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\pagestyle{fancy} \lhead{{\sf Homework 4 $|$ Due February 7 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Visual Algebra $|$
    Spring 2025 $|$ M.~Macauley}}

\begin{document}

%$\;$

\begin{enumerate}

  %%-------------------------------------------------------------------
  
\item Cayley graphs for the four semidirect products of $C_8$ with
  $C_2$ are shown below, with a different labeling scheme on their nodes.
  
  %%
  \[
  \hspace*{-8mm}
  \begin{tikzpicture}[scale=1.6,auto]
    \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt, 
      minimum size=7mm]
    \tikzstyle{R} = [draw, very thick, eRed,-stealth,bend right=10]
    \tikzstyle{R-in} = [draw, very thick, eRed,-stealth,bend right=12]
    \tikzstyle{R-out} = [draw, very thick, eRed,-stealth,bend right=15]
    \tikzstyle{R2} = [draw, very thick, eRed,-stealth,bend right=15]
    \tikzstyle{R3} = [draw, very thick, eRed,-stealth,bend left=12]
    \tikzstyle{every node}=[font=\small]
    \begin{scope}[shift={(0,5.25)}]
      \node (s) at (0:1.1) [v] {$w$};
      \node (rs) at (45:1.15) [v] {$x$};
      \node (r2s) at (90:1.15) [v] {$y$};
      \node (r3s) at (135:1.15) [v] {$z$};
      \node (r4s) at (180:1.15) [v] {$-w$};
      \node (r5s) at (225:1.15) [v] {$-x$};
      \node (r6s) at (270:1.15) [v] {$-y$};
      \node (r7s) at (315:1.15) [v] {$-z$};
      %%
      \node (1) at (0:2) [v] {$1$};
      \node (r) at (45:2) [v] {$a$};
      \node (r2) at (90:2) [v] {$b$};
      \node (r3) at (135:2) [v] {$c$};
      \node (r4) at (180:2) [v] {$-1$};
      \node (r5) at (225:2) [v] {$-a$};
      \node (r6) at (270:2) [v] {$-b$};
      \node (r7) at (315:2) [v] {$-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={(5.5,5.25)}]
      \node (s) at (0:1.1) [v] {$w$};
      \node (rs) at (45:1.15) [v] {$x$};
      \node (r2s) at (90:1.15) [v] {$y$};
      \node (r3s) at (135:1.15) [v] {$z$};
      \node (r4s) at (180:1.15) [v] {$-w$};
      \node (r5s) at (225:1.15) [v] {$-x$};
      \node (r6s) at (270:1.15) [v] {$-y$};
      \node (r7s) at (315:1.15) [v] {$-z$};
      %%
      \node (1) at (0:2) [v] {$1$};
      \node (r) at (45:2) [v] {$a$};
      \node (r2) at (90:2) [v] {$b$};
      \node (r3) at (135:2) [v] {$c$};
      \node (r4) at (180:2) [v] {$-1$};
      \node (r5) at (225:2) [v] {$-a$};
      \node (r6) at (270:2) [v] {$-b$};
      \node (r7) at (315:2) [v] {$-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] {$w$};
      \node (rs) at (45:1.15) [v] {$x$};
      \node (r2s) at (90:1.15) [v] {$y$};
      \node (r3s) at (135:1.15) [v] {$z$};
      \node (r4s) at (180:1.15) [v] {$-w$};
      \node (r5s) at (225:1.15) [v] {$-x$};
      \node (r6s) at (270:1.15) [v] {$-y$};
      \node (r7s) at (315:1.15) [v] {$-z$};
      %%
      \node (1) at (0:2) [v] {$1$};
      \node (r) at (45:2) [v] {$a$};
      \node (r2) at (90:2) [v] {$b$};
      \node (r3) at (135:2) [v] {$c$};
      \node (r4) at (180:2) [v] {$-1$};
      \node (r5) at (225:2) [v] {$-a$};
      \node (r6) at (270:2) [v] {$-b$};
      \node (r7) at (315:2) [v] {$-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] {$w$};
      \node (rs) at (45:1.15) [v] {$x$};
      \node (r2s) at (90:1.15) [v] {$y$};
      \node (r3s) at (135:1.15) [v] {$z$};
      \node (r4s) at (180:1.15) [v] {$-w$};
      \node (r5s) at (225:1.15) [v] {$-x$};
      \node (r6s) at (270:1.15) [v] {$-y$};
      \node (r7s) at (315:1.15) [v] {$-z$};
      %%
      \node (1) at (0:2) [v] {$1$};
      \node (r) at (45:2) [v] {$a$};
      \node (r2) at (90:2) [v] {$b$};
      \node (r3) at (135:2) [v] {$c$};
      \node (r4) at (180:2) [v] {$-1$};
      \node (r5) at (225:2) [v] {$-a$};
      \node (r6) at (270:2) [v] {$-b$};
      \node (r7) at (315:2) [v] {$-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}
  \end{tikzpicture}
  \]
  
