by α ↦ M_{∗}(α) is a group homomorphism, and the axiom that M_{∗}(α) = 1 when α is inner, means that in fact we have an action of Out(H) on M(H).

In some situations it is a technical advantage to have an alternative but equivalent definition of global Mackey functors and inflation functors, as R-additive functors defined on certain categories Ω_{R} and Ω_{R}^{+} which we now define. The ingredients of this approach may be found in the work of Lewis, May and McClure [6, 10] and tom Dieck [4, p.278]. As a first step we define categories ω and ω^{+}, in both of which the objects are taken to be all finite groups. A morphism G → H in ω is a G × H-set X, taken up to G × Hset isomorphism, which is free on restriction to each of G and H. It is convenient to regard G as acting from the left and H as acting from the right, and we may write _{G}X_{H} to indicate this situation. Composition of morphisms is now an amalgamated product

GX_{H} ×_{H}_{H}Y_{K} = X × Y/ ∼ where (xh, y) ∼ (x, hy) for all x ∈ X, y ∈ Y and h ∈ H. In

ω^{+} we take the morphisms G → H to be isomorphism classes of G × H sets _{G}X_{H} which are free on restriction to H. Composition is defined in the same way as for ω. In both ω and ω^{+} the morphisms have the structure of an abelian monoid given by the binary operation of disjoint union: _{G}X_{H} +_{G} Y_{H} =_{G} X_{H} ∪_{G} Y_{H}. In fact the morphisms G → H form a free abelian monoid with basis the transitive G × H-sets with the appropriate free property. This is because each G × H-set is uniquely expressible as the disjoint union of its orbits. We may thus embed Hom_{ω}(G, H) and Hom_{ω^{+}}(G, H) into free abelian groups ℤ Hom_{ω}(G, H) and ℤ Hom_{ω^{+}}(G, H) by the usual universal construction. We now define the categories Ω_{R} and Ω_{R}^{+}. These again have the finite groups as their objects, and we take

Hom_{Ω_{R}}(G, H) = ℤ Hom_{ω}(G, H) ⊗ R
Hom_{Ω_{R}^{+}}(G, H) = ℤ Hom_{ω^{+}}(G, H) ⊗ R.

The group Hom_{Ω_{ℤ}^{+}}(G, H) appears in connection with the Segal conjecture ([6], [1], [8]) where it is denoted A(G, H). The special case A(G, G) is called the double Burnside ring. We will find it convenient to introduce the notation A_{R}(G, H) for Hom_{Ω_{R}^{+}}(G, H), so that A_{ℤ}(G, H) = A(G, H).

We may now give alternative definitions of our Mackey functors as follows: a global Mackey functor is an R-additive functor M : Ω_{R} → R-mod. An inflation functor is an Radditive functor M : Ω_{R}^{+} → R-mod. We remark immediately that because Ω_{R} is isomorphic to its opposite category it does not matter if we take M to be covariant or contravariant in defining a global Mackey functor. On the other hand this does make a difference with Ω_{R}^{+}: a covariant R-additive functor Ω_{R}^{+} → R-mod gives us an object of Mack_{∗}, while the contravariant functors give objects of Mack^{∗}.

To explain the connection between the different definitions we show first how each R-additive functor Ω_{R}^{+} → R-mod gives rise to an inflation functor in the sense previously defined. Suppose F : Ω_{R}^{+} → R-mod is a contravariant R-additive functor. We define an inflation functor by putting M(G) = F (G) for all groups G. If α : H → G is a monomorphism and γ : H → G is any group homomorphism we factor α as H⟶^{α′}H′↪^{ι}G where α′ is an isomorphism and put M_{∗}(α) = F (_{G}G_{H}) where H acts on G via (α′)^{−1} and M^{∗}(γ) = F (_{H}G_{G}) where H acts on G via γ.

To go in the opposite direction we show that a functor defined on Ω_{R}^{+} may be recovered from an inflation functor, we first make some comments about the morphisms in ω and ω^{+}. Any morphism in ω or ω^{+} is a sum of transitive G × H-sets, each of which is isomorphic to a set of the form (G×H)/Δ where Δ ≤ G×H. As explained in [1], the condition that this set is free on restriction to H is equivalent to the requirement that Δ = {(g, ϕ(g)) | g ∈ K} where K is some subgroup of G and ϕ is some homomorphism K → H. The condition that the set is also free on restriction to G is now equivalent to the requirement that ϕ is a monomorphism. One sees that (G × H)/Δ ≅ _{G}G_{K} ×_{K}_{K}H_{H} where K acts on the left of H via ϕ. Now given an inflation functor M in the first sense we define F (G) = M(G) for all groups G, and put F (G × H/Δ) = M_{∗}(ι)M^{∗}(ϕ) where ι : K ↪ G is inclusion. Extending this definition by R-linearity we obtain a contravariant functor Ω_{R}^{+} → R-mod. The translation procedure we have just described works for inflation functors and Ω_{R}^{+}, but if we simply require that all our group homomorphisms are monomorphisms we obtain the translation between global Mackey functors as first defined and R-additive functors defined on Ω_{R}. To verify that these translations are valid, we check that the relations satisfied by the Mackey functor morphisms are also satisfied in ω and ω^{+}, and that all relations on morphisms in ω and ω^{+} are deducible from these.

The construction of the categories Ω_{R} and Ω_{R}^{+} gives rise to a way of looking at the functors defined on them as modules over certain algebras. Let us first replace Ω_{R} and Ω_{R}^{+} by skeletal subcategories \overline{Ω_{R}} and \overline{Ω_{R}^{+}} in which the objects are representatives of the isomorphism types of groups. We put

μ_{R} = ⨁_{G,H} Hom\overline{Ω_{R}}(G, H)
μ_{R}^{+} = ⨁_{G,H} Hom\overline{Ω_{R}^{+}}(G, H).

By defining the product of two morphisms to be their composite if they can be composed, and zero otherwise, these two R-modules acquire the structure of R-algebras. For any ring Λ we define Λ-mod to be the category of Λ-modules. The following is immediate, as in [16].

(1.1) PROPOSITION. Mack_{R} is equivalent to μ_{R}-mod, and Mack_{R}^{∗} is equivalent to μ_{R}^{+op}-mod.

We take the opposite ring μ_{R}^{+op} here because the objects in Mack_{R}^{∗} are contravariant functors on Ω_{R}^{+}. We call μ_{R} the global Mackey algebra over R. Recall from [16] that in the context of Mackey functors defined on a fixed group G we also defined an algebra μ_{R}(G) called the Mackey algebra which plays a rôle analogous to that of μ_{R} and μ_{R}^{+} with respect to the category Mack_{R}(G) of Mackey functors defined on G. For each finite group G there is an R-algebra homomorphism μ_{R}(G) → μ_{R} (which will never be unital, because μ_{R} has no identity) specified as follows. Using the notation of [16], μ_{R}(G) has as a basis the products I_{^{g}L}^{K}c_{g}R_{L}^{H} where H, K ≤ G, g is a double coset representative from K\G/H and L ≤ H ∩ K^{g} is taken up to H ∩ K^{g}-conjugacy. We define

\begin{matrix}μ_{R}(G) & → μ_{R} \\ I_{^{g}L}^{K}c_{g}R_{L}^{H} & ↦ (H × K)/Δ \end{matrix}

where Δ = {(h, ^{g}h) | h ∈ L}. Now the restriction functor Mack_{R} → Mack_{R}(G) corresponds, when we regard functors as modules, to restricting the action of μ_{R} along the homomorphism μ_{R}(G) → μ_{R} and multiplying by 1 ∈ μ_{R}(G).

(1.2) PROPOSITION. The restriction functor Mack_{R} → Mack_{R}(G) has a left adjoint.

Proof. In module notation the left adjoint is M ↦ μ_{R}⊗_{μ_{R}(G)}M, where M is a Mackey functor defined on G.

We mention also that by a similar argument the restriction functor Mack_{R}^{∗} → Mack_{R} has a left adjoint. While this approach gives an easy proof of the theoretical existence of the left adjoint, it is not so easy to describe it in explicit terms. A major part of our work will be concerned with obtaining explicit formulae for the values of related adjoints on a certain special type of Mackey functor.

2. The classification of simple functors

We will show that both the simple global Mackey functors and also the simple inflation functors are in bijection with the set of pairs (H, V) where H is a group and V is a simple R[Out(H)]-module, both taken up to isomorphism. Many of the arguments for these two cases run in parallel, but we will see when we produce explicit formulae that the detailed structure of these two types of functor is different. To describe the simple functors we will use the terminology that a minimal group for either a global Mackey functor or an inflation functor M is a group H such that M(H) ≠ 0 but M(K) = 0 for every proper subgroup K of H.

(2.1) PROPOSITION. Let S be either a simple global Mackey functor or a simple inflation functor. Then S has a unique isomorphism class of minimal groups H. Furthermore, S(H) is a simple R[Out(H)]-module.

Proof. We give the proof explicitly in the case of inflation functors, and at the same time prove the result for global Mackey functors by indicating in italics the words in the proof that must be changed. In every case the word in italics is a homomorphism which must be supposed in addition to be a monomorphism to obtain the proof for global Mackey functors. Thus the word homomorphism must be changed to monomorphism, and epimorphism must be changed to isomorphism.

Let H be any minimal group of S, chosen to have minimal order. Let W ⊆ S(H) be an R[Out(H)]-submodule of S(H). For each group K consider

T (K) = {x ∈ S(K) | S^{∗}(α)(x) ∈ W for all homomorphisms α : H → K}.

We claim that T is a subfunctor of S. first, if H⟶^{α}J⟶^{θ}K are any homomorphisms and x ∈ T (K) then S^{∗}(α)S^{∗}(θ)(x) = S^{∗}(θα)(x) ∈ W shows that S^{∗}(θ) : T (K) → T (J). If θ happens to be an isomorphism then this shows also that S_{∗}(θ) : T (J) → T (K) since S_{∗}(θ) = S^{∗}(θ^{−1}) in this case, and we can apply the previous argument to θ^{−1}. Thus it remains to show that if i : J ↪ K is an inclusion of subgroups then

I_{J}^{K} = S_{∗}(i) : T (J) → T (K).

Let y ∈ T (J) and suppose H⟶^{β}β(H)⟶^{j}K is some homomorphism expressed as a composite of an epimorphism β and an inclusion j. Then

S^{∗}(jβ)I_{J}^{K}(y) = S^{∗}(β)R_{β(H)}^{K}I_{J}^{K}(y)

= S^{∗}(β) ∑_{g∈[β(H)\K/J]} I_{β(H)∩^{g}J}^{β(H)} c_{g} R_{β(H)^{g}∩J}^{J}(y)

= S^{∗}(β) ∑_{{g∈[β(H)\K/J] \atop β(H)^{g} ⊆J}} c_{g} R_{β(H)^{g}}^{J}(y)

∈ W

since S(β(H)^{g} ∩ J) = 0 unless β(H)^{g} ⊆ J, by minimality of H, and the remaining terms are S^{∗}(γ)(y) for some homomorphism γ.

We have T (H) = W, so if W ≠ 0 then T = S by simplicity of S, and so S(H) = W in this case. Thus S(H) must be simple.

