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\usepackage{visualalgebra}  %% Put this *after* the TikZ packages

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\pagestyle{fancy} \lhead{{\sf Homework 5 $|$ Due January 28 (Monday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Visual Algebra $|$
    Fall 2026 $|$ M.~Macauley}}

\begin{document}

%$\;$

\begin{enumerate}

  %%-------------------------------------------------------------------
  
\item All of the subgroups of $D_5$ should be visually apparent from
  thinking about symmetries of a regular pentagon, shown below at
  left. At right is a Cayley graph.
  
  \vspace{-3mm}
  
  \[
  \begin{tikzpicture}[scale=1.6]
    \colorlet{color1}{actYellow}
    \colorlet{color2}{actGreen}
    \colorlet{color3}{actRed}
    \colorlet{color4}{actBlue}
    \colorlet{color5}{actPurple}
    %%
    \newdimen\r
    \r=1.1cm;
    \begin{scope}[shift={(0,0)},scale=.75]
      \coordinate (center) at (0,0);
      \draw[dashed] (18:1.25*\r)--(198:1.2*\r); 
      \draw[dashed,very thick,eBlue] (90:1.25*\r)--(270:1.2*\r);
      \draw[dashed] (162:1.25*\r)--(342:1.2*\r);
      \draw[dashed] (234:1.25*\r)--(54:1.2*\r);
      \draw[dashed] (306:1.25*\r)--(126:1.2*\r);
      \draw[color1,fill=color1,opacity=.7] (0,0)--(18:1)--(90:1)--cycle;
      \draw[color2,fill=color2,opacity=.7] (0,0)--(90:1)--(162:1)--cycle;
      \draw[color3,fill=color3,opacity=.7] (0,0)--(162:1)--(234:1)--cycle;
      \draw[color4,fill=color4,opacity=.7] (0,0)--(234:1)--(306:1)--cycle;
      \draw[color5,fill=color5,opacity=.7] (0,0)--(306:1)--(18:1)--cycle;
      \draw (18:1) \foreach \x in {90,162,234,306} { -- (\x:1) }
      -- cycle (center);
      \draw[-stealth',eRed] (1.3,-.8) to[very thick,bend right=60] (1.5,-.1);
      \node at (1.75,-.5) {\Alert{$r$}};
      \node at (-.2,1.3) {\Balert{$f$}};
      \node at (54:.5) {\large \textbf{3}};
      \node at (126:.5) {\large \textbf{4}};
      \node at (198:.5) {\large \textbf{5}};
      \node at (270:.5) {\large \textbf{1}};
      \node at (342:.5) {\large \textbf{2}};
    \end{scope}
    %%
    \begin{scope}[shift={(4,0)},scale=.75]
      \tikzstyle{every node}=[font=\footnotesize]
      \tikzstyle{r-out} = [draw, very thick, eRed,-stealth,bend right=22]
      \tikzstyle{r-in} = [draw, very thick, eRed,-stealth,bend left=20]
      \node (e) at (0:2) [v] {$1$};
      \node (r) at (72:2) [v] {$r$};
      \node (r2) at (144:2) [v] {$r^2$};
      \node (r3) at (216:2) [v] {$r^3$};
      \node (r4) at (288:2) [v] {$r^4$};
      \node (f) at (0:1) [v] {$f$};
      \node (rf) at (72:1) [v] {$rf$};
      \node (r2f) at (144:1) [v] {$r^2\!f$};
      \node (r3f) at (216:1) [v] {$r^3\!f$};
      \node (r4f) at (288:1) [v] {$r^4\!f$};
      \draw [r-out] (e) 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 (e);
      \draw [r-in] (f) to (r4f);
      \draw [r-in] (r4f) to (r3f);
      \draw [r-in] (r3f) to (r2f);
      \draw [r-in] (r2f) to (rf);
      \draw [r-in] (rf) to (f);
      \draw [bb] (e) to (f);
      \draw [bb] (r) to (rf);
      \draw [bb] (r2) to (r2f);
      \draw [bb] (r3) to (r3f);
      \draw [bb] (r4) to (r4f);
    \end{scope}
  \end{tikzpicture}
  \]

  \begin{enumerate}
  \item Construct a subgroup lattice for $D_5$. Label each edge
    from $H$ to $K$ with $[H:K]$.
  \item Find the left and right cosets of the subgroups $\<r\>$ and
    $\<f\>$.
  \item The \emph{normalizer} of $H\leq G$, denoted $N_G(H)$, is the
    union of the left cosets of $H$ that are also right cosets. Find
    the normalizer of $\<r\>$ and $\<f\>$.
  \item Two subgroups $H,K\leq G$ are \emph{conjugate} if
    $K=gHg^{-1}:=\{ghg^{-1}\mid h\in H\}$ for some $g\in G$. This
    defines an equivalence relation on the set of subgroups called
    \emph{conjugacy classes}. Partition the subgroups of $D_5$ into
    conjugacy classes.
  %\item Repeat the previous part for $D_4$. The answer will be
  %  different because $4$ is even.
  \end{enumerate}
  
