\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig,amsmath,latexsym}
\usepackage[all]{xy}
\usepackage{tikz}
\usepackage{enumerate}
\usepackage{color}
\definecolor{darkblue}{rgb}{0, 0, .6}
\definecolor{midnightblue}{rgb}{.2, .2, .7}
\definecolor{grey}{rgb}{.7, .7, .7}
\definecolor{lightgrey}{rgb}{.85, .85, .85}
\definecolor{Red}{rgb}{.9,0,0}
\definecolor{Blue}{rgb}{0,0,.9}
\definecolor{Green}{rgb}{0,.9,0}
\usetikzlibrary{arrows}
\usetikzlibrary{decorations.markings}

\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=0in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in

\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\DeclareMathOperator{\im}{Im}
\DeclareMathOperator{\id}{id}

\def\<{\langle}
\def\>{\rangle}
\def\longto{\longrightarrow}

\renewcommand\labelitemi{\rule[0.12em]{0.4em}{0.4em}}
\renewcommand\labelitemii{\normalfont\bfseries \rule[0.15em]{0.3em}{0.3em}}
\renewcommand\labelitemiii{\textasteriskcentered}
\renewcommand\labelitemiv{\textperiodcentered}

\renewcommand{\footrulewidth}{1pt}
\newenvironment{proof}[1][Proof.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{sol}[1][Solution.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{fminipage}%
{\setlength{\fboxsep}{15pt}\begin{Sbox}\begin{minipage}}%
{\end{minipage}\end{Sbox}\fbox{\TheSbox}}
\newcommand{\qed}{\hfill \ensuremath{\Box}}
\newcommand{\disp}{\displaystyle}


\pagestyle{fancy} \lhead{{\sf Homework 4 $|$ Due Wednesday, September 28
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Fall 2019 $|$ M.~Macauley}}



\begin{document}

$\;$

\noindent Read Chapter 6 of \emph{Visual Group Theory}, or Chapters
4.1, 5.4, 5.5, 7 of \emph{IBL Abstract Algebra}, and then write up
solutions to the following exercises. \bigskip

\begin{enumerate}

\item A Cayley diagram and multiplication table for the
  dihedral group $D_4$ are shown below. 

