\begin{definition}[Definition:Group Direct Product]
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 $H$.
\end{definition}