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\pagestyle{fancy} \lhead{{\sf Homework 1 $|$ Due September 1 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra $|$
    Fall 2023 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-2mm}

\begin{enumerate}
  
  %%-------------------------------------------------------------------
  
%% PROBLEM 1

\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*{-10mm}
  \scalebox{1.2}{
    \begin{tikzpicture}[scale=1.3]
      \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,Blue,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[darkgrey] at (-1.5,0) {\footnotesize\textbullet};
        \node[darkgrey] at (-.75,0) {\footnotesize\textbullet};
        \node[darkgrey] at (0,0) {\footnotesize\textbullet};
        \node[darkgrey] at (.75,0) {\footnotesize\textbullet};
        \node[darkgrey] at (1.5,0) {\footnotesize\textbullet};
        \node[darkgrey] at (2.25,0) {\footnotesize\textbullet};
        \node[darkgreen] at (3,0) {\footnotesize\textbullet};
        \node[darkgrey] at (3.75,0) {\footnotesize\textbullet};
        \node[darkgrey] at (4.5,0) {\footnotesize\textbullet};
        \node[darkgrey] at (5.25,0) {\footnotesize\textbullet};
        \node[darkgrey] at (6,0) {\footnotesize\textbullet};
        \node[darkgrey] at (6.75,0) {\footnotesize\textbullet};
        \node[darkgrey] 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{Blue}$\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{darkgreen}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}}
  \]
  %%
  %\vspace{-5mm}
  %%
  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
  \[
  \Frieze_1:=\big\<t,h,v\mid v^2=h^2=1,\,th=ht^{-1},\,tv=vt,\,hv=vh\big\>,
  \]
  where $h=h_0$. A Cayley graph is shown below. 
  