  For all four of these groups, identifying each element with its
  ``negative'' yields a ``quotient group'' of order $8$, like what we
  did with the dicyclic group $\Dic_8=Q_{16}$ in HW 2. Construct a
  Cayley table and Cayley graph for each of these quotient groups,
  using the elements \vspace{-1mm}
  \[
  \pm 1,\;\; \pm a,\;\; \pm b,\;\; \pm c,\;\; \pm w,\;\;
  \pm x,\;\; \pm y,\;\; \pm z,
  \]
  and determine to which familiar group each is isomorphic. If two
  groups give the same table and graph, you do not need to write
  this out twice.

\newpage
  

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


\item The \emph{diquaternion group} $\DQ_8$ can be constructed from
  our standard representation of $Q_8=\<R_4,S,T\>$, along with the
  reflection matrix $F$ in $D_n=\<R_n,F\>$. That is,
  \[
  \DQ_8\cong\big\<i,j,k,f\big\>\cong\left\<
  \underbrace{\begin{bmatrix} i & 0 \\ 0 & -i\end{bmatrix}}_{R=R_4},
  \underbrace{\begin{bmatrix} 0 & 1 \\ -1 & 0\end{bmatrix}}_{S},
  \underbrace{\begin{bmatrix} 0 & i \\ i & 0\end{bmatrix}}_{T=RS},
  \underbrace{\begin{bmatrix}0 & 1 \\ 1 &
      0\end{bmatrix}}_{F}\right\>.
  \]
  A Cayley graph for $\DQ_8$ generated by the \emph{Pauli matrices}
  $X,Y,Z$, is shown below (left). \vspace{-9mm}

  \[
  \hspace*{-5mm}
  \begin{tikzpicture}[scale=1.4,auto]
    \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt, 
      minimum size=5.5mm]
    \tikzstyle{v-y} = [circle, draw, fill=lightgray,inner sep=0pt, 
      minimum size=5.5mm]
    \tikzstyle{rr-bend} = [draw, very thick, eRed,bend right=13]
    \tikzstyle{bb-bend} = [draw, very thick, eBlue,bend right=13]
    %%
    \tikzstyle{every node}=[font=\scriptsize]
   \begin{scope}[shift={(0,0)}]
      \node at (0,0) {\normalsize $\DQ_8$};
      \node at (-3.75,1) {\normalsize $X=\begin{bmatrix}0&1\\1&0\end{bmatrix}$};
      \node at (-3.75,0)  {\normalsize $Y=\begin{bmatrix}0&-i\\ i&0\end{bmatrix}$};
      \node at (-3.75,-1) {\normalsize $Z=\begin{bmatrix}1&0\\0&-1\end{bmatrix}$};
      %%
      \node (s) at (0:1.1) [v] {$Z$};
      \node (rs) at (45:1.15) [v-y] {$-iY$};
      \node (r2s) at (90:1.15) [v] {$iI$};
      \node (r3s) at (135:1.15) [v-y] {$-iX$};
      \node (r4s) at (180:1.15) [v] {$-Z$};
      \node (r5s) at (225:1.15) [v-y] {$iY$};
      \node (r6s) at (270:1.15) [v] {$-iI$};
      \node (r7s) at (315:1.15) [v-y] {$iX$};
      %%
      \node (1) at (0:2) [v-y] {$I$};
      \node (r) at (45:2) [v] {$X$};
      \node (r2) at (90:2) [v-y] {$iZ$};
      \node (r3) at (135:2) [v] {$-Y$};
      \node (r4) at (180:2) [v-y] {$-I$};
      \node (r5) at (225:2) [v] {$-X$};
      \node (r6) at (270:2) [v-y] {$-iZ$};
      \node (r7) at (315:2) [v] {$Y$};
      %%
      \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}   
   \begin{scope}[shift={(5,0)}]
      \node at (0,0) {\normalsize $\DQ_8$};
      %%
      \node (s) at (0:1.1) [v] {$w$};
      \node (rs) at (45:1.15) [v-y] {$x$};
      \node (r2s) at (90:1.15) [v] {$y$};
      \node (r3s) at (135:1.15) [v-y] {$z$};
      \node (r4s) at (180:1.15) [v] {$-w$};
      \node (r5s) at (225:1.15) [v-y] {$-x$};
      \node (r6s) at (270:1.15) [v] {$-y$};
      \node (r7s) at (315:1.15) [v-y] {$-z$};
      %%
      \node (1) at (0:2) [v-y] {$1$};
      \node (r) at (45:2) [v] {$a$};
      \node (r2) at (90:2) [v-y] {$b$};
      \node (r3) at (135:2) [v] {$c$};
      \node (r4) at (180:2) [v-y] {$-1$};
      \node (r5) at (225:2) [v] {$-a$};
      \node (r6) at (270:2) [v-y] {$-b$};
      \node (r7) at (315:2) [v] {$-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}
  \]
   
