\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig, amsmath, latexsym}
\usepackage[all]{xy}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{decorations.pathreplacing}
\usetikzlibrary{patterns}
\usetikzlibrary{shapes.geometric}
%\usepackage{tikz-3dplot}
\usepackage{tkz-graph}
\usepackage{tikz-cd}
\usepackage{enumerate}
\usepackage{multicol}
\usepackage{url}

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


% Useful math macros
\def\ds{\displaystyle}
\def\bold{\bm}     % Change ``bm'' to ''mathbf'' for upright boldface.

\newcommand{\vv}[2]{\begin{bmatrix} #1 \\ #2 \end{bmatrix}}
\newcommand{\vvv}[3]{\begin{bmatrix} #1 \\ #2 \\ #3 \end{bmatrix}}
\newcommand{\vvvv}[4]{\begin{bmatrix} #1 \\ #2 \\ #3 \\ #4 \end{bmatrix}}
\newcommand{\vvvvv}[5]{\begin{bmatrix} #1 \\ #2 \\ #3 \\ #4 \\ #5 \end{bmatrix}}
\def\<{\langle}
\def\>{\rangle}
\newcommand{\ceil}[1] {\left\lceil #1 \right\rceil}
\newcommand{\floor}[1] {\left\lfloor #1 \right\rfloor}
\def\normal{\lhd}
\def\normaleq{\unlhd}
\def\nnormal{\ntriangleleft}
\def\nnormaleq{\ntrianglelefteq}
\def\Dot{\bm{.}}

%\def\subgroup{\leqslant}
%\def\nsubgroup{\nleqslant}
%\def\normal{\leqslant}
%\def\nnormal{\nleqslant}

\newcommand{\rddots}{\rotatebox{90}{$\ddots$}}
\def\setmin{-}

\def\Sum{\sum\limits}
\def\Prod{\prod\limits}
\def\Cap{\mathop\cap\limits}
\def\Cup{\mathop\cup\limits}
\def\Times{\mathop\times\limits}
\def\Oplus{\mathop\oplus\limits}
\def\Lim{\mathop\lim\limits}

\def\And{\wedge}
\def\Or{\vee}
\def\Not{\neg}

\def\longto{\longrightarrow}
\def\into{\hookrightarrow}
\def\longinto{\longhookrightarrow}

\def\A{\mathbb{A}}
\def\C{\mathbb{C}}
\def\F{\mathbb{F}}
\def\HH{\mathbb{H}}
\def\N{\mathbb{N}}
\def\Q{\mathbb{Q}}
\def\R{\mathbb{R}}
\def\Z{\mathbb{Z}}


\DeclareMathOperator{\Id}{Id}
\DeclareMathOperator{\Lcm}{lcm}
\DeclareMathOperator{\del}{\nabla}
\DeclareMathOperator{\Trace}{tr}
\DeclareMathOperator{\Span}{Span}
\DeclareMathOperator{\Image}{Im}
\DeclareMathOperator\Perm{Perm}
\DeclareMathOperator\Char{char}
\DeclareMathOperator\Aut{Aut}
\DeclareMathOperator\Inn{Inn}
\DeclareMathOperator\Out{Out}
\DeclareMathOperator\Orb{Orb}
\DeclareMathOperator\Stab{Stab}
\DeclareMathOperator\Fix{Fix}
\DeclareMathOperator\Syl{Syl}
\DeclareMathOperator\Pre{Pre}
\DeclareMathOperator\Ker{Ker}
\DeclareMathOperator\Sym{Sym}
\DeclareMathOperator\Mat{Mat}
\DeclareMathOperator{\cl}{cl}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\SL}{SL}
\DeclareMathOperator\PGL{PGL}
\DeclareMathOperator\PSL{PSL}
\DeclareMathOperator\PSU{PSU}
\DeclareMathOperator\PSP{PsP}
\DeclareMathOperator\Heis{Heis}
\DeclareMathOperator\Pauli{Pauli}
\DeclareMathOperator\Dih{D}
\DeclareMathOperator\Dic{Dic}
\DeclareMathOperator\DQ{DQ}
\DeclareMathOperator\SD{SD}
\DeclareMathOperator\SA{SA}
\DeclareMathOperator\Mod{Mod}
\DeclareMathOperator\Gal{Gal}
\DeclareMathOperator\Deg{Deg}

\DeclareMathOperator\Rect{\mathbf{Rect}}
\DeclareMathOperator\Tri{\mathbf{Tri}}
\DeclareMathOperator\Coin{\mathbf{Coin}}
\DeclareMathOperator\BOX{\mathbf{Box}}
\DeclareMathOperator\Light{\mathbf{Light}}
\DeclareMathOperator\Sq{\mathbf{Sq}}