  \tikzstyle{v-tiny} = [circle, draw,
    fill=lightgrey,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]
  \[
  \hspace*{-15mm}
  \begin{tikzpicture}[scale=1.65]
    \tikzstyle{every node}=[font=\footnotesize]
    \node (100) at (0,0) [v-tiny] {};
    \node (110) at (1,0) [v-tiny] {};
    \node (101) at (0,1) [v-tiny] {};
    \node (111) at (1,1) [v-tiny] {};
    \node (000) at (.5,.5) [v-tiny] {};
    \node (010) at (1.5,.5) [v-tiny] {};
    \node (001) at (.5,1.5) [v-tiny] {};
    \node (011) at (1.5,1.5) [v-tiny] {};
    %%
    \node (120) at (2,0) [v-tiny] {};
    \node (130) at (3,0) [v-tiny] {};
    \node (121) at (2,1) [v-tiny] {};
    \node (131) at (3,1) [v-tiny] {};
    \node (020) at (2.5,.5) [v-tiny] {};
    \node (030) at (3.5,.5) [v-tiny] {};
    \node (021) at (2.5,1.5) [v-tiny] {};
    \node (031) at (3.5,1.5) [v-tiny] {};
    %%
    \node (140) at (4,0) [v-tiny] {$h$}; \node (141) at (4,1) [v-tiny] {$1$};
    \node (040) at (4.5,.5) [v-tiny] {}; \node (041) at (4.5,1.5) [v-tiny]{$v$};
    %%
    \node (150) at (5,0) [v-tiny] {}; \node (151) at (5,1) [v-tiny] {$t$};
    \node (050) at (5.5,.5) [v-tiny] {}; \node (051) at (5.5,1.5) [v-tiny]{};
    %%
    \node (160) at (6,0) [v-tiny] {}; \node (161) at (6,1) [v-tiny] {};
    \node (060) at (6.5,.5) [v-tiny] {}; \node (061) at (6.5,1.5) [v-tiny] {};
    %%
    \node (170) at (7,0) [v-tiny] {}; \node (171) at (7,1) [v-tiny] {};
    \node (070) at (7.5,.5) [v-tiny] {}; \node (071) at (7.5,1.5) [v-tiny] {};
    %%
    \node (180) at (8,0) [v-tiny] {}; \node (181) at (8,1) [v-tiny] {};
    \node (080) at (8.5,.5) [v-tiny] {}; \node (081) at (8.5,1.5) [v-tiny] {};
    %%
    %% 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 [r-faded] (000) -- (0-10);
    \draw [r] (1-11) -- (101); \draw [r] (100) -- (1-10);
    %% Back square (cube 1)
    \draw [r] (001) -- (011); \draw [r-faded] (010) -- (000);
    \draw [bb-faded] (001) -- (000); \draw [bb-faded] (011) -- (010); 
    %% Side edges square (cube 1)
    \draw [oo] (101) -- (001); \draw [oo-faded] (100) -- (000);
    \draw [oo] (111) -- (011); \draw [oo-faded] (110) -- (010);
    %% Front square (cube 1)
    \draw [r] (101) -- (111); \draw [r] (110) -- (100);
    \draw [bb] (101) -- (100); \draw [bb] (111) -- (110);
    %% Edges between cubes 1 & 3
    \draw [r] (011) -- (021); \draw [r-faded] (020) -- (010);
    \draw [r] (111) -- (121); \draw [r] (120) -- (110);
    %% Back square (cube 2)
    \draw [r] (021) -- (031); \draw [r-faded] (030) -- (020);
    \draw [bb-faded] (021) -- (020); \draw [bb-faded] (031) -- (030); 
    %% Side edges square (cube 3)
    \draw [oo] (121) -- (021); \draw [oo-faded] (120) -- (020);
    \draw [oo] (131) -- (031); \draw [oo-faded] (130) -- (030);
    %% Front square (cube 3)
    \draw [r] (121) -- (131); \draw [r] (130) -- (120);
    \draw [bb] (121) -- (120); \draw [bb] (131) -- (130);
    %% Back square (cube 3)
    \draw [r] (031) -- (041); \draw [r-faded] (040) -- (030);
    \draw [bb-faded] (031) -- (030); \draw [bb-faded] (041) -- (040); 
    %% Side edges square (cube 4)
    \draw [oo] (141) -- (041); \draw [oo-faded] (140) -- (040);
    %% Front square (cube 4)
    \draw [r] (131) -- (141); \draw [r] (140) -- (130);
    \draw [bb] (131) -- (130); \draw [bb] (141) -- (140);
    %% Back square (cube 4)
    \draw [r] (041) -- (051); \draw [r-faded] (050) -- (040);
    \draw [bb-faded] (041) -- (040); \draw [bb-faded] (051) -- (050); 
    %% Side edges square (cube 5)
    \draw [oo] (151) -- (051); \draw [oo-faded] (150) -- (050);
    \draw [oo] (151) -- (051); \draw [oo-faded] (150) -- (050);
    %% Front square (cube 5)
    \draw [r] (141) -- (151); \draw [r] (150) -- (140);
    \draw [bb] (141) -- (140); \draw [bb] (151) -- (150);
    %% Back square (cube 5)
    \draw [r] (051) -- (061); \draw [r-faded] (060) -- (050);
    \draw [bb-faded] (051) -- (050); \draw [bb-faded] (061) -- (060); 
    %% Side edges square (cube 6)
    \draw [oo] (161) -- (061); \draw [oo-faded] (160) -- (060);
    \draw [oo] (161) -- (061); \draw [oo-faded] (160) -- (060);
    %% Front square (cube 6)
    \draw [r] (151) -- (161); \draw [r] (160) -- (150);
    \draw [bb] (151) -- (150); \draw [bb] (161) -- (160);
    %% Back square (cube 6)
    \draw [r] (061) -- (071); \draw [r-faded] (070) -- (060);
    \draw [bb-faded] (061) -- (060); \draw [bb-faded] (071) -- (070); 
    %% Side edges square (cube 7)
    \draw [oo] (171) -- (071); \draw [oo-faded] (170) -- (070);
    \draw [oo] (171) -- (071); \draw [oo-faded] (170) -- (070);
    %% Front square (cube 7)
    \draw [r] (161) -- (171); \draw [r] (170) -- (160);
    \draw [bb] (161) -- (160); \draw [bb] (171) -- (170);
    %% Back square (cube 7)
    \draw [r] (071) -- (081); \draw [r-faded] (080) -- (070);
    \draw [bb-faded] (071) -- (070); \draw [bb-faded] (081) -- (080); 
    %% Side edges square (cube 8)
    \draw [oo] (181) -- (081); \draw [oo-faded] (180) -- (080);
    \draw [oo] (181) -- (081); \draw [oo-faded] (180) -- (080);
    %% Front square (cube 8)
    \draw [r] (171) -- (181); \draw [r] (180) -- (170);
    \draw [bb] (171) -- (170); \draw [bb] (181) -- (180);   
    %% Half-edges (cube 9)
    \draw [r] (081) -- (091); \draw [r-faded] (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 $h_\ell$, or the
    vertical reflection $v$. Label the vertices of this Cayley graph
    with elements written in this form. \bigskip

