Proposition 14

Let G be a group with operators and $(H_i)_{i \in I}$ a finite family of stable subgroups of G. For G to be the restricted sum of the family of subgroups $(H_i)$, it is necessary and sufficient that each $H_i$ be normal and that G be the product of the quotient groups $(G/H^i)$, where $H^i$ is the subgroup generated by the $H_j$ for $j \neq i$.

The condition is obviously necessary. Conversely, suppose G is the product of the $K_i = G/H^i$ and let G be identified with the product of the $K_i$. Then $H_i$ is identified with a subgroup of $K_i$, so that, for $i \neq j$, every element of $H_i$ is permutable with every element of $H_j$; on the other hand, $H^i$ is identified with the product of the $K_j$ for $j \neq i$, hence $H_i = K_i$ for all $i$ and G is the direct product of the $H_i$.