Finite groups, commuting probability, and coprime automorphisms

TL;DR

This paper explores how the probability of commuting elements in certain subgroups of finite groups with coprime automorphisms influences the group's structure, establishing bounds and conditions for specific subgroup properties.

Contribution

It introduces new bounds on the structure of finite groups based on commuting probabilities and automorphism actions, extending understanding of group automorphisms and subgroup interactions.

Findings

Bounded index of $F_2([G,A])$ in $[G,A]$ under commuting probability conditions

Finite groups with automorphisms are bounded-by-abelian-by-bounded if certain commuting probabilities are high

Results connect subgroup commuting probabilities with the overall structure of the group

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 $\epsilon>0$. We show that, if for any distinct primes $p,q\in\pi(G)$ there is an $A$-invariant Sylow $p$-subgroup $P$ and an $A$-invariant Sylow $q$-subgroup $Q$ of $G$ for which $Pr([P,A],[Q,A])\ge\epsilon$, then $F_2([G,A])$ has $\epsilon$-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 $\Pr([P,A], [P,A]^x)\geq\epsilon$ for all $x\in G$, then $G$ is bounded-by-abelian-by-bounded (Theorem 1.4).