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 give $H_1 \circ H_2$ as the normal product of $H_1$ by $H_2$.

Other sources use the term semidirect product.

Some authors call it 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.