On commuting automorphisms of finite $p$-groups

Abstract

Let $G$ be a group. An automorphism $\alpha$ of $G$ is called a commuting automorphism if $[\alpha(x), x] = 1$ for all $x \in G$. Let $A(G)$ be the set of all commuting automorphisms of $G$. A group $G$ is said to be an $A(G)$-group if $A(G)$ forms a subgroup of $\mathit{Aut}(G)$. We give some sufficient conditions on a finite $p$-group $G$ such that $G$ is an $A(G)$-group. As an application we prove that a finite $p$-group $G$ of coclass 2 for an odd prime $p$ is an $A(G)$-group. Also we classify non-$A(G)$ groups $G$ of order $p^{5}$.