If K is a group with no subgroup isomorphic to H then every homomorphism α : H → K must factor through a group of smaller size than H, on which S vanishes, and so S^{∗}(α)S(K) = 0. Thus T (K) = S(K) no matter what the choice of W. In particular this holds when W = 0, in which case T is a proper subfunctor of S, so T is the zero functor. Thus S(K) = T (K) = 0, and we deduce that the unique minimal group of S is H (up to isomorphism).

The next stage in the classification of the simple global Mackey functors and inflation functors is to show for each pair (H, V) consisting of a group H and a simple R[Out(H)]-module V that there is a simple global Mackey functor S with minimal group H and S(H) ≅ V, and that there is a simple inflation functor 𝕊 with minimal group H and 𝕊(H) ≅ V. The construction depends on the properties of a pair of adjoint functors which we now describe.

For each group H there are functors

F : Mack → R[Out(H)]-mod

and

F : Mack^{∗} → R[Out(H)]-mod

both given by

M ↦ \overline{M}(H)

where \overline{M}(H) = M(H) / ∑_{K<H} I_{K}^{H}M(K). We now construct functors

G : R[Out(H)]-mod → Mack

and

G : R[Out(H)]-mod → Mack^{∗}

which are the right adjoints of the functors F. We use the same symbols F and G in the two cases of global Mackey functors and inflation functors because in each case the formulae which define the functors are the same. We will in fact define an inflation functor G(V) = J_{H,V} ∈ Mack^{∗} whose restriction as a global Mackey functor is G(V) ∈ Mack. I am grateful to Peter Symonds for pointing out the approach presented here, which is more straightforward and elegant than my original construction.

We first define J_{H,V} on groups L which are isomorphic to H, by putting J_{H,V} (L) = V as a set. We now define J_{H,V} on isomorphisms of these groups. For each group L ≅ H let us fix an isomorphism α : H → L. Suppose that we are given two such groups and fixed isomorphisms α_{i} : H → L_{i}, i = 1, 2, and also some isomorphism γ : L_{1} → L_{2} . We define

(J_{H,V})_{∗}(γ)(v) = (α_{2}^{−1}γα_{1}) · v

and (J_{H,V})^{∗}(γ) = (J_{H,V})_{∗}(γ)^{−1}. In particular this specification makes J_{H,V} (L) into an R[Aut(L)]-module, and hence an R[Out(L)]-module, which is simply V with the action transported by α : H → L. To shorten the notation we will denote this module simply by αV.

We now make the general definition of J_{H,V}. For each group G we form the direct sum

⨁_{{L≤G \atop α:H≅L}} ^{α}V

taken over all subgroups of G isomorphic to H, where for each such subgroup we have chosen an isomorphism α. There is an action of G on this direct sum given as follows. For elements v ∈ ^{α}V and g ∈ G conjugation gives an isomorphism c_{g} : L → ^{g}L and the effect of g on v is (J_{H,V})_{∗}(c_{g})(v), which lies in the summand corresponding to ^{g}L. To define J_{H,V} on G we take the fixed points

J_{H,V} (G) = ( ⨁_{{L≤G \atop α:H≅L}} ^{α}V )^{G}.

If ι : K ↪ G is an inclusion of a subgroup we define J_{H,V ∗}(ι) = I_{K}^{G} to be the composite map

I_{K}^{G} : ( ⨁_{{L≤K \atop α:H≅L}} ^{α}V )^{K} ↪ ( ⨁_{{L≤G \atop α:H≅L}} ^{α}V )^{K} ⟶^{tr_{K}^{G}}( ⨁_{{L≤G \atop α:H≅L}} ^{α}V )^{G}.

If γ : K → G is any group homomorphism we define

J_{H,V}^{∗} (γ) : ( ⨁_{{L≤G \atop α:H≅L}} ^{α}V )^{G} → ( ⨁_{{L′≤K \atop β:H≅L′}} ^{β}V )^{K}

to have components J_{H,V}^{∗} (γ|_{L′}) : ^{α}V → ^{β}V if γ(L′) = L, and 0 otherwise. In case K is a subgroup of G and γ is inclusion, this map is given by projection into those summands αV for which the corresponding subgroup L = α(H) is a subgroup of K. In case γ : K → G is an group isomorphism then J_{H,V}^{∗} (γ) is determined on the summands ^{α}V by the isomorphisms γ|_{L} : L → γ(L), in the way in which J_{H,V}^{∗} had previously been defined on isomorphisms. We define (J_{H,V})_{∗}(γ) = (J_{H,V})^{∗}(γ)^{−1}. By this means (J_{H,V})_{∗} is defined by functoriality on all monomorphisms of groups, since a monomorphism is always expressible as the composite of an isomorphism and an inclusion, and on these maps (J_{H,V})_{∗} has been defined.

This construction we have just described is really an extension of the one performed in [14, Sect. 4]. We refer to this paper for the verification of the Mackey formula (4′), and the fact that the other axioms for an inflation functor are satisfied is a straightforward check. We also see that J_{H,V} is functorial in V, so we have a functor

G : R[Out(H)]-mod → Mack^{∗}
V ↦ J_{H,V}

and also a functor G : R[Out(H)]-mod → Mack

given by restricting J_{H,V} to be a global Mackey functor. Notice from the definition that J_{H,V} (H) = V and J_{H,V} (K) vanishes unless K has a subgroup isomorphic to H.

(2.2) PROPOSITION.

(i) The functor G : V ↦ J_{H,V} ∈ Mack^{∗} is right adjoint to the functor F : Mack^{∗} → R[Out(H)]-mod.

(ii) The functor G : V ↦ J_{H,V} ∈ Mack is right adjoint to the functor F : Mack → R[Out(H)]-mod.

Proof. (i) We construct the unit η and counit ϵ of the adjunction and first we consider η. For each inflation functor M we define a natural transformation η : M → J_{H,\overline{M}(H)} whose effect at a group G is

η : M(G) ⟶^{(q_{L}R_{L}^{G})} J_{H,\overline{M}(H)}(G) = ( ⨁_{{L≤G \atop H≅L}} \overline{M}(L))^{G}.

The notation here is that q_{L} : M(L) → \overline{M}(L) is the natural quotient, and so η is the map whose components are q_{L}R_{L}^{G}. We should verify that this is indeed a morphism of inflation functors, and we show that the two squares

\begin{matrix}M(G) & ⟶^{(q_{L}R_{L}^{G})} & (⨁_{{L≤G \atop H≅L}} \overline{M}(L))^{G} \\ ^{M^{∗}(γ)}↓ & & ↓^{J_{H,\overline{M}(H)}^{∗}(γ)} \\ M(K) & ⟶^{(q_{L′} R_{L′}^{K})} & (⨁_{{L′≤K \atop H≅L′}} \overline{M}(L′))^{K} \end{matrix} and \begin{aligned}M(G) & ⟶^{(q_{L}R_{L}^{G})} & (⨁_{{L≤G \atop H≅L}} \overline{M}(L))^{G} \\ ↑ & tr_{K}^{G} \\ ^{I_{K}^{G}} ↑ & & (⨁_{{L≤G \atop H≅L}} \overline{M}(L))^{K} \\ & & ⋃ \\ M(K) & ⟶^{(q_{L}R_{L}^{K})} & (⨁_{{L≤K \atop H≅L}} \overline{M}(L))^{K} \end{aligned}

commute, where γ : K → G is any group homomorphism in the first square, and K is any subgroup of G in the second square. Commutativity of the first square is immediate from the definition of J_{H,\overline{M}(H)}^{∗}(γ). The fact that the second square commutes is less obvious, but the argument is the same as the one which appears in Theorem 4.1 of [14]. We have

q_{L}R_{L}^{G}I_{K}^{G} = q_{L}( ∑_{g∈[L\G/K]} I_{L∩^{g}K}^{L} c_{g} R_{L^{g}∩K}^{K})

= ∑_{{g∈[L\G/K] \atop L≤^{g}K}} q_{L} c_{g} R_{L^{g}}^{K}

= ∑_{{g∈[G/K] \atop L≤^{g}K}} q_{L} c_{g} R_{L^{g}}^{K}

since a double coset LgK satisfying L ≤ ^{g}K is equal to a coset gK, and using the fact that q_{L} kills all properly induced terms. This is the composition of morphisms round one side of the square, and the composition round the other way is

∑_{g∈[G/K]} c_{g} ∑_{{L′≤K \atop L′≅H}} q_{L′} R_{L′}^{K} = ∑_{{g∈[G/K] \atop L′≤K}} q_{^{g}L′} c_{g} R_{L′}^{K}.

The component of this with respect to a subgroup H ≅ L ≤ G is

∑_{{g∈[G/K] \atop L=^{g}L′≤^{g}K}} q_{L} c_{g} R_{L^{g}}^{K}

as required.

We now define the counit ϵ. At each R[Out(H)]-module V, ϵ is simply the identity map ϵ : \overline{J_{H,V}} (H) → V. We next need to check that η and ϵ are natural transformations of functors

η : 1 → GF, ϵ : FG → 1.

The case of ϵ is immediate since in fact FG is the identity functor. As for η, we have to show that if θ : M → N is a morphism of global Mackey functors then

M(G) ⟶^{(q_{L}R_{L}^{G})} ( ⨁_{H≅L≤G} \overline{M}(L))^{G}
θ_{G}↓  ↓_{(\overline{θ}_{L})}
N(G) ⟶^{(q_{L}R_{L}^{G})} ( ⨁_{H≅L≤G} \overline{N}(L))^{G}

commutes, where \overline{θ}_{L} is the morphism induced on \overline{M}(L) by θ. This follows from the fact that θ commutes with the maps R_{L}^{G}.

We finally check that the two composites

F⟶^{Fη}FGF⟶^{ϵ_{F}}F G⟶^{η_{G}}GFG⟶^{Gϵ}G

are the identity. This is easy, and we leave the details to the reader.

(ii) Exactly the same unit and counit demonstrate that we have an adjunction.

(2.3) PROPOSITION. Let V be a simple R[Out(H)]-module.

(i) As an inflation functor, every non-zero subfunctor of J_{H,V} contains J_{H,V} (H). Hence the subfunctor ⟨J_{H,V} (H)⟩_{Mack}^{∗} generated by J_{H,V} (H) in the category Mack^{∗} is the unique minimal subfunctor of J_{H,V}, and hence is a simple inflation functor.

(ii) As a global Mackey functor, every non-zero subfunctor of J_{H,V} contains J_{H,V} (H). Hence the subfunctor ⟨J_{H,V} (H)⟩_{Mack} generated by J_{H,V} (H) in the category Mack is the unique minimal subfunctor of J_{H,V}, and hence is a simple global Mackey functor.

Proof. The proofs of (i) and (ii) are the same. Let 0 ≠ M ⊆ J_{H,V} be a subfunctor. The inclusion map corresponds by adjointness to a non-zero homomorphism \overline{M}(H) → J_{H,V} (H) = V which must be an epimorphism since V is simple. Since M(H) ⊆ V we must have M(H) = V and so M(H) ⊇ J_{H,V} (H). The final assertion is immediate.

