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

Abstract

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 k ( G ) ⩾ 1 4 , and shows that G is abelian whenever k ( G ) | G | < 7 4 . It is also shown that κ(G)|G| depends only on the isoclinism class of G. Specializing to the symmetric group Sn, the paper shows that κ(Sn)⩽C/n2 for an explicitly determined constant C. This bound leads to an elementary proof of a result of Flajolet et al., that κ(Sn) ∼ A/n2 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.