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\pagestyle{fancy} \lhead{{\sf Homework 11 $|$ Due April 4 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Visual Algebra $|$
    Spring 2025 $|$ M.~Macauley}}

\begin{document}

%$\;$

\begin{enumerate}

  %%-------------------------------------------------------------------
  
\item Carry out the following steps for the groups $C_7\rtimes C_3$
  and $C_9\rtimes C_3$, whose Cayley graphs are shown below.
  \[
  \hspace*{-12mm}
  \begin{tikzpicture}[scale=1.25]
    \tikzstyle{R7} = [draw, very thick, eRed,-stealth,bend right=15]
    \tikzstyle{R9} = [draw, very thick, eRed,-stealth,bend right=10]
    \tikzstyle{R2} = [draw, very thick, eRed,-stealth,bend right=20]
    \tikzstyle{R3} = [draw, very thick, eRed,-stealth,bend left=8]
    \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt, minimum size=5.5mm]
    \tikzstyle{every node}=[font=\scriptsize]
    %%
    \begin{scope}[shift={(0,0)},scale=1]
      \node (s2) at (0:1) [v] {$s^2$};
      \node (rs2) at (51.45:1) [v] {$rs^2$};
      \node (r2s2) at (102.9:1) [v] {$r^2\!s^2$};
      \node (r3s2) at (154.3:1) [v] {$r^3\!s^2$};
      \node (r4s2) at (205.7:1) [v] {$r^4\!s^2$};
      \node (r5s2) at (257.1:1) [v] {$r^5\!s^2$};
      \node (r6s2) at (308.6:1) [v] {$r^6\!s^2$};
      %%
      \node (s) at (0:2) [v] {$s$};
      \node (rs) at (51.45:2) [v] {$rs$};
      \node (r2s) at (102.9:2) [v] {$r^2\!s$};
      \node (r3s) at (154.3:2) [v] {$r^3\!s$};
      \node (r4s) at (205.7:2) [v] {$r^4\!s$};
      \node (r5s) at (257.1:2) [v] {$r^5\!s$};
      \node (r6s) at (308.6:2) [v] {$r^6\!s$};
      %%
      \node (1) at (0:3) [v] {$1$};
      \node (r) at (51.45:3) [v] {$r$};
      \node (r2) at (102.9:3) [v] {$r^2$};
      \node (r3) at (154.3:3) [v] {$r^3$};
      \node (r4) at (205.7:3) [v] {$r^4$};
      \node (r5) at (257.1:3) [v] {$r^5$};
      \node (r6) at (308.6:3) [v] {$r^6$};
      %%
      \draw [R7] (1) to (r);
      \draw [R7] (r) to (r2);
      \draw [R7] (r2) to (r3);
      \draw [R7] (r3) to (r4);
      \draw [R7] (r4) to (r5);
      \draw [R7] (r5) to (r6);
      \draw [R7] (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}
    %%
    \begin{scope}[shift={(7,0)}]
      \node (1) at (0:3) [v] {$1$};
      \node (r) at (40:3) [v] {$r$};
      \node (r2) at (80:3) [v] {$r^2$};
      \node (r3) at (120:3) [v] {$r^3$};
      \node (r4) at (160:3) [v] {$r^4$};
      \node (r5) at (200:3) [v] {$r^5$};
      \node (r6) at (240:3) [v] {$r^6$};
      \node (r7) at (280:3) [v] {$r^7$};
      \node (r8) at (320:3) [v] {$r^8$};
      %%
