Direct Product of Groups

Given groups $G$ and $H$ , their (external) direct product is the set Cartesian product $G\times H$ equipped with componentwise multiplication:

(g_1,h_1)\cdot(g_2,h_2) := (g_1g_2,\;h_1h_2).

With this operation, $G\times H$ is a group, with identity $(e_G,e_H)$ and inverse $(g,h)^{-1}=(g^{-1},h^{-1})$ .

Remarks

The maps $\pi_G:G\times H\to G$ and $\pi_H:G\times H\to H$ given by projection onto components are group homomorphisms. Conversely, to give a homomorphism $K\to G\times H$ is equivalent to giving a pair of homomorphisms $K\to G$ and $K\to H$ .

Examples

$\mathbb{Z}\times\mathbb{Z}$ is the free abelian group of rank $2$ under addition.
$C_2\times C_2$ is the Klein four-group (an abelian group of order $4$ ).
If $G$ and $H$ are finite, then $|G\times H| = |G|\cdot |H|$ .