INTRODUCTION

If G is a finite group, and A a G-ring, then generators and relations for H*(G,A), the cohomology of G with coefficients in A, may be obtained via the Hochschild-Serre spectral sequence. Examples include. Prerequisite to such a calculation is the existence of a proper normal subgroup of G, so that for simple groups this method is not available. Perhaps, in such cases, a more appropriate method is to extract the p-part of the cohomology ([H*(G,A)]p) from the cohomology ring of the Sylow subgroup for each prime p dividing ∣G∣, the order of G ([1, p.259]). In general, this requires a knowledge of the intersection of the Sylow p-subgroups, but has the following two simplifications. Let Gp be the Sylow p-subgroup of G, and Φp the group of automorphisms of Gp induced by inner automorphisms of G.

Lemma 1 ([12, Lemma 1]). If Gpis abelian, then [H*(G,A)] consists of those elements of H*(Gp,A) which are fixed under the action of Φpon H*(Gp,A).

Lemma 2 ([12, Theorem 2]). If p is odd and Gpis cyclic, then the p-period of G is twice the order of Φp.