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 Φ(G) or Ω1(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 Hn(G/N,Z(N))≠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◁G with G/N non-cyclic.