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\pagestyle{fancy} \lhead{{\sf Homework 4 $|$ Due September 23 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra $|$
    Fall 2022 $|$ M.~Macauley}}

\begin{document}

%$\;$ \vspace{-6mm}

\begin{enumerate}
  
  %%========================= PROBLEM 1================================
  
\item In this multi-part adventure, we will explore actions
  of the group $G=D_6=\<r,f\>$ on various sets of objects, and its
  automorphism group. A Cayley graph, cycle graph, and subgroup
  lattice are shown below. \vspace{-6mm}
  
  \newcommand\Hf{6.495} % height of D_6 
  \newcommand\He{5.196} % height of C_6  
  \newcommand\Hd{3.897} % height of V_4 
  \newcommand\Hc{2.598} % height of C_3  
  \newcommand\Hb{1.299} % height of C_2 
  \newcommand\Ha{0} % height of C_1   
  \tikzstyle{R6-out} = [draw, very thick, Red,-stealth,bend right=18]
  \tikzstyle{R6-in} = [draw, very thick, Red,-stealth,bend left=12]
  \[
  \hspace*{-8mm}
  \tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=6.5mm] 
  \begin{tikzpicture}
    %%
    %% ------------- Cycle graph of D_6 ---------------------------
    %%
    \begin{scope}[shift={(0,7.5)},scale=.66]
      \tikzstyle{v-r} = [circle, draw, fill=lred,inner sep=0pt,
        minimum size=5.5mm]
      \tikzstyle{v-g} = [circle, draw, fill=lightgreen,inner sep=0pt,
        minimum size=5.5mm]
      \tikzstyle{v-b} = [circle, draw, fill=lblue,inner sep=0pt,
        minimum size=5.5mm]
      \tikzstyle{v-p} = [circle, draw, fill=conj-purple,inner sep=0pt, 
        minimum size=5.5mm]
      \tikzstyle{v-o} = [circle, draw, fill=conj-orange,inner sep=0pt, 
        minimum size=5.5mm]
      \tikzstyle{v-gr} = [circle, draw, fill=midgrey,inner sep=0pt, 
        minimum size=5.5mm]
      \tikzstyle{cy2} = [draw,thick]
      \tikzstyle{every node}=[font=\footnotesize]
      \node (1) at (90:2) [v-gr] {$1$};
      \node (r) at (150:2) [v-r] {$r$};
      \node (r2) at (210:2) [v-o] {$r^2$};
      \node (r3) at (270:2) [v-p] {$r^3$};
      \node (r4) at (330:2) [v-o] {$r^4$};
      \node (r5) at (30:2) [v-r] {$r^5$};
      \node (f) at (-3,4) [v-b] {$f$};
      \node (r2f) at (-1.8,4) [v-b] {$r^2\!f$};
      \node (r4f) at (-.6,4) [v-b] {$r^4\!f$};
      \node (rf) at (.6,4) [v-g] {$rf$};
      \node (r3f) at (1.8,4) [v-g] {$r^3\!f$};
      \node (r5f) at (3,4) [v-g] {$r^5\!f$};
      \draw [cy2] (1) to [bend right=15] (r);
      \draw [cy2] (r) to [bend right=15] (r2);
      \draw [cy2] (r2) to [bend right=15] (r3);
      \draw [cy2] (r3) to [bend right=15] (r4);
      \draw [cy2] (r4) to [bend right=15] (r5);
      \draw [cy2] (r5) to [bend right=15] (1);
      \draw [cy2] (1) to (f);
      \draw [cy2] (1) to (rf);
      \draw [cy2] (1) to (r2f);
      \draw [cy2] (1) to (r3f);
      \draw [cy2] (1) to (r4f);
      \draw [cy2] (1) to (r5f);
    \end{scope}
    %%
    %% ------------- Cayley graph of D_6 ---------------------------
    %%    
    \begin{scope}[shift={(0,2.5)},scale=1.45]
      \tikzstyle{every node}=[font=\small]
      \node (1) at (0:2) [v-small] {$1$};
      \node (r) at (60:2) [v-small] {$r$};
      \node (r2) at (120:2) [v-small] {$r^2$};
      \node (r3) at (180:2) [v-small] {$r^3$};
      \node (r4) at (240:2) [v-small] {$r^4$};
      \node (r5) at (300:2) [v-small] {$r^5$};
      \node (f) at (0:1) [v-small] {$f$};
      \node (rf) at (60:1) [v-small] {$rf$};
      \node (r2f) at (120:1) [v-small] {$r^2\!f$};
      \node (r3f) at (180:1) [v-small] {$r^3\!f$};
      \node (r4f) at (240:1) [v-small] {$r^4\!f$};
      \node (r5f) at (300:1) [v-small] {$r^5\!f$};
      \path[bb] (1) to (f);
      \path[bb] (r) to (rf);
      \path[bb] (r2) to (r2f);
      \path[bb] (r3) to (r3f);
      \path[bb] (r4) to (r4f);
      \path[bb] (r5) to (r5f);
      \path[R6-out] (1) to (r);
      \path[R6-out] (r) to (r2);
      \path[R6-out] (r2) to (r3);
      \path[R6-out] (r3) to (r4);
      \path[R6-out] (r4) to (r5);
      \path[R6-out] (r5) to (1);
      \path[R6-in] (f) to (r5f);
      \path[R6-in] (r5f) to (r4f);
      \path[R6-in] (r4f) to (r3f);
      \path[R6-in] (r3f) to (r2f);
      \path[R6-in] (r2f) to (rf);
      \path[R6-in] (rf) to (f);
    \end{scope}
    %%
    %% ------------- Subgroup lattice of D_6 ------------------------
    %%
    \begin{scope}[shift={(7.25,0)},scale=1.6]
      \tikzstyle{every node}=[font=\small]
      \node(D6) at (0,\Hf) {$D_6$};
      \node(r2-f) at (-1,\He) {$\<r^2,f\>$};
      \node(r2-rf) at (0,\He) {$\<r^2,rf\>$}; 
      \node(r) at (-2,\He) {$\<r\>$}; 
      \node(r3-f) at (2.4,\Hd) {\footnotesize $\<r^3\!,\!f\>$};
      \node(r3-rf) at (3.1,\Hd) {\footnotesize $\<r^3\!,rf\>$};
      \node(r3-r2f) at (3.8,\Hd) {\footnotesize $\<r^3\!,\!r^2\!f\>$};
      \node(r2) at (-2,\Hc) {$\<r^2\>$};
      \node(r3) at (0,\Hb) {$\<r^3\>$};
      \node(f) at (.6,\Hb) {$\<f\>$};
      \node(r4f) at (1.2,\Hb) {$\<r^4\!f\>$};      
      \node(r2f) at (1.8,\Hb) {$\<r^2\!f\>$};      
      \node(r3f) at (2.4,\Hb) {$\<r^3\!f\>$};
      \node(rf) at (3.1,\Hb) {$\<rf\>$};
      \node(r5f) at (3.8,\Hb) {$\<r^5\!f\>$};
      \node (1) at (0,\Ha) {$\<1\>$};
      \draw[faded] (D6) -- (r2-f);
      \draw[faded] (D6) -- (r);
      \draw[faded] (D6) to (r2-rf);
      \draw[faded] (D6) to (r3-f); 
      \draw[faded] (D6) -- (r3-rf);
      \draw[faded] (D6) to (r3-r2f); 
      \draw[faded] (r2-f) to (f); 
      \draw[faded] (r2-f) -- (r2f);
      \draw[faded] (r2-f) to (r4f);
      \draw[faded] (r2-f) to (r2);
      \draw[faded] (r) -- (r2);
      \draw[faded] (r) -- (r3);
      \draw[faded] (r2-rf) -- (r2);
      \draw[faded] (r2-rf) to (rf);
      \draw[faded] (r2-rf) to (r3f);
      \draw[faded] (r2-rf) to (r5f);
      \draw[faded] (r3-f) -- (f);
      \draw[faded] (r3-f) to (r3);
      \draw[faded] (r3-f) to (r3f); 
      \draw[faded] (r3-rf) to (rf);
      \draw[faded] (r3-rf) to (r3);
      \draw[faded] (r3-rf) to (r4f); 
      \draw[faded] (r3-r2f) to (r2f);
      \draw[faded] (r3-r2f) to (r3);
      \draw[faded] (r3-r2f) to (r5f);
      \draw[faded] (r2) -- (1);
      \draw[faded] (f) -- (1);
      \draw[faded] (rf) to (1);
      \draw[faded] (r2f) to (1);
      \draw[faded] (r3f) to (1);
      \draw[faded] (r4f) to (1);
      \draw[faded] (r5f) to (1); 
      \draw[faded] (r3) -- (1);
    \end{scope}
  \end{tikzpicture}
  \]

