The probability that $x^n$ and $y$ commute in a compact group.
Thu, 01/28/2010 - 16:01 — SomNambul
link: http://arxiv.org/abs/1001.4856
Abstract
Abstract
For a compact group $G$ and a fixed positive natural number $n$ let $p$
denote the Haar measure of the set of all pairs $(x,y)$ in $G\times G$ for
which $[x^n,y]=1$. It is shown that $p=0$ if the identity component $G_0$ of
$G$ is noncommutative, and if $G$ is a Lie group, then the two conditions are
equivalent. Further, $p=1$ if and only if $x^n$ is central for all $x\in G$.
References to the history are given at the end of the discussion.
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