  %%-------------------------------------------------------------------
  
\item Cayley graph of the smallest non-abelian group of odd order,
  $G=C_7\rtimes C_3$, is shown below, highlighting its semidirect
  product structure. \vspace{-5mm}
  
  \[
  \begin{tikzpicture}[scale=1.2]
    \tikzstyle{R1} = [draw, very thick, eRed,-stealth,bend right=15]
    \tikzstyle{R2} = [draw, very thick, eRed,-stealth,bend right=20]
    \tikzstyle{every node}=[font=\footnotesize]
    %%%%
    \begin{scope}[shift={(0,0)},scale=.8]
      \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt, 
        minimum size=2mm]
      \tikzstyle{r-thin} = [draw, eRed,thick,-stealth]
      \begin{scope}[shift={(0,3)}]
        \draw[lightgray, fill=lightgray] (0,0) circle (.75cm);
        \node (1) at (0:.6) [v] {};
        \node (r) at (51.42:.6) [v] {};
        \node (r2) at (102.86:.6) [v] {};
        \node (r3) at (154.29:.6) [v] {};
        \node (r4) at (205.71:.6) [v] {};
        \node (r5) at (257.14:.6) [v] {};
        \node (r6) at (308.57:.6) [v] {};
        \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] {};
        \node (r) at (51.42:.6) [v] {};
        \node (r2) at (102.86:.6) [v] {};
        \node (r3) at (154.29:.6) [v] {};
        \node (r4) at (205.71:.6) [v] {};
        \node (r5) at (257.14:.6) [v] {};
        \node (r6) at (308.57:.6) [v] {};
        \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] {};
        \node (r) at (51.42:.6) [v] {};
        \node (r2) at (102.86:.6) [v] {};
        \node (r3) at (154.29:.6) [v] {};
        \node (r4) at (205.71:.6) [v] {};
        \node (r5) at (257.14:.6) [v] {};
        \node (r6) at (308.57:.6) [v] {};
        \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={(7.5,0)},scale=1]
      \node (s2) at (0:1) [v] {$s^2$};
      \node (rs2) at (51.45:1) [v] {$rs^2$};
      \node (r2s2) at (102.9:1) [v] {$r^2\!s^2$};
      \node (r3s2) at (154.3:1) [v] {$r^3\!s^2$};
      \node (r4s2) at (205.7:1) [v] {$r^4\!s^2$};
      \node (r5s2) at (257.1:1) [v] {$r^5\!s^2$};
      \node (r6s2) at (308.6:1) [v] {$r^6\!s^2$};
      %%
      \node (s) at (0:2) [v] {$s$};
      \node (rs) at (51.45:2) [v] {$rs$};
      \node (r2s) at (102.9:2) [v] {$r^2\!s$};
      \node (r3s) at (154.3:2) [v] {$r^3\!s$};
      \node (r4s) at (205.7:2) [v] {$r^4\!s$};
      \node (r5s) at (257.1:2) [v] {$r^5\!s$};
      \node (r6s) at (308.6:2) [v] {$r^6\!s$};
      %%
      \node (1) at (0:3) [v] {$1$};
      \node (r) at (51.45:3) [v] {$r$};
      \node (r2) at (102.9:3) [v] {$r^2$};
      \node (r3) at (154.3:3) [v] {$r^3$};
      \node (r4) at (205.7:3) [v] {$r^4$};
      \node (r5) at (257.1:3) [v] {$r^5$};
      \node (r6) at (308.6:3) [v] {$r^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 left] (s2) to (1);
      \draw [b] (r) to (rs); \draw [b] (rs) to (rs2);
      \draw [b,bend left] (rs2) to (r);
      \draw [b] (r2) to (r2s); \draw [b] (r2s) to (r2s2);
      \draw [b,bend left] (r2s2) to (r2);
      \draw [b] (r3) to (r3s); \draw [b] (r3s) to (r3s2);
      \draw [b,bend left] (r3s2) to (r3);
      \draw [b] (r4) to (r4s); \draw [b] (r4s) to (r4s2);
      \draw [b,bend left] (r4s2) to (r4);
      \draw [b] (r5) to (r5s); \draw [b] (r5s) to (r5s2);
      \draw [b,bend left] (r5s2) to (r5);
      \draw [b] (r6) to (r6s); \draw [b] (r6s) to (r6s2);
      \draw [b,bend left] (r6s2) to (r6);
    \end{scope}
    \end{tikzpicture}
%    \caption{The Cayley graph of the semidirect product $C_7\rtimes_\theta C_3$, the smallest nonabelian group of odd order.}\label{fig2:C7xC3-semi}
  \]
  \begin{enumerate}
  \item On a blank Cayley graph, label nodes with the order of the
    corresponding elements. Then construct a cycle graph, labeled by
    group elements.
  \item Construct a subgroup lattice and label each edge with the
    corresponding index.
  \item Find the left and right cosets of the subgroups $\<r\>$ and
    $\<s\>$, and their normalizers.
  \item Partition the subgroups into conjugacy classes, and denote
    this on your lattice.
  \end{enumerate}

