Definition:Internal Group Direct Product/Definition by Unique Expression

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 $H_2$ :


 * $(1): \quad \struct {H_1, \circ {\restriction_{H_1} } }$ and $\struct {H_2, \circ {\restriction_{H_2} } }$ are both normal subgroups of $\struct {G, \circ}$


 * $(2): \quad$ every element of $G$ can be expressed uniquely in the form:
 * $g = h_1 \circ h_2$

Also see

 * Equivalence of Definitions of Internal Group Direct Product


 * Internal Direct Product Theorem