conjugacy class ConjugacyClass 2002-02-07 13:35:05 2002-07-25 19:28:51 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 Two elements $g$ and $g'$ of a group $G$ are said to be {\em conjugate} if there exists $h \in G$ such that $g' = hgh^{-1}$. Conjugacy of elements is an equivalence relation, and the equivalence classes of $G$ are called {\em conjugacy classes}. Two subsets $S$ and $T$ of $G$ are said to be {\em conjugate} if there exists $g \in G$ such that $$ T = \{gsg^{-1} \mid s \in S\} \subset G. $$ In this situation, it is common to write $gSg^{-1}$ for $T$ to denote the fact that everything in $T$ has the form $gsg^{-1}$ for some $s \in S$. We say that two subgroups of $G$ are conjugate if they are conjugate as subsets.