  
  %%----------------------------- PART (a) ----------------------------
  
  \begin{enumerate}
  \item Consider the right action of $D_6$ on following set of $31$
    ``binary hexagons,'' where $r$ rotates each one $60^\circ$
    counterclockwise, and $f$ flips it across a vertical
    axis. \vspace{-2mm}
    \[
    \hspace*{-10mm}
    \begin{tikzpicture}[scale=.58]
      \tikzstyle{every node}=[font=\small]
      \begin{scope}[shift={(-3.75,0)}]
        \tikzstyle{every node}=[font=\normalsize]
        \node at (0,0) {$S=\Bigg\{$};
      \end{scope}
      %%
      \begin{scope}[shift={(-1.5,0)}]
        \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
        \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(1.5,0)}]
        \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$1$}; \node at (90:.6) {$1$}; \node at (150:.6) {$1$};
        \node at (210:.6) {$1$}; \node at (270:.6) {$1$}; \node at (330:.6) {$1$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(4.5,0)}]
        \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
        \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(7.5,0)}]
        \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
        \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(10.5,0)}]
        \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
        \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(13.5,0)}]
        \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
        \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(16.5,0)}]
        \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
        \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(0,-3)}]
        \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
        \node at (210:.6) {$1$}; \node at (270:.6) {$1$}; \node at (330:.6) {$1$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(3,-3)}]
        \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
        \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$1$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(6,-3)}]
        \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$1$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
        \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(9,-3)}]
        \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$1$}; \node at (90:.6) {$1$}; \node at (150:.6) {$1$};
        \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(12,-3)}]
        \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$1$};
        \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(15,-3)}]
        \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
        \node at (210:.6) {$1$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(1.5,-6)}]
        \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
        \node at (210:.6) {$1$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(4.5,-6)}]
        \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
        \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$1$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(7.5,-6)}]
        \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
        \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(10.5,-6)}]
        \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$1$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
        \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(13.5,-6)}]
        \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$1$};
        \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(16.5,-6)}]
        \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
        \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(0,-9)}]
        \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
        \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$1$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(3,-9)}]
        \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
        \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(6,-9)}]
        \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$1$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
        \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(9,-9)}]
        \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$1$};
        \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(12,-9)}]
        \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
        \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(15,-9)}]
        \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
        \node at (210:.6) {$1$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(1.5,-12)}]
        \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
        \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$1$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(4.5,-12)}]
        \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
        \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(7.5,-12)}]
        \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$1$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
        \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(10.5,-12)}]
        \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$1$};
        \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(13.5,-12)}]
        \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
        \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
        \node at (1.5,-.2) {$,$};
      \end{scope}
      %%
      \begin{scope}[shift={(16.5,-12)}]
        \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
        \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
        \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
        \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
        \node at (210:.6) {$1$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
      \end{scope}
      %%
      \begin{scope}[shift={(18.25,-12)}]
        \node at (0,0) {$\Bigg\}$};
      \end{scope}
    \end{tikzpicture}
    \]
    \begin{enumerate}[(i)]
    \item Draw the action graph and construct the ``\emph{fixed point
      table}'', which has a checkmark in row $g$ and column $s$ if
      $\phi(g)\Dot s=s$.
    \item Next to each $s\in S$ on your action graph, write
      $\stab(s)$, the stabilizer subgroup, using its generators.
    \item Find $\fix(g)$ for each $g\in D_6$.
    \item Find $\Ker(\phi)$ and $\Fix(\phi)$.
    \end{enumerate}
    
    
  %%----------------------------- PART (b) ----------------------------
  
\item Let $G=D_6$ act on its subgroups by conjugation. Construct the
  action graph by superimposing it on the subgroup lattice, and then
  construct the fixed point table. Label the nodes by stabilizer, and
  find $\Ker(\phi)$ and $\Fix(phi)$.


  %%----------------------------- PART (c) ----------------------------
  
\item There is a \emph{Galois correspondence} between conjugacy
  classes of subgroups of $G$ and transitive actions of $G$,
  defined by collapsing the right cosets of $H$. An example of this
  correspondence for $D_4=\<r,f\>$ is shown below. Carry out this construction
  for $D_6=\<r,f\>$.
  