\newpage
  
  %%-------------------------------------------------------------------

\item In this problem, you will construct the semidirect product
  $C_9\rtimes C_3$. Recall that $\Aut(C_9)$ was constructed on the
  previous assignment.
  \begin{enumerate}
  \item Find all possible labeling maps $\theta\colon
    C_3\to\Aut(C_9)$.
  \item Construct a nonabelian semidirect product of $C_9=\<r\>$ with
    $C_3=\<s\>$, using a labeling map that makes the Cayley graph
    less tangled. Include a Cayley graph of $C_3$ with the nodes
    labeled by $\theta(s^j)$, and a Cayley graph of
    $C_9\rtimes_\theta C_3$, with the nodes labeled by $r^is^j$.
  \item Repeat Problem~2, but for the group $G=C_9\rtimes C_3$.
    It is helpful to know that it has four subgroups of
    order $9$ and four subgroups of order $3$.
  \end{enumerate}

  %%-------------------------------------------------------------------
  
\item Consider two semidirect products of $C_5$ with $C_4$, whose
  Cayley graphs are shown below.

  \vspace{-3mm}

  \[
  \hspace*{-2em}
  \begin{tikzpicture}[scale=1]
    %%%%
    \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt, 
      minimum size=6mm]
    \tikzstyle{B2} = [draw, eBlue, -stealth']       % Blue --->
    %%%%%%%%%%%%
    \begin{scope}[shift={(0,0)}]
      \tikzstyle{every node}=[font=\scriptsize]
      \node at (2.25,2.5) {\normalsize $\Alert{C_5}\!\rtimes_{\theta_1}\!\!\Balert{C_4}$};
      \begin{scope}[shift={(0,4.5)}]
        \node (1) at (0:1) [v] {$1$};
        \node (r) at (72:1) [v] {$a$};
        \node (r2) at (144:1) [v] {$a^2$};
        \node (r3) at (216:1) [v] {$a^3$};
        \node (r4) at (288:1) [v] {$a^4$};
      \end{scope}
      \begin{scope}[shift={(0,0)}]
        \node (s) at (0:1) [v] {$b$};
        \node (rs) at (72:1) [v] {$ab$};
        \node (r2s) at (144:1) [v] {$a^2\!b$};
        \node (r3s) at (216:1) [v] {$a^3\!b$};
        \node (r4s) at (288:1) [v] {$a^4\!b$};
      \end{scope}
      \begin{scope}[shift={(5,0)}]
        \node (s2) at (0:1) [v] {$b^2$};
        \node (rs2) at (72:1) [v] {$ab^2$};
        \node (r2s2) at (144:1) [v] {$a^2\!b^{2}$};
        \node (r3s2) at (216:1) [v] {$a^3\!b^{2}$};
        \node (r4s2) at (288:1) [v] {$a^4\!b^{2}$};
      \end{scope}
      \begin{scope}[shift={(5,4.5)}]
        \node (s3) at (0:1) [v] {$b^3$};
        \node (rs3) at (72:1) [v] {$ab^{\!3}$};
        \node (r2s3) at (144:1) [v] {$a^2\!b^3$};
        \node (r3s3) at (216:1) [v] {$a^3\!b^{3}$};
        \node (r4s3) at (288:1) [v] {$a^4\!b^{3}$};
      \end{scope}
      \draw [B2] (1) to (s);
      \draw [B2] (r) to[bend left=18] (rs);
      \draw [B2] (r2) to[bend right=21] (r2s); 
      \draw [B2] (r3) to[bend left=22] (r3s);
      \draw [B2] (r4) to[bend right=18] (r4s);
      %%
      \draw [B2] (s) to (s2);
      \draw [B2] (rs) to (rs2);
      \draw [B2] (r2s) to (r2s2);
      \draw [B2] (r3s) to (r3s2);
      \draw [B2] (r4s) to (r4s2);
      %%
      \draw [B2] (s2) to (s3);
      \draw [B2] (rs2) to[bend right=18] (rs3);
      \draw [B2] (r2s2) to[bend left=21] (r2s3);
      \draw [B2] (r3s2) to[bend right=22] (r3s3);
      \draw [B2] (r4s2) to[bend left=18] (r4s3);
      %%
      \draw [B2] (s3) to (1); 
      \draw [B2] (rs3) to (r);
      \draw [B2] (r2s3) to (r2);
      \draw [B2] (r3s3) to (r3);
      \draw [B2] (r4s3) to (r4);
    %%
