Abstract
Let \(G\) be a group. An automorphism \(\alpha\) of \(G\) is called a commuting automorphism if \(\alpha(x)x= x \alpha(x)\) for all \(x \in G\) . The set of all commuting automorphisms of \(G\) is denoted by \(A(G)\) . The set \(A(G)\) does not necessarily form a subgroup of the automorphism group of \(G\) . If \(A(G)\) is a subgroup of the automorphism group of \(G\) , then we say that \(G\) is an \(A\) -group. Garg [Math. Notes, 106(2), 296–298] stated that if \(G\) is a finite nonabelian \(p\) -group (where \(p\) is an odd prime) such that \(G/Z(G)\) is a metacyclic group, then \(G\) is an \(A\) -group if and only if \(G\) is of nilpotency class \(2\) . We identify that this statement is flawed and provide the correct statement. Moreover, we generalize the result by proving that if \(G\) is a finite nonabelian group such that \(G/Z(G)\) is a metacyclic group, then \(G\) is an \(A\) -group.
