hw14-subgroup-solution: +solutions to problems 1 and 3

2021/solution/hw14-subgroup-solution/README.md

@@ -0,0 +1 @@
+# README.md

2021/solution/hw14-subgroup-solution/hw-preamble.tex

@@ -0,0 +1,199 @@
+% hw-preamble.tex
+
+% geometry for A4 paper
+% See https://tex.stackexchange.com/a/119912/23098
+\geometry{
+  left=20.0mm,
+  top=20.0mm,
+  bottom=20.0mm,
+  textwidth=130mm, % main text block
+  marginparsep=5.0mm, % gutter between main text block and margin notes
+  marginparwidth=50.0mm % width of margin notes
+}
+
+% for colors
+\usepackage{xcolor} % usage: \color{red}{text}
+% predefined colors
+\newcommand{\red}[1]{\textcolor{red}{#1}} % usage: \red{text}
+\newcommand{\blue}[1]{\textcolor{blue}{#1}}
+\newcommand{\teal}[1]{\textcolor{teal}{#1}}
+\newcommand{\R}{\mathbb{R}}
+
+\usepackage{todonotes}
+
+% heading
+\usepackage{sectsty}
+\setcounter{secnumdepth}{2}
+\allsectionsfont{\centering\huge\rmfamily}
+
+% for Chinese
+\usepackage{xeCJK}
+\usepackage{zhnumber}
+\setCJKmainfont[BoldFont=FandolSong-Bold.otf]{FandolSong-Regular.otf}
+
+% for fonts
+\usepackage{fontspec}
+\newcommand{\song}{\CJKfamily{song}}
+\newcommand{\kai}{\CJKfamily{kai}}
+
+% To fix the ``MakeTextLowerCase'' bug:
+% See https://github.com/Tufte-LaTeX/tufte-latex/issues/64#issuecomment-78572017
+% Set up the spacing using fontspec features
+\renewcommand\allcapsspacing[1]{{\addfontfeature{LetterSpace=15}#1}}
+\renewcommand\smallcapsspacing[1]{{\addfontfeature{LetterSpace=10}#1}}
+
+% for url
+\usepackage{hyperref}
+\hypersetup{colorlinks = true,
+  linkcolor = teal,
+  urlcolor  = teal,
+  citecolor = blue,
+  anchorcolor = blue}
+
+\newcommand{\me}[4]{
+    \author{
+      {\bfseries 姓名：}\underline{#1}\hspace{2em}
+      {\bfseries 学号：}\underline{#2}\hspace{2em}\\[10pt]
+      {\bfseries 评分：}\underline{#3\hspace{3em}}\hspace{2em}
+      {\bfseries 评阅：}\underline{#4\hspace{3em}}
+  }
+}
+
+% Please ALWAYS Keep This.
+\newcommand{\noplagiarism}{
+  \begin{center}
+    \fbox{\begin{tabular}{@{}c@{}}
+      请独立完成作业，不得抄袭。\\
+      若得到他人帮助, 请致谢。\\
+      若参考了其它资料，请给出引用。\\
+      鼓励讨论，但需独立书写解题过程。
+    \end{tabular}}
+  \end{center}
+}
+
+% \newcommand{\goal}[1]{
+%   \begin{center}{\fcolorbox{blue}{yellow!60}{\parbox{0.50\textwidth}{\large
+%     \begin{itemize}
+%       \item 体会``思维的乐趣''
+%       \item 初步了解递归与数学归纳法
+%       \item 初步接触算法概念与问题下界概念
+%     \end{itemize}}}}
+%   \end{center}
+% }
+
+% Each hw consists of four parts:
+\newcommand{\beginrequired}{\hspace{5em}\section{作业 (必做部分)}}
+\newcommand{\beginoptional}{\section{作业 (选做部分)}}
+\newcommand{\beginot}{\section{Open Topics}}
+\newcommand{\begincorrection}{\section{订正}}
+\newcommand{\beginfb}{\section{反馈}}
+
+% for math
+\usepackage{amsmath, mathtools, amsfonts, amssymb}
+\newcommand{\set}[1]{\{#1\}}
+\newcommand{\ps}[1]{\mathcal{P}(#1)}
+
+% define theorem-like environments
+\usepackage[amsmath, thmmarks]{ntheorem}
+
+\theoremstyle{break}
+\theorempreskip{2.0\topsep}
+\theorembodyfont{\song}
