Definition:Internal Group Direct Product

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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 $\struct {G, \circ}$
$(2): \quad$ every element of $G$ can be expressed uniquely in the form:
$g = h_1 \circ h_2$


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.