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.