Some results on automorphisms of finite $p$-groups

Abstract

Let $G$ be a group and $Aut^{\Phi}(G)$ denote the normal subgroup of $Aut(G)$ consisting of all automorphisms centralizing $G/\Phi(G)$. As a well-known result in a finite $p$-group $G$, the group $Aut^{\Phi}(G)$ is inner if and only if $G$ is extraspecial or elementary abelian. In the present paper, first we give a new proof of this result. We then introduce a new subgroup of $Aut^{\Phi}(G)$ for a finite $p$-group $G$, and give some basic aspects of this subgroup.