Sectionally indecomposable groups

We introduce the notion of sectional indecomposability and study it for finite groups: a group $H$ is sectionally indecomposable if, whenever $H$ is a section of a direct product $A \times B$, then $H$ is already a section of $A$ or of $B$. We show that the study of sectionally indecomposable finite groups reduces to the monolithic case. Our main result is a complete characterisation of sectional indecomposability for monolithic primitive groups: such a group $G$ with $N = \mathrm{soc}(G)$ is sectionally indecomposable if and only if either $N$ is non-abelian, or $N$ is a $p$-group and $O_{p'}(G/N) \neq 1$. The proof relies on the introduction of the notion of an $H$-Frattini module and on the theory of the universal $p$-Frattini cover, together with a result of Griess--Schmid. As a corollary, every monolithic primitive solvable group is sectionally indecomposable. We also discuss the non-primitive case, which appears significantly harder, and highlight open questions concerning monolithic $p$-groups.