external direct product of groups

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

$(g,h)(g^{\prime },h^{\prime })=(g{g}^{\prime },h{h}^{\prime })$

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 $G=G_1\times \mathrm{\dots }\times G_n$ be the set of all ordered n-tuples $\{(g_1,g_2\mathrm{\dots },g_n)\mid g_i\in G_i\}$ and define the group operation by componentwise multiplication as before.