# Definition:Internal Group Direct Product

## Definition

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 $G$.

## General Definition

Let $\sequence {H_n} = \struct {H_1, \circ {\restriction_{H_1} } }, \ldots, \struct {H_n, \circ {\restriction_{H_n} } }$ be a (finite) sequence of subgroups of a group $\struct {G, \circ}$

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

### Definition 1

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

- $\displaystyle C: \prod_{k \mathop = 1}^n H_k \to G: \map C {h_1, \ldots, h_n} = \prod_{k \mathop = 1}^n h_k$

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

### Definition 2

- $(1): \quad$ Each $H_1, H_2, \ldots, H_n$ is a normal subgroup of $G$

- $(2): \quad$ Each element $g$ of $G$ can be expressed uniquely in the form:
- $g = h_1 \circ h_2 \circ \cdots \circ h_n$

- where $h_i \in H_i$ for all $i \in \set {1, 2, \ldots, n}$.

## Decomposition

The set of subgroups $\left({G_1, \circ {\restriction_{G_1}}}\right), \left({G_2, \circ {\restriction_{G_2}}}\right), \ldots, \left({G_n, \circ {\restriction_{G_n}}}\right)$ whose group direct product is isomorphic with $\left({G, \circ}\right)$ is called a **decomposition** of $G$.

## Also known as

Some authors refer to the **internal group direct product** $H_1 \times H_2$ as the **normal product** of $H_1$ by $H_2$.

Other sources use the term **semidirect product**.

Some authors call it just the **group direct product**, but it should not be confused with the external group direct product.

Although this is just a more specific example of the internal direct product of general algebraic structures, it is usually defined and treated separately because of its considerable conceptual importance.

## Examples

### $C_2 \times C_3$ is Internal Group Direct Product of $C_6$

The direct product of the cyclic groups $C_2$ and $C_3$ is isomorphic to the cyclic groups $C_6$.

Hence it is seen to be an internal group direct product.

## Also see

- Results about
**(internal) group direct products**can be found here.