  \begin{enumerate}
  \item Find a presentation for $\DQ_8=\<X,Y,Z\>$.
  \item Construct a Cayley graph for $\DQ_8=\<R,S,F\>$, and find a
    group presentation. Extra credit will be given for the
    best-looking construction.
  \item Carry out the ``quotient process'' as was done for $C_8\times
    C_2$, $D_8$, $\SD_8$, and $\SA_8$ in the previous problem; see the
    graph on the right.
  \end{enumerate}
    

\item  For this problem, the use of an online matrix calculator,
  like \url{https://matrixcalc.org}, that can handle complex
  exponential inputs, is strongly recommended.
  \begin{enumerate}
  \item The dicyclic group $\Dic_n$ is only defined when $n$ is
    even. However, if we try to define
    \[
    \Dic_3:=\left\<\begin{bmatrix}\zeta_3&0\\0&\overline{\zeta}_3\end{bmatrix},
    \begin{bmatrix}0&1\\-1&0\end{bmatrix}\right\>,
      \]
      then the result is still a group. Determine which group this is,
      with justification.
      
    \item The diquaternion and semidihedral groups, $\DQ_n$ and
      $\SD_n$, are only defined when $n=2^m$. However, we can still define
      \[
      \DQ_6:=\left\<\begin{bmatrix}\zeta_6&0\\0&\overline{\zeta}_6\end{bmatrix},
      \begin{bmatrix}0&1\\-1&0\end{bmatrix},
      \begin{bmatrix}0&1\\1&0\end{bmatrix}\right\>,\qquad      
      \SD_3:=\left\<\begin{bmatrix}\zeta_3&0\\0&-\overline{\zeta}_3\end{bmatrix},
      \begin{bmatrix}0&1\\1&0\end{bmatrix}\right\>.
      \]
      What groups are these? Construct a Cayley graph for each. 
  \end{enumerate}
  
\newpage
  
  %%-------------------------------------------------------------------

  
\item The automorphism group of $C_n$ is isomorphic to $U_n$, the
  multiplicative group of integers modulo $n$, from HW 2, \#2. For
  each of the following, construct a Cayley graph of $\Aut(C_n)$
  with the nodes labeled by re-wirings, and a Cayley table for this
  group. Then determine its isomorphism type.
  
