Download this article
 Download this article For screen
For printing
Recent Issues

Volume 18
Issue 6, 1039–1219
Issue 5, 847–1038
Issue 4, 631–846
Issue 3, 409–629
Issue 2, 209–408
Issue 1, 1–208

Volume 17, 12 issues

Volume 16, 10 issues

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
Editors' interests
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author index
To appear
Other MSP journals
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

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.

finite groups, commuting probability, $p$-elements
Mathematical Subject Classification
Primary: 20D20
Received: 16 December 2021
Revised: 6 April 2022
Accepted: 6 July 2022
Published: 26 May 2023
Timothy C. Burness
School of Mathematics
University of 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
Gabriel Navarro
Departament de Matemàtiques
Universitat de València

Open Access made possible by participating institutions via Subscribe to Open.