Exponents and the cohomology of finite groups

Abstract

We will provide an example of a p-group G which has elements of order p 3 in some of its integral cohomology groups but which also has the property that p 2 annihilates ¯ H i (G; Z) for all sufficiently highi. This provides a counterexample to a conjecture of A. Adem which stated that if a finite group K has an element of order p n in one of its integral cohomology groups then it has such an element in infinitely many of its cohomology groups.