Cohomologically trivial modules over finite groups of prime power order

We show the existence of cohomologically trivial Q-module A, where Q = G/Phi (G), A = Z(Phi(G)), G is a finite non-abelian p-group,Phi(G) is the Frattini subgroup of G, Z(Phi(G)) is the center of Phi(G), and Q acts on A by conjugation, i.e., z(g Phi)(G) : = z(g) = g(-1) zg for all g is an element of G and all z is an element of Z(Phi(G)). This means that the Tate cohomology groups H(n)(Q, A) are all trivial for any n is an element of Z. Our main result answers Problem 17.2 of [V.D. Mazurov, E.I. Khukhro (Eds.). The Kourovka Notebook. Unsolved Problems in Group Theory, seventeenth edition, Russian Academy of Sciences, Siberian Division, Institute of Mathematics, Novosibirsk, 20101 proposed by P. Schmid. (C) 2011 Elsevier Inc. All rights reserved.