# Definition:Internal Group Direct Product

## Definition

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 $G$.

### Definition by Unique Expression

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 $\struct {G, \circ}$

- $(2): \quad$ every element of $G$ can be expressed uniquely in the form:
- $g = h \circ k$

- where $h \in H$ and $k \in K$.

## 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 by Isomorphism

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

- the mapping $\ds \phi: \prod_{k \mathop = 1}^n H_k \to G$ from the finite Cartesian product $\struct {H_1, \circ {\restriction_{H_1} } } \times \cdots \times \struct {H_n, \circ {\restriction_{H_n} } }$ to $\struct {G, \circ}$ defined as:

- $\ds \forall k \in \set {1, 2, \ldots, n}: \forall s_k \in H_k: \map \phi {h_1, \ldots, h_n} = \prod_{k \mathop = 1}^n h_k$

- is a group isomorphism.

### Definition by Subset Product

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

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

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

- $(3): \quad$ For all $k \in \set {1, 2, \ldots, n}$: $H_k \cap \paren {H_1 \circ H_2 \circ \cdots \circ H_{k - 1} \circ H_{k + 1} \circ \cdots \circ H_n} = H_k \set e$ where $e$ is the identity element of $G$.

### Definition by Unique Expression

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

- $(1): \quad$ 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 $\struct {H_1, \circ {\restriction_{H_1} } }, \struct {H_2, \circ {\restriction_{H_2} } }, \ldots, \struct {H_n, \circ {\restriction_{H_n} } }$ whose group direct product is isomorphic with $\struct {G, \circ}$ is called a **decomposition** of $G$.

## Also known as

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

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.

### $D_4$: Internal Group Direct Product is $\set e \times D_4$

Consider the dihedral group $D_4$, which is the symmetry group of the square.

Suppose $D_4$ is the internal group direct product of two subgroups.

Then those two subgroups are $\set e$ and $D_4$ itself, where $e$ is the identity element of $D_4$.

## Also see

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