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 is an element of G are conjugate, when g and h are chosen independently and uniformly at random. The paper classifies those groups G such that kappa(G) >= 1/4, and shows that G is abelian whenever kappa(G)vertical bar G vertical bar < 7/4. It is also shown that kappa(G)vertical bar G vertical bar depends only on the isoclinism class of G. Specializing to the symmetric group S-n, the paper shows that kappa(S-n)<= C/n(2) for an explicitly determined constant C. This bound leads to an elementary proof of a result of Flajolet et al., that kappa(S-n) similar to A/n(2) as n ->infinity 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.