Stabilizer 2002-01-21 22:30:49 2003-03-23 04:31:46 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 $G$ be a group, $X$ a set, and $\cdot: G \times X \longrightarrow X$ a group action. For any subset $S$ of $X$, the {\em stabilizer} of $S$, denoted $\operatorname{Stab}(S)$, is the subgroup $$ \operatorname{Stab}(S) := \{g \in G \mid g\cdot s \in S \text{for all }\ s \in S\}. $$ The stabilizer of a single point $x$ in $X$ is often denoted $G_x$.