Definition:Internal Group Direct Product/Definition by Isomorphism

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

where $\circ {\restriction_{H_1} }, \circ {\restriction_{H_2} }$ are the restrictions of $\circ$ to $H_1, H_2$ respectively.

The group $\struct {G, \circ}$ is the internal group direct product of $H_1$ and $H_2$ the mapping $\phi: H_1 \times H_2 \to G$ defined as:


 * $\forall h_1 \in H_1, h_2 \in H_2: \map \phi {h_1, h_2} = h_1 \circ h_2$

is a group isomorphism from the cartesian product $\struct {H_1, \circ {\restriction_{H_1} } } \times \struct {H_2, \circ {\restriction_{H_2} } }$ onto $\struct {G, \circ}$.

Also see

 * Equivalence of Definitions of Internal Group Direct Product


 * Internal Direct Product Theorem