  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.