      \node (s) at (0:2) [v] {$s$};
      \node (rs) at (40:2) [v] {$rs$};
      \node (r2s) at (80:2) [v] {$r^2\!s$};
      \node (r3s) at (120:2) [v] {$r^3\!s$};
      \node (r4s) at (160:2) [v] {$r^4\!s$};
      \node (r5s) at (200:2) [v] {$r^5\!s$};
      \node (r6s) at (240:2) [v] {$r^6\!s$};
      \node (r7s) at (280:2) [v] {$r^7\!s$};
      \node (r8s) at (320:2) [v] {$r^8\!s$};
      %%
      \node (s2) at (0:1) [v] {$s^2$};
      \node (rs2) at (40:1) [v] {$rs^2$};
      \node (r2s2) at (80:1) [v] {$r^2\!s^2$};
      \node (r3s2) at (120:1) [v] {$r^3\!s^2$};
      \node (r4s2) at (160:1) [v] {$r^4\!s^2$};
      \node (r5s2) at (200:1) [v] {$r^5\!s^2$};
      \node (r6s2) at (240:1) [v] {$r^6\!s^2$};
      \node (r7s2) at (280:1) [v] {$r^7\!s^2$};
      \node (r8s2) at (320:1) [v] {$r^8\!s^2$};
      %%
      \draw [R9] (1) to (r);
      \draw [R9] (r) to (r2);
      \draw [R9] (r2) to (r3);
      \draw [R9] (r3) to (r4);
      \draw [R9] (r4) to (r5);
      \draw [R9] (r5) to (r6);
      \draw [R9] (r6) to (r7);
      \draw [R9] (r7) to (r8);
      \draw [R9] (r8) to (1);
      %%
      \draw [R3] (s) to (r7s);
      \draw [R3] (r7s) to (r5s);
      \draw [R3] (r5s) to (r3s);
      \draw [R3] (r3s) to (rs);
      \draw [R3] (rs) to (r8s);
      \draw [R3] (r8s) to (r6s);
      \draw [R3] (r6s) to (r4s);
      \draw [R3] (r4s) to (r2s);
      \draw [R3] (r2s) to (s);
      %%
      \draw [r] (s2) to (r4s2);
      \draw [r] (r4s2) to (r8s2);
      \draw [r] (r8s2) to (r3s2);
      \draw [r] (r3s2) to (r7s2);
      \draw [r] (r7s2) to (r2s2);
      \draw [r] (r2s2) to (r6s2);
      \draw [r] (r6s2) to (rs2);
      \draw [r] (rs2) to (r5s2);
      \draw [r] (r5s2) 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);
      \draw [b] (r7) to (r7s); \draw [b] (r7s) to (r7s2);
      \draw [b,bend left] (r7s2) to (r7);
      \draw [b] (r8) to (r8s); \draw [b] (r8s) to (r8s2);
      \draw [b,bend left] (r8s2) to (r8);
    \end{scope}
  \end{tikzpicture}
  \]
  \begin{enumerate}
  \item Let $G$ act on its subgroups by conjugation. Draw the action
    graph superimposed on the subgroup lattice. Find $\stab(H)$ for
    each $H\leq G$, $\Ker(\phi)$, and $\Fix(\phi)$. \bigskip
    %Which collections of subgroups arise as $\fix(g)$ for some $g\in G$, and why?
  \item Let $G$ act on the right cosets of $H=\<s\>$, via the homomorphism
    \[
    \phi\colon G\longto\Perm(S)\,,\qquad \phi(g)=\mbox{\footnotesize the
      permutation that sends each $Hx\mapsto Hxg$.}
    \]
    Construct the action graph. Find
    $\stab(Hx)$ for each right coset, $\Ker(\phi)$, and $\Fix(\phi)$.\
  \end{enumerate}

