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We show that a compact group $G$ has finite conjugacy classes, i.e., is anFC-group if and only if its center $Z(G)$ is open if and only if its commutatorsubgroup $G'$ is finite. Let $d(G)$ denote the Haar measure of the set of allpairs $(x,y)$ in $G \times G$ for which $[x,y] = 1$; this, formally, is theprobability that two randomly picked elements commute. We prove that $d(G)$ isalways rational and that it is positive if and only if $G$ is an extension ofan FC-group by a finite group. This entails that $G$ is abelian by finite. Theproofs involve measure theory, transformation groups, Lie theory of arbitrarycompact groups, and representation theory of compact groups. Examples andreferences to the history of the discussion are given at the end of the paper.

The probability that $x$ and $y$ commute in a compact group