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., z g Φ ( G ) : = z g = g − 1 z g for all g ∈ G and all z ∈ Z ( Φ ( G ) ) . This means that the Tate cohomology groups H n ( Q , A ) are all trivial for any n ∈ 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, 2010] proposed by P. Schmid.