On the autoconjugate probability of finite groups

Abstract

Let G be a finite group and let $\kappa (G)$ denote the probability that two randomly selected elements of G are conjugate. Two elements of G are called autoconjugate if one is the image of other under an automorphism of G. Let $P_a(G)$ denote the probability that two randomly selected elements of G are autoconjugate. We calculate $P_a(G)$ for some finite groups and some lower bounds of $P_a(G)$. We also prove that $P_a(G)=\kappa (G)$ if and only if $\text{Aut}(G)={\text{Aut}}_c(G)$, where ${\text{Aut}}_c(G)$ denotes the group of all class-preserving automorphisms of G. This gives another perspective on a question of Avinoam Mann.

Keywords:

• Autocommutator subgroup
• automorphism

2020 Mathematics Subject Classification:

• 20F28
• 20P05

Additional information

Funding

Research of first author is supported by SERB, DST Grant no. MTR/2017/000581. Research of third author is supported by SERB, DST Grant no. MTR/2019/000220.