Category:Internal Group Direct Products

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This category contains results about Internal Group Direct Products.
Definitions specific to this category can be found in Definitions/Internal Group Direct Products.


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 $G$.