\DeclareMathOperator\Frieze{\mathbf{Frz}}

% Colors 
\usepackage{xcolor}
\usepackage{color}
\definecolor{darkblue}{rgb}{0, 0, .6}
\definecolor{grey}{rgb}{.7, .7, .7}
\newcommand{\balert}[1]{\textcolor{blue}{#1}}
\newcommand{\galert}[1]{\textcolor{darkgreen}{#1}}
\newcommand{\Palert}[1]{\textcolor{amethyst}{#1}}
\newcommand{\Alert}[1]{\textcolor{Red}{#1}}
\newcommand{\Balert}[1]{\textcolor{Blue}{#1}}
\newcommand{\Galert}[1]{\textcolor{darkgreen}{#1}}


%\hypersetup{colorlinks=true,linkcolor=darkblue,anchorcolor=darkblue,
%  citecolor=darkblue,pagecolor=darkblue,urlcolor=darkblue,
%  pdftitle={},pdfauthor={}
%}                               %% THIS CAUSES COMPILER ERROR
\colorlet{lightred}{red!30!white}
\colorlet{lightblue}{blue!30!white}
\colorlet{lightyellow}{yellow!30!white}
\colorlet{keylime}{green!10!white}
\colorlet{darkgreen}{green!50!black}
\colorlet{midgreen}{green!70!black}
\colorlet{lightgreen}{darkgreen!30!white}
\colorlet{lightpurple}{violet!60!white}
\colorlet{darkgrey}{black!70}
\colorlet{midgrey}{black!50}
\definecolor{Red}{rgb}{.9,0,0}
\definecolor{Blue}{rgb}{0,0,.9}
\definecolor{Green}{rgb}{0,.9,0}

\definecolor{conj-purple}{HTML}{E4B7FF}
\definecolor{conj-orange}{HTML}{FFD4B7}

\colorlet{lred}{red!50!white}
\definecolor{lblue}{rgb}{0.60, 0.75, 1}
\colorlet{lgreen}{green!40!white}
\colorlet{lyellow}{yellow!30!white}
%\definecolor{lpurple}{rgb}{0.6 0.35 0.8}
\definecolor{lpurple}{rgb}{0.7 0.45 0.9}
\colorlet{lorange}{orange!50!white}
\definecolor{lightgrey}{rgb}{.85, .85, .85}
\definecolor{faded}{rgb}{.75,.75,.75}
\definecolor{midgrey}{rgb}{.5,.5,.5}
%\definecolor{midgrey}{rgb}{.35, .35, .35}
%\definecolor{amethyst}{rgb}{0.6 0.4 0.8}
\definecolor{amethyst}{rgb}{0.54 0.17 0.89}
\definecolor{amber}{rgb}{1.0 0.49 0.0}

% COLORS FOR MULTIPLICATION TABLES  
\colorlet{tred}{red!70!white}
\colorlet{tblue}{blue!60!white}     %(higher number is more faded)
\colorlet{tgreen}{green!75!black}   %(higher number is brighter)
\colorlet{tyellow}{yellow!60!white}
\colorlet{tlime}{green!80!white}
\colorlet{tpink}{pink}
\colorlet{tpurple}{lpurple}
\colorlet{torange}{orange}

\newcommand*{\n}{5}   % counter for Cayley tables

%% CAYLEY DIAGRAM GLOBAL DEFAULTS
\tikzstyle{v} = [circle, draw, fill=lightgrey,inner sep=0pt, 
  minimum size=6mm]
\tikzstyle{r} = [draw, very thick, Red, -stealth]        % Red --->
\tikzstyle{b} = [draw, very thick, Blue, -stealth]       % Blue --->
\tikzstyle{g} = [draw, very thick, darkgreen, -stealth]  % Green --->
\tikzstyle{p} = [draw, very thick, amethyst, -stealth]     % Purple --->
\tikzstyle{rr} = [draw, very thick, Red]                 % Red ----
\tikzstyle{bb} = [draw, very thick, Blue]                % Blue ----
\tikzstyle{gg} = [draw, very thick, darkgreen]           % Green ----
\tikzstyle{oo} = [draw, very thick, orange]           % Purple ----

\tikzstyle{R} = [draw, very thick, Red, -stealth']        % Red --->
\tikzstyle{B} = [draw, very thick, Blue, -stealth']       % Blue --->
\tikzstyle{G} = [draw, very thick, darkgreen, -stealth']  % Green --->

\tikzset{r/.style={draw, very thick, Red, -stealth}}        % Red --->
\tikzset{R/.style={draw, very thick, Red, -stealth'}}        % Red --->

\tikzset{v/.style={circle, draw, fill=lightgrey,inner sep=0pt, 
  minimum size=6mm}}

\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 3 $|$ Due September 16 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra $|$
    Fall 2022 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-6mm}

\begin{enumerate}
  
  %%-------------------------------------------------------------------

\item Establish the following isomorphisms by defining an explict map
  and proving that it is a bijective homomorphism.
  \begin{enumerate}
  \item $H\cong xHx^{-1}$, for any subgroup $H\leq G$ and fixed $x\in G$.
  \item $(\Q^*,\cdot\,)\cong(\Q^+,\cdot\,)\times C_2$, where
    $C_2=\{1,-1\}$. \medskip
  \end{enumerate}
  
  %%-------------------------------------------------------------------

\item Let $G$ be the \emph{semiabelian group} of order 16, defined by
  the presentation
  \[
  \SA_8=\big\<r,s\mid r^8=s^2=1, srs=r^5\big\>,
  \]
  A Cayley diagram and subgroup lattice are shown below.
  %All subgroups except $\<s\>$ and $\<r^4s\>$ are normal.