      \draw [r] (1) to (r); \draw [r] (r) to (r2); \draw [r] (r2) to (r3);
      \draw [r] (r3) to (r4); \draw [r] (r4) to (1);
      %%
      \draw [r] (s) to (r2s); \draw [r] (r2s) to (r4s); \draw [r] (r4s) to (rs);
      \draw [r] (rs) to (r3s); \draw [r] (r3s) to (s);
      %%
      \draw [r] (s2) to (r4s2); \draw [r] (r4s2) to (r3s2);
      \draw [r] (r3s2) to (r2s2); \draw [r] (r2s2) to (rs2);
      \draw [r] (rs2) to (s2);
      %%
      \draw [r] (s3) to (r3s3); \draw [r] (r3s3) to (rs3);
      \draw [r] (rs3) to (r4s3); \draw [r] (r4s3) to (r2s3);
      \draw [r] (r2s3) to (s3);
    \end{scope}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        \begin{scope}[shift={(9,0)}]
      \tikzstyle{every node}=[font=\scriptsize]
      \node at (2.35,2.5) {\normalsize $\Alert{C_5}\!\rtimes_{\theta_2}\!\!\Balert{C_4}$};
      \begin{scope}[shift={(0,4.5)}]
        \node (1) at (0:1) [v] {$1$};
        \node (r) at (72:1) [v] {$a$};
        \node (r2) at (144:1) [v] {$a^2$};
        \node (r3) at (216:1) [v] {$a^3$};
        \node (r4) at (288:1) [v] {$a^4$};
      \end{scope}
      \begin{scope}[shift={(0,0)}]
        \node (s) at (0:1) [v] {$b$};
        \node (rs) at (72:1) [v] {$ab$};
        \node (r2s) at (144:1) [v] {$a^2\!b$};
        \node (r3s) at (216:1) [v] {$a^3\!b$};
        \node (r4s) at (288:1) [v] {$a^4\!b$};
      \end{scope}
      \begin{scope}[shift={(5,0)}]
        \node (s2) at (0:1) [v] {$b^2$};
        \node (rs2) at (72:1) [v] {$ab^2$};
        \node (r2s2) at (144:1) [v] {$a^2\!b^{2}$};
        \node (r3s2) at (216:1) [v] {$a^3\!b^{2}$};
        \node (r4s2) at (288:1) [v] {$a^4\!b^{2}$};
      \end{scope}
      \begin{scope}[shift={(5,4.5)}]
        \node (s3) at (0:1) [v] {$b^3$};
        \node (rs3) at (72:1) [v] {$ab^{\!3}$};
        \node (r2s3) at (144:1) [v] {$a^2\!b^{3}$};
        \node (r3s3) at (216:1) [v] {$a^3\!b^{3}$};
        \node (r4s3) at (288:1) [v] {$a^4\!b^{3}$};
      \end{scope}
      \draw [B2] (1) to (s);
      \draw [B2] (r) to[bend left=18] (rs);
      \draw [B2] (r2) to[bend right=21] (r2s); 
      \draw [B2] (r3) to[bend left=22] (r3s);
      \draw [B2] (r4) to[bend right=18] (r4s);
      %%
      \draw [B2] (s) to (s2);
      \draw [B2] (rs) to (rs2);
      \draw [B2] (r2s) to (r2s2);
      \draw [B2] (r3s) to (r3s2);
      \draw [B2] (r4s) to (r4s2);
      %%
      \draw [B2] (s2) to (s3);
      \draw [B2] (rs2) to[bend right=18] (rs3);
      \draw [B2] (r2s2) to[bend left=21] (r2s3);
      \draw [B2] (r3s2) to[bend right=22] (r3s3);
      \draw [B2] (r4s2) to[bend left=18] (r4s3);
      %%
      \draw [B2] (s3) to (1); 
      \draw [B2] (rs3) to (r);
      \draw [B2] (r2s3) to (r2);
      \draw [B2] (r3s3) to (r3);
      \draw [B2] (r4s3) to (r4);
      %%
      \draw [r] (1) to (r); \draw [r] (r) to (r2); \draw [r] (r2) to (r3);
      \draw [r] (r3) to (r4); \draw [r] (r4) to (1);
      %%
      \draw [r] (s) to (r4s); \draw [r] (r4s) to (r3s);
      \draw [r] (r3s) to (r2s); \draw [r] (r2s) to (rs);
      \draw [r] (rs) to (s);
      %%
      \draw [r] (s2) to (rs2); \draw [r] (rs2) to (r2s2); \draw [r] (r2s2) to (r3s2);
      \draw [r] (r3s2) to (r4s2); \draw [r] (r4s2) to (s2);
      %%
      \draw [r] (s3) to (r4s3); \draw [r] (r4s3) to (r3s3);
      \draw [r] (r3s3) to (r2s3); \draw [r] (r2s3) to (rs3);
      \draw [r] (rs3) to (s3); 
   \end{scope}
    \end{tikzpicture}
\]

