\documentclass[12pt]{article}

%% Math packages
%%
\usepackage{fancyhdr,fancybox,amsmath,amsthm,amssymb}
% Removes the "Too many math alphabets used in version normal" error.
\newcommand\hmmax{0}
\newcommand\bmmax{0}
\usepackage[new]{old-arrows}
\usepackage{cancel}
\usepackage{mathdots}
\usepackage{venndiagram}
\usepackage{mathrsfs}          % Math script font

% Graphics
%%
\graphicspath{{./}{figs/}}
\usepackage{graphicx}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{decorations.pathreplacing}
\usetikzlibrary{patterns}
\usetikzlibrary{shapes.geometric}
\usetikzlibrary{matrix}
\usepackage{tikz-3dplot}
\usepackage{tkz-graph}
\usepackage{tikz-cd}

%% Colors (most are in colors.tex file)
%%
\usepackage{xcolor}
\usepackage{color}
\usepackage{visualalgebra}  %% Put this *after* the TikZ packages

%% Page layout packages
%%
\usepackage{url}
\usepackage{multicol}
\usepackage[numbers,square,sort&compress]{natbib}

%% For HW assignments
\usepackage{fancyhdr,fancybox}
\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=0in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in
\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}


%% Font and formatting packages
%%
\usepackage[english]{babel}    % Removing this causes compiler error
\usepackage{alltt}             % Like verbatim, but excludes \ and { }
%\usepackage{enumerate}         % [shortlabels] option??
\usepackage{comment}
\usepackage{soul}              % strikeout text
\usepackage{bm}                % Bold math
\usepackage[T1]{fontenc}
\usepackage{relsize}


\pagestyle{fancy} \lhead{{\sf Homework 1 $|$ Due January 17 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Visual Algebra $|$
    Spring 2025 $|$ M.~Macauley}}

\begin{document}

%$\;$

\begin{enumerate}

  %%--------------------------------------------------------------------

  %% PROBLEM 1
  
\item The eight symmetries of a square form a group that we will call
  $\Sq$, generated by a $90^\circ$ counterclockwise rotation
  $r$, and a horizontal flip $f$. A Cayley graph is shown below.
  \[
  \begin{tikzpicture}
    \begin{scope}[scale=1.3]
      \draw [dashed,very thick,eBlue] (.5,-1) to (.5,2);
      \draw [dashed] (-1,.5) to (2,.5);
      \draw [dashed] (-.75,1.75) to (1.75,-.75);
      \draw [dashed] (-.75,-.75) to (1.75,1.75);
      \path[fill=actRed] (0,.5) rectangle ++(.5,.5); 
      \path[fill=actYellow] (.5,.5) rectangle ++(.5,.5);
      \path[fill=actGreen] (0,0) rectangle ++(.5,.5);
      \path[fill=actBlue] (.5,0) rectangle ++(.5,.5);
      \draw (.25,.75) node{$\mathbf{1}$}; \draw (.75,.75) node{$\mathbf{2}$};
      \draw (.25,.25) node{$\mathbf{4}$}; \draw (.75,.25) node{$\mathbf{3}$};
      \draw (0,0) rectangle (1,1);
      \draw[-stealth',eRed] (2.1,0) to[very thick,bend right=60] (2.1,.8);
      \node at (2.5,.5) {$\color{xRed}r$};
      \node at (.3,2) {$\color{xBlue}f$};      
    \end{scope}
    %%
    \begin{scope}[shift={(8,.75)},scale=1]
      \node (e) at (-2,2) [v] {{\small $1$}};
      \node (f) at (-1,1) [v] {{\small $f$}};
      \node (r) at (-2,-2) [v] {{\small $r$}};
      \node (rf) at (-1,-1) [v] {{\small $rf$}};
      \node (r2) at (2,-2) [v] {{\small $r^2$}};
      \node (r2f) at (1,-1) [v] {{\small $r^2\!f$}};
      \node (r3) at (2,2) [v] {{\small $r^3$}};
      \node (r3f) at (1,1) [v] {{\small $r^3\!f$}};
      \draw [r] (e) to (r);
      \draw [r] (r) to (r2);
      \draw [r] (r2) to (r3);
      \draw [r] (r3) to (e);
      \draw [r] (f) to (r3f);
      \draw [r] (r3f) to (r2f);
      \draw [r] (r2f) to (rf);
      \draw [r] (rf) to (f);
      \draw [bb] (e) to (f);
      \draw [bb] (r) to (rf);
      \draw [bb] (r2) to (r2f);
      \draw [bb] (r3) to (r3f);
    \end{scope}
  \end{tikzpicture}
  \]
  
  \begin{enumerate}
  \item For each axis of reflection, express the symmetry across it in
    terms of $r$ and $f$.
  \item Find all \emph{minimal} generating sets. [\emph{Hint}: There are 12.]   
  \item Let $s=f$ and $t=r^3f=fr$. Draw a Cayley graph using $s$ and
    $t$ as generators.
  \item Write a presentation of the form $\Sq=\<r,f\mid\cdots\>$.
  \item Write a presentation of the form $\Sq=\<s,t\mid\cdots\>$.
  \item Construct a \emph{Cayley table} for this group, ordered
    $1,r,r^2,r^3,f,rf,r^2f,r^3f$. Describe how the rotations and
    reflections are ``clustered'' in this table.
  \end{enumerate}
  
  %%--------------------------------------------------------------------
  
  %% PROBLEM 2
  
\item The Cayley graphs of two groups of size $12$ are shown
  below. \vspace{-1mm}
  \[
  \begin{tikzpicture}
    %%
    \tikzstyle{every node}=[font=\footnotesize]
    %%
    \begin{scope}[shift={(0,0)},scale=1.7]
      \node (e) at (0,2.5) [v] {$1$};
      \node (a) at (1,2.5) [v] {$s$};
      \node (aa) at (2,2.5) [v] {$s^2$};
      \node (aaa) at (3,2.5) [v] {$s^3$};
      %%
      \node (b) at (0,1.25) [v] {$t$};
      \node (ba) at (1,1.25) [v] {$ts$};
      \node (aab) at (2,1.25) [v] {$s^2\!t$};
        \node (abb) at (3,1.25) [v] {$st^2$};
        %%
        \node (bb) at (0,0) [v] {$t^2$};
        \node (bba) at (1,0) [v] {$t^2\!s$};
        \node (aabb) at (2,0) [v] {\scriptsize $s^2\!t^2$};
        \node (ab) at (3,0) [v] {$st$};
        %%
        \draw [r] (e) to (a); \draw [r] (a) to (aa);
        \draw [r] (aa) to (aaa); \draw [r] (aaa) to [bend right=25] (e);
        %%
        \draw [r] (b) to (ba); \draw [r] (ba) to (aab);
        \draw [r] (aab) to (abb); \draw [r] (abb) to [bend right=25] (b);
        %%
        \draw [r] (bb) to (bba); \draw [r] (bba) to (aabb);
        \draw [r] (aabb) to (ab); \draw [r] (ab) to [bend left=25] (bb);
        %%
        \draw [b] (e) to (b); \draw [b] (b) to (bb);
        \draw [b] (bb) to [bend left=20] (aa);
        %%
        \draw [b] (aa) to (aab); \draw [b] (aab) to (aabb);
        \draw [b] (aabb) to [bend left=20] (e);
        %%
        \draw [b] (bba) to (ba); \draw [b] (ba) to (a);
        \draw [b] (a) to [bend left=20] (ab);
        %%
        \draw [b] (ab) to (abb); \draw [b] (abb) to (aaa);
        \draw [b] (aaa) to [bend left=20] (bba);
      \end{scope}
      %%
      \begin{scope}[shift={(9,0)},scale=1.1]
        \tikzstyle{every node}=[font=\small]
        \node (e) at (0,4) [v] {$e$};
        \node (x) at (1.5,4) [v] {$x$};
        \node (c) at (3,4) [v] {$c$};
        \node (d) at (4.5,4) [v] {$d$};
        \node (a) at (0,2) [v] {$a$};
        \node (b) at (1.5,2) [v] {$b$};
        \node (dd) at (3,2) [v] {$d^2$};
        \node (bb) at (4.5,2) [v] {$b^2$};
        \node (aa) at (0,0) [v] {$a^2$};
        \node (cc) at (1.5,0) [v] {$c^2$};
        \node (z) at (3,0) [v] {$z$};
        \node (y) at (4.5,0) [v] {$y$};
        \draw [bb] (e) to (x);
        \draw [bb] (a) to (c);
        \draw [bb] (b) to (d);
        \draw [bb] (aa) to (bb);
        \draw [bb] (cc) to (dd);
        \draw [bb] (z) to (y);
        \draw [r] (e) to (a);
        \draw [r] (a) to (aa);
        \draw [r] (aa) to [bend left] (e);
        \draw [r] (x) to (b);
        \draw [r] (b) to (cc);
        \draw [r] (cc) to [bend left] (x);
        \draw [r] (c) to (dd);
        \draw [r] (dd) to (z);
        \draw [r] (z) to [bend right] (c);
        \draw [r] (d) to (bb);
        \draw [r] (bb) to (y);
        \draw [r] (y) to [bend right] (d);
      \end{scope}
    \end{tikzpicture}
  \] \vspace{-12mm}  
  \begin{enumerate}
  \item Create a Cayley table for each group. (For consistency, please
    order the elements in the first group by
    $1,t^2,s^2t,t,s^2,s^2t^2,s,st^2,t^2s,st,s^3,ts$,    
    and those in second by $e,x,y,z,a,b,c,d,a^2,b^2,c^2,d^2$.)
  \item Find the inverse of each element. 
  \item Find the \emph{order} of each $g$: the minimal $k>0$ such
    that $g^k=e$, denoted $|g|$.
  \item Write a presentation for each group.
  \item Determine whether or not these two groups are
    isomorphic. Justify your answer.
  \item Squint your eyes. Do you see any patterns in these tables? 
  \end{enumerate}
  
