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

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 $\kappa (G)\ge 1\mathrm{/}4$, and shows that $G$ is abelian whenever $\kappa(G)|G| < 7/4$\mathrm{.}Itisalsoshownthat$\kappa(G)|G|$ depends only on the isoclinism class of $G$. Specialising to the symmetric group $S_n$, the paper shows that $\kappa(S_n) \leq C/n^2$foranexplicitlydeterminedconstant$C$. This bound leads to an elementary proof of a result of Flajolet \emph{et al}, that $\kappa(S_n) \sim A/n^2$as$n\rightarrow \infty$forsomeconstant$A$. The same techniques provide analogous results for $\rho (S_n)$, the probability that two elements of the symmetric group have conjugates that commute.