  \begin{center}
  \colorlet{lightgrey}{black!15}
  \tikzstyle{v} = [circle, draw, fill=lightgrey,inner sep=0pt, 
    minimum size=5mm]
  \tikzstyle{b} = [draw, very thick, blue] 
  \tikzstyle{r} = [draw, very thick, Red, -stealth]
  \tikzstyle{H} = [draw,very  thick, blue]
  \tikzstyle{V} = [draw, very thick, red]
  \colorlet{lred}{red!70!white}
  \colorlet{lblue}{blue!65!white}
  \colorlet{lime}{green!80!white}
  \colorlet{lyellow}{yellow!70!white}
  \definecolor{rouge}{rgb}{0.79 0.0 0.1}
  \definecolor{lgreen}{rgb}{0.25 0.7 0.15}
  \definecolor{lpurple}{rgb}{0.6 0.35 0.8}
  %%  
  \begin{tikzpicture}[scale=.6,auto] 
    \node (1) at (135:2.5) [v] {};
    \node (i) at (45:2.5) [v] {};
    \node (-1) at (315:2.5) [v] {};
    \node (-i) at (225:2.5) [v] {};
    \node (k) at (135:5.5) [v] {};
    \node (-j) at (45:5.5) [v] {};
    \node (-k) at (315:5.5) [v] {};
    \node (j) at (225:5.5) [v] {};
    \draw [r] (1) to (i);
    \draw [r] (-k) to (-j);
    \draw [r] (-1) to (-i);
    \draw [r] (k) to (j);
    \draw [r] (j) to (-k);
    \draw [r] (-i) to (1);
    \draw [r] (-j) to (k);
    \draw [r] (i) to (-1);
    \draw [b] (1) to (k);
    \draw [b] (i) to (-j);
    \draw [b] (-i) to (j);
    \draw [b] (-k) to (-1);
    %\node at (0,0) {$D_4$};
  \end{tikzpicture}
  \hspace{25mm}
  \newcommand*{\n}{9}%
  \begin{tikzpicture}[scale=0.6]
    \path[fill=lred] (0,8) rectangle ++(.9,1);
    \path[fill=orange] (0,7) rectangle ++(.9,1);
    \path[fill=lyellow] (0,6) rectangle ++(.9,1);
    \path[fill=lime] (0,5) rectangle ++(.9,1);
    \path[fill=lgreen] (0,4) rectangle ++(.9,1);
    \path[fill=lblue] (0,3) rectangle ++(.9,1);
    \path[fill=lpurple] (0,2) rectangle ++(.9,1);
    \path[fill=pink] (0,1) rectangle ++(.9,1);
    %%
    \path[fill=lred] (1,9.1) rectangle ++(1,.9);
    \path[fill=orange] (2,9.1) rectangle ++(1,.9);
    \path[fill=lyellow] (3,9.1) rectangle ++(1,.9);
    \path[fill=lime] (4,9.1) rectangle ++(1,.9);
    \path[fill=lgreen] (5,9.1) rectangle ++(1,.9);
    \path[fill=lblue] (6,9.1) rectangle ++(1,.9);
    \path[fill=lpurple] (7,9.1) rectangle ++(1,.9);
    \path[fill=pink] (8,9.1) rectangle ++(1,.9);
    %
    \path[fill=lred] (1,8) rectangle ++(1,1);
    \path[fill=lred] (2,5) rectangle ++(1,1);
    \path[fill=lred] (3,6) rectangle ++(1,1);
    \path[fill=lred] (4,7) rectangle ++(1,1);
    \path[fill=lred] (5,4) rectangle ++(1,1);
    \path[fill=lred] (6,3) rectangle ++(1,1);
    \path[fill=lred] (7,2) rectangle ++(1,1);
    \path[fill=lred] (8,1) rectangle ++(1,1);
    %%
    \path[fill=orange] (1,7) rectangle ++(1,1);
    \path[fill=orange] (2,8) rectangle ++(1,1);
    \path[fill=orange] (3,5) rectangle ++(1,1);
    \path[fill=orange] (4,6) rectangle ++(1,1);
    \path[fill=orange] (5,3) rectangle ++(1,1);
    \path[fill=orange] (6,2) rectangle ++(1,1);
    \path[fill=orange] (7,1) rectangle ++(1,1);
    \path[fill=orange] (8,4) rectangle ++(1,1);
    %%
    \path[fill=lyellow] (1,6) rectangle ++(1,1);
    \path[fill=lyellow] (2,7) rectangle ++(1,1);
    \path[fill=lyellow] (3,8) rectangle ++(1,1);
    \path[fill=lyellow] (4,5) rectangle ++(1,1);
    \path[fill=lyellow] (5,2) rectangle ++(1,1);
    \path[fill=lyellow] (6,1) rectangle ++(1,1);
    \path[fill=lyellow] (7,4) rectangle ++(1,1);
    \path[fill=lyellow] (8,3) rectangle ++(1,1);
    %%
    \path[fill=lime] (1,5) rectangle ++(1,1);
    \path[fill=lime] (2,6) rectangle ++(1,1);
    \path[fill=lime] (3,7) rectangle ++(1,1);