  \item Now, consider the generating set $\Frieze_1=\big\<g,h,r\>$,
    where $g=g_1$ and $r=r_0$. Construct a Cayley graph, write down
    the presentation, and then repeat Part~(a) for this Cayley graph. \bigskip
  
\item Characterize the \emph{conjugacy classes} of the reflections and
  rotations by determining which symmetries $t^iht^{-i}$ and
  $t^irt^{-i}$ are for each $i\in\Z$. Then determine 
  \[
  \cl_G(h):=\big\{xhx^{-1}\mid x\in\Frieze_1\big\},\qquad\text{and}\qquad
  \cl_G(r):=\big\{xrx^{-1}\mid x\in\Frieze_1\big\}.
  \]

  
\item Characterize the conjugacy classes of the translations and glide
  reflections. That is, find
  \[
  \cl_G(t^i):=\big\{xt^ix^{-1}\mid x\in\Frieze_1\big\},\qquad\text{and}\qquad
  \cl_G(g_i):=\big\{xg_ix^{-1}\mid x\in\Frieze_1\big\}.
  \]
  
  \end{enumerate}

  \newpage 

  %%--------------------------------------------------------------------
  
\item Let $G$ be a finite group.
  \begin{enumerate}
  \item Show that for any $n>2$, the number of elements in $G$ of
    order $n$ is even.
  \item If $|G|$ is even, show that $G$ has an element of order $2$.
  \item Give explicit examples to show how the conclusions of the
    previous parts can fail if the hyoptheses are not met: $n=2$ for
    Part~(a) and odd $|G|$ for Part~(b).
  \item Show that if $g^2=1$ for all $g\in G$, then $G$ must be
    abelian. Does this hold for infinite groups? \smallskip
  \end{enumerate}
  
  %%--------------------------------------------------------------------

\item List all fifteen abelian groups of order $432=2^4\cdot 3^3$
  writing each one as a product of cyclic groups of prime power
  order. Then, determine which group it is isomorphic to of the form
  $\Z_{n_1}\times\cdots\times\Z_{n_k}$, where $n_{i+1}\mid n_i$; this
  is the ``elementary divisor'' form. \smallskip
  
  %%--------------------------------------------------------------------

\item Draw the Cayley graph of the group $G=\<a,b,c\mid
  a^2=b^3=c^3=abc=1\>$ on the skeleton of the icosahedron, shown
  below, and label the nodes with elements written using $a$, $b$,
  and $c$.
  
  \tikzstyle{v-tiny} = [circle, draw, fill=lightgrey,inner sep=0pt, 
    minimum size=4mm]
  \tikzstyle{gr} = [draw, thick, darkgrey]
  \[
  \begin{tikzpicture}[scale=1.5,auto]
    \tikzstyle{every node}=[font=\footnotesize]
    \begin{scope}[shift={(0,0)}]
      \node (m1) at (90:2.75) [v-tiny] {};
      \node (m2) at (90:1) [v-tiny] {$1$};
      \node (m3) at (-90:.5) [v-tiny] {};
      \node (m4) at (-90:1) [v-tiny] {};
      \node (l1) at (210:2.75) [v-tiny] {};
      \node (l2) at (210:1) [v-tiny] {};
      \node (l3) at (150:1) [v-tiny] {};
      \node (l4) at (150:.5) [v-tiny] {};
      \node (r1) at (330:2.75) [v-tiny] {};
      \node (r2) at (330:1) [v-tiny] {};
      \node (r3) at (30:1) [v-tiny] {};
      \node (r4) at (30:.5) [v-tiny] {};
      \draw [gr] (m1) to (r1); \draw [gr] (r1) to (r3); \draw [gr] (r3) to (m1);
      \draw [gr] (m2) to (l4); \draw [gr] (l4) to (l3); \draw [gr] (l3) to (m2);
      \draw [gr] (m3) to (r4); \draw [gr] (r4) to (r2); \draw [gr] (r2) to (m3);
      \draw [gr] (m4) to (l1); \draw [gr] (l1) to (l2); \draw [gr] (l2) to (m4);
      \draw [gr] (m1) to (l3); \draw [gr] (l3) to (l1); \draw [gr] (l1) to (m1);
      \draw [gr] (m2) to (r3); \draw [gr] (r3) to (r4); \draw [gr] (r4) to (m2);
      \draw [gr] (m3) to (l2); \draw [gr] (l2) to (l4); \draw [gr] (l4) to (m3);
      \draw [gr] (m4) to (r2); \draw [gr] (r2) to (r1); \draw [gr] (r1) to (m4);
      \draw [gr] (l2) to (l3); \draw [gr] (r2) to (r3);
      \draw [gr] (m1) to (m2); \draw [gr] (m3) to (m4);
      \draw [gr] (l1) to (r1); \draw [gr] (l4) to (r4);  
    \end{scope}
  \end{tikzpicture}
  \]
  There are five groups of order $12$: the abelian groups $C_{12}$ and
  $C_6\times C_2$, the dihedral group $D_6$, the alternating group
  $A_4$, and the dicyclic group $\Dic_6$. Determine which group $G$ is
  isomorphic to, and then re-draw this Cayley graph with the nodes
  labeled with elements of that group. \smallskip

