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\tikzstyle{R} = [draw, very thick, Red, -stealth']        % Red --->
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\pagestyle{fancy} \lhead{{\sf Homework 5 $|$ Due February 16 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Visual Algebra $|$
    Spring 2024 $|$ 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}
  
  \colorlet{color1}{lyellow}
  \colorlet{color2}{lgreen}
  \colorlet{color3}{lred}
  \colorlet{color4}{lblue}
  \colorlet{color5}{lpurple}
  \[
  \begin{tikzpicture}[scale=1.6]
    \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,Blue] (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',Red] (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, Red,-stealth,bend right=22]
      \tikzstyle{r-in} = [draw, very thick, Red,-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}

  \tikzstyle{R1} = [draw, very thick, Red,-stealth,bend right=15]
\tikzstyle{R2} = [draw, very thick, Red,-stealth,bend right=20]
\tikzstyle{v-tiny} = [circle, draw, fill=lightgrey,inner sep=0pt,
  minimum size=6mm]
\tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt,
  minimum size=6mm]
\tikzstyle{v-big} = [circle, draw, fill=lightgrey,inner sep=0pt,
  minimum size=6mm]
\[
  \begin{tikzpicture}[scale=1.2]
   \tikzstyle{every node}=[font=\footnotesize]
    %%%%
    \begin{scope}[shift={(0,0)},scale=.8]
   \tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt, 
    minimum size=2mm]
     \tikzstyle{r-thin} = [draw, Red,thick,-stealth]
      \begin{scope}[shift={(0,3)}]
        \draw[lightgray, fill=lightgray] (0,0) circle (.75cm);
        \node (1) at (0:.6) [v-small] {};
        \node (r) at (51.42:.6) [v-small] {};
        \node (r2) at (102.86:.6) [v-small] {};
        \node (r3) at (154.29:.6) [v-small] {};
        \node (r4) at (205.71:.6) [v-small] {};
        \node (r5) at (257.14:.6) [v-small] {};
        \node (r6) at (308.57:.6) [v-small] {};
        \draw [r-thin] (1) to (r); 
        \draw [r-thin] (r) to (r2); 
        \draw [r-thin] (r2) to (r3);
        \draw [r-thin] (r3) to (r4); 
        \draw [r-thin] (r4) to (r5);
        \draw [r-thin] (r5) to (r6);
        \draw [r-thin] (r6) to (1);
        \draw [b,ultra thick] (0,-.75) to (0,-2.25);
      \end{scope}
      %%
      \begin{scope}[shift={(0,0)}]
        \draw[lightgray, fill=lightgray] (0,0) circle (.75cm);
        \node (1) at (0:.6) [v-small] {};
        \node (r) at (51.42:.6) [v-small] {};
        \node (r2) at (102.86:.6) [v-small] {};
        \node (r3) at (154.29:.6) [v-small] {};
