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 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. © 2009 Elsevier Inc. All rights reserved.