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.