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.