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

Author： Blackburn Simon R.   Britnell John R.   Wildon Mark  

Publisher： Oxford University Press

ISSN： 0024-6107

Source： Journal of the London Mathematical Society, Vol.86, Iss.3, 2012-12, pp. : 755-778

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Abstract

Let G be a finite group, and let (G) be the probability that elements G, hG are conjugate, when G and h are chosen independently and uniformly at random. The paper classifies those groups G such that (G), and shows that G is abelian whenever . It is also shown that (G)G depends only on the isoclinism class of G.Specializing to the symmetric group Sn, the paper shows that (Sn)C/n2 for an explicitly determined constant C. This bound leads to an elementary proof of a result of Flajolet et al., that (Sn) A/n2 as n for some constant A. The same techniques provide analogous results for (Sn), the probability that two elements of the symmetric group have conjugates that commute.

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