Paper detail

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

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