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

Let $G$ be a finite group, let $p$ prime and ${\rm Pr}_p(G)$ the probability that two random $p$-elements of commute. In this paper we prove Pr}_p(G) > (p^2+p-1)/p^3$ if only has normal abelian Sylow $p$-subgroup, which generalizes previous results on widely studied commuting group. This bound is best possible in sense for each there are groups with = classify all such groups. Our proof based bounding proportion commute fixed $p$-element $G \setminus \textbf{O}_p(G)$, turn relies recent work first authors point ratios primitive permutation