  %%--------------------------------------------------------------------
  
  %% PROBLEM 3
  
\item In this problem, we will define two variations of the $\Coin_2$
  group from lecture. We will consider two types of tiles, and declare the
  following to be the ``\emph{home state}'' of each:
  \[
  \begin{tikzpicture}
    \begin{scope}[shift={(0,0)}]
      \path[fill=darkgray] (0,0) rectangle ++(.5,.5);
      \path[fill=white] (.5,0) rectangle ++(.5,.5);
      \path[fill=darkgray] (1,0) rectangle ++(.5,.5);
      \draw (0,0) rectangle (1.5,.5);
      \draw (.25,.24) node{\color{white}$\mathbf{1}$};
      \draw (.75,.24) node{\color{black}$\mathbf{2}$};
      \draw (1.25,.24) node{\color{white}$\mathbf{3}$};
    \end{scope}
    %%
    \begin{scope}[shift={(4,.25)}]
      \path[fill=darkgray] (0,0) rectangle ++(.5,.5);
      \path[fill=white] (.5,0) rectangle ++(.5,.5);
      \path[fill=white] (0,-.5) rectangle ++(.5,.5);
      \path[fill=darkgray] (.5,-.5) rectangle ++(.5,.5);
      \draw (0,-.5) rectangle (1,.5);
      \draw (.25,.24) node{\color{white}$\mathbf{1}$};
      \draw (.75,.24) node{\color{black}$\mathbf{2}$};
      \draw (.75,-.24) node{\color{white}$\mathbf{3}$};
      \draw (.25,-.24) node{\color{black}$\mathbf{4}$};
    \end{scope}
  \end{tikzpicture}
  \]
  Our first group is $\Coin_3=\<c,t\>$, where $c$ ``\emph{cyclicaly
    shifts}'' the entries, and $t$ ``\emph{toggles}'' the color of the
  leftmost square:
  \[
  \begin{tikzpicture}
    \begin{scope}[shift={(0,0)}]
      \path[fill=darkgray] (0,0) rectangle ++(.5,.5);
      \path[fill=white] (.5,0) rectangle ++(.5,.5);
      \path[fill=darkgray] (1,0) rectangle ++(.5,.5);
      \draw (0,0) rectangle (1.5,.5);
      \draw (.25,.24) node{\color{white}$\mathbf{1}$};
      \draw (.75,.24) node{\color{black}$\mathbf{2}$};
      \draw (1.25,.24) node{\color{white}$\mathbf{3}$};
      \draw [p] (1.5,.25) -- (3.5,.25) node[midway,above]{$c$};
    \end{scope}
    %%
    \begin{scope}[shift={(3.5,0)}]
      \path[fill=darkgray] (0,0) rectangle ++(.5,.5);
      \path[fill=darkgray] (.5,0) rectangle ++(.5,.5);
      \path[fill=white] (1,0) rectangle ++(.5,.5); 
      \draw (0,0) rectangle (1.5,.5);
      \draw (.25,.24) node{\color{white}$\mathbf{3}$};
      \draw (.75,.24) node{\color{white}$\mathbf{1}$};
      \draw (1.25,.24) node{\color{black}$\mathbf{2}$};
    \end{scope}
    %%
    \begin{scope}[shift={(8,0)}]
      \path[fill=darkgray] (0,0) rectangle ++(.5,.5);
      \path[fill=white] (.5,0) rectangle ++(.5,.5);
      \path[fill=darkgray] (1,0) rectangle ++(.5,.5);
      \draw (0,0) rectangle (1.5,.5);
      \draw (.25,.24) node{\color{white}$\mathbf{1}$};
      \draw (.75,.24) node{\color{black}$\mathbf{2}$};
      \draw (1.25,.24) node{\color{white}$\mathbf{3}$};
      \draw [gg] (1.5,.25) -- (3.5,.25) node[midway,above]{$t$};
    \end{scope}
    %%
    \begin{scope}[shift={(11.5,0)}]
      \path[fill=white] (0,0) rectangle ++(.5,.5);
      \path[fill=white] (.5,0) rectangle ++(.5,.5);
