Definition:Internal Group Direct Product/Definition by Isomorphism

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 $K$ 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 $\struct {G, \circ}$.

Also see

 * Equivalence of Definitions of Internal Group Direct Product


 * Internal Direct Product Theorem