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COMMUTING PROBABILITY OF COMPACT GROUPS

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

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: [email protected]
*
e-mail: [email protected]
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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
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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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

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