Category:Examples of Internal Group Direct Products
This category contains examples of Internal Group Direct 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.
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 $G$.
Pages in category "Examples of Internal Group Direct Products"
The following 6 pages are in this category, out of 6 total.