Definition:Internal Group Direct Product/General Definition

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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:

$\ds C: \prod_{k \mathop = 1}^n H_k \to G: \map C {h_1, \ldots, h_n} = \prod_{k \mathop = 1}^n h_k$

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

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.

Also see

  • Results about internal group direct products can be found here.