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\documentclass[12pt,thmsa]{report}
\input tcilatex
\begin{document} \mingb
\pagestyle{headings}
\title{\yuan 近\ \, 世\ \, 代\ \, 數\ \\ Modern Algebra}
\author{\fs Elbert A. Walker 艾伯特\thanks{author 作者} }
\date{6 October 1994}
\maketitle
\pskaib
\begin{abstract}
This document describes by example the use of {\tt asaetr.sty}, a \LaTeX\ style that somewhat conforms to the style used in the {\em Transactions of the ASAE}\/. \cite{dunford} An accompanying BiBTeX style file is also used to format the bibliography in a style similar to that used by ASAE.
\end{abstract}
\ming
\tableofcontents%
\listoffigures%
\listoftables%
\chapter[Group Theory 特 論 ]{\yuan 群 論 特 論 Group Theory}
\section[導 論]{\kaib 導 論 Introduction}
\chMagnification 1200
群論是一門領域甚廣的學科，幾乎任何數學訓練都需要它。本章將簡單介紹一些% 古典的 非-Abelian 群論。在{\underbar{第二章}}，我們將介紹同構，正規子群， 商群，直和等基本觀念。 ...
\li 群論是一門領域甚廣的學科，幾乎任何數學訓練都需要它。本章將簡單介紹一些% 古典的 非-Abelian 群論。在{\underbar{第二章}}，我們將介紹同構，正規子群， 商群，直和等基本觀念。 ...
\mingb
\section{\kaib Jordan-Holder 定理}
Let $G$ be a finite group. If $G$ is not simple, then $G$ has a normal subgroup $G_1\neq G$ such that $G/G_1$ is simple\textit{.} Just let $G_1$ be a maximal normal subgroup in the finite group $G$\textit{.} Similarly, $G_1$ has a normal subgroup $G_2$ such that $G_1/G_2$ is simple. Thus we get a descending chain \[ G=G_0\supset G_1\supset G_2\supset \cdots \supset G_n=\{e\} \] such that $G_{i+1}$ is normal in $G_i$, and $G_i/G_{i+1}$ is simple. Now suppose that \[ G=H_0\supset H_1\supset H_2\supset \ldots \supset H_m=\{e\} \] is another such chain of subgroups. The Jordan-Holder Theorem asserts that $% m=n$, and that there is a one-to-one correspondence between the factor groups $G_i/G_{i+1}$ and $H_j/H_{j+1}$ such that corresponding factor groups are isomorphic. We will prove this remarkable theorem, but first some notation and terminology are needed. \textbf{The groups considered are not necessarily finite.}
\begin{definition} \ \, 設 $G$ 為一群。 $G$ 的一正規級數就是一如下之群的鍊鎖: \[ G=G_0\supset G_1\supset G_2\supset \ldots \supset G_n=\{e\} \] 其中，對於每一 $i$,\ $\,$ $G_{i+1}$ 皆為 $G_i$ 之正規子群。 The groups $G_i/G_{i+1}$ are the factor groups of the chain. The length of the chain is the number of strict inclusions in the chain. A normal series is a composition series if each factor group $G_i/G_{i+1}$ is a simple group $% \neq \{e\}$. Two normal series are equivalent if there is a one-to-one correspondence between their factor groups such that corresponding factor groups are isomorphic.
