Commuting probability for the Sylow subgroups of a profinite group

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).