Finite groups, commuting probability, and coprime automorphisms
Eloisa Detomi, Robert M. Guralnick, Marta Morigi, Pavel Shumyatsky
TL;DR
This work studies how commuting probabilities interact with coprime automorphism actions to constrain the structure of finite groups. By linking Pr([P,A],[Q,A]) for A-invariant Sylow subgroups to the upper Fitting series, the authors prove that F_2([G,A]) has an ε-bounded index in [G,A], under a natural probabilistic assumption across all prime pairs. They further show that if G=[G,A] and a uniform probability bound holds for all primes p, the group is bounded-by-abelian-by-bounded, revealing strong rigidity under coprime actions. The results are established through a combination of coprime-action theory, commuting-probability techniques, and the classification of finite simple groups, with detailed analysis in the soluble, simple, cyclic-A, and general-A cases. Collectively, the paper extends Neumann-type boundedness phenomena to the setting of coprime automorphisms, yielding new structural constraints and highlighting how local probabilistic data controls global group structure.
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 $Pr(H,K)$. Suppose that a finite group $G$ admits a group of coprime automorphisms $A$ and let $ε>0$. We show that, if for any distinct primes $p,q\inπ(G)$ there is an $A$-invariant Sylow $p$-subgroup $P$ and an $A$-invariant Sylow $q$-subgroup $Q$ of $G$ for which $Pr([P,A],[Q,A])\geε$, then $F_2([G,A])$ has $ε$-bounded index in $[G,A]$ (Theorem 1.2). Here $F_2(K)$ stands for the second term of the upper Fitting seris of a group $K$. We also show that, if $G=[G,A]$ and for any prime $p$ dividing the order of $G$ there is an $A$-invariant Sylow $p$-subgroup $P$ such that $\Pr([P,A], [P,A]^x)\geqε$ for all $x\in G$, then $G$ is bounded-by-abelian-by-bounded (Theorem 1.4).