\begin{enumerate}
\item Construct a cycle graph for each group, with the nodes labeled
  by group elements.
\item The subgroup lattices of these two groups are shown below.
  Re-draw them with the subgroups written by generators.
  \[
  \hspace*{-3em}
  \begin{tikzpicture}[shorten >= -2pt, shorten <= -2pt,scale=1.6]
    \tikzstyle{every node}=[font=\small]
    \begin{scope}[shift={(0,0)},scale=1]
      \node (G) at (0,4) {$G$};
      \node(s2-t) at (-.15,3) {$D_5$};
      \node (t) at (-1.4,2.2) {$C_5$};
      \node (s) at (-.7,2) {$C_4$};
      \node (st) at (.25,2) {$C_4$};
      \node (s3t) at (1.15,2) {$C_4$};
      \node (ts) at (1.8,2) {$C_4$};
      \node (ts3) at (2.5,2) {$C_4$};
      \node(s2) at (-.45,1.1) {$C_2$}; 
      \node(ts2) at (.25,1.1) {$C_2$}; 
      \node(sts) at (1.15,1.1) {$C_2$}; 
      \node(s2t2) at (1.8,1.1) {$C_2$};
      \node(s2t) at (2.5,1.1) {$C_2$};
      \node (1) at (0,0) {$\<1\>$};
      \draw[f,bend right=5] (G) to (s); 
      \draw[f] (G) to (st); \draw[f] (G) to (s3t);
      \draw[f] (G) to (ts); \draw[f] (G) to (ts3); \draw[f] (G) to (s2-t);
      %%
      \draw[f] (s2-t) to (t); \draw[f] (t) to (1);
      \draw[f] (s2-t) to (s2); \draw[f] (s2-t) to (ts2);
      \draw[f] (s2-t) to (sts);
      \draw[f] (s2-t) to (s2t2); 
      \draw[f,bend left=10] (s2-t) to (s2t);
      %%
      \draw[f] (s) to (s2); \draw[f] (st) to (ts2); \draw[f] (s3t) to (sts);
      \draw[f] (ts) to (s2t2); \draw[f] (ts3) to (s2t); 
      %%
      \draw[f] (1) to (s2); \draw[f] (1) to (ts2); \draw[f] (1) to (sts);
      \draw[f] (1) to (s2t2); \draw[f] (1) to (s2t);
    \end{scope}
    %%
    \begin{scope}[shift={(6,0)},scale=1]
      \node(G) at (0,4) {$G$};
      \node(r) at (-1.5,3) {$C_{10}$};
      \node (r2) at (-1.75,2.2) {$C_5$};
      \node(s) at (-.45,2) {$C_4$}; 
      \node(sr) at (.2,2) {$C_4$}; 
      \node(sr2) at (.85,2) {$C_4$}; 
      \node(r2s) at (1.5,2) {$C_4$};
      \node(rs) at (2.15,2) {$C_4$};
      \node (r5) at (0,1) {$C_2$};
      \node (1) at (0,0) {$\<1\>$};
      \draw[f] (G) to (r); \draw[f] (r) to (r5);
      \draw[f] (G) to (s); \draw[f] (G) to (sr); 
      \draw[f] (G) to (sr2); \draw[f] (G) to (r2s); \draw[f] (G) to (rs);
      \draw[f] (r5) to (s); \draw[f] (r5) to (sr); \draw[f] (r5) to (sr2);
      \draw[f] (r5) to (r2s); \draw[f] (r5) to (rs);
      \draw[f] (1) to (r5); \draw[f] (1) to (r2); \draw[f] (r) to (r2); 
    \end{scope}
\end{tikzpicture}
  \]
  
