Commuting probability for conjugate subgroups of a finite group

Given two subgroups H,K of a finite group G, the probability that a pair of random elements from H and K commutes is denoted by (H,K). We address the following question. Let P be a p-subgroup of a finite group G and assume that (P,P^x)≥>0 for every x∈ G. Is the order of P modulo O_p(G) bounded in terms of e only? With respect to this question, we establish several positive results but show that in general the answer is negative. In particular, we prove that if the composition factors of G which are isomorphic to simple groups of Lie type in characteristic p, have Lie rank at most n, then the order of P modulo O_p(G) is bounded in terms of n and e only. If P is a Sylow p-subgroup of G, then the order of P modulo O_p(G) is bounded in terms e only. Some other results of similar flavour are established. We also show that if (P_1,P_2)>0 for every two Sylow p-subgroups P_1,P_2 of a profinite group G, then Op,p'(G) is open in G.