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

Let $G$ be a finite group, let $p$ be a prime and let ${\mathrm{P}\mathrm{r}}_p(G)$ be the probability that two random $p$-elements of $G$ commute. In this paper we prove that ${\mathrm{P}\mathrm{r}}_p(G)>(p^2+p-1)\mathrm{/}{p}^3$ 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 ${\rm Pr}_p(G) = (p^2+p-1)/p^3$ 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 $G\setminus {\mathbf{\text{O}}}_p(G)$, which in turn relies on recent work of the first two authors on fixed point ratios for finite primitive permutation groups.