On the commuting probability of ππ‐elements in finite groups

Let be a finite group, be a set of primes, and be the smallest prime in . In this work, we prove that possesses a normal and abelian Hall ‐subgroup if and only if the probability that two random ‐elements of commute is larger than . We also prove that if is a ‐element not lying in , then the proportion of ‐elements commuting with is at most .