Exponents and the cohomology of finite groups

Abstract

We will provide an example of a p-group G which has elements of order p3 in some of its integral cohomology groups but which also has the property that p2 annihilates H̄i(G; Z) for all sufficiently high i. This provides a counterexample to a conjecture of A. Adem which states that if a finite group K has an element of order pn in one of its integral cohomology groups,then it has such an element in infinitely many of its cohomology groups.