We define

𝕊_{H,V} = ⟨J_{H,V} (H)⟩_{Mack}^{∗}
S_{H,V} = ⟨J_{H,V} (H)⟩_{Mack}

to be the unique simple subfunctors of J_{H,V}, as an inflation functor, and as a global Mackey functor. Because of the structure of J_{H,V} we may immediately say that H is the unique minimal group of both 𝕊_{H,V} and S_{H,V}, and that 𝕊_{H,V} (H) = V = S_{H,V} (H). We will go further than this and produce explicit formulae for both 𝕊_{H,V} (G) and S_{H,V} (G) which demonstrate that 𝕊_{H,V} and S_{H,V} are different functors, but first we show that we do have a complete list of the simple functors.

(2.4) THEOREM.

(i) The simple inflation functors are precisely the 𝕊_{H,V}, one for each pair (H, V) where H is a finite group and V is a simple R[Out(H)]-module, both taken up to isomorphism.

(ii) The simple global Mackey functors are precisely the S_{H,V}, one for each pair (H, V) where H is a finite group and V is a simple R[Out(H)]-module, both taken up to isomorphism.

Proof. We prove (ii), the proof of (i) being similar. We have shown that the S_{H,V} are simple, and it remains to show that these are all of the simple global Mackey functors. Let S be a simple global Mackey functor, H the unique minimal subgroup of S, and V = S(H) the corresponding simple R[Out(H)]-module. The isomorphism S(H) = \overline{S}(H) → V corresponds by adjointness to a non-zero map S → J_{H,V}. Since S is simple this gives an isomorphism to the unique simple subfunctor, S ≅ S_{H,V}.

We now embark on an explicit description of the functors we have constructed.

(2.5) LEMMA. Let X be a class of groups closed under taking isomorphisms and taking subgroups.

(i) Let M be an inflation functor. Then the subfunctor N of M generated by the M(H) with H ∈ X has the form

N(G) = ∑_{G↩^{i}J→^{π}H∈X} i_{∗}π^{∗}M(H)

where the sum is taken over inclusions of subgroups i and epimorphisms π. (ii) Let M be a global Mackey functor. Then the subfunctor N of M generated by the

M(H) with H ∈ X has the form

N(G) = ∑_{G↩^{i}H∈X} i_{∗}M(H)

where the sum is taken over inclusions of subgroups i.

Proof. (i) The right hand side is certainly contained in the left, and we have to show that the specification given by the right hand side does indeed give an inflation functor. To do this we show that the right hand side is stable under j_{∗} and γ^{∗} whenever j : G ↪ G′′ is an inclusion of groups and γ : G′ → G is any homomorphism of groups. Now j_{∗}i_{∗}π^{∗}M(H) = (ji)_{∗}π^{∗}M(H) shows stability under j_{∗}. With γ it suffices to consider separately the cases when γ is an inclusion of groups and when γ is an epimorphism, since an arbitrary γ may be expressed as a composite of these. When γ is an inclusion γ^{∗}i_{∗}π^{∗} may be written as a sum of terms i′_{∗}γ^{′∗}π^{∗} = i′_{∗}(πγ′)^{∗} using the Mackey formula, where now γ′ is a monomorphism of the form γ′ : G′ ∩ ^{g}J⟶^{c_{g}}G^{′g} ∩ J ↪ G. We now have i′_{∗}(πγ′)^{∗}M(H) ⊆ i′_{∗}(π|_{G^{′g}∩J}c_{g})^{∗}M(π(G^{′g} ∩ J)) and here π(G^{′g} ∩ J) ∈ X.

When γ is an epimorphism then

γ^{∗}i_{∗}π^{∗} = i′_{∗}(γ|_{J′})^{∗}π^{∗} = i′_{∗}(πγ|_{J′})^{∗}

where J′ is the preimage of J in G′ under γ and the maps are as follows:

\begin{matrix}G′ & ⟶^{γ} & G \\ ↑_{i′} & & ↑^{i} \\ J′ & ⟶^{γ|_{J′}} & J \end{matrix}

This shows that our expression of N(G) is stable under j_{∗} and γ^{∗} and hence gives an inflation functor.

(ii) The proof is similar to the proof of (i). We observe first that the right hand side is contained in the left. Then we show that the right hand side is stable under j_{∗} and j^{∗} whenever j is a monomorphism of groups. The arguments here are the same as those used to prove part (i), where the more general possibility of j being an arbitrary homomorphism was considered in the case of j^{∗}.

(2.6) THEOREM.

(i) The functors J_{H,V} are given explicitly by

J_{H,V} (G) = ⨁_{α:H≅L≤G up to G−conjugacy}(^{α}V)^{N_{G}(L)}.

(ii) The simple global Mackey functors are given explicitly by

S_{H,V} (G) = ⨁_{α:H≅L≤G up to G−conjugacy} tr_{L}^{N_{G}(L)}(^{α}V).

(iii) The simple inflation functors are given by

𝕊_{H,V} (G) = ∑_{ι:K↪G π:K→H split epi} ι_{∗}π^{∗}(V)

as a submodule of J_{H,V} (G). For a given subgroup K ≤ G and a split epimorphism π : K → H, if we write ι_{∗}π^{∗}(v) = ∑_{α:H≅L≤G up to G−conjugacy} f_{α}(v) as a sum of components with respect to the decomposition given in (i), where v ∈ V, then

f_{α}(v) = ( ∑_{{g∈[K\T (L,K)] \atop B∩^{g}L=1}} θ_{g}^{−1}) · v

where T (L, K) = {g ∈ G | ^{g}L ⊆ K}, B = Ker π and

θ_{g} : H→^{α}L→^{c_{g}}^{g}L ↪ K→^{π}H.

Note in (iii) that for a particular subgroup L = α(H) of G, if no G-conjugate of L is contained in K then f_{α}(v) = 0 since the sum which appears in the expression for this is empty. It may help in dealing with the notation here to observe that ∑_{{g∈[K\T (L,K)] \atop B∩^{g}L=1}} θ_{g}^{−1} is an element of the group ring R[Aut(H)], which represents an element of R[Out(H)].

Proof. (i) The definition says that

J_{H,V} (G) = ( ⨁_{α:H≅L≤G} ^{α}V)^{G}.

Let us fix a particular subgroup L ≤ G isomorphic to H. The summands indexed by the conjugates of L form a summand

\begin{aligned}( ⨁_{{α′:H≅L′≤G \atop L′∼_{G}L}} ^{α′}V)^{G} & ≅ (^{α}V ↑_{N_{G}(L)}^{G})^{G} \\ & ≅ (^{α}V)^{N_{G}(L)}. \end{aligned}

which gives the result. We obtain the first isomorphism because the term in parentheses on the left is the direct sum of subspaces permuted by G, of which one is ^{α}V with stabilizer N_{G}(L). The second isomorphism is a standard identification.

(ii) We have by 2.5(ii) that

S_{H,V} (G) = ∑_{α:H≅L≤G} I_{L}^{G}(^{α}V)

as a subset of J_{H,V} (G), and it is a question of examining the maps I_{L}^{G}. They were defined as a composite which in this instance is

αV ⟶^{1}^{α}V ⟶^{tr_{L}^{G}}( ⨁_{α′:H≅L′≤G} ^{α′}V)^{G}.

This has image contained in the summand determined by the conjugates of L, and identifying this with (^{α}V ↑_{N_{G}(L)}^{G})^{G} ≅ ^{α}V^{N_{G}(L)} as in part (i), the composite map is tr_{L}^{N_{G}(L)} : ^{α}V → ^{α}V^{N_{G}(L)}. Hence I_{L}^{G}(^{α}V) ≅ tr_{L}^{N_{G}(L)} ^{α}V. We notice that if L is conjugate to L_{1} in G then I_{L_{1}}^{G}(^{α_{1}}V) = I_{L}^{G}(^{α}V) as subsets of J_{H,V} (G), and if L, L_{1} are subgroups of G which are isomorphic to H but are not conjugate in G then I_{L_{1}}^{G}(^{α_{1}}V) and I_{L}^{G}(^{α}V) are contained in different summands in the expression for J_{H,V} (G). Thus our expression for S_{H,V} (G) is a direct sum taken over representatives of the G-conjugacy classes of subgroups L, and our identification of these summands gives the desired result.

(iii) To prove the first assertion we apply 2.5(i) with X consisting of all groups isomorphic to subgroups of H and M = J_{H,V}. Since J_{H,V} (L) = 0 for L ∈ X unless L ≅ H we have by that result that

𝕊_{H,V} (G) = ∑_{G↩^{i}K→^{π}H} ι_{∗}π^{∗}(V).

From the definition of the functor J_{H,V} we have π^{∗} = J_{H,V}^{∗} (π) = 0 unless there is a subgroup L ≤ K mapped isomorphically onto H by π. Thus if J_{H,V}^{∗} (π) ≠ 0 the group K must be a semidirect product K = B ⋊ L, and π is a split epimorphism.

We now examine the component f_{α} of the map ι_{∗}π^{∗}. From the definition,

π^{∗} : V → ( ⨁_{{L≤K \atop α:H≅L}} ^{α}V )^{K}

is the map

∑_{L′ complements B in K} π|_{L′}^{∗}.

On applying ι_{∗}, only those complements to B in K which are G-conjugate to L contribute to the L summand of J_{H,V} (G). The set of all G-conjugates of L in K has as a set of representatives the ^{g}L where g ∈ [T (L, K)/N_{G}(L)] and so the component of π^{∗} corresponding to the complements to B in K which are G-conjugate to L is

∑_{{g∈[T (L,K)/N_{G}(L)] \atop B∩^{g}L=1}} π|_{^{g}L}^{∗}.

Let k ∈ K and consider a complement ^{kg}L to B in K. The diagram

gL ⟶^{π} H
c_{k}↓  ↓^{c_{π(k)}}
kgL ⟶^{π} H

commutes, and c_{π(k)} acts trivially on V since it is an inner automorphism, so (π|_{^{kg}L})^{∗} = c_{k}π|_{^{g}L}^{∗}. Thus

\begin{aligned}∑_{{g∈[T (L,K)/N_{G}(L)] \atop B∩^{g}L=1}} π|_{^{g}L}^{∗} & = ∑_{{g∈[K\T (L,K)/N_{G}(L)] \atop B∩^{g}L=1}} ∑_{k∈[K/K∩^{g}N_{G}(L)]} π|_{^{kg}L}^{∗} \\ & = ∑_{{g∈[K\T (L,K)/N_{G}(L)] \atop B∩^{g}L=1}} tr_{K∩^{g}N_{G}(L)}^{K} π|_{^{g}L}^{∗}. \end{aligned}

Now the L component in J_{H,V} (G) of ι_{∗}π^{∗} is

tr_{K}^{G} ∑_{{g∈[K\T (L,K)/N_{G}(L)] \atop B∩^{g}L=1}} tr_{K∩^{g}N_{G}(L)}^{K} π|_{^{g}L}^{∗} = ∑_{{g∈[K\T (L,K)/N_{G}(L)] \atop B∩^{g}L=1}} tr_{K∩^{g}N_{G}(L)}^{G} π|_{^{g}L}^{∗}

= ∑_{{g∈[K\T (L,K)/N_{G}(L)] \atop B∩^{g}L=1}} tr_{K^{g}∩N_{G}(L)}^{G} c_{g^{−1}}π|_{^{g}L}^{∗}

= tr_{N_{G}(L)}^{G} ∑_{{g∈[K\T (L,K)/N_{G}(L)] \atop B∩^{g}L=1}} tr_{K^{g}∩N_{G}(L)}^{N_{G}(L)} c_{g^{−1}}π|_{^{g}L}^{∗}

= tr_{N_{G}(L)}^{G} ∑_{{g∈[K\T (L,K)] \atop B∩^{g}L=1}} c_{g^{−1}}π|_{^{g}L}^{∗}

∈ ( ⨁_{{L≤G \atop α:H≅L}} ^{α}V )^{G}.