    \path[fill=lime] (4,8) rectangle ++(1,1);
    \path[fill=lime] (5,1) rectangle ++(1,1);
    \path[fill=lime] (6,4) rectangle ++(1,1);
    \path[fill=lime] (7,3) rectangle ++(1,1);
    \path[fill=lime] (8,2) rectangle ++(1,1);
    %%
    \path[fill=lgreen] (1,4) rectangle ++(1,1);
    \path[fill=lgreen] (2,3) rectangle ++(1,1);
    \path[fill=lgreen] (3,2) rectangle ++(1,1);
    \path[fill=lgreen] (4,1) rectangle ++(1,1);
    \path[fill=lgreen] (5,8) rectangle ++(1,1);
    \path[fill=lgreen] (6,5) rectangle ++(1,1);
    \path[fill=lgreen] (7,6) rectangle ++(1,1);
    \path[fill=lgreen] (8,7) rectangle ++(1,1);
    %%
    \path[fill=lblue] (1,3) rectangle ++(1,1);
    \path[fill=lblue] (2,2) rectangle ++(1,1);
    \path[fill=lblue] (3,1) rectangle ++(1,1);
    \path[fill=lblue] (4,4) rectangle ++(1,1);
    \path[fill=lblue] (5,7) rectangle ++(1,1);
    \path[fill=lblue] (6,8) rectangle ++(1,1);
    \path[fill=lblue] (7,5) rectangle ++(1,1);
    \path[fill=lblue] (8,6) rectangle ++(1,1);
    %%
    \path[fill=lpurple] (1,2) rectangle ++(1,1);
    \path[fill=lpurple] (2,1) rectangle ++(1,1);
    \path[fill=lpurple] (3,4) rectangle ++(1,1);
    \path[fill=lpurple] (4,3) rectangle ++(1,1);
    \path[fill=lpurple] (5,6) rectangle ++(1,1);
    \path[fill=lpurple] (6,7) rectangle ++(1,1);
    \path[fill=lpurple] (7,8) rectangle ++(1,1);
    \path[fill=lpurple] (8,5) rectangle ++(1,1);
    %%
    \path[fill=pink] (1,1) rectangle ++(1,1);
    \path[fill=pink] (2,4) rectangle ++(1,1);
    \path[fill=pink] (3,3) rectangle ++(1,1);
    \path[fill=pink] (4,2) rectangle ++(1,1);
    \path[fill=pink] (5,5) rectangle ++(1,1);
    \path[fill=pink] (6,6) rectangle ++(1,1);
    \path[fill=pink] (7,7) rectangle ++(1,1);
    \path[fill=pink] (8,8) rectangle ++(1,1);
    %%
    \foreach \i in {1,...,\n} {
      \draw [very thin] (\i,1) -- (\i,\n); 
      \draw [very thin] (\i,\n+.1) -- (\i,\n+1); 
      \draw [very thin] (1,\i) -- (\n,\i); 
      \draw [very thin] (0,\i) -- (.9,\i); 
    } 
    \draw [very thin] (1,\n+.1) rectangle (\n,\n+1);
    \draw [very thin] (0,1) rectangle (.9,\n); 
    \node at (0.5,8.5) {\footnotesize $e$};
    \node at (0.5,7.5) {\footnotesize $r$};
    \node at (0.5,6.5) {\footnotesize $r^2$};
    \node at (0.5,5.5) {\footnotesize $r^3$}; 
    \node at (0.5,4.5) {\footnotesize $f$}; 
    \node at (0.5,3.5) {\footnotesize $rf$};
    \node at (0.5,2.5) {\footnotesize $\!r^2\!f$};
    \node at (0.5,1.5) {\footnotesize $\!r^3\!f$};
    %% 
    \node at (1.5,9.5) {\footnotesize $e$};
    \node at (2.5,9.5) {\footnotesize $r$};
    \node at (3.5,9.5) {\footnotesize $r^2$};
    \node at (4.5,9.5) {\footnotesize $r^3$}; 
    \node at (5.5,9.5) {\footnotesize $f$}; 
    \node at (6.5,9.5) {\footnotesize $rf$};
    \node at (7.5,9.5) {\footnotesize $r^2\!f$};
    \node at (8.5,9.5) {\footnotesize $r^3\!f$};
    %%
    \node at (1.5,8.5) {\footnotesize $e$};
    \node at (1.5,7.5) {\footnotesize $r$};
    \node at (1.5,6.5) {\footnotesize $r^2$};
    \node at (1.5,5.5) {\footnotesize $r^3$}; 
    \node at (1.5,4.5) {\footnotesize $f$}; 
    \node at (1.5,3.5) {\footnotesize $rf$};
    \node at (1.5,2.5) {\footnotesize $r^2\!f$};
    \node at (1.5,1.5) {\footnotesize $r^3\!f$};
    %%
    \node at (2.5,8.5) {\footnotesize $r$};
    \node at (2.5,7.5) {\footnotesize $r^2$};
    \node at (2.5,6.5) {\footnotesize $r^3$};
    \node at (2.5,5.5) {\footnotesize $e$}; 
    \node at (2.5,4.5) {\footnotesize $r^3\!f$}; 
