external direct product of groups DirectProduct2 2002-02-19 07:56:27 2005-08-26 19:47:24 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The \emph{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.