Finally the isomorphism

⨁_{H α ; ≅L≤G up to G−conjugacy} ^{α}V^{N_{G}(L)} → ( ⨁_{{L≤G \atop α:H≅L}} ^{α}V )^{G}

on the L component is just v ↦ tr_{N_{G}(L)}^{G} v and so composing with the inverse of this isomorphism we obtain that the L component f_{α} of ι_{∗}π^{∗} is

∑_{{g∈[K\T (L,K)] \atop B∩^{g}L=1}} c_{g^{−1}}π|_{^{g}L}^{∗}.

Now c_{g^{−1}}π|_{^{g}L}^{∗} here denotes the homomorphism V → ^{α}V obtained by contravariant functoriality from the group homomorphism L⟶^{c_{g}}^{g}L ↪ K⟶^{π}H using the isomorphism α : H → L. The functorial definition gives precisely c_{g^{−1}}π|_{^{g}L}^{∗}(v) = θ_{g}^{−1} · v where v ∈ V and θ_{g} is defined in the statement of this theorem. Thus the component f_{α} of ι_{∗}π^{∗} is as claimed.

(2.7) COROLLARY.

(i) S_{H,V} (G) ≠ 0 if and only if there is a subgroup L ≤ G isomorphic to H via α : H → L so that tr_{L}^{N_{G}(L)}(^{α}V) ≠ 0.

(ii) 𝕊_{H,V} (G) ≠ 0 if and only if there are subgroups L, K and B of G with L isomorphic to H via α and K = B ⋊ L so that ∑_{{g∈[K\T (L,K)] \atop B∩^{g}L=1}} θ_{g}^{−1} does not annihilate V.

We see from this that it is much easier to compute the global Mackey functors S_{H,V} (G) than the inflation functors 𝕊_{H,V} (G), and in Section 4 we will compute a number of values of S_{H,V} (G). It is entirely by reason of this ease of computation that we use global Mackey functors in our application to the computation of cohomology, although in principle the computations could be done with inflation functors as well. The description of 𝕊_{H,V} which we have just given is, however, exactly what we need to reprove the results of Martino-Priddy [8] and Benson-Feshbach [1] concerning decomposition of classifying spaces, which we will do in Section 6.

One may give more straightforward sufficient conditions for when these simple functors vanish. The result in the case of inflation functors is really the same as a condition which has been obtained by Benson and Feshbach [1, Proposition 5.9] in an apparently different context. We will show later that their situation really amounts to the same thing as ours, and our result can be deduced from theirs by translating the concepts.

Let L be a subgroup of G and let W be a R[Out(L)]-module. Then W becomes a RN_{G}(L)-module via the homomorphisms N_{G}(L) → Aut(L) → Out(L). We define Stab_{G}(W) to be the kernel of this action of N_{G}(L) on W. Thus

Stab_{G}(W) = {g ∈ N_{G}(L) | g · w = w for all w ∈ W }.

Since C_{G}(L) always acts trivially on L by conjugation we have L · C_{G}(L) ≤ Stab_{G}(W).

(2.8) PROPOSITION. Let G and H be p-groups, let R be a field of characteristic p and let V be a simple R[Out(H)]-module.

(i) If S_{H,V} (G) ≠ 0 then there exists a subgroup L ≤ G with an isomorphism α : H → L such that Stab_{G}(^{α}V) = L (i.e. N_{G}(L)/L acts faithfully on ^{α}V). In particular C_{G}(L) ≤ L.

(ii) If 𝕊_{H,V} (G) ≠ 0 then there exists a subgroup L ≤ G with an isomorphism α : H → L such that Stab_{G}(^{α}V) = C_{G}(L) · L = L × X for some subgroup X of G.

Proof. (i) From 2.6(ii) we deduce that if S_{H,V} (G) ≠ 0 then

tr_{L}^{N_{G}(L)} = tr_{Stab_{G}(^{α}V)}^{N_{G}(L)} tr_{L}^{Stab_{G}(^{α}V)} ≠ 0

for some subgroup L. Hence tr_{L}^{Stab_{G}(^{α}V)} ≠ 0 and since G is a p-group it follows that Stab_{G}(^{α}V) = L.

(ii) Using notation introduced in 2.6(iii) we have

𝕊_{H,V} (G) = ∑_{ι:K↪G π:K→H split epi}(∑_{α} f_{α})V,

for this expression to be non-zero there must be a subgroup K and an isomorphism α : H → L for some subgroup L of G such that f_{α} ≠ 0. Now

f_{α} = ∑_{{g∈[K\T (L,K)/N_{G}(L)] \atop B∩^{g}L=1}} tr_{N_{K^{g}} (L)}^{N_{G}(L)} c_{g^{−1}}π|_{^{g}L}^{∗}

and so tr_{N_{K^{g}} (L)}^{N_{G}(L)} must be non-zero for some g. Now K^{g} is a semidirect product K^{g} = B^{g} ⋊L and so N_{K^{g}}(L) = X × L where X = C_{B^{g}}(L). Thus N_{K^{g}}(L) ≤ C_{G}(L) · L ≤ Stab_{G}(^{α}V) and since

tr_{N_{K^{g}} (L)}^{N_{G}(L)} = tr_{Stab_{G}(^{α}V)}^{N_{G}(L)} tr_{N_{K^{g}} (L)}^{Stab_{G}(^{α}V)}

it follows that tr_{N_{K^{g}} (L)}^{Stab_{G}(^{α}V)} ≠ 0. Because G is a p-group and Stab_{G}(^{α}V) acts trivially on αV we deduce that N_{K^{g}}(L) = Stab_{G}(^{α}V), which in turn must coincide with X × L and C_{G}(L) · L.

(2.9) COROLLARY. Let G and H be p-groups, let R be a field of characteristic p, and let R also denote the trivial R[Out(H)]-module.

(i) S_{H,R}(G) ≠ 0 if and only if H ≅ G. (ii) If 𝕊_{H,R}(G) ≠ 0 then there is a subgroup L of G isomorphic to H for which N_{G}(L) =

L × X for some subgroup X.

Proof. We have Stab_{G}(^{α}V) = N_{G}(L) for every subgroup L of G isomorphic to H, and (ii) is immediate from 2.8(ii). In case (i), if S_{H,R}(G) ≠ 0 we have by 2.8(i) that N_{G}(L) = L, which forces L = G. Conversely, if H ≅ G we have S_{H,R}(G) = R, as remarked before 2.4.

3. The composition factors of a global Mackey functor

In this section we consider only global Mackey functors, because that is the context in which we will apply the theory about to be developed. A similar theory may be developed for inflation functors, and we leave this to the interested reader.

We will be dealing with global Mackey functors which do not have composition series in the conventional sense and so we need to give some meaning to the notion of composition factor for a global Mackey functor. To begin, we will say that a simple global Mackey functor S is a composition factor of M if there exist subfunctors M_{0} ⊂ M_{1} ⊆ M so that M_{1}/M_{0} ≅ S. To extend this notion, we fix a group G and say that a global Mackey functor has a composition series over G if there is a series of subfunctors

0 = T_{0} ⊆ B_{1} ⊂ T_{1} ⊆ ⋯ ⊆ B_{m} ⊂ T_{m} ⊆ B_{m+1} = M

such that

T_{i}/B_{i} is simple and non-zero on restriction to G for all i = 1, …, m, and (B_{i+1}/T_{i})↓_{G}= 0 for all i = 0, …, m.

When such a series exists we will call the set of simple functors T_{i}/B_{i} together with their multiplicities, the composition factors of M over G. We are thus pulling out just those composition factors of M which do not vanish on G.

(3.1) PROPOSITION. Suppose that M is a global Mackey functor which has a composition series over G. Then any other composition series over G for M has the same length and the composition factors over G (taken with multiplicities) are the same.

Proof. By a standard argument using the Zassenhaus isomorphism theorem, any two series

0 ⊆ B_{1} ⊂ T_{1} ⊆ ⋯ ⊆ B_{m} ⊂ T_{m} ⊆ M

have refinements which are of equal length and in which the sets of quotients are the same (up to isomorphism, taken with multiplicities). It is impossible to refine the simple quotients T_{i}/B_{i} any further, and so in any refinement of the above series, the set of quotients which do not vanish on G remains the same. From this we see that in any other composition series for M over G the composition factors over G are the same, and in particular there is the same number of them.

We are now concerned with the existence of composition series over G, and for this we will use a characterisation of simple global Mackey functors. Let X be a class of groups closed under isomorphism and taking subgroups. If M is a global Mackey functor we introduce the notation Ker R_{X} to denote the subfunctor of M defined by

(Ker R_{X})(K) = ⋂_{{X∈X \atop X≤K}} Ker R_{X}^{K}.

(3.2) THEOREM. Let S be a global Mackey functor, let H be a minimal group for S and let X be the class of groups isomorphic to subgroups of H. Then S is simple if and only if the following three conditions are satisfied.

(i) Ker R_{X} = 0, (ii) S = ⟨S(H)⟩, (iii) S(H) is a simple R[Out(H)]-module.

Proof. Assume S is simple. Since (Ker R_{X})(H) = 0, Ker R_{X} is a proper subfunctor, hence is zero. The other two conditions have already been established.

Assume now that the three conditions hold and let T be a non-zero subfunctor of S. Let K be a group for which T (K) ≠ 0. By (i)

(Ker R_{X})(K) = ⋂_{H≅L≤K} Ker R_{L}^{K} = 0

and so there exists a subgroup L ≤ K isomorphic to H for which T (K) ⊈ Ker R_{L}^{K}. Thus R_{L}^{K}(T (K)) ≠ 0 and so T (L) ≠ 0 and hence S(H) ⊇ T (H) ≠ 0. Since S(H) is a simple module we have S(H) = T (H). Finally we deduce S = T from (ii).

(3.3) THEOREM. Let G be a finite group and M a global Mackey functor over R. The following are equivalent:

(i) M has a composition series over G, (ii) for all subgroups K ≤ G, M(K) has a composition series as an R-module, (iii) M ↓_{G} has a composition series as a Mackey functor for G.

Proof. The implication (i) ⇒ (ii) is easy, since the restriction of a composition series over G to K gives a filtration of M(K) in which the finitely many successive quotients are evaluations of simple global Mackey functors S_{H,V} (K). From our description of the simple functors, these all have finite length over some field which is a quotient of R.

