Commuting probability for conjugate subgroups of a finite group

TL;DR

This work analyzes the commuting probability $\mathrm{Pr}(H,K)$ for subgroups of finite groups, focusing on $p$-subgroups $P$ with $\mathrm{Pr}(P,P^x)\ge \varepsilon$ for all $x$. It develops structural reductions to control $|P/O_p(G)|$, proving boundedness when composition factors of Lie type in characteristic $p$ have bounded Lie rank and when $P$ is a Sylow $p$-subgroup, while showing negative instances in general. The authors establish a profinite analogue: if $\mathrm{Pr}(P_1,P_2)>0$ for all Sylow $p$-subgroups in a profinite $G$, then $O_{p,p'}(G)$ is open and $O_p(G)$ is virtually abelian. Collectively, the results link local commuting constraints to global group structure, with explicit bounds depending on $\varepsilon$ and, in some cases, the Lie rank $n$.

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.

Theorems & Definitions (29)

• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof