DivisibleGroup 2003-07-23 15:48:38 2003-07-24 10:22:40 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} \usepackage{amsthm} % 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 \newcommand{\mc}{\mathcal} \newcommand{\mb}{\mathbb} \newcommand{\mf}{\mathfrak} \newcommand{\ol}{\overline} \newcommand{\ra}{\rightarrow} \newcommand{\la}{\leftarrow} \newcommand{\La}{\Leftarrow} \newcommand{\Ra}{\Rightarrow} \newcommand{\nor}{\vartriangleleft} \newcommand{\Gal}{\text{Gal}} \newcommand{\GL}{\text{GL}} \newcommand{\Z}{\mb{Z}} \newcommand{\R}{\mb{R}} \newcommand{\Q}{\mb{Q}} \newcommand{\C}{\mb{C}} \newcommand{\<}{\langle} \renewcommand{\>}{\rangle} An abelian group $D$ is said to be divisible if for any $x\in D$, $n\in\Z^+$, there exists an element $x'\in D$ such that $nx'=x$. Some noteworthy facts: \begin{itemize} \item An abelian group is \PMlinkname{injective}{InjectiveModule} (as a $\Z$-module) if and only if it is divisible. \item Every group is isomorphic to a subgroup of a divisible group. \item Any divisible abelian group is isomorphic to the direct sum of its torsion subgroup and $n$ copies of the group of rationals (for some cardinal number $n$). \end{itemize}