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

TL;DR

This paper characterizes when the probability that two random p-elements commute in a finite group exceeds a specific bound, linking it to the structure of the group's Sylow p-subgroup, and classifies groups achieving equality.

Contribution

It generalizes known results by establishing a precise condition involving Sylow p-subgroups for the commuting probability of p-elements in finite groups.

Findings

The probability exceeds (p^2+p-1)/p^3 iff the Sylow p-subgroup is normal and abelian.

The bound is sharp; groups achieving equality are classified.

The proof uses fixed point ratios in permutation groups.

Abstract

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