+\theoremseparator{}
+\newtheorem{problem}{题目}[subsection]
+\renewcommand{\theproblem}{\arabic{problem}}
+\newtheorem{definition}{定义}[subsection]
+\renewcommand{\thedefinition}{\arabic{definition}}
+\newtheorem{theorem}{定理}[subsection]
+\renewcommand{\thetheorem}{\arabic{theorem}}
+\newtheorem{ot}{Open Topics}
+
+\theorempreskip{3.0\topsep}
+\theoremheaderfont{\kai\bfseries}
+\theoremseparator{：}
+\theorempostwork{\bigskip\hrule}
+\newtheorem*{solution}{解答}
+\theorempostwork{\bigskip\hrule}
+\newtheorem*{revision}{订正}
+
+\theoremstyle{plain}
+\newtheorem*{cause}{错因分析}
+\newtheorem*{remark}{注}
+
+\theoremstyle{break}
+\theorempostwork{\bigskip\hrule}
+\theoremsymbol{\ensuremath{\Box}}
+\newtheorem*{proof}{证明}
+
+% \newcommand{\ot}{\blue{\bf [OT]}}
+
+% for figs
+\renewcommand\figurename{图}
+\renewcommand\tablename{表}
+
+% for fig without caption: #1: width/size; #2: fig file
+\newcommand{\fig}[2]{
+  \begin{figure}[htbp]
+    \centering
+    \includegraphics[#1]{#2}
+  \end{figure}
+}
+% for fig with caption: #1: width/size; #2: fig file; #3: caption
+\newcommand{\figcap}[3]{
+  \begin{figure}[htbp]
+    \centering
+    \includegraphics[#1]{#2}
+    \caption{#3}
+  \end{figure}
+}
+% for fig with both caption and label: #1: width/size; #2: fig file; #3: caption; #4: label
+\newcommand{\figcaplbl}[4]{
+  \begin{figure}[htbp]
+    \centering
+    \includegraphics[#1]{#2}
+    \caption{#3}
+    \label{#4}
+  \end{figure}
+}
+% for margin fig without caption: #1: width/size; #2: fig file
+\newcommand{\mfig}[2]{
+  \begin{marginfigure}
+    \centering
+    \includegraphics[#1]{#2}
+  \end{marginfigure}
+}
+% for margin fig with caption: #1: width/size; #2: fig file; #3: caption
+\newcommand{\mfigcap}[3]{
+  \begin{marginfigure}
+    \centering
+    \includegraphics[#1]{#2}
+    \caption{#3}
+  \end{marginfigure}
+}
+
+\usepackage{fancyvrb}
+
+% for algorithms
+\usepackage[]{algorithm}
+\usepackage[]{algpseudocode} % noend
+% See [Adjust the indentation whithin the algorithmicx-package when a line is broken](https://tex.stackexchange.com/a/68540/23098)
+\newcommand{\algparbox}[1]{\parbox[t]{\dimexpr\linewidth-\algorithmicindent}{#1\strut}}
+\newcommand{\hStatex}[0]{\vspace{5pt}}
+\makeatletter
+\newlength{\trianglerightwidth}
+\settowidth{\trianglerightwidth}{$\triangleright$~}
+\algnewcommand{\LineComment}[1]{\Statex \hskip\ALG@thistlm \(\triangleright\) #1}
+\algnewcommand{\LineCommentCont}[1]{\Statex \hskip\ALG@thistlm%
+  \parbox[t]{\dimexpr\linewidth-\ALG@thistlm}{\hangindent=\trianglerightwidth \hangafter=1 \strut$\triangleright$ #1\strut}}
+\makeatother
+
+% for footnote/marginnote
+% see https://tex.stackexchange.com/a/133265/23098
+\usepackage{tikz}
+\newcommand{\circled}[1]{%
+  \tikz[baseline=(char.base)]
+  \node [draw, circle, inner sep = 0.5pt, font = \tiny, minimum size = 8pt] (char) {#1};
+}
+\renewcommand\thefootnote{\protect\circled{\arabic{footnote}}}
+
+\newcommand{\score}[1]{{\bf [#1 分]}}

