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:


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

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 $$\left({G, \circ}\right)$$.