On the commuting probability of \(\pi\)-elements in finite groups (Q6615862)

The author first defines a particular probability of two randomly chosen \(\pi\)-elements of \(G\) to commute, which is denoted by \(P_{\pi}(G)\). Then he proves three very interesting combinatorial results, one of which is the following:\N\NTheorem C. Let \(G\) be a finite group, let \(\pi\) be a set of primes where \(p\) is the smallest member of \(\pi\). The group \(G\) has a normal and abelian Hall \(\pi\)-subgroup if and only \(P_{\pi}(G)> \frac{p^{2}+p-1}{p^{3}}\).\N\NHe also demonstrates with nice concrete examples that the bound is indeed sharp.