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

JR Britnell, SR Blackburn, MJ Wildon*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

15 Citations (Scopus)

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 ρ(Sn), the probability that two elements of the symmetric group have conjugates that commute. © 2012 London Mathematical Society.
Translated title of the contributionThe probability that a pair of elements of a finite group are conjugate
Original languageEnglish
Pages (from-to)755-778
Number of pages24
JournalJournal of the London Mathematical Society
Volume86
Issue number3
Early online date25 Jul 2012
DOIs
Publication statusPublished - 1 Dec 2012

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