# 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 $\struct {H_1, \circ {\restriction_{H_1} } }, \struct {H_2, \circ {\restriction_{H_2} } }$ be subgroups of a group $\struct {G, \circ}$

where $\circ {\restriction_{H_1} }, \circ {\restriction_{H_2} }$ are the restrictions of $\circ$ to $H_1, H_2$ respectively.

### Definition 1

The group $\struct {G, \circ}$ is the **internal group direct product of $H_1$ and $H_2$** if and only if the mapping:

- $C: H_1 \times H_2 \to G: \map C {h_1, h_2} = h_1 \circ h_2$

is a group isomorphism from the (group) direct product $\struct {H_1, \circ {\restriction_{H_1} } } \times \struct {H_2, \circ {\restriction_{H_2} } }$ onto $\struct {G, \circ}$.

### Definition 2

The group $\struct {G, \circ}$ is the **internal group direct product of $H_1$ and $H_2$** if and only if:

- $(1): \quad \struct {H_1, \circ {\restriction_{H_1} } }$ and $\struct {H_2, \circ {\restriction_{H_2} } }$ are both normal subgroups of $\struct {G, \circ}$

### Definition 3

The group $\struct {G, \circ}$ is the **internal group direct product of $H_1$ and $H_2$** if and only if:

- $(1): \quad \struct {H_1, \circ {\restriction_{H_1} } }$ and $\struct {H_2, \circ {\restriction_{H_2} } }$ are both normal subgroups of $\struct {G, \circ}$

- $(2): \quad G$ is the subset product of $H_1$ and $H_2$, that is: $G = H_1 \circ H_2$

- $(3): \quad$ $H_1 \cap H_2 = \set e$ where $e$ is the identity element of $G$.

## Subcategories

This category has the following 3 subcategories, out of 3 total.

### D

### I

## 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 Direct Product Theorem/General Result
- Internal Group Direct Product Commutativity
- Internal Group Direct Product Commutativity/General Result
- 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