  \begin{enumerate}
  \item $\Aut(C_8)=\<\nu,\mu\>$,\; defined by \hspace{5mm}   
    $\begin{tikzpicture}[scale=1.6,baseline=-.5ex]
    \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt, 
      minimum size=5mm]
    \tikzstyle{every node}=[font=\footnotesize]
    %%
    \begin{scope}[shift={(0,0)}]
      \node (1) at (0:1) [v] {$1$};
      \node (r) at (45:1) [v] {$r$};
      \node (r2) at (90:1) [v] {$r^2$};
      \node (r3) at (135:1) [v] {$r^3$};
      \node (r4) at (180:1) [v] {$r^4$};
      \node (r5) at (225:1) [v] {$r^5$};
      \node (r6) at (270:1) [v] {$r^6$};
      \node (r7) at (315:1) [v] {$r^7$};
      \draw [r] (1) to (r3); \draw [r] (r3) to (r6); \draw [r] (r6) to (r);
      \draw [r] (r) to (r4); \draw [r] (r4) to (r7); \draw [r] (r7) to (r2);
      \draw [r] (r2) to (r5); \draw [r] (r5) to (1);
      \node at (0,0) {\normalsize $\nu$};
    \end{scope}
    %%
    \begin{scope}[shift={(3,0)}]
      \node (1) at (0:1) [v] {$1$};
      \node (r) at (45:1) [v] {$r$};
      \node (r2) at (90:1) [v] {$r^2$};
      \node (r3) at (135:1) [v] {$r^3$};
      \node (r4) at (180:1) [v] {$r^4$};
      \node (r5) at (225:1) [v] {$r^5$};
      \node (r6) at (270:1) [v] {$r^6$};
      \node (r7) at (315:1) [v] {$r^7$};
      \draw [r] (1) to (r5); \draw [r] (r5) to (r2); \draw [r] (r2) to (r7);
      \draw [r] (r7) to (r4); \draw [r] (r4) to (r); \draw [r] (r) to (r6);
      \draw [r] (r6) to (r3); \draw [r] (r3) to (1);
      \node at (0,0) {\normalsize $\mu$};
    \end{scope}
  \end{tikzpicture}$

    \medskip
    
  \item $\Aut(C_9)=\<\varphi\>$,\; defined by \hspace{5mm}
    $\begin{tikzpicture}[scale=1.6,baseline=-.5ex]
    \tikzstyle{v} = [circle,draw,fill=lightgray,inner sep=0pt,minimum size=5mm]
    \tikzstyle{every node}=[font=\footnotesize]
    %%
    \begin{scope}[shift={(6,0)}]
      \node (1) at (0:1) [v] {$1$};
      \node (r) at (40:1) [v] {$r$};
      \node (r2) at (80:1) [v] {$r^2$};
      \node (r3) at (120:1) [v] {$r^3$};
      \node (r4) at (160:1) [v] {$r^4$};
      \node (r5) at (200:1) [v] {$r^5$};
      \node (r6) at (240:1) [v] {$r^6$};
      \node (r7) at (280:1) [v] {$r^7$};
      \node (r8) at (320:1) [v] {$r^8$};
      \draw [r] (1) to (r2); \draw [r] (r2) to (r4); \draw [r] (r4) to (r6);
      \draw [r] (r6) to (r8); \draw [r] (r8) to (r); \draw [r] (r) to (r3);
      \draw [r] (r3) to (r5); \draw [r] (r5) to (r7); \draw [r] (r7) to (1);
      \node at (0,0) {\normalsize $\varphi$};
    \end{scope}
  \end{tikzpicture}$
    
    \medskip
    
  \item $\Aut(C_{10})=\<\tau\>$,\; defined by \hspace{5mm}
    $\begin{tikzpicture}[scale=1.6,baseline=-.5ex]
    \tikzstyle{v} = [circle,draw,fill=lightgray,inner sep=0pt,minimum size=5mm]
    \tikzstyle{every node}=[font=\footnotesize]
    %%
    \begin{scope}[shift={(9,0)}]
      \node (1) at (0:1) [v] {$1$};
      \node (r) at (36:1) [v] {$r$};
      \node (r2) at (72:1) [v] {$r^2$};
      \node (r3) at (108:1) [v] {$r^3$};
      \node (r4) at (144:1) [v] {$r^4$};
      \node (r5) at (180:1) [v] {$r^5$};
      \node (r6) at (216:1) [v] {$r^6$};
      \node (r7) at (252:1) [v] {$r^7$};
      \node (r8) at (288:1) [v] {$r^8$};
      \node (r9) at (324:1) [v] {$r^9$};
      \draw [r] (1) to (r3); \draw [r] (r3) to (r6); \draw [r] (r6) to (r9);
      \draw [r] (r9) to (r2); \draw [r] (r2) to (r5); \draw [r] (r5) to (r8);
      \draw [r] (r8) to (r); \draw [r] (r) to (r4); \draw [r] (r4) to (r7);
      \draw [r] (r7) to (1);
      \node at (0,0) {\normalsize $\tau$};
    \end{scope}
  \end{tikzpicture}$
    
