Products of finite connected subgroups

Abstract

For a non-empty class of groups $\mathcal{L}$, a finite group $G=AB$ is said to be an $\mathcal{L}$-connected product of the subgroups $A$ and $B$ if $⟨a,b⟩\in \mathcal{L}$ for all $a\in A$ and $b\in B$. In a previous paper, we prove that for such a product, when $\mathcal{L}=\mathcal{S}$ is the class of finite soluble groups, then $[A,B]$ is soluble. This generalizes the theorem of Thompson which states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper our result is applied to extend to finite groups previous research in the soluble universe. In particular, we characterize connected products for relevant classes of groups; among others the class of metanilpotent groups and the class of groups with nilpotent derived subgroup. Also we give local descriptions of relevant subgroups of finite groups.