On the commuting probability of $\pi$-elements in finite groups

Abstract

Let $G$ be a finite group, let $\pi$ be a set of primes and let $p$ be the smallest prime in $\pi$. In this work, we prove that $G$ possesses a normal and abelian Hall $\pi$-subgroup if and only if the probability that two random $\pi$-elements of $G$ commute is larger than $\frac{p^2+p-1}{p^3}$. We also prove that if $x$ is a $\pi$-element not lying in $O_{\pi}(G)$, then the proportion of $\pi$-elements commuting with $x$ is at most $1/p$.