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 ∈ G are conjugate, when g and h are chosen independently and uniformly at random. The paper classifies those groups G such that κ(G) ≥ 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 Sn, the paper shows that κ(Sn) ≤ C/n for an explicitly determined constant C. This bound leads to an elementary proof of a result of Flajolet et al, that κ(Sn) ∼ A/n as n → ∞ for some constant A. The same techniques provide analogous results for ρ(Sn), the probability that two elements of the symmetric group have conjugates that commute.