# Definition:Internal Group Direct Product/Definition by Subset 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.

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 $G$.

## 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

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

## Sources

- 1967: John D. Dixon:
*Problems in Group Theory*... (previous) ... (next): Introduction: Notation