For the converse implication (ii) ⇒ (i), we proceed by induction on

ℓ(M) = ∑_{K≤G} length_{R}(M(K)),

the induction starting when this number is 0. We suppose, therefore, that ℓ(M) > 0 and that the result holds for global Mackey functors with smaller values of ℓ. Let Y consist of all groups isomorphic to subgroups of G and let B_{1} = Ker R_{Y} as a subfunctor of M. Then B_{1} ↓_{G}= 0, and we show that the quotient M/B_{1} has a simple subfunctor which is a composition factor over G. To construct this simple subfunctor, let J ≤ G be a subgroup of maximal order such that

⋂_{K {< \atop ≠} J} Ker R_{K}^{J} ≠ 0

and let 0 ≠ V ≤ M(J) be a simple R[Out(J)]-submodule of the above intersection. Since every element of V is sent to 0 on restriction to all proper subgroups of J, the subfunctor of M generated by V has the form

⟨V ⟩(K) = ∑_{J≅J_{1}≤K} Im(I_{J_{1}}^{K})

and in particular ⟨V ⟩ vanishes unless K contains a subgroup isomorphic to J. We put T_{1} = B_{1} + ⟨V ⟩, and show that T_{1}/B_{1} is simple by verifying the three conditions in our characterisation of simple global Mackey functors.

Since J is a minimal group for ⟨V ⟩ and B_{1} vanishes on subgroups of G, we see that J is also a minimal group for T_{1}/B_{1}, and furthermore (T_{1}/B_{1})(J) = ⟨V ⟩(J) = V is a simple R[Out(J)]-module. It is clear that T_{1}/B_{1} is generated by its value at J. Thus it remains to show that if X denotes the class of groups isomorphic to subgroups of J then Ker R_{X} = 0 for the functor T_{1}/B_{1}.

Consider an element x + B_{1}(K) ∈ Ker R_{X} (K) for some subgroup K, where x ∈ ⟨V ⟩(K). We show that x ∈ B_{1}(K), which will complete our argument that T_{1}/B_{1} is simple. Suppose to the contrary that x ∉ B_{1}(K), so there exists L ∈ Y with R_{L}^{K}(x) ≠ 0. Since ⟨V ⟩ vanishes on groups which do not contain a subgroup isomorphic to J, L contains such a subgroup. We show that for some subgroup J_{1} of L isomorphic to J, R_{J_{1}}^{K}(x) ≠ 0. The argument is by induction on |L|. If |L| = |J| then L ≅ J and R_{L}^{K}(x) ≠ 0 already, so the induction starts. Supposing now that |L| > |J| and the result holds for smaller values of |L|, by maximality of |J| there exists a proper subgroup L_{1} of L with R_{L_{1}}^{K}(x) = R_{L_{1}}^{L}R_{L}^{K}(x) ≠ 0, and necessarily L_{1} still contains a subgroup isomorphic to J. By induction we deduce that R_{J_{1}}^{L_{1}}R_{L_{1}}^{K}(x) = R_{J_{1}}^{K}(x) ≠ 0 for some subgroup J_{1} ≤ L, isomorphic to J. This contradicts the earlier hypothesis that x + B_{1}(K) ∈ Ker R_{X} (K), and so we deduce that x ∈ B_{1}(K), which shows that T_{1}/B_{1} is simple.

We have now constructed the start of a chain B_{1} ⊂ T_{1} ⊆ M where T_{1}/B_{1} is a composition factor over G and B_{1} ↓_{G}= 0. Now ℓ(M/T_{1}) < ℓ(M), so by induction M/T_{1} has a composition series over G, and we deduce that M does also.

The equivalence of (ii) and (iii) follows from [16].

As a way of recording the composition factors of a global Mackey functor we introduce an abelian group Π which is the product of copies of ℤ indexed by the simple global Mackey functors. Thus an element of Π is a family of integers (n_{H,V}) indexed by pairs (H, V), which on linearly ordering the pairs (H, V) we may write as a sequence. Suppose that M is a global Mackey functor with the property that for every finite group G, M(G) has a composition series as an R-module. In this situation we may associate the element [M] ∈ Π, which is defined to be the sequence whose (H, V) term [M]_{H,V} is the multiplicity of S_{H,V} in a composition series for M over H. This is also the multiplicity of S_{H,V} in a composition series for M over G, where G is any group containing a subgroup isomorphic to H. Thus the composition factors of M over G are precisely the S_{H,V} with multiplicity [M]_{H,V}, ranging over groups H which are isomorphic to subgroups of G. This means that the composition factors of M are determined by knowing [M].

At this point we will assume that R is an artinian ring and form the product

Π′ = ∏_{H} G_{0}(R[Out(H)])

of the Grothendieck groups of R[Out(H)]-modules taken over all finite groups H (up to isomorphism). A typical element may be written (a_{H}), where for each finite group H, a_{H} ∈ G_{0}(k[Out(H)]). Because of our hypothesis on R, each Grothendieck group has a ℤ-basis consisting of the simple R[Out(H)]-modules and so we may write

a_{H} = ∑_{V} n_{H,V} [V]

where V ranges over the simple R[Out(H)]-modules. Thus (a_{H}) may be represented by a family of integers (n_{H,V}) indexed by pairs (H, V), and since the elements of the group Π defined previously may be represented in the same way, we see that as abstract groups Π ≅ Π′.

We define a homomorphism ψ : Π → Π′ by

ψ((n_{H,V})) = ( ∑_{(H,V)} n_{H,V} [S_{H,V} (K)])

the expression on the right being a family of terms indexed by groups K, and where the term in parentheses on the right is an element of G_{0}(R[Out(K)]). The sum here is apparently infinite since it is taken over all pairs (H, V), but in fact there are only finitely many terms in the sum for each K, since only those groups H which are isomorphic to subgroups of K contribute.

(3.4) LEMMA. The morphism ψ is a monomorphism.

Proof. Suppose that the indexing pairs (H, V) are linearly ordered in any fashion so that |H| is non-decreasing, and that the groups which index the factors in Π′ have the induced linear order. Let a = (a_{H,V}) and b = (b_{H,V}) be distinct sequences of integers such that ψ(a) = ψ(b). Write a = c + a_{1}, b = c + b_{1} where c is the longest common initial subsequence of both a and b, and a_{1}, b_{1} are different in their first non-zero place. So

\begin{aligned}a_{1} & = (0, …, 0, a_{H,V} , …) \\ b_{1} & = (0, …, 0, b_{H,V} , …) \end{aligned}

with a_{H,V} ≠ b_{H,V} and ψ(a_{1}) = ψ(a) − ψ(c) = ψ(b) − ψ(c) = ψ(b_{1}). Since S_{H,V} (H) = V the definition of ψ gives that

ψ(0, …, 0, a_{H,V} , …) = (0, …, 0, a_{H,V} [V] + …, …),

so that the coefficient of [V] in the first non-zero entry is a_{H,V}, and similarly with ψ(b_{1}). We thus deduce that ψ(a_{1}) cannot equal ψ(b_{1}), a contradiction, and so ψ is a monomorphism.

As a consequence of this lemma, in order to check that a global Mackey functor M has composition factors given by the sequence (n_{H,V}), we only need check that ψ([M]) = ψ((n_{H,V})). The advantage of this is that the Kth component ψ([M])_{K} is just [M(K)], and so we simply have to determine the module M(K) as an element of G_{0}(R[Out(K)]). Notice that in order to compute a composition factor multiplicity n_{H,V} it suffices to compute the values of [M(K)] where K is isomorphic to a subgroup of H. This results from the fact that the restriction of ψ to the product of those factors indexed by such groups K is again a monomorphism to the product of the corresponding factors in Π′, by exactly the same argument as in the lemma.

We may exercise some further control over the groups H which appear in the parameterisation of the composition factors of a global Mackey functor M using essentially the relative projectivity as Mackey functors of the restrictions M ↓_{G}. For this we assume familiarity with the classification of the simple Mackey functors S_{H,V} given in [15].

(3.5) PROPOSITION. Let M be a global Mackey functor and X a class of groups closed under isomorphisms. Suppose that for every finite group G, the Mackey functor composition factors of M ↓_{G} are all of the form S_{H,V} with H ∈ X. Then every composition factor of M as a global Mackey functor has the form S_{H,V} with H ∈ X.

Proof. Let S_{H,V} be a composition factor of M as a global Mackey functor. Thus there are subfunctors M_{0} ⊂ M_{1} ⊆ M with M_{1}/M_{0} ≅ S_{H,V}. On restricting this filtration of M to H we obtain Mackey functors M_{0} ↓_{H}⊂ M_{1} ↓_{H}⊆ M ↓_{H}, and now

(M_{1} ↓_{H})/(M_{0} ↓_{H}) = S_{H,V} ↓_{H}

is a Mackey functor which is non-zero only on H. From the structure of the simple Mackey functors [15] we deduce that the Mackey functor composition factors of (M_{1} ↓_{H})/(M_{0} ↓_{H}) must have the form S_{H,W}, and these are also composition factors of M ↓_{H}. By hypothesis we have H ∈ X.

4. The computation of global Mackey functors

We will now show how to use the notions of composition factor and composition series to compute the values of specific Mackey functors. In particular we will take as examples the global Mackey functors H^{r}(G, T) where the coefficient module T has trivial G-action. In order that the Grothendieck groups G_{0}(R[Out(H)]) should have a basis consisting of the simple modules we will assume in this section that the ring R over which our global Mackey functors are defined is an artinian ring. However, as far as the cohomology groups H^{r}(G, T) with r ≥ 1 are concerned we need not require that T itself be artinian. This is because H^{r}(G, T) is annihilated by |G| and so the cohomology may turn out to be a finitely generated module for an artinian ring, even though T is not. We have in mind the example of H^{r}(G, ℤ), which is a finitely generated module for the artinian ring R = ℤ/|G|ℤ.

In general our method proceeds as follows. We wish to compute a certain value M(G) of a global Mackey functor, and suppose that M has a composition series over G. Thus we have a filtration

0 = T_{0} ⊆ B_{1} ⊂ T_{1} ⊆ ⋯ ⊆ B_{m} ⊂ T_{m} ⊆ B_{m+1} = M

where each T_{i}/B_{i} is a simple composition factor over G and B_{i+1}/T_{i} vanishes on G. On evaluation at G this gives a filtration

0 = B_{1}(G) ⊆ T_{1}(G) = B_{2}(G) ⋯ ⊆ T_{m−1}(G) = B_{m}(G) ⊆ T_{m}(G) = M(G)

in which there are finitely many quotients which are evaluations of simple functors S_{H,V} (G). We may conclude that if S_{H,V} occurs with multiplicity n_{H,V} as a composition factor of M then

M(G) = ∑_{H,V} n_{H,V} S_{H,V} (G) ∈ G_{0}(R[Out(G)]),

the sum being taken over pairs (H, V) with H isomorphic to a subgroup of G. By determining the multiplicies n_{H,V} using the morphism ψ of the last section and using the explicit formula 2.6(ii) for S_{H,V} (G) we obtain an expression for M(G) in the Grothendieck group.

The success of this method depends on how easily these quantities may be computed, and since the explicit formula for S_{H,V} (G) is not so bad, the difficulty rests primarily with the n_{H,V}. To simplify matters in computing the n_{H,V} we will try to find a class X of groups to which Proposition 3.5 applies. Since it is only necessary to compute n_{H,V} in case S_{H,V} (G) ≠ 0 we look for conditions for this to happen, and use 2.8 and 2.9 in conjunction with the following lemma.

(4.1) LEMMA. Let G be any group, H a p-group and P a Sylow p-subgroup of G. If S_{H,V} (P) = 0 then S_{H,V} (G) = 0.

