A Note On Commuting Automorphisms of Some Finite $p$-Groups

An automorphism $\alpha$ of a group $G$ is called a commuting automorphism if each element $x$ in $G$ commutes with its image $\alpha(x)$ under $\alpha$. Let $A(G)$ denote the set of all commuting automorphisms of $G$. Rai [Proc. Japan Acad., Ser. A 91 (5), 57–60 (2015)] has given some sufficient conditions on a finite $p$-group $G$ such that $A(G)$ is a subgroup of $Aut(G)$ and, as a consequence, has proved that, in a finite $p$-group $G$ of co-class 2, where $p$ is an odd prime, $A(G)$ is a subgroup of $Aut(G)$. We give here very elementary and short proofs of main results of Rai.