  \tikzstyle{cy2-r} = [draw,thick,bend right=12]
  \tikzstyle{cy2} = [draw,thick]
  \tikzstyle{lat} = [draw,very thick,lightgrey]
  \tikzstyle{v-small} = [circle, draw, fill=midgrey,inner sep=0pt,
    minimum size=5.5mm]
  \tikzstyle{r-thin} = [draw,Red,-stealth]
  \tikzstyle{rr-thin} = [draw,Red,]
  \tikzstyle{bb-thin} = [draw,Blue]
  \renewcommand\Hd{9.75} % height of D_4 
  \renewcommand\Hc{6.5} % height of C_4
  \renewcommand\Hb{3.25} % height of C_2 
  \renewcommand\Ha{0} % height of C_1   
  \[
  \hspace*{-10mm}
  \begin{tikzpicture}[scale=1]
    \tikzstyle{v-r} = [circle, draw, fill=lred,inner sep=0pt,
      minimum size=2mm]
    \tikzstyle{v-g} = [circle, draw, fill=lightgreen,inner sep=0pt,
      minimum size=2mm]
    \tikzstyle{v-b} = [circle, draw, fill=lblue,inner sep=0pt,
      minimum size=2mm]
    \tikzstyle{v-p} = [circle, draw, fill=conj-purple,inner sep=0pt, 
      minimum size=2mm]
    \tikzstyle{v-o} = [circle, draw, fill=conj-orange,inner sep=0pt, 
      minimum size=2mm]
    \tikzstyle{v-gr} = [circle, draw, fill=midgrey,inner sep=0pt,
      minimum size=2mm]
    \tikzstyle{v-tiny} = [circle, draw, fill=white,inner sep=0pt,
      minimum size=2mm]
    \begin{scope}[shift={(0,0)}]
      %%
      %%------------------ Conjugacy poset of D_4 -------------------
      %%
      \begin{scope}[shift={(0,\Hd)},scale=.33]
        \draw[very thick,rounded corners,lightgrey] (-2.75,-2.25)
        rectangle (2.75,3.6);
        \tikzstyle{every node}=[font=\scriptsize]
        \node (1) at (90:1.6) [v-gr] {};
        \node (r) at (180:1.6) [v-r] {};
        \node (r2) at (270:1.6) [v-p] {};
        \node (r3) at (0:1.6) [v-r] {};
        \node (f) at (-1.8,3) [v-b] {};
        \node (r2f) at (-.6,3) [v-b] {};
        \node (rf) at (.6,3) [v-g] {};
        \node (r3f) at (1.8,3) [v-g] {};
        \draw [cy2-r] (1) to [bend right=30] (r);
        \draw [cy2-r] (r) to [bend right=30] (r2);
        \draw [cy2-r] (r2) to [bend right=30] (r3);
        \draw [cy2-r] (r3) to [bend right=30] (1);
        \draw [cy2] (1) to (f);
        \draw [cy2] (1) to (rf);
        \draw [cy2] (1) to (r2f);
        \draw [cy2] (1) to (r3f);
        \node at (0,0) {\small $D_4$};
        \node (D4-bot) at (0,-2) {};
        \node (D6-bot) at (0,-2.5) {};
      \end{scope}
      %%
      \begin{scope}[shift={(-2.5,\Hc)},scale=.33]
        \draw[very thick,rounded corners,lightgrey] (-2.75,-2.25)
        rectangle (2.75,3.6);
        \node (1) at (90:1.6) [v-gr] {};
        \node (r) at (180:1.6) [v-tiny] {};
        \node (r2) at (270:1.6) [v-p] {};
        \node (r3) at (0:1.6) [v-tiny] {};
        \node (f) at (-1.8,3) [v-b] {};
        \node (r2f) at (-.6,3) [v-b] {};
        \node (rf) at (.6,3) [v-tiny] {};
        \node (r3f) at (1.8,3) [v-tiny] {};
        \draw [cy2-r] (1) to [bend right=30] (r);
        \draw [cy2-r] (r) to [bend right=30] (r2);
        \draw [cy2-r] (r2) to [bend right=30] (r3);
        \draw [cy2-r] (r3) to [bend right=30] (1);
        \draw [cy2] (1) to (f);
        \draw [cy2] (1) to (rf);
        \draw [cy2] (1) to (r2f);
        \draw [cy2] (1) to (r3f);
        \node at (0,0) {\scriptsize $\<r^2\!,f\>$};
        \node (r2-f-top) at (0,4) {};
        \node (r2-f-bot) at (0,-2.5) {};
      \end{scope}
      %%
      \begin{scope}[shift={(0,\Hc)},scale=.33]
        \draw[very thick,rounded corners,lightgrey] (-2.75,-2.25)
        rectangle (2.75,3.6);
        \node (1) at (90:1.6) [v-gr] {};
        \node (r) at (180:1.6) [v-r] {};
        \node (r2) at (270:1.6) [v-p] {};
        \node (r3) at (0:1.6) [v-r] {};
        \node (f) at (-1.8,3) [v-tiny] {};
        \node (r2f) at (-.6,3) [v-tiny] {};
        \node (rf) at (.6,3) [v-tiny] {};
        \node (r3f) at (1.8,3) [v-tiny] {};
        \draw [cy2-r] (1) to [bend right=30] (r);
        \draw [cy2-r] (r) to [bend right=30] (r2);
        \draw [cy2-r] (r2) to [bend right=30] (r3);
        \draw [cy2-r] (r3) to [bend right=30] (1);
        \draw [cy2] (1) to (f);
        \draw [cy2] (1) to (rf);
        \draw [cy2] (1) to (r2f);
        \draw [cy2] (1) to (r3f);
        \node at (0,0) {\footnotesize $\<r\>$};
        \node (r-top) at (0,4) {};
        \node (r-bot) at (0,-2.5) {};
      \end{scope}
      %%
      \begin{scope}[shift={(2.5,\Hc)},scale=.33]
        \draw[very thick,rounded corners,lightgrey] (-2.75,-2.25)
        rectangle (2.75,3.6);
        \node (1) at (90:1.7) [v-gr] {};
        \node (r) at (180:1.7) [v-tiny] {};
        \node (r2) at (270:1.7) [v-p] {};
        \node (r3) at (0:1.7) [v-tiny] {};
        \node (f) at (-1.8,3) [v-tiny] {};
        \node (r2f) at (-.6,3) [v-tiny] {};
        \node (rf) at (.6,3) [v-g] {};
        \node (r3f) at (1.8,3) [v-g] {};
        \draw [cy2-r] (1) to [bend right=30] (r);
        \draw [cy2-r] (r) to [bend right=30] (r2);
        \draw [cy2-r] (r2) to [bend right=30] (r3);
        \draw [cy2-r] (r3) to [bend right=30] (1);
        \draw [cy2] (1) to (f);
        \draw [cy2] (1) to (rf);
        \draw [cy2] (1) to (r2f);
        \draw [cy2] (1) to (r3f);
        \node at (0,0) {\scriptsize $\<r^2\!\!,\!rf\>$};
        \node (r2-rf-top) at (0,4) {};
        \node (r2-rf-bot) at (0,-2.5) {};
      \end{scope}
      %%
      \begin{scope}[shift={(-2.5,\Hb)},scale=.33]
        \draw[very thick,rounded corners,lightgrey] (-2.75,-2.25)
        rectangle (2.75,3.6);
        \node (1) at (90:1.6) [v-gr] {};
        \node (r) at (180:1.6) [v-tiny] {};
        \node (r2) at (270:1.6) [v-tiny] {};
        \node (r3) at (0:1.6) [v-tiny] {};
        \node (f) at (-1.8,3) [v-b] {};
        \node (r2f) at (-.6,3) [v-tiny] {};
        \node (rf) at (.6,3) [v-tiny] {};
        \node (r3f) at (1.8,3) [v-tiny] {};
        \draw [cy2-r] (1) to [bend right=30] (r);
        \draw [cy2-r] (r) to [bend right=30] (r2);
        \draw [cy2-r] (r2) to [bend right=30] (r3);
        \draw [cy2-r] (r3) to [bend right=30] (1);
        \draw [cy2] (1) to (f);
        \draw [cy2] (1) to (rf);
        \draw [cy2] (1) to (r2f);
        \draw [cy2] (1) to (r3f);
        \node [midgrey] at (-2.,-1.75) {\tiny $\times 2$};
        \node at (0,0) {\footnotesize $\<f\>$};
        \node (f-top) at (0,4) {};
        \node (f-bot) at (0,-2.5) {};    
      \end{scope}
      %%
      \begin{scope}[shift={(0,\Hb)},scale=.33]
        \draw[very thick,rounded corners,lightgrey] (-2.75,-2.25)
        rectangle (2.75,3.6);
        \node (1) at (90:1.6) [v-gr] {};
        \node (r) at (180:1.6) [v-tiny] {};
        \node (r2) at (270:1.6) [v-p] {};
        \node (r3) at (0:1.6) [v-tiny] {};
        \node (f) at (-1.8,3) [v-tiny] {};
        \node (r2f) at (-.6,3) [v-tiny] {};
        \node (rf) at (.6,3) [v-tiny] {};
        \node (r3f) at (1.8,3) [v-tiny] {};
        \draw [cy2-r] (1) to [bend right=30] (r);
        \draw [cy2-r] (r) to [bend right=30] (r2);
        \draw [cy2-r] (r2) to [bend right=30] (r3);
        \draw [cy2-r] (r3) to [bend right=30] (1);
        \draw [cy2] (1) to (f);
        \draw [cy2] (1) to (rf);
        \draw [cy2] (1) to (r2f);
        \draw [cy2] (1) to (r3f);
        \node at (0,0) {\footnotesize $\<r^2\>$};
        \node (r2-top) at (0,4) {};
        \node (r2-bot) at (0,-2.5) {};
      \end{scope}
      %%
      \begin{scope}[shift={(2.5,\Hb)},scale=.33]
        \draw[very thick,rounded corners,lightgrey] (-2.75,-2.25)
        rectangle (2.75,3.6);
        \node (1) at (90:1.6) [v-gr] {};