    \node at (2.5,3.5) {\footnotesize $f$};
    \node at (2.5,2.5) {\footnotesize $rf$};
    \node at (2.5,1.5) {\footnotesize $r^2\!f$};
    %%
    \node at (3.5,8.5) {\footnotesize $r^2$};
    \node at (3.5,7.5) {\footnotesize $r^3$};
    \node at (3.5,6.5) {\footnotesize $e$};
    \node at (3.5,5.5) {\footnotesize $r$}; 
    \node at (3.5,4.5) {\footnotesize $r^2\!f$}; 
    \node at (3.5,3.5) {\footnotesize $r^3\!f$};
    \node at (3.5,2.5) {\footnotesize $f$};
    \node at (3.5,1.5) {\footnotesize $rf$};
    %%
    \node at (4.5,8.5) {\footnotesize $r^3$};
    \node at (4.5,7.5) {\footnotesize $e$};
    \node at (4.5,6.5) {\footnotesize $r$};
    \node at (4.5,5.5) {\footnotesize $r^2$}; 
    \node at (4.5,4.5) {\footnotesize $rf$}; 
    \node at (4.5,3.5) {\footnotesize $r^2\!f$};
    \node at (4.5,2.5) {\footnotesize $r^3\!f$};
    \node at (4.5,1.5) {\footnotesize $f$};
    %%
    \node at (5.5,8.5) {\footnotesize $f$};
    \node at (5.5,7.5) {\footnotesize $rf$};
    \node at (5.5,6.5) {\footnotesize $r^2\!f$};
    \node at (5.5,5.5) {\footnotesize $r^3\!f$}; 
    \node at (5.5,4.5) {\footnotesize $e$}; 
    \node at (5.5,3.5) {\footnotesize $r$};
    \node at (5.5,2.5) {\footnotesize $r^2$};
    \node at (5.5,1.5) {\footnotesize $r^3$};
    %%
    \node at (6.5,8.5) {\footnotesize $rf$};
    \node at (6.5,7.5) {\footnotesize $r^2\!f$};
    \node at (6.5,6.5) {\footnotesize $r^3\!f$};
    \node at (6.5,5.5) {\footnotesize $f$}; 
    \node at (6.5,4.5) {\footnotesize $r^3$}; 
    \node at (6.5,3.5) {\footnotesize $e$};
    \node at (6.5,2.5) {\footnotesize $r$};
    \node at (6.5,1.5) {\footnotesize $r^2$};
    %%
    \node at (7.5,8.5) {\footnotesize $r^2\!f$};
    \node at (7.5,7.5) {\footnotesize $r^3\!f$};
    \node at (7.5,6.5) {\footnotesize $f$};
    \node at (7.5,5.5) {\footnotesize $rf$}; 
    \node at (7.5,4.5) {\footnotesize $r^2$}; 
    \node at (7.5,3.5) {\footnotesize $r^3$};
    \node at (7.5,2.5) {\footnotesize $e$};
    \node at (7.5,1.5) {\footnotesize $r$};
    %%
    \node at (8.5,8.5) {\footnotesize $r^3\!f$};
    \node at (8.5,7.5) {\footnotesize $f$};
    \node at (8.5,6.5) {\footnotesize $rf$};
    \node at (8.5,5.5) {\footnotesize $r^2\!f$}; 
    \node at (8.5,4.5) {\footnotesize $r$}; 
    \node at (8.5,3.5) {\footnotesize $r^2$};
    \node at (8.5,2.5) {\footnotesize $r^3$};
    \node at (8.5,1.5) {\footnotesize $e$};
  \end{tikzpicture}
\end{center} \vspace{-1mm}
Section 2 of the class lecture notes describes two algorithms for
expressing a group $G$ of order $n$ as a set of permutations in
$S_n$. One algorithm uses the Cayley diagram and the other uses the
multiplication table. In this problem, you will explore this a bit
further. \vspace{-1mm}
\begin{enumerate}
\item Label the vertices of the Cayley diagram from the set
  $\{1,\dots,8\}$ and use this to construct a permutation group
  isomorphic to $D_4$, and sitting inside $S_8$. 
\item Label the entries of the multiplication table from the set
  $\{1,\dots,8\}$ and use this to construct a permutation group
  isomorphic to $D_4$, and sitting inside $S_8$. 
\item Are the two groups you got in Parts~(a) and (b) the same? (The
  answer will depend on your choice of labeling.) If ``yes'', then
  repeat Part~(a) with a different labeling to yield a different
  group. If ``no'', then repeat Part~(a) with a different labeling to
  yield the group you got in Part~(b). \bigskip
\end{enumerate}



