FaithfulGroupAction 2003-10-15 01:31:06 2005-07-26 23:16:46 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 $A$ be a $G$-set, that is, a set acted upon by a group $G$ with action $\psi:G\times A\to A$. Then for any $g\in G$, the map $m_g\colon A\to A$ defined by \[m_g(x)= \psi(g,x)\] is a permutation of $A$ (in other words, a bijective function from $A$ to itself) and so an element of $S_A$. We can even get an homomorphism from $G$ to $S_A$ by the rule $g\mapsto m_g$. If for any pair $g,h\in G$ $g\neq h$ we have $m_g\neq m_h$, in other words, the homomorphism $g\to m_g$ being injective, we say that the action is faithful.