symmetric group SymmetricGroup2 2003-11-10 17:44:02 2003-11-29 21:41:34 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 Let $X$ be a set. Let $S(X)$ be the set of permutations of $X$ (i.e. the set of bijective functions on $X$). Then the act of taking the composition of two permutations induces a group structure on $S(X)$. We call this group the {\it symmetric group} and it is often denoted ${\rm Sym}(X)$. When $X$ has a finite number $n$ of elements, we often refer to the symmetric group as $S_n$, and describe the elements by using cycle notation.