Cohomologically trivial modules over finite groups of prime power order

Abstract

We show the existence of cohomologically trivial Q-module A, where Q=G/Φ(G), A=Z(Φ(G)), G is a finite non-abelian p-group, Φ(G) is the Frattini subgroup of G, Z(Φ(G)) is the center of Φ(G), and Q acts on A by conjugation, i.e., zgΦ(G):=zg=gΦ1zg for all gΦG and all zΦZ(Φ(G)). This means that the Tate cohomology groups Hn(Q,A) are all trivial for any nΦZ. 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, 2010] proposed by P. Schmid. © 2011 Elsevier Inc.