On the commuting probability of p-elements in a finite group

Burness, Timothy C.; Guralnick, Robert; Moretó, Alexander; Navarro, Gabriel

doi:10.2140/ant.2023.17.1209

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On the commuting probability of $p$-elements in a finite group

Timothy C. Burness, Robert Guralnick, Alexander Moretó and Gabriel Navarro

Vol. 17 (2023), No. 6, 1209–1229

DOI: 10.2140/ant.2023.17.1209

Abstract

Let $G$ be a finite group, let $p$ be a prime and let $\Pr_{p} (G)$ be the probability that two random $p$ -elements of $G$ commute. In this paper we prove that $\Pr_{p} (G) > (p^{2} + p - 1) ∕ p^{3}$ if and only if $G$ has a normal and abelian Sylow $p$ -subgroup, which generalizes previous results on the widely studied commuting probability of a finite group. This bound is best possible in the sense that for each prime $p$ there are groups with $\Pr_{p} (G) = (p^{2} + p - 1) ∕ p^{3}$ and we classify all such groups. Our proof is based on bounding the proportion of $p$ -elements in $G$ that commute with a fixed $p$ -element in $G ∖ O_{p} (G)$ , which in turn relies on recent work of the first two authors on fixed point ratios for finite primitive permutation groups.

Keywords

finite groups, commuting probability, $p$-elements

Mathematical Subject Classification

Primary: 20D20

Milestones

Received: 16 December 2021

Revised: 6 April 2022

Accepted: 6 July 2022

Published: 26 May 2023

Authors

Timothy C. Burness
School of Mathematics University of Bristol Bristol United Kingdom

Robert Guralnick
Department of Mathematics University of Southern California Los Angeles, CA United States

Alexander Moretó
Departament de Matemàtiques Universitat de València Burjassot València Spain

Gabriel Navarro
Departament de Matemàtiques Universitat de València Burjassot València Spain

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Vol. 17, No. 6, 2023