some remarks on commuting fixed point free automorphisms of groups

In this article we will find necessary and sufficient conditions for a fixed point free automorphism (fpf automorphism) of a group to be a commuting automorphism. for a given prime we find the smallest order of a non abelian p-group admitting a commuting fixed point free automorphism. we prove that a group of order p³ having a commuting fpf automorphism, has a restricted structure. moreover, we prove that if a finite group admits a fpf automorphism of order 4, then the converse of laffey’s result holds in g1.