A SPLITTING PRINCIPLE FOR MODULAR GROUP REPRESENTATIONS

PETER SYMONDS

Abstract. We have shown previously how a complex representation of a finite group can be split into a virtual sum of representations induced from 1-dimensional representations in a natural way (sometimes known as explicit Brauer induction). Here we consider the modular case.

1. Introduction

In [11] we showed how a complex representation of a finite group could be split into a virtual sum of representations induced up from 1-dimensional representations of subgroups in a natural way, by considering the projective space of the representation. Here we consider the modular case. Of course, by using a Brauer lift we could reduce the problem to the complex case, but this approach misses the special features of the modular case. In particular we want to be able to control the vertices of the subrepresentations involved.

The aim is to write a representation V of a finite group G as a virtual sum

V ≅ ∑ l_{J,ϕ} Ind_{J}^{G} ϕ,

where J runs through the subgroups and ϕ is a 1-dimensional representation of J. This should be natural with respect to restriction to subgroups, which is defined on the right hand side by applying the double coset formula for Res Ind formally to each term and collecting like terms. In order to make this explicit we introduce certain constructions on Mackey functors in section 2. In contrast to the complex case, we need to allow the coefficients to be rational, with denominator a power of the characteristic p.

The complex case was first treated by Snaith [9] and Boltje [2]. A more geometric approach was given by the author [11]. For more details and applications see [10].

Following [11] we see that such a formula can be obtained by considering the projective space associated to V and its canonical line bundle. In fact there are two versions of projective space that we can use in characteristic p. Either the usual version ℙ(V), or one in which the points correspond to both a line through the origin and a complementary hyperplane, which we denote by ℙ_{⊕}(V).

Both versions yield a natural and additive formula which is correct at the level of Brauer characters. However, if V is indecomposable with vertex P then the ℙ_{⊕} version only sums over J with Sylow p-subgroup conjugate to a subgroup of P.

We would also like to have equality in the Green ring, but since the right hand side is a priori a virtual trivial source module this is clearly only possible if V is a trivial source module. We show that in fact there is always equality in the ℙ_{⊕}(V) version if V is a trivial source module, and that furthermore in this case the coefficients are integers.

These are old notes, but with renewed interest in the subject they seem worth recording. Some of these results have also been obtained by Boltje [3] in a different context, but the geometric situation presented here is of interest in itself and applies quite generally to any equivariant vector bundle. In particular we see the importance of the Brauer map in the modular case.

2. Constructions on Mackey functors

We will always work over some fixed ring R, usually ℤ or ℚ, and all groups will be finite. A Mackey functor M over a group G asigns an R-module M(H) to each subgroup H of G. These are linked by restriction maps, conjugation maps and transfers (or induction). The maps obey certain identities, in particular the Mackey double coset formula for decomposing transfer followed by restriction. The primordial example is the character ring of a group. Another example that will be important for us is a(G), the Green ring of finite dimensional kG-modules under direct sum, with the usual restriction and induction maps. For more details see [6],

[7] or [14].

Let C be a weak coefficient system on a finite group G (see [12]), that is, C is a contravariant functor from the category with objects the subgroups of G and morphisms the inclusion and conjugation maps to the category of abelian groups. C is called a coefficient system if H acts trivially on C(H) by conjugation for all H ≤ G. Define

Ĉ(H) = ⨁_{J≤H} C(J).

An element ϕ of C(J) ⊂ Ĉ(H) will be denoted by (J, ϕ)^{H}. For g ∈ G define g(J, ϕ)^{H} = (^{g}J,^{g} ϕ)^{^{g}H}, where ^{g}ϕ(^{g}j) = ϕ(j). The conjugation maps in C combine to yield a map ĉ_{g} : Ĉ(^{g}H) → Ĉ(H), where ĉ_{g}(J, ϕ)^{H} = ^{g^{−1}}(j, ϕ)^{H}.