  %%---------------------------------------------------------------
  
\item Loosely speaking, the upcoming Sylow theorems will us that (1) all $p$-subgroups come in a single ``$p$-subgroup tower'', (2) the ``top'' of these towers are a single conjugacy class, and (3) the size of this class is $1$ mod $p$. This is illustrated below with the groups of order $12$. 
  \[
  \hspace*{-9mm}
  \begin{tikzpicture}[scale=.85,auto,xscale=.91]
    \newcommand\Hf{6.495} % height of C_12 
    \newcommand\He{5.196} % height of C_6  
    \newcommand\Hd{3.897} % height of C_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{every node}=[font=\footnotesize]
    %%
    \begin{scope}[shift={(-.1,7)},scale=1]
      \node(G) at (0,\Hf) {$C_{12}$};
      \draw[cosetBlue,fill=cosetBlue] (1,\Hd) circle [x radius=.55cm, y radius=.4cm];
      \draw[cosetRed,fill=cosetRed] (-1,\Hc) circle [x radius=.55cm, y radius=.4cm];
       \draw[cosetRed,fill=cosetRed] (.55,0)--(-.55,0)--(-1.55,\Hc)--(-.45,\Hc)--
     (.55,0);
     \draw[cosetBlue,fill=cosetBlue] (.55,0)--(-.55,0)--(.45,\Hd)--(1.55,\Hd)--
     (.55,0);
       \draw[cosetRed,fill=cosetRed,opacity=.6] (.55,0)--(-.55,0)--(-1.55,\Hc)--(-.45,\Hc)--
     (.55,0);
       \draw[cosetPurple,fill=cosetPurple] (0,0) circle [x radius=.55cm, y radius=.4cm];
        \node(C6) at (-.33,\He) {$C_6$};
        \node(C4) at (1,\Hd) {\Balert{$C_4$}};
        \draw[dashed] (1,\Hd) circle [x radius=.5cm, y radius=.35cm];
        \node(C3) at (-1,\Hc) {\Alert{$C_3$}};
        \draw[dashed] (-1,\Hc) circle [x radius=.5cm, y radius=.35cm];
        \node(C2) at (.33,\Hb) {$C_2$};
        \node(C1) at (0,0) {$C_1$};
        \draw (G) to (C6); \draw (G) to (C4); 
        \draw (C6) to (C3); \draw (C6) to (C2); 
        \draw (C4) to (C2); 
        \draw (C3) to (C1); \draw (C2) to (C1);  
      \end{scope}
      %%
      \begin{scope}[shift={(3.5,7)},scale=1]
        \node(G) at (0,\Hf) {$C_6\times C_2$};
        \draw[cosetBlue,fill=cosetBlue] (1,\Hd) circle [x radius=.55cm, y radius=.4cm];
        \draw[cosetRed,fill=cosetRed] (-1,\Hc) circle [x radius=.55cm, y radius=.4cm];
        \draw[cosetRed,fill=cosetRed] (.55,0)--(-.55,0)--(-1.55,\Hc)--(-.45,\Hc)--
     (.55,0);
     \draw[cosetBlue,fill=cosetBlue,rounded corners] (.55,0)--(-.55,0)--(0,\Hb)--(.45,\Hd)--(1.55,\Hd)--(2.1,\Hb)--
     (.55,0);
     \draw[cosetRed,fill=cosetRed,opacity=.6] (.55,0)--(-.55,0)--(-1.55,\Hc)--(-.45,\Hc)--
     (.55,0);
        \draw[cosetPurple,fill=cosetPurple] (0,0) circle [x radius=.55cm, y radius=.4cm];
        \node(12) at (-.25,\He) {$C_6$};
        \node(11) at (-1,\He) {$C_6$};
        \node(01) at (-1.75,\He) {$C_6$};
        \node(10-03) at (1,\Hd) {\Balert{$V_4$}};
        \draw[dashed] (1,\Hd) circle [x radius=.5cm, y radius=.35cm];
        \node(02) at (-1,\Hc) {\Alert{$C_3$}};
        \draw[dashed] (-1,\Hc) circle [x radius=.5cm, y radius=.35cm];
        \node(10) at (1.75,\Hb) {$C_2$};
        \node(13) at (1,\Hb) {$C_2$};
        \node(03) at (.25,\Hb) {$C_2$};
        \node(00) at (0,0) {$C_1$};
