On the commuting probability of $π$-elements in finite groups

Abstract

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