  \vspace{-7mm}
  
  \tikzstyle{R-in} = [draw, very thick, Red,-stealth,bend right=12]
  \tikzstyle{R-out} = [draw, very thick, Red,-stealth,bend right=15]
  \[
  \hspace*{-4mm}
  \begin{tikzpicture}[scale=1.55]
    \begin{scope}[shift={(0,0)},scale=1]
      \tikzstyle{every node}=[font=\footnotesize]
      \node (s) at (0:1.1) [v] {$s$};
      \node (rs) at (45:1.15) [v] {$rs$};
      \node (r2s) at (90:1.15) [v] {$r^2\!s$};
      \node (r3s) at (135:1.15) [v] {$r^3\!s$};
      \node (r4s) at (180:1.15) [v] {$r^4\!s$};
      \node (r5s) at (225:1.15) [v] {$r^5\!s$};
      \node (r6s) at (270:1.15) [v] {$r^6\!s$};
      \node (r7s) at (315:1.15) [v] {$r^7\!s$};
      %%
      \node (1) at (0:2) [v] {$1$};
      \node (r) at (45:2) [v] {$r$};
      \node (r2) at (90:2) [v] {$r^2$};
      \node (r3) at (135:2) [v] {$r^3$};
      \node (r4) at (180:2) [v] {$r^4$};
      \node (r5) at (225:2) [v] {$r^5$};
      \node (r6) at (270:2) [v] {$r^6$};
      \node (r7) at (315:2) [v] {$r^7$};
      %%
      \draw [R-out] (1) to (r);
      \draw [R-out] (r) to (r2);
      \draw [R-out] (r2) to (r3);
      \draw [R-out] (r3) to (r4);
      \draw [R-out] (r4) to (r5);
      \draw [R-out] (r5) to (r6);
      \draw [R-out] (r6) to (r7);
      \draw [R-out] (r7) to (1);
      %%
      \draw [r] (s) to (r5s);
      \draw [r] (r5s) to (r2s);
      \draw [r] (r2s) to (r7s);
      \draw [r] (r7s) to (r4s);
      \draw [r] (r4s) to (rs);
      \draw [r] (rs) to (r6s);
      \draw [r] (r6s) to (r3s);
      \draw [r] (r3s) to (s);
      %%
      \draw [bb] (1) to (s); \draw [bb] (r) to (rs);
      \draw [bb] (r2) to (r2s); \draw [bb] (r3) to (r3s);
      \draw [bb] (r4) to (r4s); \draw [bb] (r5) to (r5s);
      \draw [bb] (r6) to (r6s); \draw [bb] (r7) to (r7s);
      %%
      \node at (0,0) {\normalsize $\SA_8$}; 
    \end{scope}
    \begin{scope}[shift={(4.25,-2.15)},node distance=1cm,shorten >= -2pt, shorten <= -2pt,scale=.7]
      \tikzstyle{every node}=[font=\small]
      \node(G) at (3.5,6) {$\<r,s\>$};
      \node(rs) at (3.5,4.5) {$\<rs\>$};
      \node(r2-s) at (1.75,4.5) {$\<r^2,s\>$};
      \node(r) at (5.25,4.5) {$\<r\>$};
      \node(r4-s) at (0,3) {$\<r^4,s\>$};
      \node(r2s) at (1.75,3) {$\<r^2s\>$};
      \node(r2) at (3.5,3) {$\<r^2\>$};
      \node(s) at (-1.75,1.5) {$\<s\>$};
      \node(r4s) at (0,1.5) {$\<r^4s\>$};
      \node(r4) at (1.75,1.5) {$\<r^4\>$};
      \node(1) at (0,0) {$\<1\>$};
      \path[-] (1) edge (s);
      \path[-] (1) edge (r4s);
      \path[-] (1) edge (r4);
      \path[-] (s) edge (r4-s);
      \path[-] (r4s) edge (r4-s);
      \path[-] (r4) edge (r4-s);
      \path[-] (r4) edge (r2s);
      \path[-] (r4) edge (r2);
      \path[-] (r4-s) edge (r2-s);
      \path[-] (r2s) edge (r2-s);
      \path[-] (r2) edge (r2-s);
      \path[-] (r2) edge (r);
      \path[-] (r2) edge (rs);
      \path[-] (r2-s) edge (G);
      \path[-] (r) edge (G);
      \path[-] (rs) edge (G);
    \end{scope}
  \end{tikzpicture}
  \]
  \begin{enumerate}
  \item The subgroups $V=\<r^4,s\>$, $H=\<r^2s\>$, $K=\<r^2\>$, and
    $N=\<r^4\>$ are all normal. Highlight their cosets on a fresh Cayley
    diagram by colors.
  \item Construct a Cayley table for the quotient of $G$ by each of
    these subgroups. Then draw a Cayley diagram for each, labeling the
    nodes with elements (i.e., cosets).

