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

Abstract

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 $H^n(G/N,Z(N))\ne 0$ 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.