Definition:Internal Group Direct Product

Definition

Let $\left({H_1, \circ \restriction_{H_1}}\right), \left({H_2, \circ \restriction_{H_2}}\right)$ be subgroups of a group $\left({G, \circ}\right)$

where $\circ \restriction_{H_1}, \circ \restriction_{H_2}$ are the restrictions of $\circ$ to $H_1, H_2$ respectively.

The group $\left({G, \circ}\right)$ is the internal group direct product of $H_1$ and $H_2$ iff the mapping:

$C: H_1 \times H_2 \to G: C \left({\left({h_1, h_2}\right)}\right) = h_1 \circ h_2$

is a group isomorphism from the cartesian product $\left({H_1, \circ \restriction_{H_1}}\right) \times \left({H_2, \circ \restriction_{H_2}}\right)$ onto $\left({G, \circ}\right)$.

It can be seen that the function $C$ is the restriction of the mapping $\circ$ of $G \times G$ to the subset $H_1 \times H_2$.

General Definition

Let $\left \langle {H_n} \right \rangle = \left({H_1, \circ \restriction_{H_1}}\right), \ldots, \left({H_n, \circ \restriction_{H_n}}\right)$ be a sequence of subgroups of a group $\left({G, \circ}\right)$

where $\circ \restriction_{H_1}, \ldots, \circ \restriction_{H_n}$ are the restrictions of $\circ$ to $H_1, \ldots, H_n$ respectively.

The group $\left({G, \circ}\right)$ is the internal group direct product of $\left \langle {H_n} \right \rangle$ iff the mapping:

$\displaystyle C: \prod_{k=1}^n H_k \to G: C \left({h_1, \ldots, h_n}\right) = \prod_{k=1}^n h_k$

is a group isomorphism from the cartesian product $\left({H_1, \circ \restriction_{H_1}}\right) \times \cdots \times \left({H_n, \circ \restriction_{H_n}}\right)$ onto $\left({G, \circ}\right)$.

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.