# 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

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

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 $G=G_{1}\times\ldots\times G_{n}$ 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.

Title external direct product of groups ExternalDirectProductOfGroups 2013-03-22 12:23:17 2013-03-22 12:23:17 mathcam (2727) mathcam (2727) 8 mathcam (2727) Definition msc 20K25 direct product CategoricalDirectProduct DirectProductAndRestrictedDirectProductOfGroups