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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
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) > (p2 + p 1)p3 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) = (p2 + p 1)p3 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 Op(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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