Definition:Internal Group Direct Product/Decomposition

Definition
Let $\left({G_1, \circ {\restriction_{G_1}} }\right), \left({G_2, \circ {\restriction_{G_2}} }\right), \ldots, \left({G_n, \circ {\restriction_{G_n}} }\right)$ be subgroups of a group $\left({G, \circ}\right)$

where $\circ {\restriction_{G_1}}, \circ {\restriction_{G_2}}, \ldots, \circ {\restriction_{G_n}}$ are the operations induced by the restrictions of $\circ$ to $G_1, G_2, \ldots, G_n$ respectively.

Let $\left({G, \circ}\right)$ be the internal group direct product of $G_1$, $G_2, \ldots, G_n$.

The set of subgroups $\left({G_1, \circ {\restriction_{G_1}}}\right), \left({G_2, \circ {\restriction_{G_2}}}\right), \ldots, \left({G_n, \circ {\restriction_{G_n}}}\right)$ whose group direct product forms $\left({G, \circ}\right)$ is called a decomposition of $G$.