LeftAndRightCosetsInADoubleCoset 2008-12-06 16:15:37 2008-12-06 19:02:11 Theorem double coset % 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 \PMlinkescapeword{right} Let $H$ and $K$ be subgroups of a group $G$. Every double coset $HgK$, with $g \in G$, is a union of \PMlinkname{right}{Coset} or left cosets, since \begin{align*} HgK = \bigcup_{k \in K} Hgk\; = \bigcup_{h \in H} hgK, \end{align*} but these unions need not be disjoint. In particular, from the above equality we cannot say how many right (or left) cosets fit in a double coset. The following proposition aims to clarify this. $\,$ {\bf \PMlinkescapetext{Proposition} -} Let $H$ and $K$ be subgroups of a group $G$ and $g \in G$. We have that \begin{align*} HgK = \bigcup_{[k]\, \in\, (K \cap g^{-1}Hg) \backslash K} Hgk\; = \bigcup_{[h]\, \in\, H / (H \cap gKg^{-1})} hgK \end{align*} hold as disjoint unions. In particular, the number of right and left cosets in $HgK$ is respectively given by \begin{align*} \#(H \backslash HgK) = [K: K \cap g^{-1}Hg]\\ \#(HgK/K) =[H: H \cap gKg^{-1}] \end{align*} \subsection{Remarks} \begin{itemize} \item The number of right and left cosets in a double coset does not coincide in general, not \PMlinkescapetext{even} for double cosets of the form $HgH$. \end{itemize} \begin{thebibliography}{9} \bibitem{krieg} A. Krieg, \emph{\PMlinkescapetext{Hecke algebras}}, Mem. Amer. Math. Soc., no. 435, vol. 87, 1990. \end{thebibliography}