        \node (r) at (180:1.6) [v-tiny] {};
        \node (r2) at (270:1.6) [v-tiny] {};
        \node (r3) at (0:1.6) [v-tiny] {};
        \node (f) at (-1.8,3) [v-tiny] {};
        \node (r2f) at (-.6,3) [v-tiny] {};
        \node (rf) at (.6,3) [v-g] {};
        \node (r3f) at (1.8,3) [v-tiny] {};
        \draw [cy2-r] (1) to [bend right=30] (r);
        \draw [cy2-r] (r) to [bend right=30] (r2);
        \draw [cy2-r] (r2) to [bend right=30] (r3);
        \draw [cy2-r] (r3) to [bend right=30] (1);
        \draw [cy2] (1) to (f);
        \draw [cy2] (1) to (rf);
        \draw [cy2] (1) to (r2f);
        \draw [cy2] (1) to (r3f);
        \node [midgrey] at (-2.,-1.75) {\tiny $\times 2$};
        \node at (0,0) {\footnotesize $\<rf\>$};
        \node (rf-top) at (0,4) {};
        \node (rf-bot) at (0,-2.5) {};
      \end{scope}
      %%
      \begin{scope}[shift={(0,\Ha)},scale=.33]
        \draw[very thick,rounded corners,lightgrey] (-2.75,-2.25)
        rectangle (2.75,3.6);
        \node (1) at (90:1.6) [v-gr] {};
        \node (r) at (180:1.6) [v-tiny] {};
        \node (r2) at (270:1.6) [v-tiny] {};
        \node (r3) at (0:1.6) [v-tiny] {};
        \node (f) at (-1.8,3) [v-tiny] {};
        \node (r2f) at (-.6,3) [v-tiny] {};
        \node (rf) at (.6,3) [v-tiny] {};
        \node (r3f) at (1.8,3) [v-tiny] {};
        \draw [cy2-r] (1) to [bend right=30] (r);
        \draw [cy2-r] (r) to [bend right=30] (r2);
        \draw [cy2-r] (r2) to [bend right=30] (r3);
        \draw [cy2-r] (r3) to [bend right=30] (1);
        \draw [cy2] (1) to (f);
        \draw [cy2] (1) to (rf);
        \draw [cy2] (1) to (r2f);
        \draw [cy2] (1) to (r3f);
        \node at (0,0) {\footnotesize $\<1\>$};
        \node (1-top) at (0,4) {};
      \end{scope}
      %%
      \draw [lat] (D4-bot) to (r2-f-top);
      \draw [lat] (D4-bot) to (r-top);
      \draw [lat] (D4-bot) to (r2-rf-top);
      \draw [lat] (r2-f-bot) to (f-top);
      \draw [lat] (r2-f-bot) to (r2-top);
      \draw [lat] (r-bot) to (r2-top);
      \draw [lat] (r2-rf-bot) to (r2-top);
      \draw [lat] (r2-rf-bot) to (rf-top);
      \draw [lat] (f-bot) to (1-top);
      \draw [lat] (r2-bot) to (1-top);
      \draw [lat] (rf-bot) to (1-top);
    \end{scope}
    %%
    %%------------------ Action graphg poset of D_4 -------------------
    %%
    \begin{scope}[shift={(9,0)}]
      \tikzset{every loop/.style={min distance=10mm,looseness=15}}
      \tikzstyle{r-thin} = [draw, Red, thick, -stealth]
      \tikzstyle{rr-thin} = [draw, Red, thick]
      \tikzstyle{b-thin} = [draw, Blue, thick, -stealth]
      \tikzstyle{bb-thin} = [draw, Blue, thick]
      %%
      \begin{scope}[shift={(0,\Hd)},scale=.33]
        \tikzstyle{every node}=[font=\scriptsize]
        \node (1) at (0:0) [v-gr] {};
        \path (1) edge [b-thin,in=150,out=210,loop,-stealth] (1);
        \path (1) edge [r-thin,in=-30,out=30,loop,-stealth] (1);
        \node (D4-bot) at (0,-2) {};
      \end{scope}
      %%
      \begin{scope}[shift={(-2.5,\Hc)},scale=.33]
        \node (1) at (0:2) [v-gr] {};
        \node (r2) at (180:2) [v-p] {};
        \draw [rr-thin] (1) to (r2);
        \node (r2-f-top) at (0,2.75) {};
        \node (r2-f-bot) at (0,-1.75) {};
        \path (1) edge [b-thin,in=60,out=120,loop,-stealth] (1);
        \path (r2) edge [b-thin,in=60,out=120,loop,-stealth] (r2);
      \end{scope}
      %%
      \begin{scope}[shift={(0,\Hc)},scale=.33]
        \node (1) at (0:2) [v-gr] {};
        \node (r2) at (180:2) [v-p] {};
        \draw [rr-thin] (1) to [bend right=20] (r2);
        \draw [bb-thin] (1) to [bend left=20] (r2);
        \node (r-top) at (0,2.75) {};
        \node (r-bot) at (0,-1.75) {};
      \end{scope}
      %%
      \begin{scope}[shift={(2.5,\Hc)},scale=.33]
        \node (1) at (0:2) [v-gr] {};
        \node (r2) at (180:2) [v-p] {};
        \draw [bb-thin] (1) to (r2);
        \node (r2-rf-top) at (0,2.75) {};
        \node (r2-rf-bot) at (0,-1.75) {};
        \path (1) edge [r-thin,in=60,out=120,loop,-stealth] (1);
        \path (r2) edge [r-thin,in=60,out=120,loop,-stealth] (r2);
      \end{scope}
      %%
      \begin{scope}[shift={(-2.5,\Hb+.2)},scale=.33]
        \node (1) at (0:2) [v-gr] {};
        \node (r) at (90:2) [v-r] {};
        \node (r2) at (180:2) [v-p] {};
        \node (r3) at (270:2) [v-r] {};
        \draw [r-thin] (1) to [bend right=30] (r);
        \draw [r-thin] (r) to [bend right=30] (r2);
        \draw [r-thin] (r2) to [bend right=30] (r3);
        \draw [r-thin] (r3) to [bend right=30] (1);
        \draw [bb-thin] (r) to (r3);
        \path (1) edge [b-thin,in=-30,out=30,loop,-stealth] (1);
        \path (r2) edge [b-thin,in=150,out=210,loop,-stealth] (r2);
        \node (f-top) at (0,3) {};
        \node (f-bot) at (0,-2.75) {};
      \end{scope}
      %%
      \begin{scope}[shift={(0,\Hb+.2)},scale=.33]
        \node (1) at (0:2) [v-gr] {};
        \node (r) at (90:2) [v-r] {};
        \node (r2) at (180:2) [v-p] {};
        \node (r3) at (270:2) [v-r] {};
        \draw [rr-thin] (1) to [bend right=30] (r);
        \draw [bb-thin] (r) to [bend right=30] (r2);
        \draw [rr-thin] (r2) to [bend right=30] (r3);
        \draw [bb-thin] (r3) to [bend right=30] (1);
        \node (r2-top) at (0,3) {};
        \node (r2-bot) at (0,-2.75) {};
      \end{scope}
      %%
      \begin{scope}[shift={(2.5,\Hb+.2)},scale=.33]
        \node (1) at (0:2) [v-gr] {};
        \node (r) at (90:2) [v-r] {};
        \node (r2) at (180:2) [v-p] {};
        \node (r3) at (270:2) [v-r] {};
        \draw [r-thin] (1) to [bend right=30] (r);
        \draw [r-thin] (r) to [bend right=30] (r2);
        \draw [r-thin] (r2) to [bend right=30] (r3);
        \draw [r-thin] (r3) to [bend right=30] (1);
        \draw [bb-thin] (1) to [bend left=15] (r);
        \draw [bb-thin] (r2) to [bend left=15] (r3);
        \node (rf-top) at (0,3) {};
        \node (rf-bot) at (0,-2.75) {};
      \end{scope}
      %%
      \begin{scope}[shift={(0,\Ha)},scale=.33]
        \node (1) at (0:2.9) [v-gr] {};
        \node (r) at (90:2.9) [v-r] {};
        \node (r2) at (180:2.9) [v-p] {};
        \node (r3) at (270:2.9) [v-r] {};
        \node (f) at (0:1.4) [v-b] {};
        \node (rf) at (90:1.4) [v-g] {};
        \node (r2f) at (180:1.4) [v-b] {};
        \node (r3f) at (270:1.4) [v-g] {};
        \draw [r-thin] (1) to [bend right=30] (r);
        \draw [r-thin] (r) to [bend right=30] (r2);
        \draw [r-thin] (r2) to [bend right=30] (r3);
        \draw [r-thin] (r3) to [bend right=30] (1);
        \draw [r-thin] (f) to [bend left=22] (r3f);
        \draw [r-thin] (r3f) to [bend left=22] (r2f);
        \draw [r-thin] (r2f) to [bend left=22] (rf);
        \draw [r-thin] (rf) to [bend left=22] (f);
        \draw [bb-thin] (1) to (f);
        \draw [bb-thin] (r) to (rf);
        \draw [bb-thin] (r2) to (r2f);
        \draw [bb-thin] (r3) to (r3f);
        \node (1-top) at (0,4) {};
      \end{scope}
      %%
      \draw [lat] (D4-bot) to (r2-f-top);
      \draw [lat] (D4-bot) to (r-top);
      \draw [lat] (D4-bot) to (r2-rf-top);
      \draw [lat] (r2-f-bot) to (f-top);
      \draw [lat] (r2-f-bot) to (r2-top);
      \draw [lat] (r-bot) to (r2-top);
      \draw [lat] (r2-rf-bot) to (r2-top);
      \draw [lat] (r2-rf-bot) to (rf-top);
      \draw [lat] (f-bot) to (1-top);
      \draw [lat] (r2-bot) to (1-top);
      \draw [lat] (rf-bot) to (1-top);
    \end{scope}
  \end{tikzpicture}
  \]
  