      \path[fill=darkgray] (1,0) rectangle ++(.5,.5);
      \draw (0,0) rectangle (1.5,.5);
      \draw (.25,.24) node{\color{black}$\mathbf{1}$};
      \draw (.75,.24) node{\color{black}$\mathbf{2}$};
      \draw (1.25,.24) node{\color{white}$\mathbf{3}$};
    \end{scope}
  \end{tikzpicture}
  \]
  Our second group is $\BOX_2=\<r,s\>$, where $r$ ``\emph{rotates}''
  the squares counterclockwise, and $s$ ``\emph{swaps}'' the squares
  on the top row.
  \[
  \begin{tikzpicture}
    \begin{scope}[shift={(0,0)}]
      \path[fill=darkgray] (0,0) rectangle ++(.5,.5);
      \path[fill=white] (.5,0) rectangle ++(.5,.5);
      \path[fill=white] (0,-.5) rectangle ++(.5,.5);
      \path[fill=darkgray] (.5,-.5) rectangle ++(.5,.5);
      \draw (0,-.5) rectangle (1,.5);
      \draw (.25,.24) node{\color{white}$\mathbf{1}$};
      \draw (.75,.24) node{\color{black}$\mathbf{2}$};
      \draw (.75,-.24) node{\color{white}$\mathbf{3}$};
      \draw (.25,-.24) node{\color{black}$\mathbf{4}$};
      \draw [r] (1,0) -- (3,0) node[midway,above]{$r$};
    \end{scope}
    %%
    \begin{scope}[shift={(3,0)}]
      \path[fill=white] (0,0) rectangle ++(.5,.5);
      \path[fill=darkgray] (.5,0) rectangle ++(.5,.5);
      \path[fill=darkgray] (0,-.5) rectangle ++(.5,.5);
      \path[fill=white] (.5,-.5) rectangle ++(.5,.5);
      \draw (0,-.5) rectangle (1,.5);
      \draw (.25,.24) node{\color{black}$\mathbf{2}$};
      \draw (.75,.24) node{\color{white}$\mathbf{3}$};
      \draw (.75,-.24) node{\color{black}$\mathbf{4}$};
      \draw (.25,-.24) node{\color{white}$\mathbf{1}$};
    \end{scope}
    %%
    \begin{scope}[shift={(7,0)}]
      \path[fill=darkgray] (0,0) rectangle ++(.5,.5);
      \path[fill=white] (.5,0) rectangle ++(.5,.5);
      \path[fill=white] (0,-.5) rectangle ++(.5,.5);
      \path[fill=darkgray] (.5,-.5) rectangle ++(.5,.5);
      \draw (0,-.5) rectangle (1,.5);
      \draw (.25,.24) node{\color{white}$\mathbf{1}$};
      \draw (.75,.24) node{\color{black}$\mathbf{2}$};
      \draw (.75,-.24) node{\color{white}$\mathbf{3}$};
      \draw (.25,-.24) node{\color{black}$\mathbf{4}$};
      \draw [bb] (1,0) -- (3,0) node[midway,above]{$s$};
    \end{scope}
    %%
    \begin{scope}[shift={(10,0)}]
      \path[fill=white] (0,0) rectangle ++(.5,.5);
      \path[fill=darkgray] (.5,0) rectangle ++(.5,.5);
      \path[fill=white] (0,-.5) rectangle ++(.5,.5);
      \path[fill=darkgray] (.5,-.5) rectangle ++(.5,.5);
      \draw (0,-.5) rectangle (1,.5);
      \draw (.25,.24) node{\color{black}$\mathbf{2}$};
      \draw (.75,.24) node{\color{white}$\mathbf{1}$};
      \draw (.75,-.24) node{\color{white}$\mathbf{3}$};
      \draw (.25,-.24) node{\color{black}$\mathbf{4}$};
    \end{scope}
  \end{tikzpicture}
  \]
  Note that the square tiles don't actually need to be shaded. An
  alternate way to denote the colors of the $3\times 1$ dominos is to
  underline any number with a black background. For example, using
  this convention, the ``home state'' would be written
  $\mathbf{\underline{1}\,2\,\underline{3}}$. \medskip
  