\item Determine which group each of these is isomorphic to, and which elements $a$ and $b$ correspond to. Recall that there are only three nonabelian groups of order $20$:
  \[
  D_{10}=\big\<r,f\mid r^{10}=f^2=1,\,rfr=f\big\>,\qquad \Dic_{10}=\big\<r,s\mid r^{10}=s^4=1,\,r^5=s^2\big\>,
  \]
  \[
  \AGL_1(\Z_5)=\left\<\begin{bmatrix}1&1\\0&1\end{bmatrix},\,
 \begin{bmatrix}2&1\\0&1\end{bmatrix}\right\>\leq\GL_2(\Z_5).
   \]
   Write a presentation for both groups in this problem, in terms of
   $a$ and $b$.
   
   %%-----------------------------------------------------------------------

 \item Construct the subgroup lattice for $G=D_{10}$. It helps to think of the subgroups geometrically---there are two subgroups isomorphic to $D_5$, unique cyclic subgroups of orders $10$ and $5$, five subgroups isomorphic to $V_4$, and $11$ subgroups of order $2$. 
   
   %%-----------------------------------------------------------------------

\item For each of the diagrams below, determine whether it is the Cayley graph of a group. If yes, write a presentation and determine whether it is isomorphic to $D_{10}$, $\Dic_{10}$, or $\AGL_1(\Z_5)$. If no, explain why.
  
