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.

The group $\struct {G, \circ}$ is the internal group direct product of $\sequence {H_n}$ 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 $\struct {G, \circ}$.

Also known as
Some authors call this 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.