Abstract
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.
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