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

TL;DR

This paper characterizes when a compact group has finite conjugacy classes and explores the probability that two elements commute, showing it is always rational and positive under specific structural conditions.

Contribution

It establishes new criteria linking the structure of compact groups with properties like finite conjugacy classes and the rationality of the commuting probability.

Findings

A compact group has finite conjugacy classes iff its center is open.

The probability that two elements commute is always rational.

This probability is positive iff the group is an extension of an FC-group by a finite group.

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 $G \times G$ 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. Examples and references to the history of the discussion are given at the end of the paper.