subnormal series SubnormalSeries 2003-10-04 11:16:01 2007-04-03 11:29:49 Definition composition series normal series principal series chief series % 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{term} \PMlinkescapeword{compatible} \PMlinkescapeword{series} \PMlinkescapeword{addition} Let $G$ be a group with a subgroup $H$, and let \begin{equation} G = G_0 \rhd G_1 \rhd \cdots \rhd G_n = H \end{equation} be a series of subgroups with each $G_i$ a normal subgroup of $G_{i-1}$. Such a series is called a \emph{subnormal series} or a \emph{subinvariant series}. If in addition, each $G_i$ is a normal subgroup of $G$, then the series is called a \emph{normal series}. A subnormal series in which each $G_i$ is a maximal normal subgroup of $G_{i-1}$ is called a \emph{composition series}. A normal series in which $G_i$ is a maximal normal subgroup of $G$ contained in $G_{i-1}$ is called a \emph{principal series} or a \emph{chief series}. Note that a composition series need not end in the trivial group $1$. One speaks of a composition series (1) as a \emph{composition series from $G$ to $H$}. But the term \emph{composition series for $G$} generally means a composition series from $G$ to $1$. Similar remarks apply to principal series. Some authors use normal series as a synonym for subnormal series. This usage is, of course, not compatible with the stronger definition of normal series given above.