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COMMUTING PROBABILITY OF COMPACT GROUPS
Published online by Cambridge University Press: 24 June 2021
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
MSC classification
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 105 , Issue 1 , February 2022 , pp. 87 - 91
- Copyright
- © 2021 Australian Mathematical Publishing Association Inc.
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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