ProofOfCayleysTheorem 2002-03-03 11:14:11 2003-05-29 16:39:12 Proof Cayley's theorem 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 Let $G$ be a group, and let $S_G$ be the permutation group of the underlying set $G$. For each $g\in G$, define $\rho_g : G\rightarrow G$ by $\rho_g (h) = gh$. Then $\rho_g$ is invertible with inverse $\rho_{g^{-1}}$, and so is a permutation of the set $G$. Define $\Phi :G\rightarrow S_G$ by $\Phi (g) = \rho_g$. Then $\Phi$ is a homomorphism, since $$(\Phi (gh))(x) = \rho_{gh}(x) = ghx = \rho_g(hx) = (\rho_g\circ\rho_h)(x) = ((\Phi (g))(\Phi (h)))(x)$$ And $\Phi$ is injective, since if $\Phi (g) = \Phi (h)$ then $\rho_g = \rho_h$, so $gx = hx$ for all $x\in X$, and so $g=h$ as required. So $\Phi$ is an embedding of $G$ into its own permutation group. If $G$ is finite of order $n$, then simply numbering the elements of $G$ gives an embedding from $G$ to $S_n$.