group action GroupAction 2002-01-21 22:23:01 2006-09-01 00:41:51 Definition effective effective group action faithful faithful group action transitive transitive group action left action right action faithfully action act on acts on % 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 $X$ be a set. A left {\em group action} is a function $\cdot: G \times X \longrightarrow X$ such that: \begin{enumerate} \item $1_G \cdot x = x$ for all $x \in X$ \item $(g_1 g_2)\cdot x = g_1 \cdot (g_2 \cdot x)$ for all $g_1, g_2 \in G$ and $x \in X$ \end{enumerate} A right {\em group action} is a function $\cdot: X \times G \longrightarrow X$ such that: \begin{enumerate} \item $x \cdot 1_G = x$ for all $x \in X$ \item $x \cdot (g_1 g_2) = (x \cdot g_1) \cdot g_2$ for all $g_1, g_2 \in G$ and $x \in X$ \end{enumerate} There is a correspondence between left actions and right actions, given by associating the right action $x \cdot g$ with the left action $g \cdot x := x \cdot g^{-1}$. In many (but not all) contexts, it is useful to identify right actions with their corresponding left actions, and speak only of left actions. {\bf Special types of group actions} A left action is said to be {\em effective}, or {\em faithful}, if the function $x \mapsto g \cdot x$ is the identity function on $X$ only when $g = 1_G$. A left action is said to be {\em transitive} if, for every $x_1,x_2 \in X$, there exists a group element $g \in G$ such that $g \cdot x_1 = x_2$. A left action is {\em free} if, for every $x \in X$, the only element of $G$ that stabilizes $x$ is the identity; that is, $g \cdot x = x$ implies $g = 1_G$. Faithful, transitive, and free right actions are defined similarly.