\begin{definition}[Definition:Group Direct Product/Finite Product]
Let $\struct {G_1, \circ_1}, \struct {G_2, \circ_2}, \ldots, \struct {G_n, \circ_n}$ be groups.
Let $\ds G = \prod_{k \mathop = 1}^n G_k$ be their cartesian product.
Let $\circ$ be the operation defined on $G$ as:
:$\circ := \tuple {g_1, g_2, \ldots, g_n} \circ \tuple {h_1, h_2, \ldots, h_n} = \tuple {g_1 \circ_1 h_1, g_2 \circ_2 h_2, \ldots, g_n \circ_n h_n}$
for all ordered $n$-tuples in $G$.
The group $\struct {G, \circ}$ is called the '''(external) direct product''' of $\struct {G_1, \circ_1}, \struct {G_2, \circ_2}, \ldots, \struct {G_n, \circ_n}$.
\end{definition}