On a conjecture about automorphisms of finite p-groups

摘要

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) rank(G¢ÇZ(G)) ¹ rank(Z(G)){mathrm{rank}(G'cap Z(G))neq mathrm{rank}(Z(G))} (2) fracZ₂(G)Z(G){frac{Z_{2}(G)}{Z(G)}} is cyclic (3) C _G (Z(Φ(G))) = Φ(G) and fracZ₂(G)ÇZ(F(G))Z(G) {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.