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

Lookup NU author(s): Dr John Britnell

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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 κ(G) 14 , and shows that G is abelian whenever κ(G)|G| 74. 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 ?(Sn), the probability that two elements of the symmetric group have conjugates that commute. © 2012 London Mathematical Society.

Publication metadata

Author(s): Blackburn SR, Britnell JR, Wildon M

Publication type: Article

Publication status: Published

Journal: Journal of the London Mathematical Society

Year: 2012

Volume: 86

Issue: 3

Pages: 755-778

Print publication date: 01/12/2012

Online publication date: 25/07/2012

ISSN (print): 0024-6107

ISSN (electronic): 1469-7750

Publisher: John Wiley and Sons Ltd

URL: https://doi.org/10.1112/jlms/jds022

DOI: 10.1112/jlms/jds022

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