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

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 $\kappa (G)\ge 1/4$, 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 $\kappa (S_n)\le C/n^2$ 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 $n\to \infty$ 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.

34 pages, corrected version, to appear in Journal of the London Mathematical Society

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