2021/solution/hw14-subgroup-solution/hw14-subgroup-solution.tex

@@ -0,0 +1,175 @@
+% hw14-subgroup-solution.tex
+
+% !TEX program = xelatex
+%%%%%%%%%%%%%%%%%%%%
+% see http://mirrors.concertpass.com/tex-archive/macros/latex/contrib/tufte-latex/sample-handout.pdf
+% for how to use tufte-handout
+\documentclass[a4paper, justified]{tufte-handout}
+
+\input{hw-preamble} % feel free to modify this file if you understand LaTeX well
+%%%%%%%%%%%%%%%%%%%%
+\title{14. 群论: 子群 (14-subgroup)}
+\me{魏恒峰}{hfwei@nju.edu.cn}{}{}
+\date{2021年06月11日 发布作业 \\ 2021年06月23日 发布答案}
+%%%%%%%%%%%%%%%%%%%%
+\begin{document}
+\maketitle
+%%%%%%%%%%%%%%%%%%%%
+\noplagiarism % PLEASE DON'T DELETE THIS LINE!
+%%%%%%%%%%%%%%%%%%%%
+\begin{abstract}
+\end{abstract}
+%%%%%%%%%%%%%%%%%%%%
+\beginrequired
+%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%
+\begin{problem}[\score{4} $\star\star$]
+  设 $H \le G$。请证明,
+  \[
+    aH = H \iff a \in H \iff aH \le G
+  \]
+\end{problem}
+
+\begin{proof}
+  \begin{enumerate}[(1)]
+    \item 先证明 $aH = H \iff a \in H$。
+      \begin{itemize}
+        \item 先证明 $aH = H \implies a \in H$。\\
+          假设 $aH = H$。
+          \[
+            a = ae \in aH = H.
+          \]
+        \item 再证明 $a \in H \implies aH = H$。\\
+          假设 $a \in H$。
+          首先, 由于 $H \le G$ 满足封闭性, 所以 $aH \subseteq H$。
+          其次, 对于任意 $h \in H$, 由于 $H \le G$ 满足封闭性,
+          所以 $a^{-1}h \in H$ 且
+          \[
+            h = a (a^{-1}h) \in aH.
+          \]
+          因此, $H \subseteq aH$。
+      \end{itemize}
+    \item 再证明 $a \in H \iff aH \le G$。\\
+      \begin{itemize}
+        \item 先证明 $a \in H \implies aH \le G$。\\
+          根据 (1),
+          \[
+            a \in H \implies aH = H \implies aH \le G.
+          \]
+        \item 再证明 $aH \le G \implies a \in H$。
+          \[
+            aH \le G \implies e \in aH \implies a^{-1} \in H \implies a \in H.
+          \]
+      \end{itemize}
+  \end{enumerate}
+\end{proof}
+%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%
+\begin{problem}[\score{5 = 2 + 3} $\star\star\star$]
+  设 $\phi$ 是从群 $G$ 到 $G'$ 的同态映射。
+  请证明,
+  \begin{enumerate}[(1)]
+    \item
+      \[
+        H \le G \implies \phi(H) \le G'.
+      \]
+    \item
+      \[
+        H \triangleleft G \implies \phi(H) \triangleleft G'.
+      \]
+  \end{enumerate}
+\end{problem}
+
+\begin{proof}
+\end{proof}
+%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%
+\begin{problem}[\score{3} $\star\star$]
+  请计算
+  \[
+    \begin{pmatrix}
+      1 & 2 & 3 & 4 & 5 \\
+      1 & 3 & 4 & 5 & 2
+    \end{pmatrix}
+    \begin{pmatrix}
+      1 & 2 & 3 & 4 & 5 \\
+      3 & 2 & 4 & 1 & 5
+    \end{pmatrix},
+  \]
+  并将结果写成(不相交)轮换的乘积。
+\end{problem}
+
+\begin{solution}
+  \[
+    \begin{pmatrix}
+      1 & 2 & 3 & 4 & 5 \\
+      1 & 3 & 4 & 5 & 2
+    \end{pmatrix}
+    \begin{pmatrix}
+      1 & 2 & 3 & 4 & 5 \\
+      3 & 2 & 4 & 1 & 5
+    \end{pmatrix} =
+    \begin{pmatrix}
+      1 & 2 & 3 & 4 & 5 \\
+      4 & 3 & 5 & 1 & 2
+    \end{pmatrix}
+    = (1\; 4) (2\; 3\; 5)
+  \]
+\end{solution}
+%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%
+\begin{problem}[\score{3} $\star\star\star$]
+  考虑如下定义。
+  \begin{definition}[元素的阶]
+    设 $G$ 是有限群, $e$ 为 $G$ 的单位元, $a \in G$。
+    使 $a^{r} = e$ 成立的最小正整数称为 $a$ 的阶
+    (order)~\footnote{注意, 群的阶指的是集合 $G$ 的大小, 即 $|G|$。},
+    记作 $\text{ord}\; a = r$。
+  \end{definition}
+  设 $G$ 是有限群。请证明,
+  \[
+    \forall a \in G.\; (\text{ord}\; a) \big\vert |G|.
+  \]
+\end{problem}
+
+\begin{proof}
+\end{proof}
+%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%
+\begin{problem}[\score{5 = 2 + 1 + 2} $\star\star\star$]
+  考虑从乘法群 $\R^{\ast} = \R \setminus \set{0}$
+  到乘法群 $\R^{+}$ 的函数 $f: x \mapsto |x|$。
+  \begin{enumerate}[(1)]
+    \item 请证明, $f$ 是从 $\R^{\ast}$ 到 $\R^{+}$ 的同态。
+    \item 求 $\text{Ker}\; \phi$。
+    \item 套用群同态基本定理, 给出相应结论, 并用一两句话解释该结论。
+  \end{enumerate}
+\end{problem}
+
+\begin{proof}
+\end{proof}
+%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%
+% 如果没有需要订正的题目，可以把这部分删掉
+\begincorrection
+%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%
+% 如果没有反馈，可以把这部分删掉
+\beginfb
+
+你可以写 (也可以发邮件或者使用``教学立方'')
+\begin{itemize}
+  \item 对课程及教师的建议与意见
+  \item 教材中不理解的内容
+  \item 希望深入了解的内容
+  \item $\cdots$
+\end{itemize}
+%%%%%%%%%%%%%%%%%%%%
+\end{document}