  \begin{enumerate}
  \item Both of these groups have $24$ actions. Draw a Cayley graph
    for each, with the nodes labeled by configurations. It is helpful
    to know that the one for $\Coin_3$ can be arranged on a
    \emph{truncated cube}, whose skeleton is shown below (left). A
    Cayley graph for $\BOX_2$ can be arranged on a \emph{truncated
      octahedron}, shown below (right). But the ``home state'' at the
    yellow node.
    
    \[
    \hspace*{-12mm}
    \begin{tikzpicture}[scale=1.05]
      %%
      \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt,
        minimum size=2.5mm]
      \tikzstyle{v-yel} = [circle, draw, fill=yellow,inner sep=0pt,
        minimum size=2.5mm] 
      \tikzstyle{gr} = [draw, thick, darkgray]
      %%
      \begin{scope}[scale=.9]
        \node (a1) at (0:1) [v-yel] {}; 
        \node (a2) at (60:1) [v] {};
        \node (a3) at (120:1) [v] {};
        \node (a4) at (180:1) [v] {};
        \node (a5) at (240:1) [v] {};
        \node (a6) at (300:1) [v] {};
        \node (b1) at (2,-.58) [v] {};
        \node (b2) at (.5,1.87) [v] {};
        \node (b3) at (-.5,1.87) [v] {};
        \node (b4) at (-2,-.58) [v] {};
        \node (b5) at (-1.5,-1.44) [v] {};
        \node (b6) at (1.5,-1.44) [v] {};
        \node (c1) at (0:3) [v] {}; 
        \node (c2) at (60:3) [v] {};
        \node (c3) at (120:3) [v] {};
        \node (c4) at (180:3) [v] {};
        \node (c5) at (240:3) [v] {};
        \node (c6) at (300:3) [v] {};
        \node (d1) at (0:4) [v] {}; 
        \node (d2) at (60:4) [v] {};
        \node (d3) at (120:4) [v] {};
        \node (d4) at (180:4) [v] {};
        \node (d5) at (240:4) [v] {};
        \node (d6) at (300:4) [v] {};
        \draw [gr] (a1) to (a2); \draw [gr] (a2) to (a3); \draw [gr] (a3) to (a4);
        \draw [gr] (a4) to (a5); \draw [gr] (a5) to (a6); \draw [gr] (a6) to (a1);
        \draw [gr] (a1) to (b1); \draw [gr] (a2) to (b2); \draw [gr] (a3) to (b3);
        \draw [gr] (a4) to (b4); \draw [gr] (a5) to (b5); \draw [gr] (a6) to (b6);
        \draw [gr] (b1) to (c1); \draw [gr] (b2) to (c2); \draw [gr] (b3) to (c3);
        \draw [gr] (b4) to (c4); \draw [gr] (b5) to (c5); \draw [gr] (b6) to (c6);
        \draw [gr] (c1) to (d1); \draw [gr] (c2) to (d2); \draw [gr] (c3) to (d3);
        \draw [gr] (c4) to (d4); \draw [gr] (c5) to (d5); \draw [gr] (c6) to (d6);
        \draw [gr] (b2) to (b3); \draw [gr] (b4) to (b5); \draw [gr] (b6) to (b1);
        \draw [gr] (c1) to (c2); \draw [gr] (c3) to (c4); \draw [gr] (c5) to (c6);
        \draw [gr] (d1) to (d2); \draw [gr] (d2) to (d3); \draw [gr] (d3) to (d4);
        \draw [gr] (d4) to (d5); \draw [gr] (d5) to (d6); \draw [gr] (d6) to (d1);
      \end{scope}
      %%
      \begin{scope}[shift={(-8.25,0)}]
        \node (nw1) at (-1.5,1.5) [v-yel] {};
        \node (nw2) at (-.5,1.5) [v] {}; 
        \node (nw3) at (-1.5,.5) [v] {};
        \node (ne1) at (1.5,1.5) [v] {};`
        \node (ne2) at (1.5,.5) [v] {}; 
        \node (ne3) at (.5,1.5) [v] {};
        \node (se1) at (1.5,-1.5) [v] {};
        \node (se2) at (.5,-1.5) [v] {};
        \node (se3) at (1.5,-.5) [v] {}; 
        \node (sw1) at (-1.5,-1.5) [v] {};
        \node (sw2) at (-1.5,-.5) [v] {}; 
        \node (sw3) at (-.5,-1.5) [v] {};
        %
        \draw [gr] (nw2) to (ne3); \draw [gr] (ne2) to (se3);
        \draw [gr] (se2) to (sw3); \draw [gr] (sw2) to (nw3);
        \draw[gr](nw1)to(nw2); \draw [gr] (nw2) to (nw3); \draw [gr] (nw3) to (nw1);
        \draw[gr](ne1)to(ne2); \draw [gr] (ne2) to (ne3); \draw [gr] (ne3) to (ne1);
        \draw[gr](se1)to(se2); \draw [gr] (se2) to (se3); \draw [gr] (se3) to (se1);
        \draw[gr](sw1)to(sw2); \draw [gr] (sw2) to (sw3); \draw [gr] (sw3) to (sw1);
        %%
        \node (NW1) at (-2.25,2.25) [v] {};
        \node (NW2) at (-3.25,2.25) [v] {}; 
        \node (NW3) at (-2.25,3.25) [v] {};
        \node (NE1) at (2.25,2.25) [v] {};
        \node (NE2) at (2.25,3.25) [v] {}; 
        \node (NE3) at (3.25,2.25) [v] {};
        \node (SE1) at (2.25,-2.25) [v] {};
        \node (SE2) at (3.25,-2.25) [v] {};
        \node (SE3) at (2.25,-3.25) [v] {}; 
        \node (SW1) at (-2.25,-2.25) [v] {};
        \node (SW2) at (-2.25,-3.25) [v] {}; 
        \node (SW3) at (-3.25,-2.25) [v] {};
        %%
        \draw [gr] (NW3) to (NE2); \draw [gr] (NE3) to (SE2);
        \draw [gr] (SE3) to (SW2); \draw [gr] (SW3) to (NW2);
        %
        \draw[gr](NW1)to(NW2); \draw [gr] (NW2) to (NW3); \draw [gr] (NW3) to (NW1);
        \draw[gr](NE1)to(NE2); \draw [gr] (NE2) to (NE3); \draw [gr] (NE3) to (NE1);
        \draw[gr](SE1)to(SE2); \draw [gr] (SE2) to (SE3); \draw [gr] (SE3) to (SE1);
        \draw[gr](SW1)to(SW2); \draw [gr] (SW2) to (SW3); \draw [gr] (SW3) to (SW1);
        %%
        \draw [gr] (nw1) to (NW1); \draw [gr] (ne1) to (NE1);
        \draw [gr] (se1) to (SE1); \draw [gr] (sw1) to (SW1);
        %
      \end{scope}
    \end{tikzpicture}
    \]
  \item On a fresh copy of these graphs, color the edges of the Cayley graph and label each node by its \emph{order}. \medskip

  \item Write down a presentation for each of these groups. \medskip
    
  \item Are these groups isomorphic? Justify your answer.
    
