Hostname: page-component-6b989bf9dc-zrclq Total loading time: 0 Render date: 2024-04-14T12:45:43.062Z Has data issue: false hasContentIssue false
  • English
  • Français

COMMUTING PROBABILITY OF COMPACT GROUPS

Part of: Structure and classification of infinite or finite groups Probabilistic methods in group theory

Published online by Cambridge University Press:  24 June 2021

ALIREZA ABDOLLAHI Open the ORCID record for ALIREZA ABDOLLAHI [Opens in a new window]  and
MEISAM SOLEIMANI MALEKAN Open the ORCID record for MEISAM SOLEIMANI MALEKAN [Opens in a new window]
ALIREZA ABDOLLAHI*
Affiliation:
Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan 81746-73441, Iran and School of Mathematics, Institute for Research in Fundamental Sciences, Tehran, Iran
MEISAM SOLEIMANI MALEKAN
Affiliation:
School of Mathematics, Institute for Research in Fundamental Sciences, Tehran, Iran e-mail: msmalekan@gmail.com
*
e-mail: a.abdollahi@math.ui.ac.ir
Get access Rights & Permissions [Opens in a new window]

Abstract

For any (Hausdorff) compact group G, denote by $\mathrm{cp}(G)$ the probability that a randomly chosen pair of elements of G commute. We prove that there exists a finite group H such that $\mathrm{cp}(G)= {\mathrm{cp}(H)}/{|G:F|^2}$ , where F is the FC-centre of G and H is isoclinic to F with $\mathrm{cp}(F)=\mathrm{cp}(H)$ whenever $\mathrm{cp}(G)>0$ . In addition, we prove that a compact group G with $\mathrm{cp}(G)>\tfrac {3}{40}$ is either solvable or isomorphic to $A_5 \times Z(G)$ , where $A_5$ denotes the alternating group of degree five and the centre $Z(G)$ of G contains the identity component of G.

Keywords

commuting probabilitycompact group

MSC classification

Primary: 20E18: Limits, profinite groups
Secondary: 20P05: Probabilistic methods in group theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 1 , February 2022 , pp. 87 - 91
Check for updates
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The research of the second author was in part supported by a grant from the Institute for Research in Fundamental Sciences (IPM) (No. 1400200043). This research was supported in part by a grant from School of Mathematics, Institute for Research in Fundamental Sciences.

References

Eberhard, S., ‘Commuting probabilities of finite groups’, Bull. Lond. Math. Soc. 47(5) (2015), 796–808.CrossRefGoogle Scholar
Guralnick, R. M. and Robinson, G. R., ‘On the commuting probability in finite groups’, J. Algebra 300 (2006), 509–528.CrossRefGoogle Scholar
Gustafson, W. H., ‘What is the probability that two group elements commute?’, Amer. Math. Monthly 80 (1973), 1031–1034.CrossRefGoogle Scholar
Hall, P., ‘The classification of prime-power groups’, J. reine angew. Math. 182 (1940), 130–141.CrossRefGoogle Scholar
Hofmann, K. H. and Russo, F. G., ‘The probability that $x$ and $y$ commute in a compact group’, Math. Proc. Cambridge Philos. Soc. 153(3) (2012), 557–571.CrossRefGoogle Scholar
Hofmann, K. H. and Russo, F. G., ‘The probability that ${x}^m$ and ${y}^n$ commute in a compact group’, Bull. Aust. Math. Soc. 87(3) (2013), 503–513.CrossRefGoogle Scholar
Lescot, P., Nguyen, H. N. and Yang, Y., ‘On the commuting probability and supersolvability of finite groups’, Monatsh. Math. 174(4) (2014), 567–576.CrossRefGoogle Scholar