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' boolean AND Z(G)) not equal rank(Z(G)) (2) Z(2)(G)/Z(G) is cyclic (3) C-G(Z(Phi(G))) = Phi(G) and Z(2)(G)boolean AND 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 Phi(G) elementwise.