        \node (r4) at (205.71:.6) [v-small] {};
        \node (r5) at (257.14:.6) [v-small] {};
        \node (r6) at (308.57:.6) [v-small] {};
        \draw [r-thin] (1) to (r2); 
        \draw [r-thin] (r2) to (r4); 
        \draw [r-thin] (r4) to (r6);
        \draw [r-thin] (r6) to (r); 
        \draw [r-thin] (r) to (r3);
        \draw [r-thin] (r3) to (r5);
        \draw [r-thin] (r5) to (1);
        \draw [b,ultra thick] (0,-.75) to (0,-2.25);
        %\draw [p,ultra thick] (-.75,0) to (-2.25,0);
        %\node at (0,0) {\small $\palert{\varphi^2}$};
      \end{scope}
      %%
      \begin{scope}[shift={(0,-3)}]
      \draw[lightgray, fill=lightgray] (0,0) circle (.75cm);
        \node (1) at (0:.6) [v-small] {};
        \node (r) at (51.42:.6) [v-small] {};
        \node (r2) at (102.86:.6) [v-small] {};
        \node (r3) at (154.29:.6) [v-small] {};
        \node (r4) at (205.71:.6) [v-small] {};
        \node (r5) at (257.14:.6) [v-small] {};
        \node (r6) at (308.57:.6) [v-small] {};
        \draw [r-thin] (1) to (r4); 
        \draw [r-thin] (r4) to (r); 
        \draw [r-thin] (r) to (r5);
        \draw [r-thin] (r5) to (r2); 
        \draw [r-thin] (r2) to (r6);
        \draw [r-thin] (r6) to (r3);
        \draw [r-thin] (r3) to (1);
        \draw [b,ultra thick] (135:.75) to[bend left=35] (-.57,5.45);
        %\node at (0,0) {\small $\palert{\varphi^3}$};
      \end{scope}
    \end{scope}
   \begin{scope}[shift={(7.5,0)},scale=1]
      \node (s2) at (0:1) [v-small] {$s^2$};
      \node (rs2) at (51.45:1) [v-small] {$rs^2$};
      \node (r2s2) at (102.9:1) [v-small] {$r^2\!s^2$};
      \node (r3s2) at (154.3:1) [v-small] {$r^3\!s^2$};
      \node (r4s2) at (205.7:1) [v-small] {$r^4\!s^2$};
      \node (r5s2) at (257.1:1) [v-small] {$r^5\!s^2$};
      \node (r6s2) at (308.6:1) [v-small] {$r^6\!s^2$};
      %%
      \node (s) at (0:2) [v-small] {$s$};
      \node (rs) at (51.45:2) [v-small] {$rs$};
      \node (r2s) at (102.9:2) [v-small] {$r^2\!s$};
      \node (r3s) at (154.3:2) [v-small] {$r^3\!s$};
      \node (r4s) at (205.7:2) [v-small] {$r^4\!s$};
      \node (r5s) at (257.1:2) [v-small] {$r^5\!s$};
      \node (r6s) at (308.6:2) [v-small] {$r^6\!s$};
      %%
      \node (1) at (0:3) [v-small] {$1$};
      \node (r) at (51.45:3) [v-small] {$r$};
      \node (r2) at (102.9:3) [v-small] {$r^2$};
      \node (r3) at (154.3:3) [v-small] {$r^3$};
      \node (r4) at (205.7:3) [v-small] {$r^4$};
      \node (r5) at (257.1:3) [v-small] {$r^5$};
      \node (r6) at (308.6:3) [v-small] {$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}

