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, Pˣ) >0 for every x G. Is the order of P modulo Oₚ (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 Oₚ (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 Oₚ (G) is bounded in terms e only. Some other results of similar flavour are established. We also show that if (P₁, P₂) >0 for every two Sylow p-subgroups P₁, P₂ of a profinite group G, then O, ' (G) is open in G.
Detomi et al. (Thu,) studied this question.