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 PA(G) ≤ 5/8 for any nonabelian group. We show that PA(G) ≤ 5/8 for any noncyclic group, and then prove that this bound is not sharp. We also prove that PA(G) = P(G) if and only if Aut(G) = Autc(G), where Autc(G) denotes the group of all conjugacy class- preserving automorphisms of G. This gives another perspective on a question of Avinoam Mann.