The probability that $x$ and $y$ commute in a compact 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 $G^{\prime }$ 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.

17 pages; we have cut some points ; to appear in Math. Proc. Cambridge Phil. Soc

Cited by in corpus (9)

• On the Connectivity of the Sylow Graph of a Finite Group
• Considerations on the subgroup commutativity degree and related notions
• Probabilistic nilpotence in infinite groups
• Commuting probability and simultaneous conjugacy classes of commuting tuples in a group
• Branching rules for Unitary and Symplectic matrices
• Permutability degrees of finite groups
• Compact groups with high commuting probability of monothetic subgroups
• Compact groups with a set of positive Haar measure satisfying a nilpotent law
• Asymptotics of commuting probabilities in reductive algebraic groups