        \draw (G) to (12); \draw (G) to (11); 
        \draw (G) to (01);
        \draw (02) to (12); \draw (02) to (11); 
        \draw (02) to (01);
        \draw (10-03) to (G);
        \draw (12) to (10); 
        \draw (11) to (13); 
        \draw (01) to (03);
        \draw (00) to (02);
        \draw (10-03) to (10); \draw (10-03) to (13); 
        \draw (10-03) to (03); 
        \draw (00) to (10); \draw (00) to (13); 
        \draw (00) to (03); 
      \end{scope}
      %%
    \begin{scope}[shift={(7.5,7)}]
      \node(D6) at (0,\Hf) {$D_6$};
      \draw[cosetBlue,fill=cosetBlue] (3,\Hd) circle [x radius=1.15cm, y radius=.4cm];
      \draw[cosetRed,fill=cosetRed] (-1.25,\Hc) circle [x radius=.55cm, y radius=.4cm];
      \draw[cosetRed,fill=cosetRed] (.55,0)--(-.55,0)--(-1.8,\Hc)--(-.7,\Hc)--
     (.55,0);
     \draw[cosetBlue,fill=cosetBlue,rounded corners] (.55,0)--(-.55,0)--(-.25,\Hb)--(1.85,\Hd)--(4.15,\Hd)--(3.9,1.2)--(.55,0);
     \draw[cosetRed,fill=cosetRed,opacity=.6] (.55,0)--(-.55,0)--(-1.8,\Hc)--(-.7,\Hc)--
     (.55,0);
      \draw[cosetPurple,fill=cosetPurple] (0,0) circle [x radius=.55cm, y radius=.4cm];
      \node(r2-f) at (-.25,\He) {$D_3$};
      \node(r2-rf) at (-1,\He) {$D_3$}; 
      \node(r) at (-1.75,\He) {$C_6$}; 
      \node(r3-f) at (2.4,\Hd) {\Balert{$V_4$}};
      \node(r3-rf) at (3,\Hd) {\Balert{$V_4$}};
      \node(r3-r2f) at (3.6,\Hd) {\Balert{$V_4$}};
      \draw[dashed] (r3-rf) circle [x radius=1.1cm, y radius=.35cm];
      \node(r2) at (-1.25,\Hc) {\Alert{$C_3$}};
      \draw[dashed] (r2) circle [x radius=.5cm, y radius=.35cm];
      \node(r3) at (0,\Hb) {$C_2$};
      \node(f) at (.6,\Hb) {$C_2$};      
      \node(r4f) at (1.2,\Hb) {$C_2$};
      \node(r2f) at (1.8,\Hb) {$C_2$};
      \node(r3f) at (2.4,\Hb) {$C_2$};
      \node(rf) at (3,\Hb) {$C_2$};
      \node(r5f) at (3.6,\Hb) {$C_2$};
      \node (1) at (0,\Ha) {$C_1$};
      \draw[dashed] (r4f) circle [x radius=.93cm, y radius=.35cm];
      \draw[dashed] (rf) circle [x radius=.93cm, y radius=.35cm];
      \draw (D6) -- (r2-f);
      \draw (D6) -- (r);
      \draw (D6) to (r2-rf);
      \draw (D6) to (r3-f); 
      \draw (D6) -- (r3-rf);
      \draw (D6) to (r3-r2f); 
      \draw (r2-rf) to (f); 
      \draw (r2-rf) -- (r2f);
      \draw (r2-rf) to (r4f);
      \draw (r2-f) to (r2);
      \draw (r) -- (r2);
      \draw (r) -- (r3);
      \draw (r2-rf) -- (r2);
      \draw (r2-f) to (rf);
      \draw (r2-f) to (r3f);
      \draw (r2-f) to (r5f);
      \draw (r3-f) -- (f);
      \draw (r3-f) to (r3);
      \draw (r3-f) to (r3f); 
      \draw (r3-rf) to (rf);
      \draw (r3-rf) to (r3);
      \draw (r3-rf) to (r4f); 
      \draw (r3-r2f) to (r2f);
      \draw (r3-r2f) to (r3);
      \draw (r3-r2f) to (r5f);
      \draw (r2) -- (1);
      \draw (f) -- (1);
      \draw (rf) to (1);
      \draw (r2f) to (1);
      \draw (r3f) to (1);
      \draw (r4f) to (1);
      \draw (r5f) to (1); 
      \draw (r3) -- (1);
    \end{scope}
    %%
    \begin{scope}[shift={(13.8,7)}]
      \node(G) at (0,\Hf) {$\Dic_6$};