Whenever K ≤ L ≤ G there are restriction morphisms between the values of Ĉ given by r_{K}^{L}(J, ϕ)^{L} = (J ∩ K, res_{J∩K}^{J} ϕ)^{K}. These make Ĉ into a weak coefficient system. There are also inclusion morphisms given by i_{K}^{L}(J, ϕ)^{K} = (J, ϕ)^{L}, which give Ĉ the dual structure i.e. make it into a covariant functor on the category of conjugation and inclusion morphisms.

Define:

C^{+}(H) = H^{0}(H; Ĉ(H))

and

C_{+}(H) = H_{0}(H; Ĉ(H)).

We now define restriction and transfer maps, denoted by R and I, on these groups. We denote the restriction and transfer in cohomology by res and tr.

On C^{+}, R_{K}^{L} = (r_{K}^{L})_{∗} res_{K}^{L} and I_{K}^{L} = tr_{K}^{L}(i_{K}^{L})_{∗}.

On C_{+},

R_{K}^{L}(J, ϕ)^{L} = (r_{K}^{L})_{∗} ∑_{g∈K\L/J} ^{g}(J, ϕ)^{L} = ∑_{g∈K\L/J}(^{g}J ∩ K, res_{J∩K^{g}}^{J} ^{g}ϕ)^{K},

and I_{K}^{L}(J, ϕ)^{K} = (J, ϕ)^{L}.

The following is now straightforward to check:

Proposition 2.1. With the morphisms mentioned above, both C^{+} and C_{+} are Mackey functors.

If the coefficient system C was obtained by forgetting the transfers in a Mackey functor (also denoted C), then there is a morphism of Mackey functors α : C_{+} → C, given by α_{H}(J, ϕ)^{H} = tr_{J}^{H} ϕ.

Proposition 2.2. If C is a coefficient system then there is a morphism of Mackey functors N : C_{+} → C^{+}, given by

N_{H}(J, ϕ)^{H} = ∑_{g∈H/J} ^{g}(J, ϕ)^{H}.

It is an isomorphism if |G| is invertible in R.

Proof. From the definitions we obtain for K ≤ L:

R_{K}^{L}N_{L}(J, ϕ)^{L} = ∑_{l∈L/J}(K ∩^{l}J,^{ϕ})^{L},

and

N_{K}R_{K}^{L}(J, ϕ)^{L} = ∑_{l∈K\L/J} ∑_{k∈K∩^{l}J}(K ∩^{kl}J,^{kl}ϕ)^{K},

which are both equal.

Also:

N_{L}I_{K}^{L}(J, ϕ)^{K} = ∑_{l∈L/J} ^{l}(J, ϕ)^{H},

and

I_{K}^{L}N_{K}(J, ϕ)^{K} = ∑_{k∈K/J} ∑_{l∈L/K} ^{hk}(J, ϕ)^{L},

are again equal.

That N commutes with conjugations is clear.

Note that C^{+}(H) has a basis consisting of elements of the form

S(J, ϕ)^{H} = ∑_{h∈H/N_{H}(J,ϕ)}^{h}(J, ϕ)^{H}.

If |G| is invertible then N has an inverse M given by

M_{H}S(J, ϕ)^{H} = |N_{H}(J, ϕ)^{H}/J|^{−1}(J, ϕ)^{H}.

Remark. The definition of C_{+} goes back to Deligne [5]. The definition of C^{+} is based on Thévenaz’s twin functor [13]. It appears not to be quite the same as this or the one in [3], because of the definition of the restriction maps, but in fact the two are equivalent (see [12]). The present version corresponds better with the geometry. The functors C ↦ C_{+} and C ↦ C^{+} have various important properties (loc. cit.).

