Definition:Internal Group Direct Product/Decomposition

Definition
Let $\struct {G_1, \circ {\restriction_{G_1} } }, \struct {G_2, \circ {\restriction_{G_2} } }, \ldots, \struct {G_n, \circ {\restriction_{G_n} } }$ be subgroups of a group $\struct {G, \circ}$

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 $\struct {G, \circ}$ be the internal group direct product of $G_1$, $G_2, \ldots, G_n$.

The set of subgroups $\struct {G_1, \circ {\restriction_{G_1} } }, \struct {G_2, \circ {\restriction_{G_2} } }, \ldots, \struct {G_n, \circ {\restriction_{G_n} } }$ whose group direct product is isomorphic with $\struct {G, \circ}$ is called a decomposition of $G$.