Definition 14

Let $G$ be a group with operators and $(H_i)_{i \in I}$ a family of stable subgroups of $G$. $G$ is called the internal restricted sum (or restricted sum) of the family of subgroups $(H_i)$ if every element of $H_i$ is permutable with every element of $H_j$ for $j \neq i$ and the unique homomorphism of $\prod_{i \in I} H_i$ into $G$ whose restriction to each of the $H_i$ is the canonical injection is an isomorphism.

When $I$ is finite, we also say, by an abuse of language, internal direct product (or direct product, or product) instead of internal restricted sum. Every stable subgroup H of G for which there exists a stable subgroup H' of G such that G is the direct product of H and H' is called a direct factor of G.