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

Let G$G$ be a finite group, π$\pi$ be a set of primes, and p$p$ be the smallest prime in π$\pi$. In this work, we prove that G$G$ possesses a normal and abelian Hall π$\pi$‐subgroup if and only if the probability that two random π$\pi$‐elements of G$G$ commute is larger than p2+p−1p3$\frac{p^2+p-1}{p^3}$. We also prove that if x$x$ is a π$\pi$‐element not lying in Oπ(G)$O_{\pi }(G)$, then the proportion of π$\pi$‐elements commuting with x$x$ is at most 1/p$1/p$.