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

TL;DR

This paper characterizes the structure of finite groups based on the probability that two random $\pi$-elements commute, linking it to the existence of a normal abelian Hall $\pi$-subgroup.

Contribution

It establishes a precise probabilistic criterion for the existence of a normal abelian Hall $\pi$-subgroup in finite groups.

Findings

A normal abelian Hall $\pi$-subgroup exists if and only if the commuting probability exceeds a specific threshold.

The maximum proportion of $\pi$-elements commuting with a non-$O_{\pi}(G)$ element is at most $1/p$.

Provides a probabilistic characterization connecting group structure and element commuting behavior.

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 $1/p$.