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 $G$.
Subcategories
This category has the following 5 subcategories, out of 5 total.
Pages in category "Internal Group Direct Products"
The following 13 pages are in this category, out of 13 total.
I
- Internal and External Group Direct Products are Isomorphic
- Internal Direct Product Generated by Subgroups
- Internal Direct Product Theorem
- Internal Group Direct Product Commutativity
- Internal Group Direct Product is Injective
- Internal Group Direct Product is Injective/General Result
- Internal Group Direct Product Isomorphism
- Internal Group Direct Product of Normal Subgroups