  %%-------------------------------------------------------------------
    
\item Cayley graphs for six groups of order $16$ are shown
  below. Carry out the following steps for each.
  
  \begin{enumerate}
  \item Write down a group presentation.
  \item Identifying each element with its ``negative'' yields a
    quotient group of order $8$. Construct a Cayley table and Cayley
    graph for these quotient, using the elements
    \[
    \pm 1,\;\; \pm a,\;\; \pm b,\;\; \pm c,\;\; \pm w,\;\;
    \pm x,\;\; \pm y,\;\; \pm z,
    \]
    and determine the resulting isomorphism type. If two groups give the same
    table and graph, you do not need to write this out twice.
    
  
  
  %%
  \tikzstyle{v-big} = [circle, draw, fill=lightgrey,inner sep=0pt, 
    minimum size=7mm]
  \tikzstyle{R} = [draw, very thick, Red,-stealth,bend right=10]
  \tikzstyle{R-in} = [draw, very thick, Red,-stealth,bend right=12]
  \tikzstyle{R-out} = [draw, very thick, Red,-stealth,bend right=15]
  \tikzstyle{R2} = [draw, very thick, Red,-stealth,bend right=15]
  \tikzstyle{R3} = [draw, very thick, Red,-stealth,bend left=12]
  \tikzstyle{rr-bend} = [draw, very thick, Red,bend right=13]
\tikzstyle{bb-bend} = [draw, very thick, Blue,bend right=13]
  \[
  \hspace*{-8mm}
  \begin{tikzpicture}[scale=1.5]
    \tikzstyle{every node}=[font=\small]
    \begin{scope}[shift={(0,5)}]
      \node (s) at (0:1.1) [v-big] {$w$};
      \node (rs) at (45:1.15) [v-big] {$x$};
      \node (r2s) at (90:1.15) [v-big] {$y$};
      \node (r3s) at (135:1.15) [v-big] {$z$};
      \node (r4s) at (180:1.15) [v-big] {$-w$};
      \node (r5s) at (225:1.15) [v-big] {$-x$};
      \node (r6s) at (270:1.15) [v-big] {$-y$};
      \node (r7s) at (315:1.15) [v-big] {$-z$};
      %%
      \node (1) at (0:2) [v-big] {$1$};
      \node (r) at (45:2) [v-big] {$a$};
      \node (r2) at (90:2) [v-big] {$b$};
      \node (r3) at (135:2) [v-big] {$c$};
      \node (r4) at (180:2) [v-big] {$-1$};
      \node (r5) at (225:2) [v-big] {$-a$};
      \node (r6) at (270:2) [v-big] {$-b$};
      \node (r7) at (315:2) [v-big] {$-c$};
      %%
      \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-in] (s) to (rs);
      \draw [R-in] (rs) to (r2s);
      \draw [R-in] (r2s) to (r3s);
      \draw [R-in] (r3s) to (r4s);
      \draw [R-in] (r4s) to (r5s);
      \draw [R-in] (r5s) to (r6s);
      \draw [R-in] (r6s) to (r7s);
      \draw [R-in] (r7s) 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 $C_8\times C_2$};
    \end{scope}
    %%
    \begin{scope}[shift={(5.5,5)}]
      \node (s) at (0:1.1) [v-big] {$w$};
      \node (rs) at (45:1.15) [v-big] {$x$};
      \node (r2s) at (90:1.15) [v-big] {$y$};
      \node (r3s) at (135:1.15) [v-big] {$z$};
      \node (r4s) at (180:1.15) [v-big] {$-w$};
      \node (r5s) at (225:1.15) [v-big] {$-x$};
      \node (r6s) at (270:1.15) [v-big] {$-y$};
      \node (r7s) at (315:1.15) [v-big] {$-z$};
      %%
      \node (1) at (0:2) [v-big] {$1$};
      \node (r) at (45:2) [v-big] {$a$};
      \node (r2) at (90:2) [v-big] {$b$};
      \node (r3) at (135:2) [v-big] {$c$};
      \node (r4) at (180:2) [v-big] {$-1$};
      \node (r5) at (225:2) [v-big] {$-a$};
      \node (r6) at (270:2) [v-big] {$-b$};
      \node (r7) at (315:2) [v-big] {$-c$};
