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 $H_1 \times H_2$.

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.