Commuting probability for the Sylow subgroups of a profinite group

TL;DR

This paper investigates the commuting probabilities of Sylow subgroups in profinite groups and shows conditions under which such groups are virtually prosoluble or pronilpotent.

Contribution

It establishes new criteria linking commuting probabilities of Sylow subgroups to the structural properties of profinite groups, specifically virtual prosolubility and pronilpotency.

Findings

If certain Sylow subgroup commuting probabilities are positive, the group is virtually prosoluble.

Positive commuting probabilities between Hall subgroups imply the group is virtually pronilpotent.

Provides new characterizations of profinite groups based on subgroup commuting probabilities.

Abstract

Given two subgroups $H,K$ of a compact group $G$, the probability that a random element of $H$ commutes with a random element of $K$ is denoted by $Pr(H,K)$. We show that if $G$ is a profinite group containing a Sylow $2$-subgroup $P$, a Sylow $3$-subgroup $Q_3$ and a Sylow $5$-subgroup $Q_5$ such that $Pr(P,Q_3)$ and $Pr(P,Q_5)$ are both positive, then $G$ is virtually prosoluble (Theorem 1.1). Furthermore, if $G$ is a prosoluble group in which for every subset $\pi\subseteq\pi(G)$ there is a Hall $\pi$-subgroup $H_\pi$ and a Hall $\pi'$-subgroup $H_{\pi'}$ such that $Pr(H_\pi,H_{\pi'})>0$, then $G$ is virtually pronilpotent (Theorem 1.2).