     \draw[cosetBlue,fill=cosetBlue] (1,\Hd) circle [x radius=1.3cm, y radius=.4cm];
     \draw[cosetRed,fill=cosetRed] (-1.25,\Hc) circle [x radius=.55cm, y radius=.45cm];
     \draw[cosetRed,fill=cosetRed] (.55,0)--(-.55,0)--(-1.8,\Hc)--(-.7,\Hc)--
     (.55,0);
     \draw[cosetBlue,fill=cosetBlue,rounded corners] (.55,0)--(-.55,0)--(-.4,\Hb)--(-.35,\Hd)--(2.3,\Hd)--(.55,\Hb)--(.55,0);
     \draw[cosetRed,fill=cosetRed,opacity=.6] (.55,0)--(-.55,0)--(-1.8,\Hc)--(-.7,\Hc)--
     (.55,0);
     \draw[cosetPurple,fill=cosetPurple] (0,0) circle [x radius=.55cm, y radius=.4cm];
      \node(b) at (-1,\He) {$C_6$};
      \node(a) at (.25,\Hd) {\Balert{$C_4$}};
      \node(ab) at (1,\Hd) {\Balert{$C_4$}};
      \node(ba) at (1.75,\Hd) {\Balert{$C_4$}};
      \draw[dashed] (1,\Hd) circle [x radius=1.25cm, y radius=.4cm];
      \node(bb) at (-1.25,\Hc) {\Alert{$C_3$}};
      \draw[dashed] (-1.25,\Hc) circle [x radius=.5cm, y radius=.35cm];
      \node(aa) at (0,\Hb) {$C_2$};
      \node(1) at (0,0) {$C_1$};
      \draw (1)--(aa); \draw (1)--(bb);
      \draw(b)--(aa); \draw (b)--(bb);
      \draw(G)--(a); \draw(G)--(ab); \draw(G)--(ba); \draw(G)--(b);
      \draw(aa)--(a); \draw(aa)--(ab); \draw(aa)--(ba);
    \end{scope}
    %%
      \begin{scope}[shift={(18.2,7)}]
        \node(A4) at (0,\Hf) {$A_4$};
        \draw[cosetBlue,fill=cosetBlue] (1,\Hd) circle [x radius=.55cm, y radius=.4cm];
        \draw[cosetRed,fill=cosetRed] (-1.15,\Hc) circle [x radius=1.35cm, y radius=.45cm];
      \draw[cosetRed,fill=cosetRed] (.55,0)--(-.55,0)--(-2.5,\Hc)--(.2,\Hc)--
     (.55,0);
      \draw[cosetBlue,fill=cosetBlue,rounded corners] (.55,0)--(-.55,0)--(0,\Hb)--(.45,\Hd)--(1.55,\Hd)--(2.1,1.25)--(.55,0);
      \draw[cosetRed,fill=cosetRed,opacity=.6] (.55,0)--(-.55,0)--(-2.5,\Hc)--(.2,\Hc)--
     (.55,0);
        \draw[cosetPurple,fill=cosetPurple] (0,0) circle [x radius=.55cm, y radius=.4cm];
        \node(V4) at (1,\Hd) {\Balert{$V_4$}};
        \node(C34) at (-.25,\Hc) {\Alert{$C_3$}};
        \node(C33) at (-.85,\Hc){\Alert{$C_3$}};
        \node(C32) at (-1.45,\Hc){\Alert{$C_3$}};
        \node(C31) at (-2.05,\Hc) {\Alert{$C_3$}};
        \node(C21) at (.25,\Hb) {$C_2$};
        \node(C22) at (1,\Hb) {$C_2$};
        \node(C23) at (1.75,\Hb) {$C_2$};
        \node(1) at (0,0) {$C_1$};
        \draw (A4)--(V4);
        \draw(A4)--(C31);
        \draw(A4)--(C32);
        \draw(A4)--(C33);
        \draw(A4)--(C34);
        \draw(C31)--(1);
        \draw(C32)--(1);
        \draw(C33)--(1);
        \draw(C34)--(1);
        \draw(V4)--(C21);
        \draw(V4)--(C22);
        \draw(V4)--(C23);
        \draw(C21)--(1);
        \draw(C22)--(1);
        \draw(C23)--(1);
        \draw[dashed] (1,\Hd) circle [x radius=.5cm, y radius=.35cm];
        \draw[dashed] (-1.15,\Hc) circle [x radius=1.3cm, y radius=.4cm];
        \draw[dashed] (C22) circle [x radius=1.15cm, y radius=.4cm];
      \end{scope}
    \end{tikzpicture}
    \]
    Using the LMFDB, construct analogous diagrams for the groups of
    order $18$ and $20$. \smallskip

