Mathematics – Group Theory
Scientific paper
2010-01-27
Mathematics
Group Theory
29 pages; with respect to the previous version we have revised some points
Scientific paper
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 $G \times G$ 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. Finally, for an arbitrary pair of positive natural numbers $m$ and $n$ we investigate the effect on the group structure of the positivity of the probability that the powers $x^m$ and $y^n$ commute for two randomly picked elements $x,y \in G$. This involves additional additional complications which are studied in the case of compact Lie groups. Examples and references to the history of the discussion are given at the end of the paper.
Hofmann Karl Heinrich
Russo Francesco G.
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