The probability that \(x^m\) and \(y^n\) commute in a compact group

Authors

  • Karl H. Hofmann Technische Universitaet Darmstadt
  • Francesco G. Russo University of Palermo

Keywords:

Probability of commuting pairs, commutativity degree, FC-groups, compact groups, Haar measure.

Abstract

In a recent article [HR: The probability that x and y commute in a compact group; Math. Proc. Cambridge Phil. Soc., to appear] we calculated for a compact group $G$ the probability $d(G)$ that two randomly picked elements $x, y\in G$ satisfy $xy=yx$, and we discussed the remarkable consequences on the structure of $G$ which follow from the assumption that $d(G)$ is positive. In this note we consider two natural numbers $m$ and $n$ and the probabilty $d_{m,n}(G)$ that for two randomly selected elements $x, y\in G$ the relation $x^my^n=y^nx^m$ holds. If $m, n>1$, this situation is more complicated than in [HR] even if $G$ is a compact Lie group. If $G$ is a compact Lie group and if its identity component $G_0$ is abelian, then it follows readily that $d_{m,n}(G)$ is positive. We show here that the following condition suffices for the converse to hold in an arbitrary compact group $G$: ($*$) {\it For any nonopen closed subgroup $H$ of $G$, the sets $\{g\in G: g^k\in H\}$ for both $k=m$ and $k=n$ have Haar measure $0$.} Indeed we show that if a compact group $G$ satisfies ($*$) and if $d_{m,n}(G)>0$, then the identity component of $G$ is abelian 10.1017/S0004972712000573

Author Biography

Karl H. Hofmann, Technische Universitaet Darmstadt

Department of Mathematics Professor em. and Adjunct Professor of Mathematics Tulane University, New Orleans

Published

2013-04-29

Issue

Vol. 87 No. 3 (2013)

Section

Articles
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