  %%---------------------------------------------------------------
    
 
  
\item In this problem, we will explore the actions of the dicyclic
  group $\Dic_6$ and its automorphism group on itself and its
  subgroups by conjugation. A Cayley graph, subgroup lattice, and
  conjugacy classes are shown below.
  \[
  \hspace*{-6mm}
  \begin{tikzpicture}[scale=1.25]
    \tikzstyle{R6-out} = [draw, very thick, eRed,-stealth,bend right=18]
    \tikzstyle{R6-in} = [draw, very thick, eRed,-stealth,bend left=12]
    \tikzstyle{R-out} = [draw, very thick, eRed,-stealth,bend right=15]
    \tikzstyle{R-in} = [draw, very thick, eRed,-stealth,bend left=12]
    \tikzstyle{B} = [draw, very thick, eBlue,-stealth,bend right=25]
    %%
    \begin{scope}[shift={(0,2.1)},scale=1.05]
      \tikzstyle{every node}=[font=\small]
      \node (1) at (0:2) [v] {$1$};
      \node (r) at (60:2) [v] {$r$};
      \node (r2) at (120:2) [v] {$r^2$};
      \node (r3) at (180:2) [v] {$r^3$};
      \node (r4) at (240:2) [v] {$r^4$};
      \node (r5) at (300:2) [v] {$r^5$};
      \node (s) at (0:1) [v] {$s$};
      \node (sr5) at (60:1) [v] {$rs$};
      \node (sr4) at (120:1) [v] {$r^2\!s$};
      \node (sr3) at (180:1) [v] {$r^3\!s$};
      \node (sr2) at (240:1) [v] {$r^4\!s$};
      \node (sr) at (300:1) [v] {$r^5\!s$};
      %%
      \path[b] (1) to (s);
      \path[B] (s) to (r3);
      \path[b] (r3) to (sr3);
      \path[B] (sr3) to (1);
      %%
      \path[b] (r) to (sr5);
      \path[B] (sr5) to (r4);
      \path[b] (r4) to (sr2);
      \path[B] (sr2) to (r);
      %%
      \path[b] (r2) to (sr4);
      \path[B] (sr4) to (r5);
      \path[b] (r5) to (sr);
      \path[B] (sr) to (r2);
      \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[R-in] (sr5) to (s);
      \path[R-in] (sr4) to (sr5);
      \path[R-in] (sr3) to (sr4);
      \path[R-in] (sr2) to (sr3);
      \path[R-in] (sr) to (sr2);
      \path[R-in] (s) to (sr);
      %%
      \node at (0,0) {\small $\Dic_6$};
    \end{scope}
    %%%%
    \begin{scope}[shift={(4.5,0)},scale=.7,shorten >= -2pt, shorten <= -2pt]
      \tikzstyle{every node}=[font=\normalsize]
      \node(G) at (0,6) {$\Dic_6$};
      \node(b) at (-1.5,4.5) {$\<r\>$};
      \node(aa) at (0,1.5) {$\<r^3\>$};
      \node(bb) at (-2,2.25) {$\<r^2\>$};
      \node(a) at (.5,24/7) {$\<s\>$};
      \node(ab) at (1.75,24/7) { $\<rs\>$};
      \node(ba) at (3,24/7) {$\<r^2\!s\>$};
      \node(1) at (0,0) {$\left\<1\right\>$};
      %\draw(1)--(aa); \draw(1)--(bb);
      %\draw(b)--(aa); \draw(b)--(bb);
      %\draw(G)--(a); \draw(G)--(ab); \draw(G)--(ba); \draw(G)--(b);
      %\draw(aa)--(a); \draw(aa)--(ab); \draw(aa)--(ba);
      \draw (1) to (aa);
      \draw (1) to (bb);
      \draw (b) to (aa);
      \draw (b) to (bb);
      \draw (aa) to (a);
      \draw (aa) to (ab);
      \draw (aa) to (ba);
      \draw (G) to (a);
      \draw (G) to (ab);
      \draw (G) to (ba);
      \draw (G) to (b);   
      %\node at (0,-1) {$\Dic_{12}=\<a,b\mid a^4=b^3,\;bab=a\>$};
    \end{scope}
    \begin{scope}[shift={(7.35,.6)}]
      \tikzstyle{every node}=[font=\small]
      \draw[thin,fill=boxPurple] (0,0) rectangle (1,1.6);
      \draw[thin,fill=boxOrange] (1,0) rectangle (3,1.6);
      \draw[thin,fill=boxOrange] (0,1.6) rectangle (3,3.2);
      \draw[thin] (1,0) to (1,1.6);
      \draw[thin] (0,2.4) to (3,2.4);
      \draw[thin] (0,.8) to (3,.8);
      \node at (.5,.4) {$1$};
      \node at (1.5,.4) {$r$};
      \node at (2.5,.4) {$r^5$};
      \node at (.5,1.2) {$r^3$};
      \node at (1.5,1.2) {$r^2$};
      \node at (2.5,1.2) {$r^4$};
      \node at (.5,2) {$s$};
      \node at (1.5,2) {$r^2s$};
      \node at (2.5,2) {$r^4s$};
      \node at (.5,2.8) {$rs$};
      \node at (1.5,2.8) {$r^5s$};
      \node at (2.5,2.8) {$r^3s$};
      %\node at (1.5,-.6) {\small conjugacy classes};
    \end{scope}
    %%%
  \end{tikzpicture}
  \]