  \[
  \hspace*{-10mm}
  \begin{tikzpicture}[scale=.95,auto]
    \tikzstyle{every node}=[font=\footnotesize] 
    %% 
    \begin{scope}[shift={(0,8.5)},scale=.9]
      \node (a1) at (18:1) [v] {$r^4$};
      \node (a2) at (90:1) [v] {$1$};
      \node (a3) at (162:1) [v] {$r$};
      \node (a4) at (234:1) [v] {$r^2$};
      \node (a5) at (306:1) [v] {$r^3$};
      \node (b1) at (54:2) [v] {\scriptsize $s^3\!r^3$};
      \node (b2) at (126:2) [v] {$s^3$};
      \node (b3) at (198:2) [v] {\scriptsize $s^3\!r^2$};
      \node (b4) at (270:2) [v] {\scriptsize $s^3\!r^4$};
      \node (b5) at (342:2) [v] {$s^3\!r$};
      \node (c1) at (54:3.1) [v] {\scriptsize $s^2\!r^3$};
      \node (c2) at (126:3.1) [v] {$s^2$};
      \node (c3) at (198:3.1) [v] {\scriptsize $s^2\!r^4$};
      \node (c4) at (270:3.1) [v] {\scriptsize $s^2\!r^3$};
      \node (c5) at (342:3.1) [v] {\scriptsize $s^2\!r^2$};
      \node (d1) at (18:4.3) [v] {$sr^2$};
      \node (d2) at (90:4.3) [v] {$s$};
      \node (d3) at (162:4.3) [v] {$sr^3$};
      \node (d4) at (234:4.3) [v] {$sr$};
      \node (d5) at (306:4.3) [v] {$sr^4$};
      \draw [r] (a1) to (a2); \draw [r] (a2) to (a3); \draw [r] (a3) to (a4);
      \draw [r] (a4) to (a5); \draw [r] (a5) to (a1);
      \draw [r,bend left=40] (b1) to (b4); \draw [r,bend left=40] (b4) to (b2);
      \draw [r,bend left=40] (b2) to (b5); \draw [r,bend left=40] (b5) to (b3);
      \draw [r,bend left=40] (b3) to (b1);
      \draw [r] (c1) to (c5); \draw [r] (c5) to (c4); \draw [r] (c4) to (c3);
      \draw [r] (c3) to (c2); \draw [r] (c2) to (c1);
      \draw [r,bend right=55] (d1) to (d3); \draw [r,bend right=55] (d3) to (d5);
      \draw [r,bend right=55] (d5) to (d2); \draw [r,bend right=55] (d2) to (d4);
      \draw [r,bend right=55] (d4) to (d1);
      \draw [b] (a1) to (d1); \draw [b] (d1) to (c1);
      \draw [b] (c1) to (b1);  \draw [b] (b1) to (a1);
      \draw [b] (a2) to (d2); \draw [b] (d2) to (c2);
      \draw [b] (c2) to (b2);  \draw [b] (b2) to (a2);
      \draw [b] (a3) to (d3); \draw [b] (d3) to (c3);
      \draw [b] (c3) to (b3);  \draw [b] (b3) to (a3);
      \draw [b] (a4) to (d4); \draw [b] (d4) to (c4);
      \draw [b] (c4) to (b4);  \draw [b] (b4) to (a4);
      \draw [b] (a5) to (d5); \draw [b] (d5) to (c5);
      \draw [b] (c5) to (b5);  \draw [b] (b5) to (a5);       
    \end{scope}
    %%
    %%
    \begin{scope}[shift={(0,0)},scale=.95]
      \node (a1) at (18:1) [v] {$r^4$};
      \node (a2) at (90:1) [v] {$1$};
      \node (a3) at (162:1) [v] {$r$};
      \node (a4) at (234:1) [v] {$r^2$};
      \node (a5) at (306:1) [v] {$r^3$}; 
      \node (b1) at (18:2) [v] {\scriptsize $r^4\!s^3$};
      \node (b2) at (90:2) [v] {$s^3$};
      \node (b3) at (162:2) [v] {$rs^3$};
      \node (b4) at (234:2) [v] {\scriptsize $r^2\!s^3$};
      \node (b5) at (306:2) [v] {\scriptsize $r^3\!s^3$};
      \node (c1) at (18:3) [v] {\scriptsize $r^4\!s^2$};
      \node (c2) at (90:3) [v] {$s^2$};
      \node (c3) at (162:3) [v] {$rs^2$};
      \node (c4) at (234:3) [v] {\scriptsize $r^2\!s^2$};
      \node (c5) at (306:3) [v] {\scriptsize $r^3\!s^2$};
      \node (d1) at (18:4) [v] {$r^4\!s$};
      \node (d2) at (90:4) [v] {$s$};
      \node (d3) at (162:4) [v] {$rs$};
      \node (d4) at (234:4) [v] {$r^2\!s$};
      \node (d5) at (306:4) [v] {$r^3\!s$};
      \draw [r] (a1) to (a2); \draw [r] (a2) to (a3); \draw [r] (a3) to (a4);
      \draw [r] (a4) to (a5); \draw [r] (a5) to (a1);
      \draw [r] (b1) to (b5); \draw [r] (b5) to (b4); \draw [r] (b4) to (b3);
      \draw [r] (b3) to (b2); \draw [r] (b2) to (b1);
      \draw [r] (c1) to (c2); \draw [r] (c2) to (c3); \draw [r] (c3) to (c4);
      \draw [r] (c4) to (c5); \draw [r] (c5) to (c1);
      \draw [r] (d1) to (d5); \draw [r] (d5) to (d4); \draw [r] (d4) to (d3);
      \draw [r] (d3) to (d2); \draw [r] (d2) to (d1);
      %% 
      \draw [b,bend right=35] (a1) to (d1); \draw [b] (d1) to (c1);
      \draw [b] (c1) to (b1); \draw [b] (b1) to (a1);
      \draw [b,bend right=35] (a2) to (d2); \draw [b] (d2) to (c2);
      \draw [b] (c2) to (b2); \draw [b] (b2) to (a2);
