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The probability that x and y commute in a compact group

  • KARL H. HOFMANN (a1) and FRANCESCO G. RUSSO (a1)
Abstract
Abstract

We show that a compact group G has finite conjugacy classes, i.e., is an FC-group if and only if its center Z(G) is open if and only if its commutator subgroup G′ is finite. Let d(G) denote the Haar measure of the set of all pairs (x,y) in G×G for which [x,y]=1; this, formally, is the probability that two randomly picked elements commute. We prove that d(G) is always rational and that it is positive if and only if G is an extension of an FC-group by a finite group. This entails that G is abelian by finite. The proofs involve measure theory, transformation groups, Lie theory of arbitrary compact groups, and representation theory of compact groups. Examples and references to the history of the discussion are given at the end of the paper.

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[3] P. Diaconis Random walks on groups: characters and geometry, in: Groups St. Andrews 2001 in Oxford, Vol. I, London Math. Soc. Lecture Note Ser. 304 (Cambridge University Press, 2003), pp. 120142.

[4] P. Erdős and P. Túran On some problems of statistical group theory. Acta Math. Acad. Sci. Hung. 19 (1968), 413435.

[8] W. H. Gustafson What is the probability that two group elements commute? Amer. Math. Monthly 80 (1973), 10311304.

[9] E. Hewitt and K. Ross Abstract Harmonic Analysis. Vol. I (Springer, Berlin, 1963).

[13] P. Lescot Isoclinism classes and commutativity degrees of finite groups. J. Algebra 177 (1985), 847869.

[14] L. Lévai and L. Pyber Profinite groups with many commuting pairs or involutions. Arch. Math. (Basel) 75 (2000), 17.

[16] P. M. Neumann Two combinatorial problems in group theory. Bull. London Math. Soc. 21 (1989), 456458.

[17] A. Shalev Profinite groups with restricted centralizers. Proc. Amer. Math. Soc. 122 (1994), 12791284.

[18] T. tom Dieck Transformation Groups (de Gruyter, Berlin, 1987).

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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