  
  %%----------------------------- PART (d) ----------------------------
  
\item The automorphism group of $D_3$ is
  $\Aut(D_3)=\<\alpha,\beta\>\cong D_3$, where  
  \[
  \left\{\!\!\begin{array}{ll}
    \alpha(r)=r \\ \alpha(f)=rf
  \end{array}\right.\qquad\qquad
  \left\{\!\!\begin{array}{ll}
    \beta(r)=r^{-1} \\ \beta(f)=f
  \end{array}\right.
  \]
  Below are two Cayley graphs for the automorphism group
  $\Aut(D_3)\cong D_3$ with the nodes labeled by rewired Cayley
  graphs of $D_3=\<\Alert{r},\Balert{f}\>$ and
  $D_3=\<\Balert{s},\Galert{t}\>=\<\Balert{f},\Galert{rf}\>$.


  \tikzstyle{v-tiny} = [circle, draw, fill=grey,inner sep=0pt, minimum size=1.2mm] 
  \tikzstyle{gg-thin} = [draw, darkgreen]
  \tikzstyle{r-thin} = [draw, Red, -stealth]
  \tikzstyle{bb-thin} = [draw, Blue]
  \tikzstyle{o} = [draw, very thick,orange, -stealth]
  \tikzstyle{p} = [draw, very thick,amethyst]
  %%
  \[
  \hspace*{-16mm}
  \begin{tikzpicture}[scale=.95]
    %%
    %%------------------ Cayley graph of Aut(D_4) ---------------------
    %%
    \begin{scope}[shift={(0,0)},scale=1]
      \begin{scope}[shift={(0:1.25)},scale=.12]
        \draw[grey, fill=lightgrey] (0,0) circle (60mm);
        \node (f) at (0:2.5) [v-tiny] {};
        \node (rf) at (120:2.5) [v-tiny] {};
        \node (rrf) at (240:2.5) [v-tiny] {};
        \node (e) at (0:5.5) [v-tiny] {};
        \node (rr) at (240:5.5) [v-tiny] {};
        \node (r) at (120:5.5) [v-tiny] {};
        \draw [r-thin] (e) to [bend left] (rr);
        \draw [r-thin] (rr) to [bend left] (r);
        \draw [r-thin] (r) to [bend left] (e);
        \draw [r-thin] (f) to [bend right] (rf);
        \draw [r-thin] (rf) to [bend right] (rrf);
        \draw [r-thin] (rrf) to [bend right] (f);
        \draw [bb-thin] (e) to (f);
        \draw [bb-thin] (r) to (rf);
        \draw [bb-thin] (rr) to (rrf);
        \node (o-in_f) at (90:4.8) {};
        \node (o-out_f) at (270:4.8) {};
        \node (p_f) at (0:4.8) {};
      \end{scope}
      %%
      \begin{scope}[shift={(120:1.25)},scale=.12]
        \draw[grey, fill=lightgrey] (0,0) circle (60mm);
        \node (f) at (0:2.5) [v-tiny] {};
        \node (rf) at (120:2.5) [v-tiny] {};
        \node (rrf) at (240:2.5) [v-tiny] {};
        \node (e) at (0:5.5) [v-tiny] {};
        \node (rr) at (240:5.5) [v-tiny] {};
        \node (r) at (120:5.5) [v-tiny] {};
        \draw [r-thin] (e) to [bend left] (rr);
        \draw [r-thin] (rr) to [bend left] (r);
        \draw [r-thin] (r) to [bend left] (e);
        \draw [r-thin] (f) to [bend right] (rf);
        \draw [r-thin] (rf) to [bend right] (rrf);
        \draw [r-thin] (rrf) to [bend right] (f);
        \draw [bb-thin] (e) to [bend left] (rrf);
        \draw [bb-thin] (rr) to [bend left] (rf);
        \draw [bb-thin] (r) to [bend left] (f);
        \node (o-in_rf) at (195:4.8) {};
        \node (o-out_rf) at (30:4.8) {};
        \node (p_rf) at (120:4.8) {};
      \end{scope}
      %%
      \begin{scope}[shift={(240:1.25)},scale=.12]
        \draw[grey, fill=lightgrey] (0,0) circle (60mm);
        \node (f) at (0:2.5) [v-tiny] {};
        \node (rf) at (120:2.5) [v-tiny] {};
        \node (rrf) at (240:2.5) [v-tiny] {};
        \node (e) at (0:5.5) [v-tiny] {};
        \node (rr) at (240:5.5) [v-tiny] {};
        \node (r) at (120:5.5) [v-tiny] {};
        \draw [r-thin] (e) to [bend left] (rr);
        \draw [r-thin] (rr) to [bend left] (r);
        \draw [r-thin] (r) to [bend left] (e);
        \draw [r-thin] (f) to [bend right] (rf);
        \draw [r-thin] (rf) to [bend right] (rrf);
        \draw [r-thin] (rrf) to [bend right] (f);
        \draw [bb-thin] (e) to [bend right] (rf);
        \draw [bb-thin] (r) to [bend right] (rrf);
        \draw [bb-thin] (rr) to [bend right] (f);
        \node (o-in_r2f) at (330:4.8) {};
        \node (o-out_r2f) at (165:4.8) {};
        \node (p_r2f) at (240:4.8) {};
      \end{scope}
      %%
      %%------------------- outer ring of nodes -----------------------
      %%
      \begin{scope}[shift={(0:3.5)},scale=.12]
        \draw[grey, fill=lightgrey] (0,0) circle (60mm);
        \node (f) at (0:2.5) [v-tiny] {};
        \node (rf) at (120:2.5) [v-tiny] {};
        \node (rrf) at (240:2.5) [v-tiny] {};
        \node (e) at (0:5.5) [v-tiny] {};
        \node (rr) at (240:5.5) [v-tiny] {};
        \node (r) at (120:5.5) [v-tiny] {};
        \draw [r-thin] (e) to [bend right] (r);
        \draw [r-thin] (r) to [bend right] (rr);
        \draw [r-thin] (rr) to [bend right] (e);
        \draw [r-thin] (f) to [bend left] (rrf);
        \draw [r-thin] (rrf) to [bend left] (rf);
        \draw [r-thin] (rf) to [bend left] (f);
        \draw [bb-thin] (e) to (f);
        \draw [bb-thin] (r) to (rf);
        \draw [bb-thin] (rr) to (rrf);
        \node (o-in_1) at (270:4.8) {};
        \node (o-out_1) at (90:4.8) {};
        \node (p_1) at (180:4.8) {};
      \end{scope}
      %%
      \begin{scope}[shift={(120:3.5)},scale=.12]
        \draw[grey, fill=lightgrey] (0,0) circle (60mm);
        \node (f) at (0:2.5) [v-tiny] {};
        \node (rf) at (120:2.5) [v-tiny] {};
        \node (rrf) at (240:2.5) [v-tiny] {};
        \node (e) at (0:5.5) [v-tiny] {};
        \node (rr) at (240:5.5) [v-tiny] {};
        \node (r) at (120:5.5) [v-tiny] {};
        \draw [r-thin] (e) to [bend right] (r);
        \draw [r-thin] (r) to [bend right] (rr);
        \draw [r-thin] (rr) to [bend right] (e);
        \draw [r-thin] (f) to [bend left] (rrf);
        \draw [r-thin] (rrf) to [bend left] (rf);
        \draw [r-thin] (rf) to [bend left] (f);
        \draw [bb-thin] (e) to [bend right] (rf);
        \draw [bb-thin] (r) to [bend right] (rrf);
        \draw [bb-thin] (rr) to [bend right] (f);
        \node (o-in_r) at (30:4.8) {};
        \node (o-out_r) at (195:4.8) {};
        \node (p_r) at (300:4.8) {};
      \end{scope}
      %%
      \begin{scope}[shift={(240:3.5)},scale=.12]
        \draw[grey, fill=lightgrey] (0,0) circle (60mm);
        \node (f) at (0:2.5) [v-tiny] {};
        \node (rf) at (120:2.5) [v-tiny] {};
        \node (rrf) at (240:2.5) [v-tiny] {};
        \node (e) at (0:5.5) [v-tiny] {};
        \node (rr) at (240:5.5) [v-tiny] {};
        \node (r) at (120:5.5) [v-tiny] {};
        \draw [r-thin] (e) to [bend right] (r);
        \draw [r-thin] (r) to [bend right] (rr);
        \draw [r-thin] (rr) to [bend right] (e);
        \draw [r-thin] (f) to [bend left] (rrf);
        \draw [r-thin] (rrf) to [bend left] (rf);
        \draw [r-thin] (rf) to [bend left] (f);
        \draw [bb-thin] (e) to [bend left] (rrf);
        \draw [bb-thin] (rr) to [bend left] (rf);
        \draw [bb-thin] (r) to [bend left] (f);
        \node (o-in_r2) at (165:4.8) {};
        \node (o-out_r2) at (330:4.8) {};
        \node (p_r2) at (60:4.8) {};
      \end{scope}
      \draw [o] (o-out_1) to[bend right=45] (o-in_r);
      \draw [o] (o-out_r) to[bend right=45] (o-in_r2);
      \draw [o] (o-out_r2) to[bend right=45] (o-in_1);
      \draw [o] (o-out_f) to[bend left=30] (o-in_r2f);
      \draw [o] (o-out_r2f) to[bend left=35] (o-in_rf);
      \draw [o] (o-out_rf) to[bend left=30] (o-in_f);
      \draw [p] (p_1) to (p_f);
      \draw [p] (p_r) to (p_rf);
      \draw [p] (p_r2) to (p_r2f);
    \end{scope}
    %%
    %%------------------ 2nd Cayley graph of Aut(D_4) ------------------
    %%    
    \begin{scope}[shift={(9,0)}]
      \begin{scope}[shift={(0:1.25)},scale=.12]
        \draw[grey, fill=lightgrey] (0,0) circle (60mm);
        \node (e) at (90:5.5) [v-tiny] {};
        \node (rrf) at (30:5.5) [v-tiny] {};
        \node (rr) at (-30:5.5) [v-tiny] {};
        \node (rf) at (-90:5.5) [v-tiny] {};
        \node (r) at (-150:5.5) [v-tiny] {};
        \node (f) at (150:5.5) [v-tiny] {};
        \path[gg-thin] (e) to (rf);
        \path[bb-thin] (rf) to (r);
        \path[gg-thin] (r) to (rrf);
        \path[bb-thin] (rrf) to (rr);
        \path[gg-thin] (rr) to (f);
        \path[bb-thin] (f) to (e);
        \node (o-in_f) at (90:4.8) {};
        \node (o-out_f) at (270:4.8) {};
        \node (p_f) at (0:4.8) {};
      \end{scope}
      %%
      \begin{scope}[shift={(120:1.25)},scale=.12]
        \draw[grey, fill=lightgrey] (0,0) circle (60mm);
        \node (e) at (90:5.5) [v-tiny] {};
        \node (rrf) at (30:5.5) [v-tiny] {};
        \node (rr) at (-30:5.5) [v-tiny] {};
        \node (rf) at (-90:5.5) [v-tiny] {};
        \node (r) at (-150:5.5) [v-tiny] {};
        \node (f) at (150:5.5) [v-tiny] {};
        \path[bb-thin] (e) to (rrf);
        \path[bb-thin] (r) to (f);
        \path[bb-thin] (rr) to (rf);
        \path[gg-thin] (e) to (f);
        \path[gg-thin] (r) to (rf);
        \path[gg-thin] (rr) to (rrf);
        \node (o-in_rf) at (195:4.8) {};
        \node (o-out_rf) at (30:4.8) {};
        \node (p_rf) at (120:4.8) {};
      \end{scope}
      %%
      \begin{scope}[shift={(240:1.25)},scale=.12]
        \draw[grey, fill=lightgrey] (0,0) circle (60mm);
        \node (e) at (90:5.5) [v-tiny] {};
        \node (rrf) at (30:5.5) [v-tiny] {};
        \node (rr) at (-30:5.5) [v-tiny] {};
        \node (rf) at (-90:5.5) [v-tiny] {};
        \node (r) at (-150:5.5) [v-tiny] {};
        \node (f) at (150:5.5) [v-tiny] {};
        \path[bb-thin] (e) to (rf);
        \path[bb-thin] (r) to (rrf);
        \path[bb-thin] (rr) to (f);
        \path[gg-thin] (e) to (rrf);
        \path[gg-thin] (r) to (f);
        \path[gg-thin] (rr) to (rf);
        \node (o-in_r2f) at (330:4.8) {};
        \node (o-out_r2f) at (165:4.8) {};
        \node (p_r2f) at (240:4.8) {};
      \end{scope}
      %%
      %%----------------outer ring of nodes ----------------------------
      %%
      \begin{scope}[shift={(0:3.5)},scale=.12]
        \draw[grey, fill=lightgrey] (0,0) circle (60mm);
        \node at (0,0) {$\Id$};
        \node (e) at (90:5.5) [v-tiny] {};
        \node (rrf) at (30:5.5) [v-tiny] {};
        \node (rr) at (-30:5.5) [v-tiny] {};
        \node (rf) at (-90:5.5) [v-tiny] {};
        \node (r) at (-150:5.5) [v-tiny] {};
        \node (f) at (150:5.5) [v-tiny] {};
        \path[gg-thin] (e) to (rrf);
        \path[bb-thin] (rrf) to (rr);
        \path[gg-thin] (rr) to (rf);
        \path[bb-thin] (rf) to (r);
        \path[gg-thin] (r) to (f);
        \path[bb-thin] (f) to (e);
        \node (o-in_1) at (270:4.8) {};
        \node (o-out_1) at (90:4.8) {};
        \node (p_1) at (180:4.8) {};
      \end{scope}
      %%
      \begin{scope}[shift={(120:3.5)},scale=.12]
        \draw[grey, fill=lightgrey] (0,0) circle (60mm);
        \node (e) at (90:5.5) [v-tiny] {};
        \node (rrf) at (30:5.5) [v-tiny] {};
        \node (rr) at (-30:5.5) [v-tiny] {};
        \node (rf) at (-90:5.5) [v-tiny] {};
        \node (r) at (-150:5.5) [v-tiny] {};
        \node (f) at (150:5.5) [v-tiny] {};
        \path[gg-thin] (e) to (f);
        \path[gg-thin] (r) to (rf);
        \path[gg-thin] (rr) to (rrf);
        \path[bb-thin] (e) to (rf);
        \path[bb-thin] (r) to (rrf);
        \path[bb-thin] (rr) to (f);
        \node (o-in_r) at (30:4.8) {};
        \node (o-out_r) at (195:4.8) {};
        \node (p_r) at (300:4.8) {};
      \end{scope}
      %%
      \begin{scope}[shift={(240:3.5)},scale=.12]
        \draw[grey, fill=lightgrey] (0,0) circle (60mm);
        \node (e) at (90:5.5) [v-tiny] {};
        \node (rrf) at (30:5.5) [v-tiny] {};
        \node (rr) at (-30:5.5) [v-tiny] {};
        \node (rf) at (-90:5.5) [v-tiny] {};
        \node (r) at (-150:5.5) [v-tiny] {};
        \node (f) at (150:5.5) [v-tiny] {};
        \path[gg-thin] (e) to (rf);
        \path[gg-thin] (r) to (rrf);
        \path[gg-thin] (rr) to (f);
        \path[bb-thin] (e) to (rrf);
        \path[bb-thin] (r) to (f);
        \path[bb-thin] (rr) to (rf);
        \node (o-in_r2) at (165:4.8) {};
        \node (o-out_r2) at (330:4.8) {};
        \node (p_r2) at (60:4.8) {};
      \end{scope}
      %%
      \draw [o] (o-out_1) to[bend right=45] (o-in_r);
      \draw [o] (o-out_r) to[bend right=45] (o-in_r2);
      \draw [o] (o-out_r2) to[bend right=45] (o-in_1);
      \draw [o] (o-out_f) to[bend left=30] (o-in_r2f);
      \draw [o] (o-out_r2f) to[bend left=35] (o-in_rf);
      \draw [o] (o-out_rf) to[bend left=30] (o-in_f);
      \draw [p] (p_1) to (p_f);
      \draw [p] (p_r) to (p_rf);
      \draw [p] (p_r2) to (p_r2f);
    \end{scope}
  \end{tikzpicture}
  \]
  Construct two Cayley graphs for $\Aut(D_6)$ labeled with rewirings,
  using one or both of the following Cayley graphs. (That is, there
  are four possibilities, pick two of them to do.)