3. Vector Bundles

Let ξ be a G-vector bundle over a field k with base a G-space X. This means that ξ is a vector bundle and G is a finite group of automorphisms. For any J ≤ H ≤ G and any finite-dimensional representation ϕ of J over k, let X_{(J,ϕ)^{H}} denote the subspace of points x ∈ X such that Stab_{H} x = J and the representation of J on the fibre of ξ is isomorphic to ϕ. We call this a stratum.

We write (J, ϕ) ≤ (K, θ) to mean that J ≤ K and Res_{J}^{K} θ ≅ ϕ. Let χ be a conjugation invariant function on the (J, ϕ)^{H} satisfying the addition property

χ_{(J,ϕ)^{H}} = ∑_{\begin{matrix}(K,θ), K≤H \\ (J,ϕ)≤(K,θ) \end{matrix}} χ_{(K,θ)^{G}}.

In practice, χ will always be the Euler characteristic in compactly supported cohomology of some X_{(J,ϕ)^{H}}.

We define a characteristic class c_{H}(χ) ∈ a^{+}(H) by

c_{H}(χ) = ∑_{(J,ϕ)/H} χ_{(J,ϕ)^{H}} (J, ϕ)^{H}.

(The summation is over the (J, ϕ)^{H} with J ≤ H up to conjugation.)

The additivity property of χ ensures that this construction is natural in H in the sense that for K ≤ H ≤ G we have c_{K}(χ) = R_{K}^{H}c_{H}(χ).

If we define χ_{(J,ϕ)^{H}} = χ_{c}(X_{(J,ϕ)^{H}}) as above,then we write c_{H}(ξ) instead of c_{H}(χ). If we consider ξ as belonging to a suitable category of equivariant bundles, we can form the associated Grothendieck group M under disjoint union. This is even a Mackey functor if we define the transfer of the K vector bundle ξ over X by tr_{K}^{L} ξ = L ×_{K} ξ, a vector bundle over L ×_{K} X. Then c gives a morphism of Mackey functors over G from M to a^{+}.

Usually it is more convenient to consider instead L_{H}(ξ) = N^{−1}c_{H}(ξ) ∈ ℚ ⊗ a_{+}(H). In this case we have:

L_{H}(ξ) = ∑_{(J,ϕ)/H} \frac{χ_{c}(X_{(J,ϕ)})}{|N_{H}(J, ϕ)^{H}/J|}(J, ϕ)^{H}.

Notice that (ℚ ⊗ a)_{+} = ℚ ⊗ (a_{+}), so there is no ambiguity.

Proposition 3.1. a) If X is a G-CW-complex and χ is the usual Euler characteristic in compactly supported cohomology, then L_{H}(ξ) is integral, i.e. it has image in a_{+}(H).

b) If X is a quasi-projective variety over an algebraically closed field of characteristic p and χ is the Euler characteristic in compactly supported étale cohomology, then L_{H}(ξ) ∈ ℤ[\frac{1}{p}] ⊗ a_{+}(H).

Proof. a) Since N_{H}(J, ϕ)^{H}/J acts freely on X_{(J,ϕ)^{H}}, its order divides the Euler characteristic.

b) In [8] it is shown that if a group K acts freely then the p′ part of its order divides the Euler characteristic. Now we proceed as in part (a).

Given a p-subgroup P of G, let N = N_{G}(P) and \overline{N} = N/P. The Brauer map is defined on a kG-module M as Br_{P} M = H^{0}(P; M)/ ∑_{Q≨P} Im tr_{Q}^{P}, and this leads to a map Br_{P} : a(G) → a(\overline{N}) (see [1]). We can also define it on bundles by first restricting to the part of the base fixed by P, and then using the previous construction on the fibres. A Brauer map can also be defined on both a^{+} and a_{+} by restricting to N, projecting onto the (J, ϕ)^{N} with P ≤ J and then replacing (J, ϕ)^{N} by (J/P, Br_{P} ϕ)\overline{N}.

