Powerful p-groups have non-inner automorphisms of order p and some cohomology

In this paper we study the longstanding conjecture of whether there exists a non-inner automorphism of order p for a finite non-abelian p-group. We prove that if G is a finite non-abelian p-group such that G/Z(G) is powerful then G has a non-inner automorphism of order p leaving either Phi(G) or Omega(1)(Z(G)) elementwise fixed. We also recall a connection between the conjecture and a cohomological problem and we give an alternative proof of the latter result for odd p. by showing that the Tate cohomology H(n)(G/N, Z(N)) not equal 0 for all n >= 0, where G is a finite p-group, p is odd, G/Z(G) is p-central (i.e., elements of order p are central) and N (sic) G with G/N non-cyclic. (C) 2009 Elsevier Inc. All rights reserved.