  \end{enumerate}
  
  \newpage   
  
%%--------------------------------------------------------------------

%% PROBLEM 4

\item Consider the frieze shown below:
  \tikzstyle{up}=[shape=diamond, aspect=.5,anchor=south,fill=orange,draw]
  \tikzstyle{down}=[shape=diamond, aspect=.5,anchor=north,fill=orange,draw]
  \[
  \hspace*{-5mm}
  \scalebox{1.2}{
    \begin{tikzpicture}[scale=1.2]
      \begin{scope}[shift={(0,0)}]
        \tikzstyle{every node}=[font=\footnotesize]
        \draw (-1.5,0) node [up] {}; \draw (-1.5,0) node [down] {};
        \draw (0,0) node [up] {}; \draw (0,0) node [down] {};
        \draw (1.5,0) node [up] {}; \draw (1.5,0) node [down] {};
        \draw (3,0) node [up] {}; \draw (3,0) node [down] {};
        \draw (4.5,0) node [up] {}; \draw (4.5,0) node [down] {};
        \draw (6,0) node [up] {}; \draw (6,0) node [down] {};
        \draw (7.5,0) node [up] {}; \draw (7.5,0) node [down] {};
        \draw[thick] (-1.9,0) -- (7.9,0);
        \draw (-2.15,-.01) node {\small $\cdots$};
        \draw (8.21,-.01) node {\small $\cdots$};
        \draw[dotted] (-1.5,-1.1)--(-1.5,1.1);
        \draw[dotted] (-.75,-1.1)--(-.75,1.1);
        \draw[dotted] (0,-1.1)--(0,1.1);
        \draw[dotted] (.75,-1.1)--(.75,1.1);
        \draw[dotted] (1.5,-1.1)--(1.5,1.1);
        \draw[dotted] (2.25,-1.1)--(2.25,1.1);
        \draw[dashed,eBlue,thick] (3,-1.25)--(3,1.25);
        \draw[dotted] (3.75,-1.1)--(3.75,1.1);
        \draw[dotted] (4.5,-1.1)--(4.5,1.1);
        \draw[dotted] (5.25,-1.1)--(5.25,1.1);
        \draw[dotted] (6,-1.1)--(6,1.1);
        \draw[dotted] (6.75,-1.1)--(6.75,1.1);
        \draw[dotted] (7.5,-1.1)--(7.5,1.1);
        \node[darkgray] at (-1.5,0) {\footnotesize\textbullet};
        \node[darkgray] at (-.75,0) {\footnotesize\textbullet};
        \node[darkgray] at (0,0) {\footnotesize\textbullet};
        \node[darkgray] at (.75,0) {\footnotesize\textbullet};
        \node[darkgray] at (1.5,0) {\footnotesize\textbullet};
        \node[darkgray] at (2.25,0) {\footnotesize\textbullet};
        \node[xGreen] at (3,0) {\footnotesize\textbullet};
        \node[darkgray] at (3.75,0) {\footnotesize\textbullet};
        \node[darkgray] at (4.5,0) {\footnotesize\textbullet};
        \node[darkgray] at (5.25,0) {\footnotesize\textbullet};
        \node[darkgray] at (6,0) {\footnotesize\textbullet};
        \node[darkgray] at (6.75,0) {\footnotesize\textbullet};
        \node[darkgray] at (7.5,0) {\footnotesize\textbullet};
        \node at (-1.5,1.4) {$\ell_{-\!6}$};
        \node at (-.75,1.4) {$\ell_{-\!5}$};
        \node at (0,1.4) {$\ell_{-\!4}$};
        \node at (.75,1.4) {$\ell_{-\!3}$};
        \node at (1.5,1.4) {$\ell_{-\!2}$};
        \node at (2.25,1.4) {$\ell_{-\!1}$};
        \node at (3,1.4) {\color{xBlue}$\ell_0$};
        \node at (3.75,1.4) {$\ell_1$};
        \node at (4.5,1.4) {$\ell_2$};
        \node at (5.25,1.4) {$\ell_3$};
        \node at (6,1.4) {$\ell_4$};
        \node at (6.75,1.4) {$\ell_5$};
        \node at (7.5,1.4) {$\ell_6$};
        \node at (-1.14,-.15) {$p_{-\!6}$};
        \node at (-.5,.2) {$p_{-\!5}$};
        \node at (.36,-.15) {$p_{-\!4}$};
        \node at (1,.2) {$p_{-\!3}$};
        \node at (1.85,-.15) {$p_{-\!2}$};
        \node at (2.51,.2) {$p_{-\!1}$};
        \node at (3.3,-.15) {$\color{xGreen}p_0$};
        \node at (3.95,.2) {$p_1$};
        \node at (4.8,-.15) {$p_2$};
        \node at (5.45,.2) {$p_3$};
        \node at (6.3,-.15) {$p_4$};
        \node at (6.95,.2) {$p_5$};
        \node at (7.8,-.15) {$p_6$};
      \end{scope}
  \end{tikzpicture}}
  \]
  Let $t$ be a minimal translation to the right, $h_i$ a reflection
  across $\ell_i$, and $r_j$ a $180^\circ$ rotation around $p_j$. Let
  $v$ be the vertical reflection and $g_i=t^iv$ a glide
  reflection. A presentation for the frieze group is
  \[
  \Frieze_1:=\big\<t,r,v\mid v^2=r^2=1,\,tr=rt^{-1},\,tv=vt,\,rv=vr\big\>,
  \]
  where $r=r_0$. A Cayley graph is shown below. \vspace{-2mm}
  
