Category:Internal Group Direct Products

This category contains results about Internal Group Direct Products.
Definitions specific to this category can be found in Definitions/Internal Group Direct Products.

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.

Definition by Isomorphism

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

Definition by Subset Product

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

$(1): \quad \struct {H, \circ {\restriction_H} }$ 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 $G$.