# Definition:Internal Group Direct Product/General Definition

## 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 $i \in \set {1, 2, \ldots, n}$.

## 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.

## Also see

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