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The probability that x and y commute in a compact group

Published online by Cambridge University Press:  03 July 2012

KARL H. HOFMANN  and
FRANCESCO G. RUSSO
KARL H. HOFMANN
Affiliation:
Fachbereich Mathematik, Technische Universität, Schlossgartenstr. 7, 64289 Darmstadt, Germany. e-mail: hofmann@mathematik.tu-darmstadt.de
FRANCESCO G. RUSSO
Affiliation:
Fachbereich Mathematik, Technische Universität, Schlossgartenstr. 7, 64289 Darmstadt, Germany. e-mail: hofmann@mathematik.tu-darmstadt.de
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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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 153 , Issue 3 , November 2012 , pp. 557 - 571
Copyright
Copyright © Cambridge Philosophical Society 2012

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References

REFERENCES

[1]Bourbaki, N.Intégration (Hermann, Paris, 1963, Chap. 7 et 8).Google Scholar
[2]Bredon, G.Introduction to Compact Transformation Groups (Academic Press, New York, 1972).Google Scholar
[3]Diaconis, P.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. 120–142.CrossRefGoogle Scholar
[4]Erdős, P. and Túran, P.On some problems of statistical group theory. Acta Math. Acad. Sci. Hung. 19 (1968), 413–435.CrossRefGoogle Scholar
[5]Erfanian, A. and R.Kamyabi–Gol. On the mutually commuting n-tuples in compact groups. Int. J. Algebra 1 (2007), 251–262.CrossRefGoogle Scholar
[6]Erfanian, A. and Rezaei, R.On the commutativity degree of compact groups. Arch. Math. (Basel) 93 (2009), 201–212.Google Scholar
[7]Erfanian, A. and Russo, F.Probability of mutually commuting n-tuples in some classes of compact groups. Bull. Iran. Math. Soc. 34 (2008), 27–37.Google Scholar
[8]Gustafson, W. H.What is the probability that two group elements commute? Amer. Math. Monthly 80 (1973), 1031–1304.CrossRefGoogle Scholar
[9]Hewitt, E. and Ross, K.Abstract Harmonic Analysis. Vol. I (Springer, Berlin, 1963).Google Scholar
[10]Hofmann, K. H. and Morris, S. A.The Structure of Compact Groups (de Gruyter, Berlin, Second Edition, 2006).CrossRefGoogle Scholar
[11]Hofmann, K. H. and Morris, S. A. The Lie theory of connected Pro-Lie groups. Eur. Math. Soc. Publ. House 2007.CrossRefGoogle Scholar
[12]Hofmann, K. H. and Morris, S. A.Elements of Compact Semigroups (Charles E. Merill, Columbus, Ohio, 1966).Google Scholar
[13]Lescot, P.Isoclinism classes and commutativity degrees of finite groups. J. Algebra 177 (1985), 847–869.CrossRefGoogle Scholar
[14]Lévai, L. and Pyber, L.Profinite groups with many commuting pairs or involutions. Arch. Math. (Basel) 75 (2000), 1–7.CrossRefGoogle Scholar
[15]Neumann, B. H.On a problem of P. Erdős. J. Aust. Math. Soc. Ser. A 21 (1976), 467–472.CrossRefGoogle Scholar
[16]Neumann, P. M.Two combinatorial problems in group theory. Bull. London Math. Soc. 21 (1989), 456–458.CrossRefGoogle Scholar
[17]Shalev, A.Profinite groups with restricted centralizers. Proc. Amer. Math. Soc. 122 (1994), 1279–1284.CrossRefGoogle Scholar
[18]tom Dieck, T.Transformation Groups (de Gruyter, Berlin, 1987).CrossRefGoogle Scholar