\end{definition}
\begin{theorem}[Jordan-Holder] \ \, Any two composition series of a group are equivalent. \end{theorem}
\proof\ \ If a group has a composition series of length one, then the group is simple and any two composition series are certainly equivalent. Now suppose that a group $G$ has a composition series
\begin{equation} \label{one} G=G_0\supset G_1\supset G_2\supset \ldots \supset G_n=\{e\} \end{equation}
of length $n>1$, and that if a group has a composition series of length less than $n$, then any two composition series of that group are equivalent. Let
\begin{equation} \label{two} G=H_0\supset H_1\supset H_2\supset \ldots \supset H_m=\{e\} \end{equation} be any composition series of $G$. Consider the series
\begin{equation} \label{three} G=G_0\supset G_1\supset G_1\cap H_1\supset G_2\cap H_1\supset \ldots \supset G_n\cap H_1=\{e\} \end{equation} and
\begin{equation} \label{four} G=H_0\supset H_1\supset H_1\cap G_1\supset H_2\cap G_1\supset \ldots \supset H_m\cap G_1=\{e\} \end{equation} Since $G_{i+1}\cap H_1$ is a normal subgroup of $G_i\cap H_1$ and $% G_i\supset G_{i+1}$, the Third Isomorphism Theorem (\textbf{2.3.12}) yields \[ (G_i\cap H_1)/(G_{i+1}\cap H_1)=(G_i\cap H_1)/(G_{i+1}\cap (G_i\cap H_1))\approx \] \[ G_{i+1}(G_i\cap H_1)/G_{i+1}, \] and $G_{i+1}(G_i\cap H_1)$ is a normal subgroup of $G_i$ since it is a product of two normal subgroups. Since $G_i/G_{i+1}$ is a simple group, $% (G_{i+1}(G_i\cap H_1))/G_{i+1}$ is either $G_i/G_{i+1}$ or $G_{i+1}/G_{i+1}$% . That is, $G_{i+1}(G_i\cap H_1)$ is either $G_{i+1}$ or $G_i$. Therefore, if we remove repetitions from \[ G_1\supset (G_1\cap H_1)\supset (G_2\cap H_1)\supset \ldots \supset (G_n\cap H_1)=\{e\}, \] we get a composition series for $G_1$. By our induction hypothesis, the resulting composition series is equivalent to the composition series \[ G_1\supset G_2\supset \ldots \supset G_n=\{e\}\text{,} \] and hence (\ref{one}) and (\ref{three}) (with repetitions removed) are equivalent. $\cdots$
\section{\kaib 問 題 集 Problem set}
\begin{enumerate}
\item 試証: 若 $G$ 有一合成級數，則任何正規子群 $G$ 都有合成級數。
\item 試証: 若 $N$ 在 $G$ 中為正規，且若 $G$ 有一合成級數，則 $G/N$ 亦然。
\item Suppose that $G$ has a composition series and that $N$ is normal in $% G $. Prove that $G$ has a composition series of which $N$ is a member.
\item Suppose that $G$ has a composition series. Prove that any normal series of $G$ has a refinement that is a composition series.
\item \label{zasslemma}(\textbf{Zassenhaus's Lemma}) Let \textit{A} and $B$ be subgroups of a group $G$, and let $M$ and $N$ be normal in \textit{A} and $B$, respectively. Prove that \end{enumerate}
\chapter{\yuan 其 他}
\section{圖形與表格}
\begin{figure}[htb]
\setlength{\unitlength}{0.1mm}
\begin{center}
\begin{picture}(181,181)(0,0) \thinlines \multiput(80,80)(-20,-20){4}{\framebox(80,80){}} \thicklines \put(0,0){\framebox(180,180){}} \put(60,60){\line( 1, 1){ 60}} \end{picture} \end{center} \caption[原始之圖]{原始之圖 Primitive figure.}
\end{figure}
{ \renewcommand{\footnoterule}{}
\begin{table}[hbp]
\footnotesize \vglue0pt
\caption{出版工具 Comparison of Publishing Tools}
\begin{center} \renewcommand{\footnoterule}{} \begin{center} \renewcommand{\thefootnote}{\fnsym{footnote}}
\begin{tabular}{crr}
\thickhline Tool & \multicolumn{1}{c}
{Learning Curve %\footnote{1.0 being easiest} }
& \multicolumn{1}{c}{Support %\footnote{10.0 being the best} }\\
\thinhline
FrameMaker & 5.0 & 6.0 \\ Troff & 10.0 & 1.0 \\
\TeX %\footnote{\TeX\ is the winner!}\/
& 7.0 & 10.0 \\
\thickhline
\end{tabular}
\linethickness{0pt}
\end{center}
\end{center}
\end{table} }
\begin{thebibliography}{9}
\bibitem[杜甫 1964]{dunford} N. Dunford and J. Schwartz, 泛函分析,
\bibitem{} M. Struwe, \emph{Semilinear wave equations}, Bull. Amer. Math. Soc. \textbf{26} (1992), 53-85.
\bibitem{} W.P. Thurston, \emph{Geometry and topology of three manifolds}, Lecture notes, Princeton Univ., NJ, 1979.
\end{thebibliography}
\appendix
\chapter{使 用 ChiTeX}
\begin{theindex} \item 代數 1, 12 \item series 2, 33 \subitem convergent, 22, 55 \indexspace \end{theindex}
\end{document}