  \item Let $N=\<r^4\>$. The shaded region below shows an order-$4$
    cyclic subgroup of $G/N$, generated by the element
    $rN$, and how the union of these four cosets is the
    order-$8$ subgroup $\<r\>$ of $G$. Construct analogous tables for
    the other five non-trivial proper subgroups of $G/N$, and
    then draw the subgroup lattice of $G/N$.

    \vspace{-2mm}
    
    \definecolor{faded}{rgb}{.75, .75, .75}
    \definecolor{midgrey}{rgb}{.35, .35, .35}
    \[
    \hspace*{-5mm}
    \begin{tikzpicture}[xscale=.9]
      \tikzstyle{every node}=[font=\small,anchor=south]
    \begin{scope}[shift={(0,0)}]
      \draw[thin,fill=faded] (0,0) rectangle (2,3.2);
      \draw[thin] (0,2.4) to (4,2.4); 
      \draw[thin] (0,1.6) to (4,1.6);
      \draw[thin] (0,.8) to (4,.8);
      \draw[thin] (2,0) to (4,0);
      \draw[thin] (2,3.2) to (4,3.2);
      \draw[thin] (4,0) to (4,3.2);
      \node at (1,.1) {$N$};
      \node at (3,.1) {$sN$};
      \node at (1,.9) {$rN$};
      \node at (3,.9) {$rsN$};
      \node at (1,1.7) {$r^2N$};
      \node at (3,1.7) {$r^2sN$};
      \node at (1,2.5) {$r^3N$};
      \node at (3,2.5) {$r^3sN$};
      \node at (2,-.8) {$\<rN\>\leq G/N$};
    \end{scope}
    %%
    \begin{scope}[shift={(6,0)}]
      \draw[thin,fill=faded] (0,0) rectangle (2,3.2);
      \draw[thin] (0,2.4) to (4,2.4); 
      \draw[thin] (0,1.6) to (4,1.6);
      \draw[thin] (0,.8) to (4,.8);
      \draw[thin] (2,0) to (4,0);
      \draw[thin] (2,3.2) to (4,3.2);
      \draw[thin] (4,0) to (4,3.2);
      \node at (.5,.15) {$1$};
      \node at (1.5,.15) {$r^4$};
      \node at (2.5,.15) {$s$};
      \node at (3.5,.15) {$r^4s$};
      \node at (.5,.95) {$r$};
      \node at (1.5,.95) {$r^5$};
      \node at (2.5,.95) {$rs$};
      \node at (3.5,.95) {$r^5s$};
      \node at (.5,1.75) {$r^2$};
      \node at (1.5,1.75) {$r^6$};
      \node at (2.5,1.75) {$r^2s$};
      \node at (3.5,1.75) {$r^6s$};
      \node at (.5,2.55) {$r^3$};
      \node at (1.5,2.55) {$r^7$};
      \node at (2.5,2.55) {$r^3s$};
      \node at (3.5,2.55) {$r^7s$};
      \node at (2,-.8) {$\<r\>/N\leq G/N$};
    \end{scope}
    %%
    \begin{scope}[shift={(12,0)}]
      \draw[thin,fill=faded] (0,0) rectangle (2,3.2);
      \draw[thin] (2,0) to (4,0);
      \draw[thin] (2,3.2) to (4,3.2);
      \draw[thin] (4,0) to (4,3.2);
      \node at (.5,.15) {$1$};
      \node at (1.5,.15) {$r^4$};
      \node at (2.5,.15) {$s$};
      \node at (3.5,.15) {$r^4s$};
      \node at (.5,.95) {$r$};
      \node at (1.5,.95) {$r^5$};
      \node at (2.5,.95) {$rs$};
      \node at (3.5,.95) {$r^5s$};
      \node at (.5,1.75) {$r^2$};
      \node at (1.5,1.75) {$r^6$};
      \node at (2.5,1.75) {$r^2s$};
      \node at (3.5,1.75) {$r^6s$};
      \node at (.5,2.55) {$r^3$};
      \node at (1.5,2.55) {$r^7$};
      \node at (2.5,2.55) {$r^3s$};
      \node at (3.5,2.55) {$r^7s$};
      \node at (2,-.8) {$\<r\>\leq G$};
    \end{scope}
    %%
  \end{tikzpicture}
    \]
  \item Repeat the previous part for the subgroups $H$, $K$, and $V$,
    but include the trivial and nonproper subgroups.
  
