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 $\displaystyle G = \prod_{k=1}^n G_k$ 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.

Also see

 * Internal Group Direct Product