  \tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt, 
    minimum size=6mm]
  \tikzstyle{B2} = [draw, Blue, -stealth']       % Blue --->
  \[
  \hspace*{-2em}
  \begin{tikzpicture}[scale=1]
    %%%%
    %%%%%%%%%%%%
    %%%%%%%%%%%%
    %%%%%%%%%%%%
    \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-small] {$1$};
        \node (r) at (72:1) [v-small] {$a$};
        \node (r2) at (144:1) [v-small] {$a^2$};
        \node (r3) at (216:1) [v-small] {$a^3$};
        \node (r4) at (288:1) [v-small] {$a^4$};
      \end{scope}
      \begin{scope}[shift={(0,0)}]
        \node (s) at (0:1) [v-small] {$b$};
        \node (rs) at (72:1) [v-small] {$ab$};
        \node (r2s) at (144:1) [v-small] {$a^2\!b$};
        \node (r3s) at (216:1) [v-small] {$a^3\!b$};
        \node (r4s) at (288:1) [v-small] {$a^4\!b$};
      \end{scope}
      \begin{scope}[shift={(5,0)}]
        \node (s2) at (0:1) [v-small] {$b^2$};
        \node (rs2) at (72:1) [v-small] {$ab^2$};
        \node (r2s2) at (144:1) [v-small] {$a^2\!b^{2}$};
        \node (r3s2) at (216:1) [v-small] {$a^3\!b^{2}$};
        \node (r4s2) at (288:1) [v-small] {$a^4\!b^{2}$};
      \end{scope}
      \begin{scope}[shift={(5,4.5)}]
        \node (s3) at (0:1) [v-small] {$b^3$};
        \node (rs3) at (72:1) [v-small] {$ab^{\!3}$};
        \node (r2s3) at (144:1) [v-small] {$a^2\!b^3$};
        \node (r3s3) at (216:1) [v-small] {$a^3\!b^{3}$};
        \node (r4s3) at (288:1) [v-small] {$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-small] {$1$};
        \node (r) at (72:1) [v-small] {$a$};
        \node (r2) at (144:1) [v-small] {$a^2$};
        \node (r3) at (216:1) [v-small] {$a^3$};
        \node (r4) at (288:1) [v-small] {$a^4$};
      \end{scope}
      \begin{scope}[shift={(0,0)}]
        \node (s) at (0:1) [v-small] {$b$};
        \node (rs) at (72:1) [v-small] {$ab$};
        \node (r2s) at (144:1) [v-small] {$a^2\!b$};
        \node (r3s) at (216:1) [v-small] {$a^3\!b$};
        \node (r4s) at (288:1) [v-small] {$a^4\!b$};
      \end{scope}
      \begin{scope}[shift={(5,0)}]
        \node (s2) at (0:1) [v-small] {$b^2$};
        \node (rs2) at (72:1) [v-small] {$ab^2$};
        \node (r2s2) at (144:1) [v-small] {$a^2\!b^{2}$};
        \node (r3s2) at (216:1) [v-small] {$a^3\!b^{2}$};
        \node (r4s2) at (288:1) [v-small] {$a^4\!b^{2}$};
      \end{scope}
      \begin{scope}[shift={(5,4.5)}]
        \node (s3) at (0:1) [v-small] {$b^3$};
        \node (rs3) at (72:1) [v-small] {$ab^{\!3}$};
        \node (r2s3) at (144:1) [v-small] {$a^2\!b^{3}$};
        \node (r3s3) at (216:1) [v-small] {$a^3\!b^{3}$};
        \node (r4s3) at (288:1) [v-small] {$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}[(a)]
\item On blank Cayley graphs, label the order of each element. Then construct a cycle graph, with the nodes labeled by group elements. 
\item The subgroup lattices of these two groups are shown below, not
  necessarily in the right order. Determine which lattice corresponds
  to which Cayley graph (with justification), and then re-draw them
  with the subgroups written by generators.
  \[
  \hspace*{-3em}
  \begin{tikzpicture}[shorten >= -2pt, shorten <= -2pt,scale=1.4]
    \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[bend right=5] (G) to (s); 
      \draw (G) to (st); \draw (G) to (s3t);
      \draw (G) to (ts); \draw (G) to (ts3); \draw (G) to (s2-t);
      %%
      \draw (s2-t) to (t); \draw (t) to (1);
      \draw (s2-t) to (s2); \draw (s2-t) to (ts2);
      \draw (s2-t) to (sts);
      \draw (s2-t) to (s2t2); 
      \draw[bend left=10] (s2-t) to (s2t);
      %%
      \draw (s) to (s2); \draw (st) to (ts2); \draw (s3t) to (sts);
      \draw (ts) to (s2t2); \draw (ts3) to (s2t); 
      %%
      \draw (1) to (s2); \draw (1) to (ts2); \draw (1) to (sts);
      \draw (1) to (s2t2); \draw (1) to (s2t);
    \end{scope}
    %%
    \begin{scope}[shift={(6.75,0)},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 (G) to (r); \draw (r) to (r5);
      \draw (G) to (s); \draw (G) to (sr); 
      \draw (G) to (sr2); \draw (G) to (r2s); \draw (G) to (rs);
      \draw (r5) to (s); \draw (r5) to (sr); \draw (r5) to (sr2);
      \draw (r5) to (r2s); \draw (r5) to (rs);
      \draw (1) to (r5); \draw (1) to (r2); \draw (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.
  