Proof. By the formula for simple global Mackey functors we have

S_{H,V} (P) = ⨁_{H≅^{α}L≤P} tr_{L}^{N_{P} (L)}(^{α}V) = 0

so tr_{L}^{N_{P} (L)}(^{α}V) = 0 for all such L. Hence tr_{L}^{N_{G}(L)}(^{α}V) = 0 for all H≅^{α}L ≤ G since tr_{L}^{N_{G}(L)} = tr_{N_{P} (L)}^{N_{G}(L)} · tr_{L}^{N_{P} (L)}. So

S_{H,V} (G) = ⨁_{H≅^{α}L≤G} tr_{L}^{N_{G}(L)}(^{α}V) = 0.

We turn to the specific case of group cohomology with finitely generated trivial coefficients H^{r}(G, T). Since this cohomology group is a finite abelian group we may perform the calculation one prime at a time, computing the Sylow p-subgroup H^{r}(G, T)_{p}. Let us write M^{r}(G) = H^{r}(G, T)_{p}. It is well-known that on every group the Mackey functor which this defines is projective relative to p-subgroups, and in fact the Mackey functor composition factors of M^{r} ↓_{G} are of the form S_{H,V} where H is a p-group [16]. Thus Proposition 3.5 applies on taking X to be all p-groups, and we deduce that the composition factors of M^{r} as a global Mackey functor have the form S_{H,V} with H a p-group. Here V must necessarily be a simple module defined over a field of characteristic p which is a homomorphic image of our ground ring R

We may represent the map ψ considered in Section 3 by means of an infinite matrix Ψ with rows indexed by the finite groups (taken up to isomorphism), columns indexed by the simple global Mackey functors and entries in the Grothendieck groups G_{0}(R[Out(G)]). The entry in row G and column S_{H,V} is the module S_{H,V} (G). Using 2.8 and 2.9 we can immediately say that many entries of Ψ are zero, and we now present part of Ψ in the case when R is any field of characteristic 2.

1 C_{2} C_{4} C_{2} × C_{2}  C_{8} Q_{8}  D_{2^{n}} Q_{2^{n}} SD_{2^{n}}
1  1  1  1 2  1  1  2  1  1  1
1  1
C_{2}  1
C_{4}  1
C_{2} × C_{2}  1 2
C_{8}  1
Q_{8}  1  2
D_{2^{n}}  1 ⊕ 1  1
Q_{2^{n}}  1 ⊕ 1  1
SD_{2^{n}}  1  1  1

Most of the entries in this table are computed to be zero by Lemma 2.9. The only remaining entries in question are S_{C_{2}×C_{2},2}(P) and S_{Q_{8},2}(P), which are computed using 2.6. In this table and throughout the ensuing calculation we denote the trivial module R by 1, and the natural 2-dimensional GL(2, 2)=S_{3}-module by 2.

We now explain how to use this table to compute the value of a global Mackey functor, illustrating the method by considering the cohomology M^{r}(G) = H^{r}(G, 𝔽_{2}) of a group G with D_{2^{n}} as a Sylow 2-subgroup, n ≥ 3. By 4.1 we see from the table that

M^{r}(G) = n_{C_{2}×C_{2},2}S_{C_{2}×C_{2},2}(G) + n_{D_{2^{n}} ,1}S_{D_{2^{n}} ,1}(G) ∈ G_{0}(𝔽_{2}[Out(G)]).

The first of these coefficients n_{C_{2}×C_{2},2} is computed from the similarly obtained equation

(4.2)  M^{r}(C_{2} × C_{2}) = n_{C_{2}×C_{2},1}S_{C_{2}×C_{2},1}(C_{2} × C_{2}) + n_{C_{2}×C_{2},2}S_{C_{2}×C_{2},2}(C_{2} × C_{2}).

Here we see that n_{C_{2}×C_{2},2} equals the multiplicity of the 2-dimensional simple 𝔽_{2}[Out(C_{2}×C_{2})]-module as a composition factor of M^{r}(C_{2}×C_{2}), so that a knowledge of the cohomology of C_{2} × C_{2} gives this coefficient. Returning to the first equation, knowledge of M^{r}(G) for any group G with Sylow 2-subgroup D_{2^{n}} will now suffice to determine n_{D_{2^{n}} ,1} — for example, G = D_{2^{n}} will do. In general we have to make the requirement that S_{D_{2^{n}} ,1}(G) ≠ 0 at this step, a requirement which in this particular case is always satisfied. Having determined the two coefficients we calculate M^{r}(G) using the formula 2.6 for the simple global Mackey functors.

We may summarise the approach in general as follows: for a p-group P, n_{P,V} can be computed knowing the values of n_{H,W} for subgroups H ≤ P with S_{H,W} (P) ≠ 0 and knowing the cohomology of any group G which has P as a Sylow p-subgroup, for which S_{P,V} (G) ≠ 0. One may always take G to be P itself, but sometimes it may be more convenient to compute the cohomology of a bigger group G. Altogether one would have used as input a knowledge of the cohomology of a group with H as a Sylow p-subgroup, for every p-subgroup H under consideration. Finally, having determined the numbers n_{H,V} in this way we obtain the cohomology of any group which has P as a Sylow p-subgroup by evaluating the relevant simple global Mackey functors at that group, using 2.6.

We now use this method to compute the additive structure of the cohomology of various groups at the prime 2. It is convenient to work with the Poincaré series

P_{G}(t) = ∑_{r=0}^{∞} t^{r} dim H^{r}(G, 𝔽_{2}),

and we will use the notation A_{G}(L) = N_{G}(L)/C_{G}(L) to denote the automizer of L in G. The cohomology of the groups specified in the next theorem has previously been determined in [7] and [9] by different methods.

(4.3) THEOREM.

(a) Let G be a group with D_{2^{n}} as a Sylow 2-subgroup, n ≥ 3. Then

P_{G}(t) = \frac{1 + t^{3}}{(1 − t^{2})(1 − t^{3})} + \frac{λt}{(1 − t)(1 − t^{3})}

where λ is the number of conjugacy classes of subgroups L ≅ C_{2} × C_{2} in G for which 3 | |̸A_{G}(L)|.

(b) Let G be a group with SD_{2^{n}} as a Sylow 2-subgroup, n ≥ 4. Then

P_{G}(t) = \frac{1 + t^{5}}{(1 − t^{3})(1 − t^{4})} + \frac{λt(1 + t)}{(1 − t^{4})} + \frac{μt}{(1 − t)(1 − t^{3})}

where

λ = \{ \begin{matrix}0 & if 3 | |A_{G}(Q_{8})| \\ 1 & otherwise \end{matrix}  and  μ = \{ \begin{matrix}0 & if 3 | |A_{G}(C_{2} × C_{2})| \\ 1 & otherwise \end{matrix} .

(c) Let G be a group with Q_{2^{n}} as a Sylow 2-subgroup, n ≥ 3. Then

P_{G}(t) = \frac{1 + t^{3} + λt(1 + t)}{(1 − t^{4})}

where λ is the number of conjugacy classes of subgroups L ≅ Q_{8} in G for which 3 | |̸A_{G}(L)|. (In case n = 3, replace λ in the expression for P_{G}(t) by 2λ.)

Proof. Let us write n_{H,V}^{r} for the multiplicity of S_{H,V} as a composition factor in M^{r}. On summing over r we obtain immediately

P_{G}(t) = ∑_{H,V}(dim S_{H,V} (G) · ∑_{r=0}^{∞} t^{r}n_{H,V}^{r} ).

It is convenient to write

N_{H,V} (t) = ∑_{r=0}^{∞} t^{r}n_{H,V}^{r} .

(a) Suppose G has D_{2^{n}} as a Sylow 2-subgroup. From our earlier discussion, the only pairs (H, V) which contribute to the sum are (C_{2} × C_{2}, 2) and (D_{2^{n}}, 1), and so

P_{G}(t) = dim S_{C_{2}×C_{2},2}(G)N_{C_{2}×C_{2},2} + dim S_{D_{2^{n}} ,1}(G)N_{D_{2^{n}} ,1}.

From equation 4.2 and the fact that H^{∗}(C_{2} × C_{2}, 𝔽_{2}) is the polynomial algebra on the 2-dimensional simple 𝔽_{2}S_{3}-module, we obtain that N_{C_{2}×C_{2},2}(t) = ∑_{r=0}^{∞} t^{r}n_{C_{2}×C_{2},2}^{r} is the series in which n_{C_{2}×C_{2},2}^{r} is the multiplicity of 2 as a composition factor of the rth symmetic power of 2. This series is calculated to be

N_{C_{2}×C_{2},2}(t) = \frac{t}{(1 − t)(1 − t^{3})}

using a slight extension of Molien’s theorem (see [3]) and the theory of Brauer characters. Taking the particular case of G = D_{2^{n}} we have

P_{D_{2^{n}}} (t) = \frac{1}{(1 − t)^{2}}
= dim S_{C_{2}×C_{2},2}(D_{2^{n}})N_{C_{2}×C_{2},2}(t) + dim S_{D_{2^{n}} ,1}(D_{2^{n}})N_{D_{2^{n}} ,1}(t)
= 2N_{C_{2}×C_{2},2}(t) + N_{D_{2^{n}} ,1}(t).

We deduce that

N_{D_{2^{n}} ,1}(t) = \frac{1 + t^{3}}{(1 − t^{2})(1 − t^{3})}.

For a general group G with D_{2^{n}} as Sylow 2-subgroup we have

S_{C_{2}×C_{2},2}(G) = ⨁_{C_{2}×C_{2}≅L≤G} tr_{L}^{N_{G}(L)} 2.

There are at most two conjugacy classes of C_{2} × C_{2} subgroups in G, since D_{2^{n}} has two such classes; their normalizers in D_{2^{n}} are both D_{8}. Thus N_{G}(L)/C_{G}(L) is either C_{2} or S_{3}. Since C_{G}(L)/L has odd order and acts trivially on the 2-dimensional module, tr_{L}^{C_{G}(L)} 2 = 2. From the well-known structure of 2 as an 𝔽_{2}S_{3}-module we have dim tr_{L}^{N_{G}(L)} 2 = 1 or 0 according as N_{G}(L)/C_{G}(L) ≅ C_{2} or S_{3}. We deduce that dim S_{C_{2}×C_{2},2}(G) equals the number of conjugacy classes of C_{2} × C_{2} whose automizer has order not divisible by 3.

We compute also that

S_{D_{2^{n}} ,1}(G) = tr_{D_{2^{n}}}^{N_{G}(D_{2^{n}})} 1

has dimension 1 always, since N_{G}(D_{2^{n}})/D_{2^{n}} has odd order. By putting these pieces together we obtain the statement of part (a) of the theorem.

(b) Suppose now that G has SD_{2^{n}} as a Sylow 2-subgroup. In this case

P_{G}(t) = dim S_{C_{2}×C_{2},2}(G)N_{C_{2}×C_{2},2} + dim S_{Q_{8},2}(G)N_{Q_{8},2} + dim S_{SD_{2^{n}} ,1}(G)N_{SD_{2^{n}} ,1}.

To compute N_{Q_{8},2} we use the equation

M^{r}(Q_{8}) = S_{Q_{8},1}(Q_{8})n_{Q_{8},1}^{r} + S_{Q_{8},2}(Q_{8})n_{Q_{8},2}^{r}

which holds in G_{0}(𝔽_{2}[Out(Q_{8})]), and the structure of H^{∗}(Q_{8}, 𝔽_{2}) tells us that multiplicities n_{Q_{8},2}^{r} form the sequence 0, 1, 1, 0, 0, 1, 1, 0, . . .. Thus

N_{Q_{8},2}(t) = \frac{t(1 + t)}{1 − t^{4}}.

