The external direct product $G \times H$ of two groups $G$ and $H$ is defined to be the set of ordered pairs $(g,h)$ with $g\in G$ and $h\in H$ The group operation is defined by

$(g,h)(g',h') = (gg', hh')$ It can be shown that $G \times H$ obeys the group axioms. More generally, we can define the external direct product of $n$ groups, in the obvious way. Let $G = G_1 \times \ldots \times G_n$ be the set of all ordered n-tuples $\{(g_1, g_2 \ldots ,g_n) \mid g_i \in G_i\}$ and define the group operation by componentwise multiplication as before.