A remarkable result of Thompson states that a finite group is soluble if and only if all 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 \({{\mathcal {S}}}\)-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 about independent primes regarding the soluble graph of almost simple groups that might be interesting in its own right.

汤普森（Thompson）的一项非凡结果表明，当且仅当其两个生成的所有子组都可溶时，一个有限基团才可溶。该结果已通过多种方式进行了概括，并且它是群体理论研究的广泛领域的核心，旨在从两个生成的（或更一般而言，n生成的）子组的局部特性中获取群体的全局特性。。我们从因式分解组的角度为汤普森定理的扩展做出了贡献。更精确地讲，我们研究有限组\（G = AB \）和子组\（A，\，B \）使得\（\ langle a，b \ rangle \）对于所有\（a \ in A \）都是可溶的和\（b \ in B \）。在这种情况下，组G据说是所有有限可溶基团\（{\ mathcal {S}} \\}类的子组A和B的\（{{\ mathcal {S}}} \\）连通产品。我们的主定理指出，当且仅当[ A，  B ]可溶时，\（G = AB \）是\（{\ mathcal {S}} \\）连通。在证明的过程中，我们得出关于几乎简单的组的可溶图的独立素数的结果，这些结果本身可能会很有趣。