6533b823fe1ef96bd127ec02
RESEARCH PRODUCT
The probability that $x$ and $y$ commute in a compact group
Karl H. HofmannFrancesco G. Russosubject
Haar measureGroup (mathematics)General MathematicsCommutator subgroupactions on Hausdorff spaces20C05 20P05 43A05Center (group theory)Group Theory (math.GR)Functional Analysis (math.FA)CombinatoricsMathematics - Functional AnalysisProbability of commuting pairConjugacy classCompact groupFOS: MathematicsComponent (group theory)compact groupCharacteristic subgroupAbelian groupMathematics - Group TheoryMathematicsdescription
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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2010-01-27 |