On the commuting probability of p-elements in a finite group

Let G be a finite group, let p be a prime and let Prp(G) be the probability that two random p-elements of G commute. In this paper we prove that Prp(G) > (p2+p−1)/p3 if and only if G has a normal and abelian Sylow p-subgroup, which generalizes previous results on the widely studied commuting probability of a finite group. This bound is best possible in the sense that for each prime p there are groups with Prp(G) = (p2 + p − 1)/p3 and we classify all such groups. Our proof is based on bounding the proportion of p-elements in G that commute with a fixed p-element in G \ Op(G), which in turn relies on recent work of the first two authors on fixed point ratios for finite primitive permutation groups.