Recall the basic property that Br_{P} k(G/J) ≅ k(^{P} (G/J)) as k\overline{N}-modules, where we write ^{P} (G/J) for the fixed points of P acting on G/J on the left.

Remark. What we call Br_{P} M is by many authors denoted M[P].

Proposition 3.2. The Brauer map Br_{P} commutes with both c : M → a^{+}, with N : a_{+} → a^{+} and with α : a_{+} → a.

Proof. We will omit to indicate certain obvious restrictions of ϕ.

If we consider an equivariant stratum

X_{S(J,ϕ)^{G}} = ∐_{g∈G/N_{G}(J,ϕ)} X_{^{g}(J,ϕ)^{G}},

then

Br_{P} X_{S(J,ϕ)^{G}} = ∐_{\begin{matrix}g∈G/N_{G}(J,ϕ) \\ P ≤^{g}J \end{matrix}} X_{((N∩^{g}J)/P,Br_{P} ^{g}ϕ)\overline{N}} .

On a^{+}(G) we have

Br_{P} S(J, ϕ)^{G} = ∑_{\begin{matrix}g∈G/N_{G}(J,ϕ) \\ P ≤^{g}J \end{matrix}}((N ∩^{g}J)/P, Br_{P} ^{g}ϕ)\overline{N},

and on a_{+}(G)

Br_{P} (J, ϕ)^{G} = ∑_{g∈N\^{P} (G/J)}((N ∩^{g}J)/P, Br_{P} ^{g}ϕ)\overline{N}.

Finally, in a(G) (cf. [1])

Br_{P} Ind_{J}^{G} ϕ = ∑_{g∈n\^{P} (G/J)} Ind\overline{N} ; (N∩^{g}J)/P Br_{P} ^{g}ϕ.

We leave to the reader the easy task of calculating the six compositions and verifying the three identities claimed.

As a consequence we have the following result.

Proposition 3.3. Br_{P} L_{G}(ξ) = L_{N_{G}(P)/P} (Br_{P} ξ).

4. Projective spaces

From now on we consider only line bundles over an algebraically closed field k of characteristic p. We will mostly only be using the part of a_{+}(G) generated by 1-dimensional representations, which we denote by a_{+}^{1}(G).

Remark. The map L identifies the Grothendieck group of equivariant line bundles over finite G-sets (an extended Burnside ring) with a_{+}^{1}(G).

Given a representation V of G, there are two spaces that we shall consider: the usual projective space, ℙ(V), and also ℙ_{⊕}(V) = {(L, M) subspaces of V; dim L = 1, L ⊕ M = V }. Thus a point consists of a line through the origin in V and a complement to it. Notice that ℙ_{⊕}(V) is a quasi-projective variety since ℙ_{⊕}(V) ≅ {(u, v) ∈ ℙ(V) × ℙ(V^{∗}); ⟨u, v⟩ ≠ 0}.

Both ℙ(V) and ℙ_{⊕}(V) carry canonical line bundles ξ and ξ_{⊕} respectively, in which the fibre above the point associated to the line L in V is the line L itself.

Remark. In characteristic zero, ℙ_{⊕}(V) is equivariantly homotopy equivalent to ℙ(V) and there is little point in considering it.

We write L(V) (resp. L_{⊕}(V)) for L(ξ) (resp. L(ξ_{⊕})). Note that both lie in ℚ ⊗ a_{+}^{1}.

As a consequence of the naturality of c_{H} we have the important property:

Proposition 4.1. Both L and L_{⊕} are natural in H, i.e. L_{H}(Res_{H}^{G} V) = R_{H}^{G}L_{G}(V) and L_{⊕,H}(Res_{H}^{G} V) = R_{H}^{G}L_{⊕,G}(V).

Proposition 4.2. Both L and L_{⊕} are additive in V.