      %%
      \draw [R2] (1) to (r);
      \draw [R2] (r) to (r2);
      \draw [R2] (r2) to (r3);
      \draw [R2] (r3) to (r4);
      \draw [R2] (r4) to (r5);
      \draw [R2] (r5) to (r6);
      \draw [R2] (r6) to (r7);
      \draw [R2] (r7) to (1);
      %%
      \draw [R3] (s) to (r7s);
      \draw [R3] (r7s) to (r6s);
      \draw [R3] (r6s) to (r5s);
      \draw [R3] (r5s) to (r4s);
      \draw [R3] (r4s) to (r3s);
      \draw [R3] (r3s) to (r2s);
      \draw [R3] (r2s) to (rs);
      \draw [R3] (rs) 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 $D_8$};
    \end{scope}
    %%
    \begin{scope}[shift={(0,0)}]
      \node (s) at (0:1.1) [v-big] {$w$};
      \node (rs) at (45:1.15) [v-big] {$x$};
      \node (r2s) at (90:1.15) [v-big] {$y$};
      \node (r3s) at (135:1.15) [v-big] {$z$};
      \node (r4s) at (180:1.15) [v-big] {$-w$};
      \node (r5s) at (225:1.15) [v-big] {$-x$};
      \node (r6s) at (270:1.15) [v-big] {$-y$};
      \node (r7s) at (315:1.15) [v-big] {$-z$};
      %%
      \node (1) at (0:2) [v-big] {$1$};
      \node (r) at (45:2) [v-big] {$a$};
      \node (r2) at (90:2) [v-big] {$b$};
      \node (r3) at (135:2) [v-big] {$c$};
      \node (r4) at (180:2) [v-big] {$-1$};
      \node (r5) at (225:2) [v-big] {$-a$};
      \node (r6) at (270:2) [v-big] {$-b$};
      \node (r7) at (315:2) [v-big] {$-c$};
      %%
      \draw [R2] (1) to (r);
      \draw [R2] (r) to (r2);
      \draw [R2] (r2) to (r3);
      \draw [R2] (r3) to (r4);
      \draw [R2] (r4) to (r5);
      \draw [R2] (r5) to (r6);
      \draw [R2] (r6) to (r7);
      \draw [R2] (r7) to (1);
      %%
      \draw [r] (s) to (r3s);
      \draw [r] (r3s) to (r6s);
      \draw [r] (r6s) to (rs);
      \draw [r] (rs) to (r4s);
      \draw [r] (r4s) to (r7s);
      \draw [r] (r7s) to (r2s);
      \draw [r] (r2s) to (r5s);
      \draw [r] (r5s) 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 $\SD_8$};
    \end{scope}
    %%
    \begin{scope}[shift={(5.5,0)}]
      \node (s) at (0:1.1) [v-big] {$w$};
      \node (rs) at (45:1.15) [v-big] {$x$};
      \node (r2s) at (90:1.15) [v-big] {$y$};
      \node (r3s) at (135:1.15) [v-big] {$z$};
      \node (r4s) at (180:1.15) [v-big] {$-w$};
      \node (r5s) at (225:1.15) [v-big] {$-x$};
      \node (r6s) at (270:1.15) [v-big] {$-y$};
      \node (r7s) at (315:1.15) [v-big] {$-z$};
      %%
      \node (1) at (0:2) [v-big] {$1$};
      \node (r) at (45:2) [v-big] {$a$};
      \node (r2) at (90:2) [v-big] {$b$};
      \node (r3) at (135:2) [v-big] {$c$};
      \node (r4) at (180:2) [v-big] {$-1$};
      \node (r5) at (225:2) [v-big] {$-a$};
      \node (r6) at (270:2) [v-big] {$-b$};
      \node (r7) at (315:2) [v-big] {$-c$};
      %%
      \draw [R2] (1) to (r);
      \draw [R2] (r) to (r2);
      \draw [R2] (r2) to (r3);
      \draw [R2] (r3) to (r4);
      \draw [R2] (r4) to (r5);
      \draw [R2] (r5) to (r6);
      \draw [R2] (r6) to (r7);
      \draw [R2] (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={(0,.-5)},scale=1,auto]
      %\tikzstyle{every node}=[font=\scriptsize]
      \tikzstyle{R-out} = [draw, very thick, Red,-stealth,bend right=15]
      \tikzstyle{R-in} = [draw, very thick, Red,-stealth,bend left=12]