  \[
  \hspace*{-6mm}
  \begin{tikzpicture}[scale=1.6]
    \tikzstyle{v} = [circle, draw,fill=lightgray,inner sep=0pt,
      minimum size=4.5mm]
    \tikzstyle{r-faded} = [draw, very thick, Red!50!white,
      -stealth] \tikzstyle{bb-faded} = [draw, very thick, blue!40!white]
    \tikzstyle{oo-faded} = [draw, very thick, orange!40!white]
    \tikzstyle{gg-faded} = [draw, very thick, darkgreen!40!white]
    \tikzstyle{every node}=[font=\footnotesize]
    %%
    \node (100) at (0,0) [v] {};
    \node (110) at (1,0) [v] {};
    \node (101) at (0,1) [v] {};
    \node (111) at (1,1) [v] {};
    \node (000) at (.5,.5) [v] {};
    \node (010) at (1.5,.5) [v] {};
    \node (001) at (.5,1.5) [v] {};
    \node (011) at (1.5,1.5) [v] {};
    %%
    \node (120) at (2,0) [v] {};
    \node (130) at (3,0) [v] {};
    \node (121) at (2,1) [v] {};
    \node (131) at (3,1) [v] {};
    \node (020) at (2.5,.5) [v] {};
    \node (030) at (3.5,.5) [v] {};
    \node (021) at (2.5,1.5) [v] {};
    \node (031) at (3.5,1.5) [v] {};
    %%
    \node (140) at (4,0) [v] {$r$}; \node (141) at (4,1) [v] {$1$};
    \node (040) at (4.5,.5) [v] {}; \node (041) at (4.5,1.5) [v]{$v$};
    %%
    \node (150) at (5,0) [v] {}; \node (151) at (5,1) [v] {$t$};
    \node (050) at (5.5,.5) [v] {}; \node (051) at (5.5,1.5) [v]{};
    %%
    \node (160) at (6,0) [v] {}; \node (161) at (6,1) [v] {};
    \node (060) at (6.5,.5) [v] {}; \node (061) at (6.5,1.5) [v] {};
    %%
    \node (170) at (7,0) [v] {}; \node (171) at (7,1) [v] {};
    \node (070) at (7.5,.5) [v] {}; \node (071) at (7.5,1.5) [v] {};
    %%
    \node (180) at (8,0) [v] {}; \node (181) at (8,1) [v] {};
    \node (080) at (8.5,.5) [v] {}; \node (081) at (8.5,1.5) [v] {};
    %%
    %% Ghost nodes 
    \node (1-10) at (-.8,0) {$\mathbf{\cdots}$}; \node (1-11) at (-.8,1) {$\mathbf{\cdots}$};
    \node (0-10) at (-.3,.5) {$\mathbf{\cdots}$}; \node (0-11) at (-.3,1.5) {$\mathbf{\cdots}$};
    \node (190) at (8.8,0) {$\mathbf{\cdots}$}; \node (191) at (8.8,1) {$\mathbf{\cdots}$};
    \node (090) at (9.3,.5) {$\mathbf{\cdots}$}; \node (091) at (9.3,1.5) {$\mathbf{\cdots}$};
    %% Half-edges (cube 0)
    \draw [r] (0-11) -- (001); \draw [rFaded] (000) -- (0-10);
    \draw [r] (1-11) -- (101); \draw [r] (100) -- (1-10);
    %% Back square (cube 1)
    \draw [r] (001) -- (011); \draw [rFaded] (010) -- (000);
    \draw [ggFaded] (001) -- (000); \draw [ggFaded] (011) -- (010); 
    %% Side edges square (cube 1)
    \draw [oo] (101) -- (001); \draw [ooFaded] (100) -- (000);
    \draw [oo] (111) -- (011); \draw [ooFaded] (110) -- (010);
    %% Front square (cube 1)
    \draw [r] (101) -- (111); \draw [r] (110) -- (100);
    \draw [gg] (101) -- (100); \draw [gg] (111) -- (110);
    %% Edges between cubes 1 & 3
    \draw [r] (011) -- (021); \draw [rFaded] (020) -- (010);
    \draw [r] (111) -- (121); \draw [r] (120) -- (110);
    %% Back square (cube 2)
    \draw [r] (021) -- (031); \draw [rFaded] (030) -- (020);
    \draw [ggFaded] (021) -- (020); \draw [ggFaded] (031) -- (030); 
    %% Side edges square (cube 3)
    \draw [oo] (121) -- (021); \draw [ooFaded] (120) -- (020);
    \draw [oo] (131) -- (031); \draw [ooFaded] (130) -- (030);
    %% Front square (cube 3)
    \draw [r] (121) -- (131); \draw [r] (130) -- (120);
    \draw [gg] (121) -- (120); \draw [gg] (131) -- (130);
    %% Back square (cube 3)
    \draw [r] (031) -- (041); \draw [rFaded] (040) -- (030);
    \draw [ggFaded] (031) -- (030); \draw [ggFaded] (041) -- (040); 
    %% Side edges square (cube 4)
    \draw [oo] (141) -- (041); \draw [ooFaded] (140) -- (040);
    %% Front square (cube 4)
    \draw [r] (131) -- (141); \draw [r] (140) -- (130);
    \draw [gg] (131) -- (130); \draw [gg] (141) -- (140);
    %% Back square (cube 4)
    \draw [r] (041) -- (051); \draw [rFaded] (050) -- (040);
    \draw [ggFaded] (041) -- (040); \draw [ggFaded] (051) -- (050); 
    %% Side edges square (cube 5)
    \draw [oo] (151) -- (051); \draw [ooFaded] (150) -- (050);
    \draw [oo] (151) -- (051); \draw [ooFaded] (150) -- (050);
    %% Front square (cube 5)
    \draw [r] (141) -- (151); \draw [r] (150) -- (140);
    \draw [gg] (141) -- (140); \draw [gg] (151) -- (150);
    %% Back square (cube 5)
    \draw [r] (051) -- (061); \draw [rFaded] (060) -- (050);
    \draw [ggFaded] (051) -- (050); \draw [ggFaded] (061) -- (060); 
    %% Side edges square (cube 6)
    \draw [oo] (161) -- (061); \draw [ooFaded] (160) -- (060);
    \draw [oo] (161) -- (061); \draw [ooFaded] (160) -- (060);
    %% Front square (cube 6)
    \draw [r] (151) -- (161); \draw [r] (160) -- (150);
    \draw [gg] (151) -- (150); \draw [gg] (161) -- (160);
    %% Back square (cube 6)
    \draw [r] (061) -- (071); \draw [rFaded] (070) -- (060);
    \draw [ggFaded] (061) -- (060); \draw [ggFaded] (071) -- (070); 
    %% Side edges square (cube 7)
    \draw [oo] (171) -- (071); \draw [ooFaded] (170) -- (070);
    \draw [oo] (171) -- (071); \draw [ooFaded] (170) -- (070);
    %% Front square (cube 7)
    \draw [r] (161) -- (171); \draw [r] (170) -- (160);
    \draw [gg] (161) -- (160); \draw [gg] (171) -- (170);
    %% Back square (cube 7)
    \draw [r] (071) -- (081); \draw [rFaded] (080) -- (070);
    \draw [ggFaded] (071) -- (070); \draw [ggFaded] (081) -- (080); 
    %% Side edges square (cube 8)
    \draw [oo] (181) -- (081); \draw [ooFaded] (180) -- (080);
    \draw [oo] (181) -- (081); \draw [ooFaded] (180) -- (080);
    %% Front square (cube 8)
    \draw [r] (171) -- (181); \draw [r] (180) -- (170);
    \draw [gg] (171) -- (170); \draw [gg] (181) -- (180);   
    %% Half-edges (cube 9)
    \draw [r] (081) -- (091); \draw [rFaded] (090) -- (080);
    \draw [r] (181) -- (191); \draw [r] (190) -- (180);
  \end{tikzpicture}
  \]
  
