A note on commuting automorphisms of some finite $p$-groups

TL;DR

This paper provides simple, concise proofs for conditions under which the set of commuting automorphisms forms a subgroup in certain finite p-groups, extending previous results by Rai.

Contribution

It offers elementary proofs of Rai's results on commuting automorphisms forming subgroups in finite p-groups, especially for co-class 2 groups with odd primes.

Findings

Conditions for A(G) to be a subgroup are established.

In co-class 2 p-groups with odd p, A(G) is a subgroup.

Elementary proofs simplify understanding of commuting automorphisms.

Abstract

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 {\bf 91} (2015), no. 5, 57-60] 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.