# Thompson-like characterization of solubility for products of finite groups

## TL;DR

This paper extends Thompson's theorem on the solubility of finite groups by characterizing when a group formed as a product of subgroups is soluble based on the solubility of their commutator, with implications for factorized groups.

## Contribution

It introduces a new criterion for solubility in factorized groups, linking the group's structure to the solubility of the commutator subgroup, expanding the understanding of local-global properties in group theory.

## Key findings

- A finite group G = AB is soluble if and only if [A,B] is soluble.
- The characterization applies to groups with subgroups A, B such that all two-generated subgroups are soluble.
- Derived a new result about independent primes in the soluble graph of almost simple 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 $\langle a, b\rangle$ is soluble for all $a \in A$ and $b \in B$. In this case, the group $G$ is said to be an $\cal S$-connected product of the subgroups $A$ and $B$ for the class $\cal S$ of all finite soluble groups. Our main theorem states that $G = AB$ is $\cal S$-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.