Definition:Internal Group Direct Product/Definition by Unique Expression

Definition
Let $\struct {H, \circ {\restriction_H} }$ and $\struct {K, \circ {\restriction_K} }$ be subgroups of a group $\struct {G, \circ}$

where $\circ {\restriction_H}$ and $\circ {\restriction_K}$ are the restrictions of $\circ$ to $H, K$ respectively.

The group $\struct {G, \circ}$ is the internal group direct product of $H$ and $K$ :


 * $(1): \quad \struct {H, \circ {\restriction_H} }$ and $\struct {K, \circ {\restriction_K} }$ are both normal subgroups of $\struct {G, \circ}$


 * $(2): \quad$ every element of $G$ can be expressed uniquely in the form:
 * $g = h \circ k$
 * where $h \in H$ and $k \in K$.

Also see

 * Equivalence of Definitions of Internal Group Direct Product


 * Internal Direct Product Theorem