  \end{enumerate}

\newpage
  
  %%-------------------------------------------------------------------
  
\item Suppose $A,B\leq G$, and that $A$ normalizes $B$. That is,
  $A\leq N_G(B)$.
  \begin{enumerate}
  \item Show that $AB\leq G$.
  \item Show that $B\normaleq AB$ and $A\cap B\normaleq A$.  
  \item Show that $A/(A\cap B)\cong AB/B$.
  \item Show that if $A$ and $B$ are both normal with $G=AB$, then
    \[
    G/(A\cap B)\cong (G/A)\times(G/B).
    \]
  \end{enumerate}
  
  %%-------------------------------------------------------------------
  
\item Prove the remaining parts of the correspondence theorem. That
  is, if $N\leq H\leq G$ is a chain of subgroups and $N\normaleq G$,
  then show all of the following.
  \begin{enumerate}
  \item $H/N\normaleq G/N$ if and only if $H\normaleq G$
  \item $[G/N:H/N]=[G:H]$
  \item $H/N\cap K/N=(H\cap K)/N$
  \item $\<H/N,K/N\>=\<H,K\>/N$
  \item $H/N$ is conjugate to $K/N$ in $G/N$ if and only if $H$ is
    conjugate to $K$ in $G$. \\
  \end{enumerate}


\item The semidirect product $C_7\rtimes C_3$ can be constructed in a
  ``visual'' manner by starting with a Cayley graph of $\Aut(C_7)\cong
  U_6\cong C_6$, shown below, with the nodes labeled by
  ``rewirings'' of a Cayley graph of $C_7$.

  \tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=5mm] \tikzstyle{v-yel} = [circle, draw,
    fill=lyellow,inner sep=0pt, minimum size=5mm] \tikzstyle{r-thin} =
            [draw, thick, Red,-stealth]
  \[
  \hspace*{-12mm}
  \begin{tikzpicture}[scale=1.5,baseline=-.5ex]
    \tikzstyle{every node}=[font=\footnotesize]
    \draw [p] (18:3.5) to [bend right=12] (42:3.5);
    \draw [p] (78:3.5) to [bend right=12] (102:3.5);
    \draw [p] (138:3.5) to [bend right=12] (162:3.5);
    \draw [p] (198:3.5) to [bend right=12] (222:3.5);
    \draw [p] (258:3.5) to [bend right=12] (282:3.5);
    \draw [p] (318:3.5) to [bend right=12] (342:3.5);
    \begin{scope}[shift={(0:3)},scale=.85]
      \node (1) at (0:1) [v-yel] {$1$};
      \node (r) at (51.45:1) [v-yel] {$a$};
      \node (r2) at (102.9:1) [v-small] {$a^2$};
      \node (r3) at (154.3:1) [v-small] {$a^3$};
      \node (r4) at (205.7:1) [v-small] {$a^4$};
      \node (r5) at (257.1:1) [v-small] {$a^5$};
      \node (r6) at (308.6:1) [v-small] {$a^6$};
      \draw [r-thin] (1) to (r); \draw [r-thin] (r) to (r2); \draw [r-thin] (r2) to (r3);
      \draw [r-thin] (r3) to (r4); \draw [r-thin] (r4) to (r5); \draw [r-thin] (r5) to (r6);
      \draw [r-thin] (r6) to (1);
      \node at (0,0) {\Large Id=$\sigma_1$};
    \end{scope}
    %%\normalsize
    \begin{scope}[shift={(60:3)},scale=.85]
      \node at (2.5,.3) {\normalsize ``\emph{tripling map}'' $\sigma_3$};
      \node at (2.5,-.3) {\normalsize$U_7=\<3\>$};
      \node (1) at (0:1) [v-yel] {$1$};
      \node (r) at (51.45:1) [v-small] {$a$};
      \node (r2) at (102.9:1) [v-small] {$a^2$};
      \node (r3) at (154.3:1) [v-yel] {$a^3$};
      \node (r4) at (205.7:1) [v-small] {$a^4$};
      \node (r5) at (257.1:1) [v-small] {$a^5$};
      \node (r6) at (308.6:1) [v-small] {$a^6$};
      \draw [r-thin] (1) to (r3); \draw [r-thin] (r3) to (r6); \draw [r-thin] (r6) to (r2);
      \draw [r-thin] (r2) to (r5); \draw [r-thin] (r5) to (r); \draw [r-thin] (r) to (r4);
      \draw [r-thin] (r4) to (1);
      \node at (0,0) {\Large $\phi$};
    \end{scope}
