Commuting probability for conjugate subgroups of a finite group

TL;DR

This paper investigates the probability that elements from conjugate subgroups commute in finite groups, establishing bounds under certain conditions and exploring implications for the structure of the group.

Contribution

It provides new bounds on the order of p-subgroups modulo O_p(G) based on commuting probabilities, especially for groups with Lie type composition factors.

Findings

If the composition factors are of Lie type with bounded Lie rank, the order of P modulo O_p(G) is bounded by e and the rank.

For Sylow p-subgroups, the order modulo O_p(G) depends only on e.

Positive commuting probability between Sylow p-subgroups implies O_{p,p'}(G) is open.

Abstract

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 \pr(H,K). We address the following question. Let P be a p-subgroup of a finite group G and assume that \pr(P,P^x)\geq\e>0 for every x\in 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 \pr(P_1,P_2)>0 for every two Sylow p-subgroups P_1,P_2 of a profinite group G, then O_{p,p'}(G) is open in G.