      \tikzstyle{B} = [draw, very thick, Blue,-stealth,bend right=22]
      \node (s) at (0:1) [v-big] {$w$};
      \node (rs) at (45:1) [v-big] {$x$};
      \node (r2s) at (90:1) [v-big] {$y$};
      \node (r3s) at (135:1) [v-big] {$z$};
      \node (r4s) at (180:1) [v-big] {$-w$};
      \node (r5s) at (225:1) [v-big] {$-x$};
      \node (r6s) at (270:1) [v-big] {$-y$};
      \node (r7s) at (315:1) [v-big] {$-z$};
      %%
      \node (1) at (0:2) [v-big] {$1$};
      \node (r) at (45:2) [v-big] {$a$};
      \node (r2) at (90:2) [v-big] {$b$};
      \node (r3) at (135:2) [v-big] {$c$};
      \node (r4) at (180:2) [v-big] {$-1$};
      \node (r5) at (225:2) [v-big] {$-a$};
      \node (r6) at (270:2) [v-big] {$-b$};
      \node (r7) at (315:2) [v-big] {$-c$};
      %%
      \draw [b] (1) to (s); \draw [B] (s) to (r4);
      \draw [b] (r4) to (r4s); \draw [B] (r4s) to (1);
      \draw [b] (r) to (rs); \draw [B] (rs) to (r5);
      \draw [b] (r5) to (r5s); \draw [B] (r5s) to (r);
      \draw [b] (r2) to (r2s); \draw [B] (r2s) to (r6);
      \draw [b] (r6) to (r6s); \draw [B] (r6s) to (r2);
      \draw [b] (r3) to (r3s); \draw [B] (r3s) to (r7);
      \draw [b] (r7) to (r7s); \draw [B] (r7s) to (r3);
      %%
      \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-in] (s) to (r7s);
      \draw [R-in] (r7s) to (r6s);
      \draw [R-in] (r6s) to (r5s);
      \draw [R-in] (r5s) to (r4s);
      \draw [R-in] (r4s) to (r3s);
      \draw [R-in] (r3s) to (r2s);
      \draw [R-in] (r2s) to (rs);
      \draw [R-in] (rs) to (s);
      %%
      \node at (0,0) {\normalsize $Q_{16}$};
    \end{scope}   
    %%
    \begin{scope}[shift={(5.5,-5)}]
      \node at (0,0) {\normalsize $\DQ_8$};
      %%
      \node (s) at (0:1.1) [v-big] {$w$};
      \node (rs) at (45:1.15) [v-big] {$x$};
      \node (r2s) at (90:1.15) [v-big] {$y$};
      \node (r3s) at (135:1.15) [v-big] {$z$};
      \node (r4s) at (180:1.15) [v-big] {$-w$};
      \node (r5s) at (225:1.15) [v-big] {$-x$};
      \node (r6s) at (270:1.15) [v-big] {$-y$};
      \node (r7s) at (315:1.15) [v-big] {$-z$};
      %%
      \node (1) at (0:2) [v-big] {$1$};
      \node (r) at (45:2) [v-big] {$a$};
      \node (r2) at (90:2) [v-big] {$b$};
      \node (r3) at (135:2) [v-big] {$c$};
      \node (r4) at (180:2) [v-big] {$-1$};
      \node (r5) at (225:2) [v-big] {$-a$};
      \node (r6) at (270:2) [v-big] {$-b$};
      \node (r7) at (315:2) [v-big] {$-c$};
      %%
      \draw [rr-bend] (1) to (r);
      \draw [bb-bend] (r) to (r2);
      \draw [rr-bend] (r2) to (r3);
      \draw [bb-bend] (r3) to (r4);
      \draw [rr-bend] (r4) to (r5);
      \draw [bb-bend] (r5) to (r6);
      \draw [rr-bend] (r6) to (r7);
      \draw [bb-bend] (r7) to (1);
      %%
      \draw [bb] (s) to (r3s);
      \draw [rr] (r3s) to (r6s);
      \draw [bb] (r6s) to (rs);
      \draw [rr] (rs) to (r4s);
      \draw [bb] (r4s) to (r7s);
      \draw [rr] (r7s) to (r2s);
      \draw [bb] (r2s) to (r5s);
      \draw [rr] (r5s) to (s);
      %%
      \draw [gg] (1) to (s); \draw [gg] (r) to (rs);
      \draw [gg] (r2) to (r2s); \draw [gg] (r3) to (r3s);
      \draw [gg] (r4) to (r4s); \draw [gg] (r5) to (r5s);
      \draw [gg] (r6) to (r6s); \draw [gg] (r7) to (r7s);
      %%
   \end{scope}   
\end{tikzpicture}
  \]