    %%
    \begin{scope}[shift={(120:3)},scale=.85]
      \node at (-2.8,.3) {\normalsize ``\emph{doubling map}'' $\sigma_2$};
      \node at (-2.8,-.3) {\normalsize $3^2\equiv 2\!\pmod{7}$};
      \node (1) at (0:1) [v-yel] {$1$};
      \node (r) at (51.45:1) [v-small] {$a$};
      \node (r2) at (102.9:1) [v-yel] {$a^2$};
      \node (r3) at (154.3:1) [v-small] {$a^3$};
      \node (r4) at (205.7:1) [v-small] {$a^4$};
      \node (r5) at (257.1:1) [v-small] {$a^5$};
      \node (r6) at (308.6:1) [v-small] {$a^6$};
      \draw [r-thin] (1) to (r2); \draw [r-thin] (r2) to (r4); \draw [r-thin] (r4) to (r6);
      \draw [r-thin] (r6) to (r); \draw [r-thin] (r) to (r3); \draw [r-thin] (r3) to (r5);
      \draw [r-thin] (r5) to (1);
      \node at (0,0) {\Large $\phi^2$};
    \end{scope}
    %%
    \begin{scope}[shift={(180:3)},scale=.85]
      \node at (-2.8,.3) {\normalsize ``\emph{sextupling map}'' $\sigma_6$};
      \node at (-2.8,-.3) {\normalsize $3^3\equiv 6\!\pmod{7}$};
      \node (1) at (0:1) [v-yel] {$1$};
      \node (r) at (51.45:1) [v-small] {$a$};
      \node (r2) at (102.9:1) [v-small] {$a^2$};
      \node (r3) at (154.3:1) [v-small] {$a^3$};
      \node (r4) at (205.7:1) [v-small] {$a^4$};
      \node (r5) at (257.1:1) [v-small] {$a^5$};
      \node (r6) at (308.6:1) [v-yel] {$a^6$};
      \draw [r-thin] (1) to (r6); \draw [r-thin] (r6) to (r5); \draw [r-thin] (r5) to (r4);
      \draw [r-thin] (r4) to (r3); \draw [r-thin] (r3) to (r2); \draw [r-thin] (r2) to (r);
      \draw [r-thin] (r) to (1);      
      \node at (0,0) {\Large $\phi^3$};
    \end{scope}
    %%
    \begin{scope}[shift={(240:3)},scale=.85]
      \node at (-3,.3) {\normalsize ``\emph{quadrupling map}'' $\sigma_4$};
      \node at (-3,-.3) {\normalsize $3^4\equiv 4\!\pmod{7}$};
      \node (1) at (0:1) [v-yel] {$1$};
      \node (r) at (51.45:1) [v-small] {$a$};
      \node (r2) at (102.9:1) [v-small] {$a^2$};
      \node (r3) at (154.3:1) [v-small] {$a^3$};
      \node (r4) at (205.7:1) [v-yel] {$a^4$};
      \node (r5) at (257.1:1) [v-small] {$a^5$};
      \node (r6) at (308.6:1) [v-small] {$a^6$};
      \draw [r-thin] (1) to (r4); \draw [r-thin] (r4) to (r); \draw [r-thin] (r) to (r5);
      \draw [r-thin] (r5) to (r2); \draw [r-thin] (r2) to (r6); \draw [r-thin] (r6) to (r3);
      \draw [r-thin] (r3) to (1);      
      \node at (0,0) {$\phi^4$};
    \end{scope}
    %%
    \begin{scope}[shift={(300:3)},scale=.85]
      \node at (2.93,.3) {\normalsize ``\emph{quintupling map}'' $\sigma_5$};
      \node at (2.9,-.3) {\normalsize $3^5\equiv 5\!\pmod{7}$};      
      \node (1) at (0:1) [v-yel] {$1$};
      \node (r) at (51.45:1) [v-small] {$a$};
      \node (r2) at (102.9:1) [v-small] {$a^2$};
      \node (r3) at (154.3:1) [v-small] {$a^3$};
      \node (r4) at (205.7:1) [v-small] {$a^4$};
      \node (r5) at (257.1:1) [v-yel] {$a^5$};
      \node (r6) at (308.6:1) [v-small] {$a^6$};
      \draw [r-thin] (1) to (r5); \draw [r-thin] (r5) to (r3); \draw [r-thin] (r3) to (r);
      \draw [r-thin] (r) to (r6); \draw [r-thin] (r6) to (r4); \draw [r-thin] (r4) to (r2);
      \draw [r-thin] (r2) to (1);      
      \node at (0,0) {\Large $\phi^5$};
    \end{scope}
\end{tikzpicture}
%\caption{A Cayley graph of the automorphism group of $C_7$, with the nodes labeled by rewirings.}\label{fig2:Aut(C_7)-Cayley}
  \]