  %%
  %%------------------ Two Cayley graphs of Aut(D_6) ---------------
  %%
  \tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=4.5mm]
  \tikzstyle{o-out} = [draw, very thick, orange,-stealth,bend right=20]
  \tikzstyle{o-in} = [draw, very thick, orange,-stealth,bend left=16]
  \tikzstyle{rr-faded} = [draw, very thick, lightred]
  \tikzstyle{bb-faded} = [draw, very thick, lightblue]
  \tikzstyle{gg-faded} = [draw, very thick, lightgreen]
  \tikzstyle{yy-faded} = [draw, very thick, Gold!30!white]
  \tikzstyle{pp-faded} = [draw, very thick, amethyst!30!white]
  \tikzstyle{tt-faded} = [draw, very thick, turq!30!white]
  \[
  \hspace*{-15mm}
  \begin{tikzpicture}[scale=1.25]
    \begin{scope}[shift={(0,0)}]
      \node (1) at (0:2) [v-small] {\scriptsize Id};
      \node (r) at (60:2) [v-small] {};
      \node (r2) at (120:2) [v-small] {};
      \node (r3) at (180:2) [v-small] {};
      \node (r4) at (240:2) [v-small] {};
      \node (r5) at (300:2) [v-small] {};
      \node (f) at (0:1) [v-small] {};
      \node (rf) at (60:1) [v-small] {};
      \node (r2f) at (120:1) [v-small] {};
      \node (r3f) at (180:1) [v-small] {};
      \node (r4f) at (240:1) [v-small] {};
      \node (r5f) at (300:1) [v-small] {};
      \draw [o-out] (1) to (r);
      \draw [o-out] (r) to (r2);
      \draw [o-out] (r2) to (r3);
      \draw [o-out] (r3) to (r4);
      \draw [o-out] (r4) to (r5);
      \draw [o-out] (r5) to (1);
      \draw [o-in] (f) to (r5f);
      \draw [o-in] (r5f) to (r4f);
      \draw [o-in] (r4f) to (r3f);
      \draw [o-in] (r3f) to (r2f);
      \draw [o-in] (r2f) to (rf);
      \draw [o-in] (rf) to (f);
      \draw [pp] (1) to (f);
      \draw [pp] (r) to (rf);
      \draw [pp] (r2) to (r2f);
      \draw [pp] (r3) to (r3f);
      \draw [pp] (r4) to (r4f);
      \draw [pp] (r5) to (r5f);
      \node at (0,0) {$\Aut(D_6)$};
    \end{scope}
    %%
    \begin{scope}[shift={(4.25,-1)},scale=.8]
      \tikzstyle{every node}=[font=\footnotesize]
      \node (10) at (0,2.5) [v-small] {\scriptsize $\Id$};
      \node (11) at (1.5,2.5) [v-small] {};
      \node (12) at (3,2.5) [v-small] {};
      \node (13) at (4.5,2.5) [v-small] {};
      \node (14) at (6,2.5) [v-small] {};
      \node (15) at (7.5,2.5) [v-small] {};
      \node (00) at (0,0) [v-small] {};
      \node (01) at (1.5,0) [v-small] {};
      \node (02) at (3,0) [v-small] {};
      \node (03) at (4.5,0) [v-small] {};
      \node (04) at (6,0) [v-small] {};
      \node (05) at (7.5,0) [v-small] {};
      \draw [tt] (10) to (11); 
      \draw [pp] (11) to (12);
      \draw [tt] (12) to (13);
      \draw [pp] (13) to (14);
      \draw [tt] (14) to (15);
      \draw [pp] (15) to [bend right=20] (10);
      \draw [pp] (00) to (01);
      \draw [tt] (01) to (02);
      \draw [pp] (02) to (03);
      \draw [tt] (03) to (04);
      \draw [pp] (04) to (05);
      \draw [tt] (05) to [bend left=20] (00);
      \draw [yy] (00) to (10);
      \draw [yy] (01) to (11);
      \draw [yy] (02) to (12);
      \draw [yy] (03) to (13);
      \draw [yy] (04) to (14);
      \draw [yy] (05) to (15);
    \end{scope}
  \end{tikzpicture}
  \]
  
