Definition:Internal Group Direct Product/Decomposition

Definition
Let $\struct {H_1, \circ {\restriction_{H_1} } }, \struct {H_2, \circ {\restriction_{H_2} } }, \ldots, \struct {H_n, \circ {\restriction_{H_n} } }$ be subgroups of a group $\struct {G, \circ}$

where $\circ {\restriction_{H_1} }, \circ {\restriction_{H_2} }, \ldots, \circ {\restriction_{H_n} }$ are the operations induced by the restrictions of $\circ$ to $H_1, H_2, \ldots, H_n$ respectively.

Let $\struct {G, \circ}$ be the internal group direct product of $H_1$, $H_2, \ldots, H_n$.

The set of subgroups $\struct {H_1, \circ {\restriction_{H_1} } }, \struct {H_2, \circ {\restriction_{H_2} } }, \ldots, \struct {H_n, \circ {\restriction_{H_n} } }$ whose group direct product is isomorphic with $\struct {G, \circ}$ is called a decomposition of $G$.