  \end{enumerate}
  
\newpage
  
  %%-------------------------------------------------------------------  
  
\item Two Cayley graphs arranged on a truncated cube are shown
  below. One of these is of the symmetric group $S_4$, and the other is
  $A_4\times C_2$, which can be written as 
  \[
  A_4\times C_2=\big\{\pm\sigma\mid \sigma\in A_4\big\}.
  \]
  Determine which Cayley graphs goes with which group, and then label
  the nodes with permutations in cycle notation, written as a product
  of disjoint cycles.

  \tikzstyle{v-tiny} = [circle, draw, fill=lightgrey,inner sep=0pt, 
  minimum size=4.25mm]
\newcommand\aaa{1}\newcommand\bbb{2}\newcommand\ccc{3}\newcommand\ddd{4.6}
\[
\hspace*{-10mm}
  \begin{tikzpicture}[scale=1.25]
    \begin{scope}[shift={(0,0)}]
      \node (nw1) at (-1.5,1.5) [v-tiny] {\scriptsize $e$};
      \node (nw2) at (-.5,1.5) [v-tiny] {}; 
      \node (nw3) at (-1.5,.5) [v-tiny] {};
      \node (ne1) at (1.5,1.5) [v-tiny] {};
      \node (ne2) at (1.5,.5) [v-tiny] {}; 
      \node (ne3) at (.5,1.5) [v-tiny] {};
      \node (se1) at (1.5,-1.5) [v-tiny] {};
      \node (se2) at (.5,-1.5) [v-tiny] {};
      \node (se3) at (1.5,-.5) [v-tiny] {}; 
      \node (sw1) at (-1.5,-1.5) [v-tiny] {};
      \node (sw2) at (-1.5,-.5) [v-tiny] {}; 
      \node (sw3) at (-.5,-1.5) [v-tiny] {};
      %
      \draw [bb] (nw2) to (ne3); \draw [bb] (ne2) to (se3);
      \draw [bb] (se2) to (sw3); \draw [bb] (sw2) to (nw3);
      \draw[r](nw2)to(nw1); \draw [r] (nw3) to (nw2); \draw [r] (nw1) to (nw3);
      \draw[r](ne2)to(ne1); \draw [r] (ne3) to (ne2); \draw [r] (ne1) to (ne3);
      \draw[r](se2)to(se1); \draw [r] (se3) to (se2); \draw [r] (se1) to (se3);
      \draw[r](sw2)to(sw1); \draw [r] (sw3) to (sw2); \draw [r] (sw1) to (sw3);
      %%
      \node (NW1) at (-2.25,2.25) [v-tiny] {};
      \node (NW2) at (-3.25,2.25) [v-tiny] {}; 
      \node (NW3) at (-2.25,3.25) [v-tiny] {};
      \node (NE1) at (2.25,2.25) [v-tiny] {};
      \node (NE2) at (2.25,3.25) [v-tiny] {}; 
      \node (NE3) at (3.25,2.25) [v-tiny] {};
      \node (SE1) at (2.25,-2.25) [v-tiny] {};
      \node (SE2) at (3.25,-2.25) [v-tiny] {};
      \node (SE3) at (2.25,-3.25) [v-tiny] {}; 
      \node (SW1) at (-2.25,-2.25) [v-tiny] {};
      \node (SW2) at (-2.25,-3.25) [v-tiny] {}; 
      \node (SW3) at (-3.25,-2.25) [v-tiny] {};
      %%
      \draw [bb] (NW3) to (NE2); \draw [bb] (NE3) to (SE2);
      \draw [bb] (SE3) to (SW2); \draw [bb] (SW3) to (NW2);
      %
      \draw[r](NW2)to(NW1); \draw [r] (NW3) to (NW2); \draw [r] (NW1) to (NW3);
      \draw[r](NE2)to(NE1); \draw [r] (NE3) to (NE2); \draw [r] (NE1) to (NE3);
