2025/05/15 by Eloisa Detomi, Detomi, Eloisa, Robert M. Guralnick +5 · 1 citation
Mathematics · #20D20 #20E45 #20P05 #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.2505.10521
openalex publication_date 2025/05/15 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28
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,Px)≥\e>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 \pr(P1,P2)>0 for every two Sylow p-subgroups P1,P2 of a profinite group G, then Op,p'(G) is open in G.