\begin{definition}[Definition:Internal Group Direct Product/Definition by Subset Product]
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 $K$''' {{iff}}:
:$(1): \quad \struct {H, \circ {\restriction_H} }$ and $\struct {K, \circ {\restriction_K} }$ are both normal subgroups of $\struct {G, \circ}$
:$(2): \quad G$ is the subset product of $H$ and $K$, that is: $G = H \circ K$
:$(3): \quad$ $H \cap K = \set e$ where $e$ is the identity element of $G$.
\end{definition}