\item Find all subgroups of the following groups, and arrange them in
  a Hasse diagram, or subgroup lattice. Moreover, label each edge
  between $K\leq H$ with the index, $[H:K]$.
  \begin{enumerate}
    \item $C_{23}=\<r\mid r^{23}=1\>$;
    \item $C_{24}=\<r\mid r^{24}=1\>$;
    \item $\Z_3\times\Z_3=\{(a,b)\mid a,b\in\{0,1,2\}\}$; 
    \item $\Z_2\times\Z_2\times\Z_2=\{(a,b,c)\mid a,b,c\in\{0,1\}\}$;
      (\emph{Tip}: it's notationally easier to write elements as
      binary strings, e.g., $abc$ instead of $(a,b,c)$);
    \item $S_3=\{e,(1\,2),(2\,3),(1\,3),(1\,2\,3),(1\,3\,2)\}$;
    \item $A_4=\{e,(123),(132),(124),(142),(134),(143),(234),(243),(12)(34),(13)(24),(14)(23)\}$;
    \item $Q_8=\<i,j,k\mid i^2=j^2=k^2=ijk=-1\>$. \bigskip
  \end{enumerate}

  \newpage


  
% 6.13 (modified)
\item For each subgroup $H$ of $S_4$ described below, determine what
  well-known group it is isomorphic to. Justify your answers. 

  \begin{enumerate}
  \item $H=\<(1\,2)\,,\,(3\,4)\>$; 
  \item $H=\<(1\,2)\,(3\,4)\,,\,(1\,3)\,(2\,4)\>$;
  \item $H=\<(1\,2)\,,\,(2\,3)\>$; 
  \item $H=\<(1\,2)\,,\,(1\,3\,2\,4)\>$;  
  \item $H=\<(1\,2\,3)\,,\,(2\,3\,4)\>$. \bigskip
  \end{enumerate}



\item Prove the following, algebraically (that is, do not refer to
  Cayley diagrams):
  \begin{enumerate}[(a)]
  \item If $\mathcal{H}$ is a collection of subgroups of $G$, then
    $\displaystyle\bigcap_{H_\alpha\in\mathcal{H}}H_\alpha$ is a subgroup of $G$.
  \item For any (possibly infinite) subset $S\subseteq G$, the
    subgroup generated by $S$ is defined as
    \[
    \<S\>:=\{s_1^{e_1}s_2^{e_2}\cdots s_k^{e_k}\mid s_i\in
    S,\;e_i\in\{-1,1\}\}\,.
    \]
    That is, $\<S\>$ consists of all finite ``words'' that can be
    written using the elements in $S$ and their inverses. Note that
    the $s_i$'s need not be distinct. Prove that
    \[
    \<S\>=\bigcap_{S\subseteq H_\alpha\leq G} H_\alpha\,,
    \]
    where the intersection is taken over all subgroups of $G$ that
    contain $S$. [\emph{Hint}: To prove that $A=B$, you need to show that
      that $A\subseteq B$ and $B\subseteq A$.] \bigskip
  \end{enumerate}
  


\item For a subgroup $H\leq G$ and element $x\in G$, the set
  $xH:=\{xh\mid h\in H\}$ is a \emph{left coset} of $H$.
  \begin{enumerate}[(a)]
  \item Prove that if $x\in H$, then $xH=H$. What is the
    interpretation of this statement in terms of the Cayley diagram?
  \item Prove that if $b\in aH$, then $aH=bH$.
  \item Show that for any $x\in G$, the map 
    \[
    \varphi\colon H\longto xH\,,\qquad \varphi\colon h\longmapsto xh
    \]
    is a bijection. Conclude that all left cosets of $H$ have the same size.
  \item Conclude that $G$ is partitioned by the left cosets of $H$,
    all of which are equal size. \bigskip
  \end{enumerate}  



\item A subgroup $H$ of $G$ is \emph{normal} if $xH=Hx$ for all $x\in
  G$. Prove that if $[G:H]=2$, then $H$ is a normal subgroup of
  $G$. [\emph{Hint}: Use the results of the previous problem.]


\end{enumerate}

\end{document}


