\begin{definition}[Definition:Internal Group Direct Product/Definition by Isomorphism]
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}} the mapping $\phi: H \times K \to G$ defined as:
:$\forall h \in H, k \in K: \map \phi {h, k} = h \circ k$
is a group isomorphism from the '''(external) group direct product''' $\struct {H, \circ {\restriction_H} } \times \struct {K, \circ {\restriction_K} }$ onto $\struct {G, \circ}$.
\end{definition}