## 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/n

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/n

^{2}for an explicitly determined constant C. This bound leads to an elementary proof of a result of Flajolet et al., that κ(Sn) ∼ A/n^{2}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 contribution | The probability that a pair of elements of a finite group are conjugate |
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Original language | English |

Pages (from-to) | 755-778 |

Number of pages | 24 |

Journal | Journal of the London Mathematical Society |

Volume | 86 |

Issue number | 3 |

Early online date | 25 Jul 2012 |

DOIs | |

Publication status | Published - 1 Dec 2012 |