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

让$G$是有限群，令$p$成为质数并让${\mathrm{压力}<!--FUNCTION\; APPLICATION-->}_p(G)$是两个随机的概率$p$-要点$G$通勤。在本文中，我们证明${\mathrm{压力}<!--FUNCTION\; APPLICATION-->}_p(G)>(p^{\mathrm{2个}}+p-\mathrm{1个})∕p^{\mathrm{3个}}$当且仅当$G$有一个正常的和交换的 Sylow$p$-subgroup，它概括了先前关于广泛研究的有限群通勤概率的结果。这个界限是最好的，因为对于每个素数$p$有团体${\mathrm{压力}<!--FUNCTION\; APPLICATION-->}_p(G)=(p^{\mathrm{2个}}+p-\mathrm{1个})∕p^{\mathrm{3个}}$我们对所有这些群体进行分类。我们的证明是基于限制比例$p$-元素在$G$以固定的方式上下班$p$-元素在$G\setminus {欧}_p(G)$，这又依赖于前两位作者最近关于有限本原置换群不动点比率的工作。