  \tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=4mm] 
  \[
  \hspace*{-5mm}
  \begin{tikzpicture}[scale=.9,auto]
    \tikzstyle{every node}=[font=\tiny]
    %%
    \begin{scope}[shift={(0,8.5)}]
      \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] {$s^3\!r^3$};
      \node (b2) at (126:2) [v] {$s^3$};
      \node (b3) at (198:2) [v] {$s^3\!r^2$};
      \node (b4) at (270:2) [v] {$s^3\!r^4$};
      \node (b5) at (342:2) [v] {$s^3\!r$};
      \node (c1) at (54:3.1) [v] {$s^2\!r^3$};
      \node (c2) at (126:3.1) [v] {$s^2$};
      \node (c3) at (198:3.1) [v] {$s^2\!r^4$};
      \node (c4) at (270:3.1) [v] {$s^2\!r^3$};
      \node (c5) at (342:3.1) [v] {$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={(10,8.5)},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] {$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] {$r^2\!s^3$};
      \node (b5) at (306:2) [v] {$r^3\!s^3$};
      \node (c1) at (18:3) [v] {$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] {$r^2\!s^2$};
      \node (c5) at (306:3) [v] {$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={(0,0)},scale=1]
      \node (a1) at (72:2) [v-small] {};
      \node (ar) at (36:3.5) [v-small] {};
      \node (ar2) at (36:2) [v-small] {};
      \node (ar3) at (72:3.5) [v-small] {};
      %%
      \node (b1) at (0:2) [v-small] {};
      \node (br) at (324:3.5) [v-small] {};
      \node (br2) at (324:2) [v-small] {};
      \node (br3) at (0:3.5) [v-small] {};
      %%
      \node (c1) at (288:2) [v-small] {};
      \node (cr) at (252:3.5) [v-small] {};
      \node (cr2) at (252:2) [v-small] {};
      \node (cr3) at (288:3.5) [v-small] {};
      %%
      \node (d1) at (216:2) [v-small] {};
      \node (dr) at (180:3.5) [v-small] {};
      \node (dr2) at (180:2) [v-small] {};
      \node (dr3) at (216:3.5) [v-small] {};
      %%
      \node (e1) at (144:2) [v-small] {};
      \node (er) at (108:3.5) [v-small] {};
      \node (er2) at (108:2) [v-small] {};
      \node (er3) at (144:3.5) [v-small] {};
      %%
      \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}
    %% 
    %% 
  \tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=3.25mm] 
    \begin{scope}[shift={(10,-.4)},scale=.85]
    \node (a2) at (54:1) [v-small] {};
    \node (a4) at (126:1) [v-small] {};
    \node (a6) at (198:1) [v-small] {};
    \node (a8) at (270:1) [v-small] {};
    \node (a10) at (342:1) [v-small] {};
    \node (b1) at (18:3) [v-small] {};
    \node (b2) at (54:2) [v-small] {};
    \node (b3) at (90:3) [v-small] {};
    \node (b4) at (126:2) [v-small] {};
    \node (b5) at (162:3) [v-small] {};
    \node (b6) at (198:2) [v-small] {};
    \node (b7) at (234:3) [v-small] {};
    \node (b8) at (270:2) [v-small] {};
    \node (b9) at (306:3) [v-small] {};
    \node (b10) at (342:2) [v-small] {};
    \node (c1) at (18:4.5) [v-small] {};
    \node (c3) at (90:4.5) [v-small] {};
    \node (c5) at (162:4.5) [v-small] {};
    \node (c7) at (234:4.5) [v-small] {};
    \node (c9) at (306:4.5) [v-small] {};
    \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}

%\newpage  

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

\end{document}