  \newpage
  
  Next, we define the ``labeling map,'' a homomorphism
  \[
  \theta\colon C_3\longto\Aut(C_7),\qquad \theta\colon b^k\longmapsto\phi^{2k},
  \]
  which tells us how to ``stick in'' rewired copies of $A=C_7$ into
  ``inflated'' nodes of $B=C_3$.

  
  
%% The ``rewirings'', ``balloons'', and ``labels'' for C7 and C3????
  \[
  \tikzstyle{v-yel} = [circle, draw,
    fill=lyellow,inner sep=0pt, minimum size=8mm]
  \hspace*{-10mm}
  \tikzstyle{v-big} = [circle, draw, fill=lightgrey,inner sep=0pt, 
    minimum size=8mm]
  \begin{tikzpicture}[scale=1.6]
    %%
   \begin{scope}[shift={(0,0)},scale=.7]
    \node (1) at (3,0) [v] {\small $1$};
    \node (r) at (1.5,0) [v] {\small $b$};
    \node (r2) at (0,0) [v] {\small $b^2$};
    \draw [b] (1) to (r);
    \draw [b] (r) to (r2);
    \draw [b] (r2) to [bend right] (1);
    \node at (1.5,1.2) {\small\Balert{$C_3$}};
    \node at (1.5,.7) {``\emph{balloons}''};
    \end{scope}
    %%
    \begin{scope}[shift={(4.5,0)},scale=.9]
      \tikzstyle{every node}=[font=\small]
      \node (1) at (0:1) [v-yel] {\Palert{$\varphi^0$}};
      \node (r) at (60:1) [v-big] {{$\varphi^1$}};
      \node (r2) at (120:1) [v-yel] {\Palert{$\varphi^2$}};
      \node (r3) at (180:1) [v-big] {{$\varphi^3$}};
      \node (r4) at (240:1) [v-yel] {\Palert{$\varphi^4$}};
      \node (r5) at (300:1) [v-big] {{$\varphi^5$}};
      \draw [p] (1) to (r); \draw [p] (r) to (r2);
      \draw [p] (r2) to (r3); \draw [p] (r3) to (r4);
      \draw [p] (r4) to (r5); \draw [p] (r5) to (1);
      \node at (0,.2) {\small\Alert{$\Aut(C_7)$}};
      \node at (0,-.2) {``\emph{rewirings}''};
    \end{scope}
    %%
    \begin{scope}[shift={(7,0)},scale=.7]
      \tikzstyle{every node}=[font=\small]
      \node (1) at (3,0) [v-yel] {\Palert{$\varphi^0$}};
      \node (r) at (1.5,0) [v-yel] {\Palert{$\varphi^2$}};
      \node (r2) at (0,0) [v-yel] {\Palert{$\varphi^4$}};
      \draw [b] (1) to (r);
      \draw [b] (r) to (r2);
      \draw [b] (r2) to [bend right] (1);
      \node at (1.5,1.2) {\small\Palert{$\theta(C_3)$}};
      \node at (1.5,.7) {``\emph{labels}''};
    \end{scope}
  \end{tikzpicture}
  \]
  By connecting up the corresponding nodes, we get a Cayley graph for
  $C_7\rtimes C_3$, like the following.