Proof. Just as in the complex case ther is an action of k^{∗} on ℙ(V ⊕ W) by multiplication on the first factor only: λ(u, v) = (λu, v). This preserves the strata, has fixed point set ℙ(V) ∪ ℙ(W) and acts freely elsewhere. By [8], the order of each element of finite order in k^{∗} must divide the Euler characteristic of a non-fixed stratum, so the latter must be 0.

The same argument applies with ℙ replaced by ℙ_{⊕}.

Remark. For any equivariant bundle over a space X,

\overline{X_{(J,ϕ)^{G}}} = X_{(J,ϕ)^{J}}

and also

\overline{X_{(J,ϕ)^{G}}} = ∐_{(J,ϕ)≤(K,θ)} X_{(K,θ)^{G}}.

It follows that the coefficients of L and L_{⊕} can be calculated algebraically as follows: It suffices to know the Euler characteristics of the closures of the strata, χ(\overline{ℙ(V)_{(J,ϕ)}}), since the numbers χ(ℙ(V)_{(J,ϕ)}) can be obtained from them formally by using a Moebius function. (And similarly with the subscript ⊕.)

Proposition 4.3. a) χ(\overline{ℙ(V)_{(J,ϕ)^{G}}}) is the multiplicity of ϕ as a submodule of Res_{J}^{G} V.

b) χ(\overline{ℙ_{⊕}(V)_{(J,ϕ)^{G}}}) is the multiplicity of ϕ as a summand of Res_{J}^{G} V.

Proof. Recall that for any vector space U, χ(ℙ(U)) = dim U.

a) \overline{ℙ(V)_{(J,ϕ)^{G}}} is the projective space on the subspace of Res_{J}^{G} V generated by submodules isomorphic to ϕ, and this has the dimension stated.

b) \overline{ℙ_{⊕}(V)_{(J,ϕ)}} is the coefficient of (J, ϕ)^{J} in L_{⊕,J}(Res_{J}^{G} V). But if Res_{J}^{G} V ≅ ϕ^{r}⊕W (where ϕ is not a summand of W), then L_{⊕,J}(Res_{J}^{G} V) = r(J, ϕ)^{J}+L_{⊕,J}(W), and L_{⊕,J}(W) has no (J, ϕ)^{J} term.

Corollary 4.4. If V is indecomposable with vertex P, then L_{⊕,G}(V) only involves (J, ϕ)^{G} for which the Sylow p-subgroup of J is conjugate to a subgroup of P.

Proof. The vertex of ϕ is the Sylow p-subgroup of J. Now Res_{J}^{G} V can not have a summand with vertex not subconjugate to P.

Theorem 4.5. As kG-modules, V, αL(V) and αL_{⊕}(V) all have the same simple factors.

Proof. It is sufficient to check the Brauer characters so, by naturality, we can restrict to the cyclic p′ groups. But now V is a sum of 1-dimensional representations, and by additivity it is enough to consider the case dim V = 1. But now the result is clear.

Thus αL and αL_{⊕} are our two explicit Brauer induction formulas.

Example. Let C be the cyclic group of order p, and V_{r} the non-trivial indecomposable kC-module of dimension r, 1 ≤ r ≤ p.

Then

L_{C}(V_{r}) = (C, k)^{C} + \frac{r − 1}{p}(1, k)^{C},

and

L_{⊕,C}(V) = \begin{cases}(C, k)^{C} & if r = 1, \\ \frac{r}{p}(1, k)^{C} & if r ≥ 2. \end{cases}

Proposition 4.6. If V is a trivial source module (i.e. a summand of a permutation module) then, as virtual kN_{G}(P)/P -modules, Br_{P} α_{G}L_{⊕,G}(V) and α_{N_{G}(P)/P} L_{⊕,N_{G}(P)/P} (Br_{P} V) have the same simple factors.

