On the module structure of the mod p cohomology of a p-group (Q790942)

Let P be a finite p-group and let Q be a group acting faithfully on P. The paper studies the Q-module structure of the cohomology algebra \(H^*(P,k)\) where k denotes the field of p elements. The first author has shown that if Q is a p'-group then every simple kQ-module appears infinitely often as a direct summand of \(H^*(P,k)\). Here this result is generalized to arbitrary groups Q: There exists a number \(\ell\) such that for all \(m=0,1,2,...\) every simple kQ-module appears as a composition factor of \(\oplus^{\ell +m}_{i=m+1}H^{2i}(P,k).\) An upper bound for \(\ell\) is given in terms of P, Q and the operation of Q on P.