      \draw[r](SE2)to(SE1); \draw [r] (SE3) to (SE2); \draw [r] (SE1) to (SE3);
      \draw[r](SW2)to(SW1); \draw [r] (SW3) to (SW2); \draw [r] (SW1) to (SW3);
      %%
      \draw [bb] (nw1) to (NW1); \draw [bb] (ne1) to (NE1);
      \draw [bb] (se1) to (SE1); \draw [bb] (sw1) to (SW1);
    \end{scope}
    %%
    \begin{scope}[shift={(0,-8)}]
      \node (nw1) at (-1.5,1.5) [v-tiny] {\scriptsize $e$};
      \node (nw2) at (-.5,1.5) [v-tiny] {}; 
      \node (nw3) at (-1.5,.5) [v-tiny] {};
      \node (ne1) at (1.5,1.5) [v-tiny] {};
      \node (ne2) at (1.5,.5) [v-tiny] {}; 
      \node (ne3) at (.5,1.5) [v-tiny] {};
      \node (se1) at (1.5,-1.5) [v-tiny] {};
      \node (se2) at (.5,-1.5) [v-tiny] {};
      \node (se3) at (1.5,-.5) [v-tiny] {}; 
      \node (sw1) at (-1.5,-1.5) [v-tiny] {};
      \node (sw2) at (-1.5,-.5) [v-tiny] {}; 
      \node (sw3) at (-.5,-1.5) [v-tiny] {};
      %
      \draw [bb] (nw2) to (ne3); \draw [bb] (ne2) to (se3);
      \draw [bb] (se2) to (sw3); \draw [bb] (sw2) to (nw3);
      \draw[r](nw2)to(nw1); \draw [r] (nw3) to (nw2); \draw [r] (nw1) to (nw3);
      \draw[r](ne3)to(ne1); \draw [r] (ne2) to (ne3); \draw [r] (ne1) to (ne2);
      \draw[r](se2)to(se1); \draw [r] (se3) to (se2); \draw [r] (se1) to (se3);
      \draw[r](sw3)to(sw1); \draw [r] (sw2) to (sw3); \draw [r] (sw1) to (sw2);
      %%
      \node (NW1) at (-2.25,2.25) [v-tiny] {};
      \node (NW2) at (-3.25,2.25) [v-tiny] {}; 
      \node (NW3) at (-2.25,3.25) [v-tiny] {};
      \node (NE1) at (2.25,2.25) [v-tiny] {};
      \node (NE2) at (2.25,3.25) [v-tiny] {}; 
      \node (NE3) at (3.25,2.25) [v-tiny] {};
      \node (SE1) at (2.25,-2.25) [v-tiny] {};
      \node (SE2) at (3.25,-2.25) [v-tiny] {};
      \node (SE3) at (2.25,-3.25) [v-tiny] {}; 
      \node (SW1) at (-2.25,-2.25) [v-tiny] {};
      \node (SW2) at (-2.25,-3.25) [v-tiny] {}; 
      \node (SW3) at (-3.25,-2.25) [v-tiny] {};
      %%
      \draw [bb] (NW3) to (NE2); \draw [bb] (NE3) to (SE2);
      \draw [bb] (SE3) to (SW2); \draw [bb] (SW3) to (NW2);
      %
      \draw[r](NW1)to(NW2); \draw [r] (NW2) to (NW3); \draw [r] (NW3) to (NW1);
      \draw[r](NE1)to(NE3); \draw [r] (NE3) to (NE2); \draw [r] (NE2) to (NE1);
      \draw[r](SE1)to(SE2); \draw [r] (SE2) to (SE3); \draw [r] (SE3) to (SE1);
      \draw[r](SW1)to(SW3); \draw [r] (SW3) to (SW2); \draw [r] (SW2) to (SW1);
      %%
      \draw [bb] (nw1) to (NW1); \draw [bb] (ne1) to (NE1);
      \draw [bb] (se1) to (SE1); \draw [bb] (sw1) to (SW1);
      %
    \end{scope}
  \end{tikzpicture}
  \]


\end{enumerate}

\end{document}

