Thompson-like characterization of solubility for products of finite groups

Abstract

A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, $n$-generated) subgroups. We contribute an extension of Thompson's theorem from the perspective of factorized groups. More precisely, we study finite groups $G=AB$ with subgroups $A,B$ such that $⟨a,b⟩$ is soluble for all $a\in A$ and $b\in B$. In this case, the group $G$ is said to be an $\mathcal{S}$-connected product of the subgroups $A$ and $B$ for the class $\mathcal{S}$ of all finite soluble groups. Our main theorem states that $G=AB$ is $\mathcal{S}$-connected if and only if $[A,B]$ is soluble. In the course of the proof we derive a result of own interest about independent primes regarding the soluble graph of almost simple groups.