Cayley's theorem CayleysTheorem 2002-02-19 07:10:56 2002-04-14 18:53:28 Theorem % 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 Let $G$ be a group, then $G$ is isomorphic to a subgroup of the permutation group $S_{G}$ If $G$ is finite and of order $n$, then $G$ is isomorphic to a subgroup of the permutation group $S_{n}$ Furthermore, suppose $H$ is a proper subgroup of $G$. Let $X = \{Hg | g \in G\}$ be the set of right cosets in $G$. The map $\theta:G \to S_{X}$ given by $\theta(x)(Hg) = Hgx$ is a homomorphism. The kernel is the largest normal subgroup of $H$. We note that $|S_X| = [G : H]!$. Consequently if $|G|$ doesn't divide $[G : H]!$ then $\theta$ is not an isomorphism so $H$ contains a non-trivial normal subgroup, namely the kernel of $\theta$.