The probability that a pair of elements of a finite group are conjugate

TL;DR

This paper investigates the probability that two elements of a finite group are conjugate, classifies groups with high conjugacy probability, and analyzes asymptotic behavior in symmetric groups, revealing structural and probabilistic properties.

Contribution

It classifies finite groups with conjugacy probability at least 1/4, shows this probability depends only on isoclinism class, and derives asymptotic bounds for symmetric groups.

Findings

Groups with .25 conjugacy probability are classified.

.25 probability times group order is less than 7/4 only for abelian groups.

Asymptotic behavior of conjugacy probability in symmetric groups is established.

Abstract

Let $G$ be a finite group, and let $\kappa(G)$ be the probability that elements $g$, $h\in G$ are conjugate, when $g$ and $h$ are chosen independently and uniformly at random. The paper classifies those groups $G$ such that $\kappa(G) \geq 1/4$, and shows that $G$ is abelian whenever $\kappa(G)|G| < 7/4$. It is also shown that $\kappa(G)|G|$ depends only on the isoclinism class of $G$. Specialising to the symmetric group $S_n$, the paper shows that $\kappa(S_n) \leq C/n^2$ for an explicitly determined constant $C$. This bound leads to an elementary proof of a result of Flajolet \emph{et al}, that $\kappa(S_n) \sim A/n^2$ as $n\rightarrow \infty$ for some constant $A$. The same techniques provide analogous results for $\rho(S_n)$, the probability that two elements of the symmetric group have conjugates that commute.