  \begin{enumerate}
  \item The right action of $\Dic_6$ on itself by conjugation is
    defined by the homomorphism
    \[
    \phi\colon\Dic_6\longto\Perm(S)\,,\qquad \phi(g)=\mbox{\footnotesize the
      permutation that sends each $x\mapsto g^{-1}xg$.}
    \]
    Draw the action graph and construct the fixed point table. Find
    $\stab(s)$ for each $s\in S$, $\fix(g)$ for each $g\in G$, as well as
    $\Ker(\phi)$ and $\Fix(\phi)$.
  \item The automorphism group of $\Dic_6$ is
    $\Aut(\Dic_6)=\<\varphi_r,\varphi_s,\omega\>$ acts on
    $\Dic_6$, where $\omega$ is the outer automorphism defined by
    \[
    \omega\colon\Dic_6\longto\Dic_6,\qquad\omega(r)=r,\quad\omega(s)=s^{-1}=r^3s,
    \]
    that ``reverses'' the blue arrows. Make a diagram showing how each
    automorphism permutes the elements of $\Dic_6$. Then construct the
    action graph, fixed point table, and find $\stab(s)$, $\fix(g)$,
    $\Ker(\phi)$ and $\Fix(\phi)$.
  \item The automorphism group
    $\Aut(\Dic_6)=\<\varphi_r,\varphi_s,\omega\>$ is isomorphic to
    $D_6$. Construct a Cayley graph and subgroup lattice using these
    generators.
  \item The group $\Aut(\Dic_6)$ also acts on the conjugacy classes of
    $\Dic_6$. Construct the action graph, fixed point table, and
    find $\stab(s)$, $\fix(g)$, $\Ker(\phi)$ and $\Fix(\phi)$. \bigskip
  \end{enumerate}

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

\item Let $\phi\colon G\to\Perm(S)$ be a \emph{left} group
  action. Prove the orbit-stabilizer theorem by constructing a
  bijection between $\orb(s)$ and \emph{left} cosets of $H=\stab(s)$. Use
  analogous notational conventions from lecture, e.g., $\phi(g)\Dot s$
  instead of $s\Dot\phi(g)$. \medskip
  
