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.

Title	external direct product of groups
Canonical name	ExternalDirectProductOfGroups
Date of creation	2013-03-22 12:23:17
Last modified on	2013-03-22 12:23:17
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	8
Author	mathcam (2727)
Entry type	Definition
Classification	msc 20K25
Synonym	direct product
Related topic	CategoricalDirectProduct
Related topic	DirectProductAndRestrictedDirectProductOfGroups