Definition:Group Direct Product

Definition
Let $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ be groups.

Let $G \times H: \set {\tuple {g, h}: g \in G, h \in H}$ be their cartesian product.

The (external) direct product of $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ is the group $\struct {G \times H, \circ}$ where the operation $\circ$ is defined as:


 * $\tuple {g_1, h_1} \circ \tuple {g_2, h_2} = \tuple {g_1 \circ_1 g_2, h_1 \circ_2 h_2}$

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 $H$, and so on, 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:
 * $G \times \set {e_H}$ and $\set {e_G} \times H$

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


 * External Direct Product of Groups is Group

Generalizations

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