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 ∈ A and b ∈ B. In this case, the group G is said to be an S-connected product of the subgroups A and B for the class S of all finite soluble groups. Our main theorem states that G = AB is 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.