Commuting Probability of Compact Groups

TL;DR

This paper investigates the probability that two randomly chosen elements of a compact group commute, establishing a connection to finite groups and deriving structural results based on this probability.

Contribution

It proves a relation between the commuting probability of a compact group and a finite group, enabling transfer of finite group results to compact groups.

Findings

If cp(G)>0, then cp(G) relates to a finite group H via the FC-center.

For cp(G)>3/40, G is either solvable or isomorphic to A_5 times an abelian group.

The commuting probability of G can be explicitly computed in certain cases.

Abstract

For any (Hausdorff) compact group $G$ with the normalized Haar measure ${\mathbf m}_G$, denote by ${\rm cp}(G)$ the probability ${\mathbf m}_{G\times G}(\{(x,y)\in G\times G \;|\; xy=yx\})$ of commuting a randomly chosen pair of elements of $G$. Here we prove that if ${\rm cp}(G)>0$, then there exists a finite group $H$ such that ${\rm cp}(G)= \frac{{\rm cp}(H)}{|G:F|^2}$, where $F$ is the FC-center of $G$ i.e. the set of all elements of $G$ whose conjugacy classes are finite and $H$ is isoclinic to $F$ with ${\rm cp}(F)={\rm cp}(H)$. The latter equality enables one to transfer many existing results concerning commuting probability of finite groups to one of compact groups. For example, here for a compact group $G$ we prove that if ${\rm cp}(G)>\frac{3}{40}$ then either $G$ is solvable or, else $G\cong A_5 \times T$ for some abelian group $T$, in which case ${\rm cp}(G)=\frac{1}{12}$; where $A_5$ denotes the alternating group of degree $5$.