Since we have already determined N_{C_{2}×C_{2},2} we may obtain N_{SD_{2^{n}} ,1} from any Poincaré series P_{G}(t) where G has SD_{2^{n}} as a Sylow 2-subgroup, and we suggest either SD_{2^{n}} itself using the calculation in [5], or GL(3, 2) using Quillen’s calculation [11]. Using SD_{2^{n}} we obtain

\begin{aligned}N_{SD_{2^{n}} ,1} & = P_{SD_{2^{n}}} (t) − N_{Q_{8},2} − N_{C_{2}×C_{2},2} \\ & = \frac{1 + t^{5}}{(1 − t^{3})(1 − t^{4})} \end{aligned}

using the information on the values of simple functors on G given in the matrix Ψ. Finally we obtain the Poincaré series for an arbitrary group with SD_{2^{n}} as a Sylow 2-subgroup by evaluation the simple global Mackey functors. We find that dim S_{C_{2}×C_{2},2}(G) = μ and dim S_{Q_{8},2}(G) = λ, the numbers given in the statement of the theorem.

(c) We suppose that G has Q_{2^{n}} as a Sylow 2-subgroup, in which case

P_{G}(t) = dim S_{Q_{8},2}(G)N_{Q_{8},2} + dim S_{Q_{2}^{n},1}(G)N_{Q_{2}^{n},1}.

Taking G = Q_{2^{n}} and noting that dim S_{Q_{8},2}(Q_{2}^{n}) = 2, our existing knowledge of N_{Q_{8},2} gives

N_{Q_{2}^{n},1} = P_{Q_{2}^{n}}(t) − 2N_{Q_{8},2}

= \frac{(1 + t + t^{2})(1 + t)}{1 − t^{4}} − 2\frac{t(1 + t)}{1 − t^{4}}

= \frac{1 + t^{3}}{1 − t^{4}} .

This gives the desired result, noting that for general G with Q_{2^{n}} as Sylow 2-subgroup

dim S_{Q_{2}^{n},1}(G) = 1 and dim S_{Q_{8},2}(G) = \{ \begin{matrix}λ & if n ≥ 4, \\ 2λ & if n = 3. \end{matrix}

It is amusing to consider a power series whose evaluation at each group G is the Poincaré series P_{G}(t) for that group. This series is

P (t) = ∑_{r=0}^{∞} t^{r}M^{r},

a power series in t with coefficients in Π, and it is a way of recording the composition factors of M^{r}, for every r. Since M^{r} = ∑_{H,V} n_{H,V}^{r} S_{H,V} we evidently have P (t) = ∑_{H,V} N_{H,V} (t)S_{H,V}. The coefficient of t^{r} is a list of the composition factors of M^{r}, and on evaluation at each group G the coefficient of t^{r} is H^{r}(G, 𝔽_{2}), as an element of Π′. Summing up our calculations so far, together with some further ones, we have:

(4.4) THEOREM.

P (t) =\frac{1}{1 − t}S_{C_{2},1} + \frac{1}{1 − t}S_{C_{4},1} + \frac{1 + t^{3}}{(1 − t^{2})(1 − t^{3})}S_{C_{2}×C_{2},1} + \frac{t}{(1 − t)(1 − t^{3})}S_{C_{2}×C_{2},2}

+ \frac{1}{1 − t}S_{C_{8},1} + \frac{1 + t^{3}}{(1 − t^{2})(1 − t^{3})}S_{D_{8},1} + \frac{1 + t^{3}}{1 − t^{4}} S_{Q_{8},1} + \frac{t(1 + t)}{1 − t^{4}}S_{Q_{8},2}

+ \frac{1 − t + t^{4} − t^{7} + t^{8}}{(1 − t)(1 − t^{3})(1 − t^{7})}S_{C_{2}×C_{2}×C_{2},1} + \frac{t(1 − t^{3} + t^{4} + t^{5})}{(1 − t)(1 − t^{3})(1 − t^{7})}S_{C_{2}×C_{2}×C_{2},3a}

+ \frac{t^{2}(1 + t − t^{2} + t^{5})}{(1 − t)(1 − t^{3})(1 − t^{7})}S_{C_{2}×C_{2}×C_{2},3b} + \frac{t^{4}}{(1 − t)(1 − t^{3})(1 − t^{7})}S_{C_{2}×C_{2}×C_{2},8}

+ \frac{1 + t^{5}}{(1 − t^{3})(1 − t^{4})}S_{SD_{16},1} + \frac{1 + t^{2} + t^{3} − t^{4} + t^{5} + t^{6} + t^{8}}{(1 − t)(1 − t^{3})(1 − t^{7})} S_{D_{8}×C_{2},8} + ⋯

Remark. As well as having a use in the explicit computation of cohomology, the method just described may also be used to derive certain theoretical results such as Swan’s theorem [12] on the cohomology of p-normal groups. We omit the details.

5. Projective covers of simple functors

Throughout this section we will assume that our coefficient ring R is a field or a complete discrete valuation ring, and in this situation our main results are that the simple functors have projective covers, and that the representable functors are finite direct sums of these. The theory works both for global Mackey functors and inflation functors. When we use it in the next section we will only need the case of inflation functors, and so we restrict attention to these functors now. However, the theory for global Mackey functors is just the same and may be obtained by a minor modification of the ideas, at times achieved by merely writing Ω_{R} instead of Ω_{R}^{+}, etc.

Since inflation functors are contravariant R-additive functors Ω_{R}^{+} → R-mod, it is an immediate consequence of Yoneda’s lemma that the representable functors A_{R}( , G) = Hom_{Ω_{R}^{+}}( , G) are projective. We state Yoneda’s lemma and its immediate well-known consequences in this context.

(5.1) PROPOSITION. For each finite group G the representable functor A_{R}( , G) has the following properties.

(i) Hom_{Mack}_{R}^{∗}(A_{R}( , G), M) ≅ M(G) for each inflation functor M. (ii) In particular Hom_{Mack}_{R}^{∗}(A_{R}( , G), A_{R}( , H)) ≅ A_{R}(G, H) is a finitely generated

R-module and End_{Mack}_{R}^{∗}(A_{R}( , G)) ≅ A_{R}(G, G) is an R-algebra finitely generated as an R-module.

(iii) A_{R}( , G) is a projective object in Mack_{R}^{∗}. (iv) A_{R}( , G) is generated as an inflation functor by its value at G.

Proof. We prove only (iv). The argument of Yoneda’s lemma is that if θ ∈ A_{R}(H, G) is any element then θ = A_{R}(θ, G)(1_{G}) where 1_{G} : G → G is the identity morphism in Ω_{R}^{+}. Since the subfunctor of A_{R}( , G) generated by 1_{G} must be closed under mappings A_{R}(θ, G) we deduce that θ lies in this subfunctor.

We will also need the following preliminary lemma.

(5.2) LEMMA. End_{Mack}_{R}^{∗}(𝕊_{H,V}) ≅ End_{R[Out(H)]}(V).

Proof. We have

End_{Mack}^{∗}(𝕊_{H,V}) ≅ Hom(𝕊_{H,V} , J_{H,V})
≅ Hom(\overline{𝕊_{H,V}} (H), V)
= End_{R[Out(H)]}(V),

the first isomorphism arising from the fact that 𝕊_{H,V} is the unique simple subfunctor of J_{H,V} and the second from the adjoint property.

We define Rad A_{R}( , G) to be the intersection of the maximal subfunctors of A_{R}( , G), or in other words the intersection of the kernels of the homomorphisms from A_{R}( , G) to simple functors.

(5.3) LEMMA. Let R be a field or a discrete valuation ring. Then

A_{R}( , G)/ Rad A_{R}( , G) ≅ ⨁ 𝕊_{H,V}^{d_{H,V}}

where d_{H,V} = dim 𝕊_{H,V} (G)/ dim End(V), and in the case of a discrete valuation ring the dimensions are taken over the residue field. This functor is a finite direct sum of simple functors, and its endomorphism ring is a finite dimensional semisimple algebra.

Proof. The multiplicity with which 𝕊_{H,V} appears as a homomorphic image of A_{R}( , G) is

dim Hom(A_{R}( , G), 𝕊_{H,V})/ dim End(𝕊_{H,V}) = dim 𝕊_{H,V} (G)/ dim End(V).

The number of simples which can be homomorphic images of A_{R}( , G) is thus finite, since only 𝕊_{H,V} with H isomorphic to a subgroup of G can appear, each with finite multiplicity, and for each group H there are only finitely many isomorphism classes of simple R[Out(H)]-modules. Thus A_{R}( , G)/ Rad A_{R}( , G) is isomorphic to the direct sum of all these simples, taken with their multiplicities.

(5.4) THEOREM. Assume that the ground ring R is a field or a complete discrete valuation ring. Every simple object 𝕊_{H,V} in Mack^{∗} has a projective cover, which is a projective inflation functor ℙ_{H,V} such that ℙ_{H,V} / Rad ℙ_{H,V} ≅ 𝕊_{H,V}.

Proof. Let 𝕊_{H,V} be simple. Since 𝕊_{H,V} (H) ≠ 0 it follows from 5.3 that 𝕊_{H,V} is a direct summand of

A_{R}( , H)/ Rad A_{R}( , H)

and has the form e(A_{R}( , H)/ Rad A_{R}( , H)) for some idempotent endomorphism e.

Since A_{R}( , H) is projective, every endomorphism of A_{R}( , H)/ Rad A_{R}( , H) lifts to an endomorphism of A_{R}( , H). Thus

End(A_{R}( , H)) → End(A_{R}( , H)/ Rad A_{R}( , H))

is an epimorphism, and by 5.3 the target ring is semisimple. Now End(A_{R}( , H)) is an algebra finitely generated as an R-module and so idempotents lift from End(A_{R}( , H)/ Rad A_{R}( , H)) to End(A_{R}( , H)), preserving primitivity (see [3, 6.8]). Thus there is an idempotent endomorphism ê of A_{R}( , H) which induces e on A_{R}( , H)/ Rad A_{R}( , H), and ê is primitive. The summand ê(A_{R}( , H)) = ℙ_{H,V} is now an indecomposable projective object in Mack^{∗} with the property that ℙ_{H,V} / Rad ℙ_{H,V} = 𝕊_{H,V}, and so this is the projective cover of 𝕊_{H,V}.

At this stage we may conclude that A_{R}( , G) has a set of summands ℙ_{H,V} corresponding to the decomposition of A_{R}( , G)/ Rad A_{R}( , G) as a direct sum of simples. These are precisely the direct summands of A_{R}( , G) which have a simple object as a homomorphic image. There remains the possibility, however, that A_{R}( , G) may have some summands which have no simple homomorphic image, and this is a possibility which we have to exclude. The property of A_{R}( , G) which we use is that it is generated by its value at G.

(5.5) LEMMA. Let M ∈ Mack^{∗} be such that there is a finite set of groups Y closed under taking subgroups so that

(i) M(Y) has finite composition length as an R[Out(Y)]-module for all Y ∈ Y, (ii) M = ⟨M(Y) | Y ∈ Y⟩. Then M is an epimorphic image of a finite direct sum of indecomposable projectives ℙ_{H,V}.

