On the Probability That an Automorphism Fixes a Group Element

Abstract

Let G be a finite group. Denote by P(G) the probability that two elements of G, selected randomly and with replacement, commute, and by PA(G) the probability that a randomly chosen automorphism of G fixes a randomly chosen element of G. It is known that for any nonabelian group. We show that for any noncyclic group, and then prove that this bound is not sharp. We also prove that if and only if , where denotes the group of all conjugacy class-preserving automorphisms of G. This gives another perspective on a question of Avinoam Mann.