Definition:Group Direct Product

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

Let $G \times H : \left\{{\left({g, h}\right): g \in G, h \in H}\right\}$ be their cartesian product.

The (external) direct product of $\left({G, \circ_1}\right)$ and $\left({H, \circ_2}\right)$ is the group $\left({G \times H, \circ}\right)$ 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 $H$.

Also known as
The group direct product is referred to in some sources, when dealing with additive groups, as the (group) direct sum.

In such contexts, the symbol $G \times H$ can often be seen as $G \mathop {\dot +} H$.

On we consider all groups, whatever their nature, to be instances of the same abstract concept, and therefore do not use this notation.

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

There exist subgroups in $G \times H$ which are isomorphic with $G$ and $H$ though, namely:
 * $G \times \left\{e_H\right\}$ and $\left\{e_G\right\} \times H$

where $e_G$ and $e_H$ are identity elements of $G$ and $H$ respectively.

Also see

 * Definition:Direct Sum of Groups
 * Definition:Internal Group Direct Product

Generalizations

 * Definition:External Direct Product on general algebraic structures, of which this is a specific example.
 * Definition:Binary Product (Category Theory)