Commuting probability for the Sylow subgroups of a profinite group

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). Furthermore, if G is a prosoluble group in which for every subset pi subset of 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 2).