  %%---------------------------------------------------------------

  
\newpage

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


\item Recall that two group actions $\phi_1\colon
  G_1\longto\Perm(S_1)$ and $\phi_2\colon G_2\longto\Perm(S_2)$ are
  \emph{equivalent} if there is an isomorphism $\iota\colon G_1\to
  G_2$ and a bijection $\sigma\colon S_1\to S_2$ such that
  $\sigma\circ\phi_1(g)=\phi_2(\iota(g))\circ\sigma$ for all $g\in
  G_1$. In other words, the following diagram commutes:
  \[
  \begin{tikzcd}[column sep=large,row sep=large,scale=1.65]
    S_1 \arrow{r}{\phi_1(g)} \arrow{d}[swap]{\sigma} & S_1 \arrow{d}{\sigma} \\   
    S_2 \arrow[swap]{r}{\phi_2(\iota(g))} & S_2
  \end{tikzcd}
\]
In class, we proved that every transitive $G$-action is equivalent to $G$ acting on a set of cosets by multiplication. We did this by using the identity isomorphism $\iota\colon G\to G$ between the groups, and the bijection $\sigma\colon S\to H\!\setminus\!G$ defined by
\[
\sigma\colon S\longto H\!\setminus\!G,\qquad\sigma\colon s\Dot\phi(x)\mapsto Hx.
\]
This was inpired by the following picture:
\[
\begin{tikzpicture}[>=stealth, shorten > = 1pt, shorten < = 0pt, scale=.75, thick,fill=black]
  \begin{scope}[shift={(0,0)}, shorten >= 0pt, shorten <= 0pt,scale=.7]
    \path[fill=actOrange] (0,0) -- (0:1) -- (60:1) -- (0,0);
    \path[fill=actOrange] (0,0) -- (60:1) -- (120:1) -- (0,0);
    \path[fill=actPurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
    \path[fill=actOrange] (0,0) -- (180:1) -- (240:1) -- (0,0);
    \path[fill=actOrange] (0,0) -- (240:1) -- (300:1) -- (0,0);
    \path[fill=actOrange] (0,0) -- (300:1) -- (0:1) -- (0,0);
    \draw (0:1) -- (60:1) -- (120:1) -- (180:1) -- (240:1) -- (300:1) -- (0:1);
    \node (x-out) at (90:.866) {};
    \node at (180:2.3) {$s\Dot\phi(x)$};
    \draw[-stealth] (5.5,2.5) to node[above] {$\sigma$} (9.5,2.5);
  \end{scope}
  %%
  \begin{scope}[shift={(3,4)}, shorten >= 0pt, shorten <= 0pt,scale=.7]
    \path[fill=actRed] (0,0) -- (0:1) -- (60:1) -- (0,0);
    \path[fill=actRed] (0,0) -- (60:1) -- (120:1) -- (0,0);
    \path[fill=actRed] (0,0) -- (120:1) -- (180:1) -- (0,0);
    \path[fill=actRed] (0,0) -- (180:1) -- (240:1) -- (0,0);
    \path[fill=actBlue] (0,0) -- (240:1) -- (300:1) -- (0,0);
    \path[fill=actRed] (0,0) -- (300:1) -- (0:1) -- (0,0);
    \draw (0:1) -- (60:1) -- (120:1) -- (180:1) -- (240:1) -- (300:1) -- (0:1);
    \node (xg-in) at (210:.866) {};
    \node at (180:2.4) {$s\Dot\phi(xg)$};
  \end{scope}
  \draw [r] (x-out) to [bend left=15] node[above left] {$g$} (xg-in);
  %%
  \begin{scope}[shift={(8,0)}, shorten >= 0pt, shorten <= 0pt,scale=.7]
    \draw (-1.25,-.75) -- (1.25,-.75) -- (1.25,.75) -- (-1.25,.75) -- (-1.25,-.75);
    \node at (0,0) {$\bm{Hx}$};
    \node (Hx-out) at (90:.866) {};
  \end{scope}
  \begin{scope}[shift={(11,4)}, shorten >= 0pt, shorten <= 0pt,scale=.7]
    \draw (-.75,-1) -- (1.75,-1) -- (1.75,.5) -- (-.75,.5) -- (-.75,-1);
    \node at (.5,-.25) {$\bm{Hxg}$};
    \node (Hxg-in) at (210:.866) {};
  \end{scope}
  \draw [r] (Hx-out) to [bend left=15] node[above left] {$g$} (Hxg-in);
\end{tikzpicture}
\]
Prove an analogous statement: that every simply transitive action is equivalent to $G$ acting on itself by conjugation. The proof is not much different than just replacing each ``$H$'' with the identity element, ``$1$'' in the one we did in class.
  
    

    
\end{enumerate}

\end{document}    
    
  
 
