Definition:Group Direct Product

Let $$\left({G, \circ_1}\right)$$ and $$\left({H, \circ_2}\right)$$ be groups.

The (external) direct product of $$\left({G, \circ_1}\right)$$ and $$\left({H, \circ_2}\right)$$ is the set of ordered pairs:


 * $$\left({G \times H, \circ}\right) = \left\{{\left({g, h}\right): g \in G, h \in H}\right\}$$

where the operation $$\circ$$ is defined as:


 * $$\left({g_1, h_1}\right) \circ \left({g_2, h_2}\right) = \left({g_1 \circ_1 g_2, h_1 \circ_2 h_2}\right)$$

This is usually referred to as the group direct product of $$G$$ and $$H$$.

Comment
Although this is just a more specific example of the external direct product of general algebraic structures, it is usually defined and treated separately because of its considerable conceptual importance.

Note that $$G$$ and $$H$$ are not subsets of $$G \times H$$ and therefore are not subgroups of it either.