On modules of trivial cohomology over a finite group

Abstract

We say that a module A over a finite group G is of trivial cohomology when we have H(H, A) 0 for every integer n and every subgroup H of G. Recently the writer proved" THEOREM. Let A be a module over a finite group G. Assume that for every prime p dividing the order [G] of G there is an integer r, such that Hrp(H,, A) Hrp+I(H A) O, where H, is a p-Sylow subgroup of G. Then the G-module A is of trivial co- homology.