      \draw [b,bend right=35] (a3) to (d3); \draw [b] (d3) to (c3);
      \draw [b] (c3) to (b3); \draw [b] (b3) to (a3);
      \draw [b,bend right=35] (a4) to (d4); \draw [b] (d4) to (c4);
      \draw [b] (c4) to (b4); \draw [b] (b4) to (a4);
      \draw [b,bend right=35] (a5) to (d5); \draw [b] (d5) to (c5);
      \draw [b] (c5) to (b5); \draw [b] (b5) to (a5);
    \end{scope}
    %%
    %%  
    \begin{scope}[shift={(9,8.5)},scale=1]
      \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt,
        minimum size=4mm] 
      %%
      \node (a1) at (72:2) [v] {};
      \node (ar) at (36:3.5) [v] {};
      \node (ar2) at (36:2) [v] {};
      \node (ar3) at (72:3.5) [v] {};
      %%
      \node (b1) at (0:2) [v] {};
      \node (br) at (324:3.5) [v] {};
      \node (br2) at (324:2) [v] {};
      \node (br3) at (0:3.5) [v] {};
      %%
      \node (c1) at (288:2) [v] {};
      \node (cr) at (252:3.5) [v] {};
      \node (cr2) at (252:2) [v] {};
      \node (cr3) at (288:3.5) [v] {};
      %%
      \node (d1) at (216:2) [v] {};
      \node (dr) at (180:3.5) [v] {};
      \node (dr2) at (180:2) [v] {};
      \node (dr3) at (216:3.5) [v] {};
      %%
      \node (e1) at (144:2) [v] {};
      \node (er) at (108:3.5) [v] {};
      \node (er2) at (108:2) [v] {};
      \node (er3) at (144:3.5) [v] {};
      %%
      \draw [b] (a1) to (ar); \draw [b] (ar) to (ar2);
      \draw [b] (ar2) to (ar3); \draw [b] (ar3) to (a1);
      %%
      \draw [b] (b1) to (br); \draw [b] (br) to (br2);
      \draw [b] (br2) to (br3); \draw [b] (br3) to (b1);
      %%
      \draw [b] (c1) to (cr); \draw [b] (cr) to (cr2);
      \draw [b] (cr2) to (cr3); \draw [b] (cr3) to (c1);
      %%
      \draw [b] (d1) to (dr); \draw [b] (dr) to (dr2);
      \draw [b] (dr2) to (dr3); \draw [b] (dr3) to (d1);
      %%
      \draw [b] (e1) to (er); \draw [b] (er) to (er2);
      \draw [b] (er2) to (er3); \draw [b] (er3) to (e1);
      %% 
      \draw [gg] (a1) to (dr2); \draw [gg] (b1) to (er2); \draw [gg] (c1) to (ar2);
      \draw [gg] (d1) to (br2); \draw [gg] (e1) to (cr2);
      %%
      \draw [gg] (ar) to (br3); \draw [gg] (br) to (cr3); \draw [gg] (cr) to (dr3);
      \draw [gg] (dr) to (er3); \draw [gg] (er) to (ar3);
    \end{scope}
    %% 
    %% 
    \begin{scope}[shift={(9,0)},scale=.8]
      \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt,
        minimum size=3.25mm] 
      %%
      \node (a2) at (54:1) [v] {};
      \node (a4) at (126:1) [v] {};
      \node (a6) at (198:1) [v] {};
      \node (a8) at (270:1) [v] {};
      \node (a10) at (342:1) [v] {};
      \node (b1) at (18:3) [v] {};
      \node (b2) at (54:2) [v] {};
      \node (b3) at (90:3) [v] {};
      \node (b4) at (126:2) [v] {};
      \node (b5) at (162:3) [v] {};
      \node (b6) at (198:2) [v] {};
      \node (b7) at (234:3) [v] {};
      \node (b8) at (270:2) [v] {};
      \node (b9) at (306:3) [v] {};
      \node (b10) at (342:2) [v] {};
      \node (c1) at (18:4.5) [v] {};
      \node (c3) at (90:4.5) [v] {};
      \node (c5) at (162:4.5) [v] {};
      \node (c7) at (234:4.5) [v] {};
      \node (c9) at (306:4.5) [v] {};
      \draw [rr] (a2) to (a4); \draw [gg] (a4) to (a6); \draw [bb] (a6) to (a8);
      \draw [rr] (a8) to (a10); \draw [bb] (a10) to (a2);
      \draw [rr] (b1) to (b2); \draw [bb] (b2) to (b3); \draw [rr] (b3) to (b4);
      \draw [gg] (b4) to (b5); \draw [bb] (b5) to (b6); \draw [gg] (b6) to (b7);
      \draw [rr] (b7) to (b8); \draw [bb] (b8) to (b9); \draw [rr] (b9) to (b10);
      \draw [bb] (b10) to (b1);
      \draw [rr] (c1) to (c3); \draw [bb] (c3) to (c5); \draw [gg] (c5) to (c7);
      \draw [rr] (c7) to (c9); \draw [bb] (c9) to (c1);
      \draw [gg] (a2) to (b2); \draw [bb] (a4) to (b4); \draw [rr] (a6) to (b6);
      \draw [gg] (a8) to (b8); \draw [gg] (a10) to (b10);
      \draw [gg] (b1) to (c1); \draw [gg] (b3) to (c3); \draw [rr] (b5) to (c5);
      \draw [bb] (b7) to (c7); \draw [gg] (b9) to (c9);
    \end{scope}
  \end{tikzpicture}
\]
\end{enumerate}
  
  %%-------------------------------------------------------------------
  
\end{enumerate}

\end{document}