  
  %%----------------------------- PART (e) ----------------------------
  
\item Half of the automorphisms of $D_6$ are \emph{inner},
  which means they have the form $\varphi_x\colon g\mapsto x^{-1}gx$
  for some $x\in D_6$. Let $\omega\in\Aut(D_6)$ be the outer automorphism
  \[
  \omega\colon D_6\longto D_6,\qquad \omega(r)=r,\quad \omega(f)=rf
  \]
  of order $6$ that cyclically rotates axes of reflections of the
  square. The automorphism group is the union of cosets
  \[
  \Aut(D_6)=\Inn(D_6)\cup\Inn(D_6)\omega\cong\Inn(D_6)\rtimes\Out(D_6).
  \]
  This partition, but for $D_4$, is shown below. Construct an
  analogous diagram for $D_6$.
  
  %%
  %%------------------ All 12 automorphisms of D_6 ---------------
  %%
  \tikzstyle{gr} = [draw, very thick, gray, -stealth]
  \[
  \hspace*{-12mm}
  \scalebox{.68}{
  \begin{tikzpicture}[scale=1.75]
    \tikzstyle{every node}=[font=\large,anchor=south]
    \node at (1.5,9.65) {\LARGE $\Inn(D_4)=\<\varphi_r,\varphi_f\>$};
    \node at (7,9.65) {\LARGE $\Inn(D_4)\omega$};
    \begin{scope}[shift={(0,7.5)}]
      \node at (-1.5,.8) {\LARGE $\Id=\varphi_1$};
      \draw[very thick,lightgray] (-.5,0) rectangle (3.5,2);
      \draw[very thick, lightgray] (.5,0) to (.5,2);
      \draw[very thick, lightgray] (1.5,0) to (1.5,2);
      \draw[very thick, lightgray] (-.5,1) to (.5,1);
      \draw[very thick, lightgray] (1.5,1) to (3.5,1);
      \node (1) at (0,1.13) {$1$};
      \node (r2) at (0,.08) {$r^2$};
      \node (r) at (1,1.15) {$r$};
      \node (r3) at (1,.09) {$r^3$};
      \node (f) at (2,1.1) {$f$};
      \node (r2f) at (3,1.09) {$r^2f$};
      \node (rf) at (2,.1) {$rf$};
      \node (r3f) at (3,.09) {$r^3f$};
      \path (1) edge [gr,loop above,>=stealth] (1);
      \path (r) edge [gr,loop above,>=stealth] (r);
      \path (r2) edge [gr,loop above,>=stealth] (r2);
      \path (r3) edge [gr,loop above,>=stealth] (r3);
      \path (f) edge [gr,loop above,>=stealth] (f);
      \path (rf) edge [gr,loop above,>=stealth] (rf);
      \path (r2f) edge [gr,loop above,>=stealth] (r2f);
      \path (r3f) edge [gr,loop above,>=stealth] (r3f);
    \end{scope}
    %%
    \begin{scope}[shift={(0,5)}]
      \node at (-1.2,.8) {\LARGE \Alert{$\varphi_r$}};
      \draw[very thick,lightgray] (-.5,0) rectangle (3.5,2);
      \draw[very thick, lightgray] (.5,0) to (.5,2);
      \draw[very thick, lightgray] (1.5,0) to (1.5,2);
      \draw[very thick, lightgray] (-.5,1) to (.5,1);
      \draw[very thick, lightgray] (1.5,1) to (3.5,1);
      \node (1) at (0,1.13) {$1$};
      \node (r2) at (0,.08) {$r^2$};
      \node (r) at (1,1.15) {$r$};
      \node (r3) at (1,.09) {$r^3$};
      \node (f) at (2,1.2) {$f$};
      \node (r2f) at (3,1.19) {$r^2f$};
      \node (rf) at (2,.1) {$rf$};
      \node (r3f) at (3,.09) {$r^3f$};
      \draw [rr] (f) to (r2f);
      \draw [rr] (rf) to (r3f);
      \path (1) edge [r,loop above,>=stealth] (1);
      \path (r) edge [r,loop above,>=stealth] (r);
      \path (r2) edge [r,loop above,>=stealth] (r2);
      \path (r3) edge [r,loop above,>=stealth] (r3);
    \end{scope}
    %%
    \begin{scope}[shift={(0,2.5)}]
      \node at (-1.2,.8) {\LARGE \Balert{$\varphi_f$}};
      \draw[very thick,lightgray] (-.5,0) rectangle (3.5,2);
      \draw[very thick, lightgray] (.5,0) to (.5,2);
      \draw[very thick, lightgray] (1.5,0) to (1.5,2);
      \draw[very thick, lightgray] (-.5,1) to (.5,1);
      \draw[very thick, lightgray] (1.5,1) to (3.5,1);
      \node (1) at (0,1.13) {$1$};
      \node (r2) at (0,.08) {$r^2$};
      \node (r) at (1,1.15) {$r$};
      \node (r3) at (1,.09) {$r^3$};
      \node (f) at (2,1.1) {$f$};
      \node (r2f) at (3,1.09) {$r^2f$};
      \node (rf) at (2,.1) {$rf$};
      \node (r3f) at (3,.09) {$r^3f$};
      \draw [bb] (r) to (r3);
      \draw [bb] (rf) to (r3f);
      \path (1) edge [b,loop above,>=stealth] (1);
      \path (f) edge [b,loop above,>=stealth] (f);
      \path (r2) edge [b,loop above,>=stealth] (r2);
      \path (r2f) edge [b,loop above,>=stealth] (r2f);
    \end{scope}
    %%
    \begin{scope}[shift={(0,0)}]
      \node at (-1.2,.8) {\LARGE \Palert{$\varphi_{rf}$}};
      \draw[very thick,lightgray] (-.5,0) rectangle (3.5,2);
      \draw[very thick, lightgray] (.5,0) to (.5,2);
      \draw[very thick, lightgray] (1.5,0) to (1.5,2);
      \draw[very thick, lightgray] (-.5,1) to (.5,1);
      \draw[very thick, lightgray] (1.5,1) to (3.5,1);
      \node (1) at (0,1.13) {$1$};
      \node (r2) at (0,.08) {$r^2$};
      \node (r) at (1,1.15) {$r$};
      \node (r3) at (1,.09) {$r^3$};
      \node (f) at (2,1.3) {$f$};
      \node (r2f) at (3,1.29) {$r^2f$};
      \node (rf) at (2,.1) {$rf$};
      \node (r3f) at (3,.09) {$r^3f$};
      \draw [pp] (r) to (r3);
      \draw [pp] (f) to (r2f);
      \path (1) edge [p,loop above,>=stealth] (1);
      \path (rf) edge [p,loop above,>=stealth] (rf);
      \path (r2) edge [p,loop above,>=stealth] (r2);
      \path (r3f) edge [p,loop above,>=stealth] (r3f);
    \end{scope}
    %%
    \begin{scope}[shift={(5.5,7.5)}]
      \node at (4,.8) {\LARGE \Galert{$\omega$}};
      \draw[very thick,lightgray] (-.5,0) rectangle (3.5,2);
      \draw[very thick, lightgray] (.5,0) to (.5,2);
      \draw[very thick, lightgray] (1.5,0) to (1.5,2);
      \draw[very thick, lightgray] (-.5,1) to (.5,1);
      \draw[very thick, lightgray] (1.5,1) to (3.5,1);
      \node (1) at (0,1.13) {$1$};
      \node (r2) at (0,.08) {$r^2$};
      \node (r) at (1,1.15) {$r$};
      \node (r3) at (1,.09) {$r^3$};
      \node (f) at (2,1.3) {$f$};
      \node (r2f) at (3,1.29) {$r^2f$};
      \node (rf) at (2,.1) {$rf$};
      \node (r3f) at (3,.09) {$r^3f$};
      \draw [g] (f) to (rf); \draw [g] (rf) to (r2f);
      \draw [g] (r2f) to (r3f); \draw [g] (r3f) to (f);
      \path (1) edge [g,loop above,>=stealth] (1);
      \path (r) edge [g,loop above,>=stealth] (r);
      \path (r2) edge [g,loop above,>=stealth] (r2);
      \path (r3) edge [g,loop above,>=stealth] (r3);
    \end{scope}
    %%
    \begin{scope}[shift={(5.5,5)}]
      \node at (4.2,.8) {\LARGE \Galert{$\varphi_r\omega$}};
      \draw[very thick,lightgray] (-.5,0) rectangle (3.5,2);
      \draw[very thick, lightgray] (.5,0) to (.5,2);
      \draw[very thick, lightgray] (1.5,0) to (1.5,2);
      \draw[very thick, lightgray] (-.5,1) to (.5,1);
      \draw[very thick, lightgray] (1.5,1) to (3.5,1);
      \node (1) at (0,1.13) {$1$};
      \node (r2) at (0,.08) {$r^2$};
      \node (r) at (1,1.15) {$r$};
      \node (r3) at (1,.09) {$r^3$};
      \node (f) at (2,1.3) {$f$};
      \node (r2f) at (3,1.29) {$r^2f$};
      \node (rf) at (2,.1) {$rf$};
      \node (r3f) at (3,.09) {$r^3f$};
      \draw [g] (f) to (r3f); \draw [g] (r3f) to (r2f);
      \draw [g] (r2f) to (rf); \draw [g] (rf) to (f);
      \path (1) edge [g,loop above,>=stealth] (1);
      \path (r) edge [g,loop above,>=stealth] (r);
      \path (r2) edge [g,loop above,>=stealth] (r2);
      \path (r3) edge [g,loop above,>=stealth] (r3);
    \end{scope}
    %%
    \begin{scope}[shift={(5.5,2.5)}]
      \node at (4.2,.8) {\LARGE \Galert{$\varphi_f\omega$}};
      \draw[very thick,lightgray] (-.5,0) rectangle (3.5,2);
      \draw[very thick, lightgray] (.5,0) to (.5,2);
      \draw[very thick, lightgray] (1.5,0) to (1.5,2);
      \draw[very thick, lightgray] (-.5,1) to (.5,1);
      \draw[very thick, lightgray] (1.5,1) to (3.5,1);
      \node (1) at (0,1.13) {$1$};
      \node (r2) at (0,.08) {$r^2$};
      \node (r) at (1,1.25) {$r$};
      \node (r3) at (1,.09) {$r^3$};
      \node (f) at (2,1.3) {$f$};
      \node (r2f) at (3,1.29) {$r^2f$};
      \node (rf) at (2,.1) {$rf$};
      \node (r3f) at (3,.09) {$r^3f$};
      \draw [gg] (r) to (r3);
      \draw [gg] (f) to (rf);
      \draw [gg] (r2f) to (r3f);
      \path (1) edge [g,loop above,>=stealth] (1);
      \path (r2) edge [g,loop above,>=stealth] (r2);
    \end{scope}
    %%
    \begin{scope}[shift={(5.5,0)}]
      \node at (4.2,.8) {\LARGE \Galert{$\varphi_{rf}\omega$}};
      \draw[very thick,lightgray] (-.5,0) rectangle (3.5,2);
      \draw[very thick, lightgray] (.5,0) to (.5,2);
      \draw[very thick, lightgray] (1.5,0) to (1.5,2);
      \draw[very thick, lightgray] (-.5,1) to (.5,1);
      \draw[very thick, lightgray] (1.5,1) to (3.5,1);
      \node (1) at (0,1.13) {$1$};
      \node (r2) at (0,.08) {$r^2$};
      \node (r) at (1,1.25) {$r$};
      \node (r3) at (1,.09) {$r^3$};
      \node (f) at (2,1.3) {$f$};
      \node (r2f) at (3,1.29) {$r^2f$};
      \node (rf) at (2,.1) {$rf$};
      \node (r3f) at (3,.09) {$r^3f$};
      \draw [gg] (r) to (r3);
      \draw [gg] (f) to (r3f);
      \draw [gg] (rf) to (r2f);
      \path (1) edge [g,loop above,>=stealth] (1);
      \path (r2) edge [g,loop above,>=stealth] (r2);
    \end{scope}
  \end{tikzpicture}}
  \]