  
  \begin{enumerate}
  \item Every symmetry is either a translation $t^i$, glide reflection
    $g_j$, rotation $r_k$, horizontal reflection $h_\ell$ (note that the
    vertical reflection is $v=g_0$). Label the vertices of this Cayley graph
    with elements written in this form. \bigskip
  
  \item Determine which symmetries $t^iht^{-i}$ and $t^irt^{-i}$ are
    for each $i\in\Z$. \bigskip
    
  \item Using your answer to Part~(b) and the fact that $v$ commutes
    with every symmetry, derive an expression for $g^ihg^{-i}$ and
    $g^irg^{-i}$, for each $i\in\Z$. \bigskip
    
  \item Label the following Cayley graph for $\Frieze_2=\<g,r\>$ with
    elements of the form $g_j$, $h_k$, and $r_\ell$ for $i,j,k\in\Z$.
    
    \[
    \hspace*{-11mm}
    \begin{tikzpicture}[scale=1.65,auto]
      \tikzstyle{every node}=[font=\footnotesize]
      \begin{scope}[shift={(0,0)}]
        \node (e) at (0,1) [v] {$1$};
        \node (r) at (-1,1) [v] {};
        \node (r2) at (-2,1) [v] {};
        \node (r2-inv) at (2,1) [v] {};
        \node (r3) at (-3,1) [v] {};
        \node (r3-inv) at (3,1) [v] {};
        \node (r4) at (-4,1) [v] {};
        \node (r4-inv) at (4,1) [v] {};
        \node (r-inv) at (1,1) [v] {};
        \node (f) at (0,0) [v] {$r_1$};
        \node (rf) at (-1,0) [v] {};
        \node (r2f) at (-2,0) [v] {};
        \node (fr2) at (2,0) [v] {};
        \node (r3f) at (-3,0) [v] {};
        \node (fr3) at (3,0) [v] {};
        \node (r4f) at (-4,0) [v] {};
        \node (fr4) at (4,0) [v] {};
        \node (fr) at (1,0) [v] {};
        \draw [p] (r3-inv) to (r4-inv);
        \draw [p] (r2-inv) to (r3-inv);
        \draw [p] (r-inv) to (r2-inv);
        \draw [p] (e) to (r-inv);
        \draw [p] (r) to (e);
        \draw [p] (r2) to  (r);
        \draw [p] (r3) to  (r2);
        \draw [p] (r4) to  (r3);
        \draw [p] (rf) to (r2f);
        \draw [p] (r2f) to (r3f);
        \draw [p] (r3f) to (r4f);
        \draw [p] (f) to (rf);
        \draw [p] (fr) to (f);
        \draw [p] (fr2) to (fr);
        \draw [p] (fr3) to (fr2);
        \draw [p] (fr4) to (fr3);
        \draw [p] (-4.5,1) to (r4);
        \draw [p] (r4f) to (-4.5,0);
        \draw [p] (r4-inv) to (4.5,1);
        \draw [p] (4.5,0) to (fr4);
        \draw [gg] (e) to (f);
        \draw [gg] (r) to (rf);
        \draw [gg] (r2) to (r2f);
        \draw [gg] (r3) to (r3f);
        \draw [gg] (r4) to (r4f);
        \draw [gg] (r-inv) to (fr);
        \draw [gg] (r2-inv) to (fr2);
        \draw [gg] (r3-inv) to (fr3);
        \draw [gg] (r4-inv) to (fr4);
        \draw (4.7,1) node[xPurple] {\normalsize $\cdots$};
        \draw (4.7,0) node[xPurple] {\normalsize $\cdots$};
        \draw (-4.7,1) node[xPurple] {\normalsize $\cdots$};
        \draw (-4.7,0) node[xPurple] {\normalsize $\cdots$};  
      \end{scope}
    \end{tikzpicture}
    \]
    
  \end{enumerate}
  
  %%----------------------------------------------------------
  
\end{enumerate}

\end{document}
