Definition:Internal Group Direct Product

Definition
Let $$\left({G_1, \circ \! \restriction_{G_1}}\right), \left({G_2, \circ \! \restriction_{G_2}}\right)$$ be subgroups of a group $$\left({G, \circ}\right)$$

where $$\circ \! \restriction_{G_1}, \circ \! \restriction_{G_2}$$ are the restrictions of $$\circ$$ to $$G_1, G_2$$ respectively.

The group $$\left({G, \circ}\right)$$ is the internal group direct product of $$G_1$$ and $$G_2$$ if the mapping:


 * $$C: G_1 \times G_2 \to G: C \left({\left({g_1, g_2}\right)}\right) = g_1 \circ g_2$$

is a group isomorphism from the cartesian product $$\left({G_1, \circ \! \restriction_{G_1}}\right) \times \left({G_2, \circ \! \restriction_{G_2}}\right)$$ 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 $$G_1 \times G_2$$.

General Definition
Let $$\left({G_1, \circ \! \restriction_{G_1}}\right), \ldots, \left({G_n, \circ \! \restriction_{G_n}}\right)$$ be subgroups of a group $$\left({G, \circ}\right)$$

where $$\circ \! \restriction_{G_1}, \ldots, \circ \! \restriction_{G_n}$$ are the restrictions of $$\circ$$ to $$G_1, \ldots, G_n$$ respectively.

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


 * $$C: \prod_{k=1}^n G_k \to G: C \left({g_1, \ldots, g_n}\right) = \prod_{k=1}^n g_k$$

is a group isomorphism from the cartesian product $$\left({G_1, \circ \! \restriction_{G_1}}\right) \times \cdots \times \left({G_n, \circ \! \restriction_{G_n}}\right)$$ onto $$\left({G, \circ}\right)$$.