  %%--------------------------PARTS (f-j) -----------------------------
  
\item Find an outer automorphism $\eta\in\Aut(D_6)$ such that
  $\Aut(D_6)\cong\Inn(D_6)\rtimes\<\eta\>$, and then construct the
  subgroup lattice of $\Aut(D_6)$.
      
\item Go back to the Cayley graphs for $\Aut(D_6)$ constructed in
  Part~(d), and label the large nodes with the corresponding
  automorphisms.

\item Construct the action graph and fixed point table of the action
  of $\Aut(D_6)$ on $D_6$. Find $\Ker(\phi)$ and $\Fix(\phi)$.
  
\item Construct the action graph and fixed point table of the action
  of $\Aut(D_6)$ on the conjugacy classes of $D_6$. Find $\Ker(\phi)$
  and $\Fix(\phi)$.

\item Revisit the action graphs of $D_6$ that you constructed via the
  Galois correspondence. Have any of them not appeared in this problem
  thus far as an orbit? If so, define an action of $D_6$ that has it
  as its action graph.
  
  \end{enumerate}

  
  %%========================= PROBLEM 2 ================================

  
\item Let $N,K\leq G$. Show that if the following conditions hold:
  \begin{multicols}{3}
    \begin{enumerate}
    \item $N$ is normal in $G$
    \item $K\cap N=\{e\}$
    \item $G=KN$.
    \end{enumerate}
  \end{multicols}
  then $G\cong K\ltimes_\theta N$. Define the specific homomorphism
  $\theta\colon K\to\Aut(N)$ and show why this works.
  
  
  %%========================= PROBLEM 3 ================================
  
  
\item Show that $\Inn(G)$ is a subgroup of $\Aut(G)$, that it is
  normal, and that it is isomorphic to $G/Z(G)$. \bigskip
  
  
  %%========================= PROBLEM 4 ================================
  
  
\item Suppose a group $G$ of order $55$ acts on a set $S$ of size
  $14$. Let $s\in S$ be an arbitrary element.
  \begin{enumerate}
  \item What are the possible sizes of the orbit of $s$?
  \item What are the possible sizes of the stabilizer of $s$?
  \item Show that this action must have a fixed point.
  \item What is the fewest number of fixed points that this action can
    have? Justify your answer. \bigskip
  \end{enumerate}


  %%========================= PROBLEM 5 ================================
  
  
\item Let $\phi\colon G\to\Perm(S)$ be a group action.
  \begin{enumerate}
  \item Prove the orbit-stabilizer theorem for left actions by
    constructing a bijection between $\orb(s)$ and left cosets of
    $\stab(s)$. Use the same notational conventions from lecture. 
  \item We saw that elements in the same orbit have conjugate
    stabilizers. Explicitly, for right actions, this is
    \[
    \stab(s\Dot\phi(g))=g^{-1}\stab(s)g,\quad\text{for all $g\in G$ and $s\in
      S$},
    \]
    and is summarized by the following commutative diagram:
    \[
    \begin{tikzcd}[column sep=5em,row sep=3.5em]
      s \arrow[r,mapsto,"\phi(x)"] \arrow[d,mapsto,swap,"\phi(g)"]
      & s \arrow[d,mapsto,"\phi(g)"] \\   
      s' \arrow[r,mapsto,dashed,"\phi(g^{-1}xg)"] & s'
    \end{tikzcd}
    \]
    Formulate and prove an analogous statement for left actions.
  \end{enumerate}
    
\end{enumerate}

\end{document}








