On a conjecture about automorphisms of finite p-groups

Abstract

Let G be a nonabelian finite p-group. A longstanding conjecture asserts that G admits a noninner automorphism of order p. In this paper, we prove that if G satisfies one of the following conditions (1) \({\mathrm{rank}(G'\cap Z(G))\neq \mathrm{rank}(Z(G))}\) (2) \({\frac{Z_{2}(G)}{Z(G)}}\) is cyclic (3) C G (Z(Φ(G))) = Φ(G) and \({\frac{Z_{2}(G)\cap Z(\Phi(G))}{Z(G)} }\) is not elementary abelian of rank rs, where r = d(G) and s = rank (Z(G)), then G has a noninner central automorphism of order p which fixes Φ(G) elementwise.