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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 thatk  (  G  )  ⩾   1  4, and shows that  G  is abelian wheneverk  (  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  S   n  , the paper shows that κ( 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 κ( S   n  ) ∼  A / n  2  as  n →∞ 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.

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