ProofThatAFiniteAbelianGroupHasElementWithLvertGrvertexpG 2007-01-14 14:27:50 2007-01-14 14:27:50 Proof exponent 0 % 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 \newtheorem{thm}{Theorem} \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \newtheorem{ax}{Axiom} \begin{thm} If $G$ is a finite abelian group, then $G$ has an element of order $\exp(G)$. \end{thm} \textbf{Proof. } Write $\exp(G)=\prod p_i^{k_i}$. Since $\exp(G)$ is the least common multiple of the orders of each group element, it follows that for each $i$, there is an element whose order is a multiple of $p_i^{k_i}$, say $\lvert c_i\rvert=a_i p_i^{k_i}$. Let $d_i=c_i^{a_i}$. Then $\lvert d_i\rvert=p_i^{k_i}$. The $d_i$ thus have pairwise relatively prime orders, and thus \[\left\lvert\prod d_i\right\rvert=\prod\left\lvert d_i\right\rvert=\exp(G)\] so that $\prod d_i$ is the desired element.