\[
\begin{tikzpicture}
   \tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt, 
    minimum size=2mm]
     \tikzstyle{r-thin} = [draw, Red,thick,-stealth]
    %%%%
    \begin{scope}[shift={(0,0)}]
      \begin{scope}[shift={(0,3)}]
        \draw[lightgray, fill=lightgray] (0,0) circle (.75cm);
        \node (1) at (0:.6) [v-small] {};
        \node (r) at (51.42:.6) [v-small] {};
        \node (r2) at (102.86:.6) [v-small] {};
        \node (r3) at (154.29:.6) [v-small] {};
        \node (r4) at (205.71:.6) [v-small] {};
        \node (r5) at (257.14:.6) [v-small] {};
        \node (r6) at (308.57:.6) [v-small] {};
        \draw [r-thin] (1) to (r); 
        \draw [r-thin] (r) to (r2); 
        \draw [r-thin] (r2) to (r3);
        \draw [r-thin] (r3) to (r4); 
        \draw [r-thin] (r4) to (r5);
        \draw [r-thin] (r5) to (r6);
        \draw [r-thin] (r6) to (1);
        \draw [b,ultra thick] (0,-.75) to (0,-2.25);
      \end{scope}
      %%
      \begin{scope}[shift={(0,0)}]
        \draw[lightgray, fill=lightgray] (0,0) circle (.75cm);
        \node (1) at (0:.6) [v-small] {};
        \node (r) at (51.42:.6) [v-small] {};
        \node (r2) at (102.86:.6) [v-small] {};
        \node (r3) at (154.29:.6) [v-small] {};
        \node (r4) at (205.71:.6) [v-small] {};
        \node (r5) at (257.14:.6) [v-small] {};
        \node (r6) at (308.57:.6) [v-small] {};
        \draw [r-thin] (1) to (r2); 
        \draw [r-thin] (r2) to (r4); 
        \draw [r-thin] (r4) to (r6);
        \draw [r-thin] (r6) to (r); 
        \draw [r-thin] (r) to (r3);
        \draw [r-thin] (r3) to (r5);
        \draw [r-thin] (r5) to (1);
        \draw [b,ultra thick] (0,-.75) to (0,-2.25);
        %\draw [p,ultra thick] (-.75,0) to (-2.25,0);
        %\node at (0,0) {\small $\palert{\varphi^2}$};
      \end{scope}
      %%
      \begin{scope}[shift={(0,-3)}]
      \draw[lightgray, fill=lightgray] (0,0) circle (.75cm);
        \node (1) at (0:.6) [v-small] {};
        \node (r) at (51.42:.6) [v-small] {};
        \node (r2) at (102.86:.6) [v-small] {};
        \node (r3) at (154.29:.6) [v-small] {};
        \node (r4) at (205.71:.6) [v-small] {};
        \node (r5) at (257.14:.6) [v-small] {};
        \node (r6) at (308.57:.6) [v-small] {};
        \draw [r-thin] (1) to (r4); 
        \draw [r-thin] (r4) to (r); 
        \draw [r-thin] (r) to (r5);
        \draw [r-thin] (r5) to (r2); 
        \draw [r-thin] (r2) to (r6);
        \draw [r-thin] (r6) to (r3);
        \draw [r-thin] (r3) to (1);
        \draw [b,ultra thick] (135:.75) to[bend left=35] (-.57,5.45);
        %\node at (0,0) {\small $\palert{\varphi^3}$};
      \end{scope}
    \end{scope}
%%%%%%%%%%%%%%%%%%%%%%%%
  \begin{scope}[shift={(8.25,0)},scale=1.5]
  \tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt, 
  minimum size=6.75mm]
\tikzstyle{R1} = [draw, very thick, Red,-stealth,bend right=15]
\tikzstyle{R2} = [draw, very thick, Red,-stealth,bend right=20]   \tikzstyle{every node}=[font=\footnotesize]
    %%
      \node (s2) at (0:1) [v-small] {$b^2$};
      \node (rs2) at (51.45:1) [v-small] {$ab^2$};
      \node (r2s2) at (102.9:1) [v-small] {$a^2\!b^2$};
      \node (r3s2) at (154.3:1) [v-small] {$a^3\!b^2$};
      \node (r4s2) at (205.7:1) [v-small] {$a^4\!b^2$};
      \node (r5s2) at (257.1:1) [v-small] {$a^5\!b^2$};
      \node (r6s2) at (308.6:1) [v-small] {$a^6b^2$};
      %%
      \node (s) at (0:2) [v-small] {$b$};
      \node (rs) at (51.45:2) [v-small] {$ab$};
      \node (r2s) at (102.9:2) [v-small] {$a^2\!b$};
      \node (r3s) at (154.3:2) [v-small] {$a^3b$};
      \node (r4s) at (205.7:2) [v-small] {$a^4b$};
      \node (r5s) at (257.1:2) [v-small] {$a^5b$};
      \node (r6s) at (308.6:2) [v-small] {$a^6b$};
      %%
      \node (1) at (0:3) [v-small] {$1$};
      \node (r) at (51.45:3) [v-small] {$a$};
      \node (r2) at (102.9:3) [v-small] {$a^2$};
      \node (r3) at (154.3:3) [v-small] {$a^3$};
      \node (r4) at (205.7:3) [v-small] {$a^4$};
      \node (r5) at (257.1:3) [v-small] {$a^5$};
      \node (r6) at (308.6:3) [v-small] {$a^6$};
      %%
      \draw [R1] (1) to (r);
      \draw [R1] (r) to (r2);
      \draw [R1] (r2) to (r3);
      \draw [R1] (r3) to (r4);
      \draw [R1] (r4) to (r5);
      \draw [R1] (r5) to (r6);
      \draw [R1] (r6) to (1);
      %%
      \draw [R2] (s) to (r2s);
      \draw [R2] (r2s) to (r4s);
      \draw [R2] (r4s) to (r6s);
      \draw [R2] (r6s) to (rs);
      \draw [R2] (rs) to (r3s);
      \draw [R2] (r3s) to (r5s);
      \draw [R2] (r5s) to (s);
      %%
      \draw [r] (s2) to (r4s2);
      \draw [r] (r4s2) to (rs2);
      \draw [r] (rs2) to (r5s2);
      \draw [r] (r5s2) to (r2s2);
      \draw [r] (r2s2) to (r6s2);
      \draw [r] (r6s2) to (r3s2);
      \draw [r] (r3s2) to (s2);
      %%
      \draw [b] (1) to (s); \draw [b] (s) to (s2);
      \draw [b,bend right] (s2) to (1);
      \draw [b] (r) to (rs); \draw [b] (rs) to (rs2);
      \draw [b,bend right] (rs2) to (r);
      \draw [b] (r2) to (r2s); \draw [b] (r2s) to (r2s2);
      \draw [b,bend right] (r2s2) to (r2);
      \draw [b] (r3) to (r3s); \draw [b] (r3s) to (r3s2);
      \draw [b,bend right] (r3s2) to (r3);
      \draw [b] (r4) to (r4s); \draw [b] (r4s) to (r4s2);
      \draw [b,bend right] (r4s2) to (r4);
      \draw [b] (r5) to (r5s); \draw [b] (r5s) to (r5s2);
      \draw [b,bend right] (r5s2) to (r5);
      \draw [b] (r6) to (r6s); \draw [b] (r6s) to (r6s2);
      \draw [b,bend right] (r6s2) to (r6);
    \end{scope}
\end{tikzpicture}  
\]


  
  \begin{enumerate}
  \item Carry out analogous steps to construct Cayley graphs of
    $C_9\rtimes C_3$ and $C_3\rtimes C_6$, and then write down a
    presentation for each group. [\emph{Hint}: There are two
      homomorphisms $C_3\to\Aut(C_9)$ that will work, but one of them
      leads to a much less tangled diagram.]
  \item In each case, this group is isomorphic to the Cartesian
    product $G=A\times B$ with a different binary operation than the
    direct product. Give an explicit formula for
    $(a^i,b^j)*(a^k,b^\ell)$ using this new binary operation.
  \end{enumerate}

  
\end{enumerate}

\end{document}