Proof. We proceed by induction on ∑_{Y ∈Y} ℓ(M(Y)) where ℓ denotes composition length as an R[Out(H)]-module. When this number is 0 condition (ii) implies that M = 0, so this starts the induction. Supposing now that some M(Y) ≠ 0, pick a minimal group H for M which lies in Y. There is a non-zero homomorphism M(H) = \overline{M}(H) → V for some simple R[Out(H)]-module V, hence a non-zero homomorphism M → J_{H,V}. Thus 𝕊_{H,V} is a composition factor of M, and there is a non-zero morphism ℙ_{H,V} → M. Write M_{1} for the cokernel of this map. It is a functor also satisfying conditions (i) and (ii) but now the sum of the composition lengths for M_{1} is smaller than it was for M and so by induction there is a finite direct sum Q of indecomposable projectives ℙ_{H,V} and an epimorphism Q → M_{1}. Lift this map to a morphism Q → M using projectivity, and form ℙ_{H,V} ⊕ Q → V. This is necessarily an epimorphism.

Remark. Any epimorphic image of a finite direct sum of ℙ_{H,V} ’s also satisfies conditions (i) and (ii), so we obtain a characterisation of such epimorphic images as the functors satisfying (i) and (ii).

(5.6) THEOREM. Let R be a field or a complete discrete valuation ring. Then A_{R}( , G) is the direct sum in Mack^{∗} of indecomposable projectives ℙ_{H,V} where ℙ_{H,V} occurs with multiplicity dim 𝕊_{H,V} (G)/ dim End(V), and where V is a simple R[Out(H)]-module.

Proof. From the argument in 5.4 we see that ℙ_{H,V} does occur as a summand with the stated multiplicity, and it remains to show that no other summands occur. Such a summand, also by the argument of 5.4, would be a projective which has no simple functor as a homomorphic image. In case R is a discrete valuation ring, on dividing everything by the maximal ideal of R we would reduce to the same situation over the residue field of R, since by Nakayama’s lemma the reduction of a non-zero module is non-zero. Thus to show the non-existence of such summands we may assume that R is a field. Now A_{R}( , G) satisfies conditions (i) and (ii) of 5.5 where we take Y to consist of all subgroups of G. This is because for each group Y, A_{R}(Y, G) is finite dimensional from its definition, whence condition (i), and condition (ii) is satisfied because of 5.1(iv). It is immediate that every direct summand of A_{R}( , G) therefore also satisfies (i) and (ii) of 5.5 and so must be an epimorphic image of a direct sum of projectives ℙ_{H,V}. Such an epimorphic image would indeed have a simple functor as an epimorphic image. We deduce that there can be no remaining summand.

6. The stable decomposition of BG

We now have a description of the indecomposable summands of A_{R}( , G) in the case of a field or a discrete valuation ring, and since this functor has endomorphism ring A_{R}(G, G) these summands are in bijection with the primitive idempotents of A_{R}(G, G). In case G = P is a p-group and R = ℤ_{p} (or 𝔽_{p}) it is a consequence of Carlsson’s theorem pointed out by Lewis May and McClure [6] (see also [10]) that these idempotents are in bijection with the indecomposable stable wedge summands of (BP_{+})_{p}^{∧}. By BP_{+} we will mean the suspension spectrum of the classifying space BP after a disjoint base point has been adjoined, and we will only ever consider p-completions (BP_{+})_{p}^{∧}. For a p-group P, BP_{+} is in fact already p-complete except for a sphere spectrum summand. We regard these classifying space spectra as objects in the category S of all p-completed suspension spectra with homotopy classes of stable maps. As stated in [10, Cor. 15] , when P is a p-group the group of homomorphisms Hom_{S}((BP_{+})_{p}^{∧}, (BG_{+})_{p}^{∧}) = [BP_{+}, BG_{+}]_{p}^{∧} is isomorphic to A_{ℤ_{p}}(P, G), and when G = P we have a ring isomorphism. From this one not only sees that primitive idempotents of A_{ℤ_{p}}(P, P) are in bijection with the indecomposable summands of (BP_{+})_{p}^{∧}, but also that two summands are isomorphic in S precisely if the idempotents are conjugate, which in turn happens precisely if the corresponding indecomposable projective summands of A_{ℤ_{p}}( , P) are isomorphic. We wish to say more than this, namely to say when two summands of (BG_{+})_{p}^{∧} and (BH_{+})_{p}^{∧} are isomorphic for different groups G and H, thereby making the connection with the theory of dominant summands discussed in [1]. One easily guesses the answer for p-groups: two such stable summands of (BG_{+})_{p}^{∧} and (BH_{+})_{p}^{∧} are isomorphic precisely if the corresponding indecomposable projective summands of A_{ℤ_{p}}( , G) and A_{ℤ_{p}}( , H) are isomorphic. The proof of this relies on the following straightforward observation.

(6.1) PROPOSITION. Let p be a prime.

(i) The full subcategory A of S whose objects are the classifying spaces (BP_{+})_{p}^{∧} of finite p-groups P is equivalent to the full subcategory B of Mack_{ℤ_{p}}^{∗} whose objects are the representable functors A_{ℤ_{p}}( , P), where again P ranges over finite p-groups.

(ii) The full subcategory C of S whose objects are stable summands of the classifying spaces (BP_{+})_{p}^{∧} with P a p-group is equivalent to the full subcategory D of Mack_{ℤ_{p}}^{∗} whose objects are the indecompsable projectives ℙ_{H,V} with H a p-group.

When G is an arbitrary finite group it is well-known (see e.g. [1]) that (BG_{+})_{p}^{∧} is isomorphic to a summand of (BP_{+})_{p}^{∧} where P is a Sylow p-subgroup of G. Thus the category C just defined in 6.1(ii) equals the full subcategory of S whose objects are the stable summands of all (BG_{+})_{p}^{∧}, ranging over all finite groups G.

Proof. (i) We define a functor A → B by (BP_{+})_{p}^{∧} ↦ A_{ℤ_{p}}( , P) on objects, and on morphisms by means of the isomorphism

Hom_{S}((BP_{+})_{p}^{∧}, (BQ_{+})_{p}^{∧}) ≅ A_{ℤ_{p}}(P, Q) ≅ Hom_{Mack}^{∗}(A_{ℤ_{p}}( , P), A_{ℤ_{p}}( , Q)).

There is an inverse functor given on objects simply by A_{ℤ_{p}}( , P) ↦ (BP_{+})_{p}^{∧}. One has to check that the isomorphism between the Hom sets is functorial, and indeed composition of morphisms in A and in B does correspond to the operation

A_{ℤ_{p}}(P, Q) × A_{ℤ_{p}}(Q, K) → A_{ℤ_{p}}(P, K)

described in [10] or [1].

(ii) One may construct the category C from A as follows. The objects of C may be identified with the idempotents e ∈ End_{S}((BP_{+})_{p}^{∧}), as P varies. Now if f ∈ End_{S}((BQ_{+})_{p}^{∧}) is another idempotent then Hom_{C}(e, f) = f · A_{ℤ_{p}}(P, Q) · e.

By an exactly similar construction we may obtain D from B. Since A and B are isomorphic, we deduce that C and D are isomorphic.

(6.2) THEOREM. For any finite group G the indecomposable stable summands of (BG_{+})_{p}^{∧} are in bijection with those summands ℙ_{H,V} of A_{ℤ_{p}}( , G) for which H is a p-group. Here V is a simple 𝔽_{p}[Out(H)]-module, and the multiplicity of a summand parametrized by (H, V) as a summand of (BG_{+})_{p}^{∧} equals the multiplicity of ℙ_{H,V} as a summand of A_{ℤ_{p}}( , G), which equals dim_{𝔽_{p}} 𝕊_{H,V} (G) / dim_{𝔽_{p}} End(V). This number may be computed using 5.5 and 5.6. In particular, for each indecomposable summand X of a p-completed classifying space (BG_{+})_{p}^{∧} there is a unique isomorphism class of groups H minimal such that X is a summand of (BH_{+})_{p}^{∧}, and H is a p-group. Such a summand is parametrized by a pair (H, V) and its multiplicity as a summand of (BH_{+})_{p}^{∧} is dim V/ dim End_{𝔽_{p} Out(H)}(V).

Proof. To deal with the first part of the statement, let P be a Sylow p-subgroup of G. As indicated in [1], the maps Bi : (BP_{+})_{p}^{∧} → (BG_{+})_{p}^{∧} and the transfer map (BG_{+})_{p}^{∧} → (BP_{+})_{p}^{∧} are equivalences between (BG_{+})_{p}^{∧} and a summand of (BP_{+})_{p}^{∧}. Thus the composite of these maps is an endomorphism of (BP_{+})_{p}^{∧} which has as its image a summand isomorphic to (BG_{+})_{p}^{∧}. In the category of functors this corresponds to the composite

A_{ℤ_{p}}( , P)⟶^{_{P} G_{G}}A_{ℤ_{p}}( , G)⟶^{_{G}G_{P}}A_{ℤ_{p}}( , P)

as is also explained in [1], although here the middle term A_{ℤ_{p}}( , G) no longer corresponds to (BG_{+})_{p}^{∧}. The map denoted _{P} G_{G} is surjective on evaluation at all p-subgroups Q of G since in the interpretation of A_{ℤ_{p}}(Q, G) as being the free ℤ_{p}-module with basis the equivalence classes of pairs (K, ϕ) where K ≤ Q and ϕ : K → G (see [6], [10] or [1]), it is clear that ϕ can always be factored ϕ : K → P → G, so that (K, ϕ) is the image of an element of A_{ℤ_{p}}(P, G). Similarly this factorisation implies that _{G}G_{P} is injective on A_{ℤ_{p}}(Q, G) since equivalence of a pair (K, ϕ) as a map to G implies equivalence as a map to P.

The above argument shows that all summands of A_{ℤ_{p}}( , G) of the form ℙ_{H,V} with H a p-group are in the image of _{P} G_{G}, since these summands are generated by their values at p-groups. On the other hand the image of this map is isomorphic to a summand of A_{ℤ_{p}}( , P), and so can be a direct sum only of such ℙ_{H,V}.

The remaining parts of the theorem now follow from 5.6 and 2.1.

This theorem together with the information about the simple inflation functors given in Section 2 encapsulates many of the principal results of [1] and [8]. The theory of ‘dominant summands’ of [1] translates into the statement that the classifying space summand parametrized by the pair (H, V) is not a summand of (BG_{+})_{p}^{∧} unless H is isomorphic to a subgroup of G, and this summand does appear with non-zero multiplicity as a summand of BH_{+}, as stated in 6.2. From 2.7(ii) we obtain the criterion of [1] that the summand parametrized by (H, V) appears with non-zero multiplicity in (BG_{+})_{p}^{∧} if and only if one of the expressions ∑ θ_{g}^{−1} considered there acts in a non-zero fashion on V. From 2.8(ii) we obtain a necessary condition for the summand parametrized by (H, V) to appear as a summand of (BG_{+})_{p}^{∧}

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School of Mathematics, University of Minnesota, Minneapolis, MN 55455

email: webb@math.umn.edu