Definition:Group Direct Product

Definition
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 $$H$$.

Generalized Definition
Let $$\left({G_1, \circ_1}\right), \left({G_2, \circ_2}\right), \ldots, \left({G_n, \circ_n}\right)$$ be groups.

Let $$G = \prod_{k=1}^n G_k$$ be as defined in generalized cartesian product.

The operation induced on $$G$$ by $$\circ_1, \ldots, \circ_n$$ is the operation $$\circ$$ defined by:


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

for all ordered $n$-tuples in $$G$$.

The group $$\left({G, \circ}\right)$$ is called the (external) direct product of $$\left({G_1, \circ_1}\right), \left({G_2, \circ_2}\right), \ldots, \left({G_n, \circ_n}\right)$$.

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