Proof. For a line bundle, Br_{P} is just restriction to the part of the base space fixed by P.Thus in view of 3.3, we need χ(\overline{ℙ_{⊕}(V)^{P}}_{(J,ϕ)^{N_{G}(P)/P}}) = χ(\overline{ℙ_{⊕}(V [P])}_{(J,ϕ)^{N_{G}(P)/P}}) for each J ≤ N_{G}(P)/P. We check this by restricting V to J′, the inverse image of J in N_{G}(P). The result then follows from the following lemma.

Lemma 4.7. ϕ^{r} is a summand of Res_{N_{G}(P)}^{G} V if and only if it is a summand of Br_{P} V.

Proof. One implication is clear. For the other, use [1] theorem 3.2, or the method of lifting idempotents in the endomorphism rings.

Corollary 4.8. If V is a trivial source module then V = αL_{⊕}(V) in ℚ ⊗ a(G).

Proof. Both sides have the same simple components after the Brauer map at any p-subgroup P by 4.6, so they are equal, by a version of Conlon’s Induction Theorem (see [4]).

Although in characteristic zero L does not commute with induction, in the modular case it does behave well in certain cases.

Lemma 4.9. If H ⊴ G is of p-power index, then L_{⊕,G}(Ind_{H}^{G} V) = I_{H}^{G}L_{⊕,H}(V).

Proof. It is sufficient to prove the case of index p, by induction on a chain of subgroups. We can suppose that V is indecomposable. If U ≰ H, Res_{U}^{G} Ind_{H}^{G} V = Ind_{U∩H}^{U} V, which is indecomposable. Thus both sides of the equation lie in the part of ℚ ⊗ a(G) in which the coefficient of (J, ϕ)^{G} is 0 when J ≰ H. But R_{H}^{G} is injective on this, and

R_{H}^{G}L_{⊕,G}(Ind_{H}^{G} V) = L_{⊕,H}(Res_{H}^{G} Ind_{H}^{G} V) = ∑_{g∈G/H} L_{⊕,H}(^{g}V) = R_{H}^{G}I_{H}^{G}L_{⊕,H}(V).

5. Integrality

Theorem 5.1. If V is a trivial source module then L_{⊕,G}(V) ∈ a_{+}^{1}(G), i.e. the coefficients are integers.

The proof will take up the whole of this section.

Let us make the coefficients in L_{⊕}(V) explicit by writing

L_{⊕,G}(V) = ∑_{(J,ϕ)/G} l_{V} (J, ϕ)^{G}(J, ϕ)^{G}.

Suppose that G and V form a counterexample to the theorem with minimal possible |G|, and that V is indecomposable. Let (J, ϕ)^{G} have maximum |J| among the (J, ϕ)^{G} such that l_{V} (J, ϕ)^{G} is not integral.

Lemma 5.2. In the circumstances above, (J, ϕ) is normal in G and J is of p-power index in G.

Proof. For H ≤ G,

L_{⊕,H}(Res_{H}^{G} V) = R_{H}^{G}L_{⊕,G}(V) = R_{H}^{G} ∑_{(K,θ)/G} l_{V} (K, θ)^{G}(K, θ)^{G}

= ∑_{(K,θ)/G} ∑_{g∈H\G/K} l_{V} (K, θ)^{G}(H ∩^{g} K,^{g} θ)^{H}.

If H = N_{G}(J, ϕ), then the coefficient of (J, ϕ)^{H} is l_{V} (J, ϕ)^{G} + ∑ l_{V} (I, η), where the summation is over terms with |I| ≩ |J| so, by hypothesis, is a sum of integers. Thus l_{V} (J, ϕ)^{H} is not integral and so we must have H = G.