    \medskip
    
  \item $\Aut(C_{16})=\<\rho,\sigma\>$,\; defined by
    \[
    \begin{tikzpicture}[scale=1.4,auto]
      \tikzstyle{v}=[circle,draw,fill=lightgray,inner sep=0pt,
        minimum size=5.5mm] \tikzstyle{every node}=[font=\scriptsize]
      %%
      \begin{scope}[shift={(0,0)}]
        \node (1) at (0:2) [v] {$1$};
        \node (r) at (22.5:2) [v] {$r$};
        \node (r2) at (45:2) [v] {$r^2$};
        \node (r3) at (67.5:2) [v] {$r^3$};
        \node (r4) at (90:2) [v] {$r^4$};
        \node (r5) at (112.5:2) [v] {$r^5$};
        \node (r6) at (135:2) [v] {$r^6$};
        \node (r7) at (157.5:2) [v] {$r^7$};
        \node (r8) at (180:2) [v] {$r^8$};
        \node (r9) at (-157.5:2) [v] {$r^9$};
        \node (r10) at (-135:2) [v] {$r^{10}$};
        \node (r11) at (-112.5:2) [v] {$r^{1\!1}$};
        \node (r12) at (-90:2) [v] {$r^{1\!2}$};
        \node (r13) at (-67.5:2) [v] {$r^{1\!3}$};
        \node (r14) at (-45:2) [v] {$r^{1\!4}$};
        \node (r15) at (-22.5:2) [v] {$r^{1\!5}$};
        \draw [r] (1) to (r3); \draw [r] (r3) to (r6); \draw [r] (r6) to (r9);
        \draw [r] (r9) to (r12); \draw [r] (r12) to (r15);
        \draw [r] (r15) to (r2); \draw [r] (r2) to (r5);
        \draw [r] (r5) to (r8); \draw [r] (r8) to (r11);
        \draw [r] (r11) to (r14); \draw [r] (r14) to (r); \draw [r] (r) to (r4);
        \draw [r] (r4) to (r7); \draw [r] (r7) to (r10);
        \draw [r] (r10) to (r13); \draw [r] (r13) to (1);
        \node at (0,0) {\normalsize $\rho$};
      \end{scope}
      %%
      \begin{scope}[shift={(5,0)}]
        \node (1) at (0:2) [v] {$1$};
        \node (r) at (22.5:2) [v] {$r$};
        \node (r2) at (45:2) [v] {$r^2$};
        \node (r3) at (67.5:2) [v] {$r^3$};
        \node (r4) at (90:2) [v] {$r^4$};
        \node (r5) at (112.5:2) [v] {$r^5$};
        \node (r6) at (135:2) [v] {$r^6$};
        \node (r7) at (157.5:2) [v] {$r^7$};
        \node (r8) at (180:2) [v] {$r^8$};
        \node (r9) at (-157.5:2) [v] {$r^9$};
        \node (r10) at (-135:2) [v] {$r^{1\!0}$};
        \node (r11) at (-112.5:2) [v] {$r^{1\!1}$};
        \node (r12) at (-90:2) [v] {$r^{1\!2}$};
        \node (r13) at (-67.5:2) [v] {$r^{1\!3}$};
        \node (r14) at (-45:2) [v] {$r^{1\!4}$};
        \node (r15) at (-22.5:2) [v] {$r^{1\!5}$};
        \draw [r] (1) to (r7); \draw [r] (r7) to (r14); \draw [r] (r14) to (r5);
        \draw [r] (r5) to (r12); \draw [r] (r12) to (r3);
        \draw [r] (r3) to (r10); \draw [r] (r10) to (r);
        \draw [r] (r) to (r8); \draw [r] (r8) to (r15);
        \draw [r] (r15) to (r6); \draw [r] (r6) to (r13);
        \draw [r] (r13) to (r4); \draw [r] (r4) to (r11);
        \draw [r] (r11) to (r2); \draw [r] (r2) to (r9);
        \draw [r] (r9) to (1);
        \node at (0,0) {\normalsize $\sigma$};
      \end{scope}
    \end{tikzpicture}
    \]

  \end{enumerate}
    
    
  
  %%-------------------------------------------------------------------
  
\end{enumerate}

\end{document}