Now take a subgroup S with J ≤ S ≤ G and such that S/J is a Sylow p-subgroup of G/J. Setting H = S in the previous formula we see that the coefficient of (J, ϕ)^{S} in L_{⊕,S}(Res_{S}^{G} V) is

∑_{(K,θ)/G} ∑_{\begin{matrix}g∈S\G/K \\ (S∩^{g}K,^{g}θ)=_{S}(J,ϕ) \end{matrix}} l_{V} (K, θ)^{G} ≡ ∑_{(K,θ)/G} ∑_{\begin{matrix}g∈S\G/K \\ (^{g}K,^{g}θ)=_{S}(J,ϕ) \end{matrix}} l_{V} (K, θ)^{G} (mod ℤ)

≡ ∑_{(K,θ)/G} ∑_{\begin{matrix}g∈S\G/K \\ (K,θ)=(J,ϕ) \end{matrix}} l_{V} (K, θ)^{G} (mod ℤ)

since (J, ϕ) is normal

≡ ∑_{g∈S\G/J} l_{V} (J, ϕ)^{G} (mod ℤ)

≡ |G/S|l_{V} (J, ϕ)^{S} (mod ℤ),

since S\G/J = S\G. This must equal l_{V} (J, ϕ)^{S}, so if S ≠ G it is an integer, by minimality of G.

Since |G/S| is coprime to p and the only possible denominator in l_{V} (J, ϕ)^{S} is a power of p, we must have S = G.

Lemma 5.3. Let P be the vertex of V. Then P is a Sylow p-subgroup of G.

Proof. By 4.4 we may assume that J ∩ P is a Sylow p-subgroup of J. Since V is a summand of Ind_{P}^{G} k ≅ Ind_{P J}^{G} Ind_{P}^{P J} k, it must also be a summand of Ind_{P J}^{G} U for some indecomposable trivial source kP J-module U.

But P J is of p-power index in G, so by Green’s Indecomposability Theorem, Ind_{P J}^{G} U is indecomposable and thus isomorphic to V. By 4.9, the coefficient of (J, ϕ)^{G} in L_{⊕,G}(V) is the same as that of (J, ϕ)^{P J} in L_{⊕,P J}(U), so by minimality, P J = G. But |P J/P | = |J/J ∩ P |, which is coprime to p.

Lemma 5.4. If k is a summand of Res_{J}^{G} V, then V ≅ k.

Proof. V is a summand of Ind_{P}^{G} k, and notice that Ind_{P}^{G} k has exactly one summand isomorphic to k.

Res_{J}^{G} V is a summand of Res_{J}^{G} Ind_{P}^{G} k ≅ Ind_{J∩P}^{J} k, by the previous lemma. Again this has only one trivial summand, so it must be the restriction of the one over G.

The image of ϕ must lie in a finite subfield of k so there is some power of p, say p^{r}, such that ϕ(j)^{p^{r}} = ϕ(j) for all j ∈ J. We may assume that r is big enough that |J| divides p^{r}. Now define an extension Φ of ϕ to G by Φ(g) = ϕ(g^{p^{r}}) for g ∈ G.

Let Φ^{∗} denote the dual of Φ. If V is a counterexample, then so is Φ^{∗} ⊗ V, since l_{Φ⊗V} (K, θ)^{H} = l_{V} (K, Φ ⊗ θ)^{H}. But l_{V} (J, Φ)^{G} = l_{Φ^{∗}⊗V} (J, k)^{G}, so we can assume that Φ = k.

But l_{V} (J, k)^{G} can be expressed as a linear combination of the multiplicities of θ as a summand of Res_{K}^{G} V for pairs (K, θ) ≥ (J, k). If k is not a summand of Res_{J}^{G} V then clearly these are all zero, and l_{V} (J, k)^{G} = 0, contradicting the assumption that it is not an integer. Thus k is a summand of Res_{J}^{G}, and by the previous lemma V = k. But clearly k is not a counterexample. This contradiction completes the proof of 5.1.

Example. From 4.4 and 5.1 we can obtain the known fact that every projective module is virtually a sum with integer coefficients of modules induced up from p′-subgroups.

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Department of Mathematics, U.M.I.S.T., P.O. Box 88, Manchester M60 1QD, England

E-mail address: Peter.Symonds@umist.ac.uk