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\pagestyle{fancy} \lhead{{\sf Homework 6 $|$ Due February 21 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Visual Algebra $|$
    Spring 2025 $|$ M.~Macauley}}

\begin{document}

%$\;$

\begin{enumerate}

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

  \item Below are three Cayley graphs of $A_4$, each highlighting the left
  cosets of a different subgroup. We can take $a=(123)$, $b=(134)$,
  $x=(12)(34)$, and $z=(13)(24)$. \vspace{-5mm}
  
  \[
  \hspace*{-3mm}
  \begin{tikzpicture}[scale=1.05]
    \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt,
      minimum size=5mm]
    \tikzstyle{every node}=[font=\scriptsize]
    %%
    \begin{scope}[shift={(0,-1.1)},scale=.7] 
      \draw[cosetBlue, fill=cosetBlue] (.75,.75) circle (15mm);
      \draw[cosetBlue, fill=cosetBlue] (4.25,.75) circle (15mm);
      \draw[cosetBlue, fill=cosetBlue] (2.5,3.75) circle (15mm);
      \node (e) at (1.75,4.5) [v] {$e$};
      \node (x) at (3.25,4.5) [v] {$x$};
      \node (z) at (1.75,3) [v] {$z$};
      \node (y) at (3.25,3) [v] {$y$};
      \node (a) at (3.5,1.5) [v] {$a$};
      \node (c) at (5,1.5) [v] {$c$};
      \node (d) at (3.5,0) [v] {$d$};
      \node (b) at (5,0) [v] {$b$};
      \node (dd) at (1.5,0) [v] {$d^2$};
      \node (bb) at (1.5,1.5) [v] {$b^2$};
      \node (aa) at (0,1.5) [v] {$a^2$};
      \node (cc) at (0,0) [v] {$c^2$};
      \draw [r] (x) to [bend left=50] (b);
      \draw [r] (e) to [bend left=60] (a);
      \draw [r] (z) to [bend left=60] (c);
      \draw [r] (y) to [bend left=60] (d);
      \draw [r] (a) to [bend left] (aa);
      \draw [r] (b) to [bend left=35] (cc);
      \draw [r] (c) to [bend left=65] (dd);
      \draw [r] (d) to [bend left=40] (bb);
      \draw [r] (aa) to [bend left] (e);
      \draw [r] (bb) to [bend left=40] (y);
      \draw [r] (cc) to [bend left=25] (x);
      \draw [r] (dd) to [bend left] (z);
      \node at (1.75,4.5) [v] {$e$};
      \node at (.85,4.5) {\normalsize$\bm{N}$};
      \draw [bb] (e) to (x);
      \draw [bb] (z) to (y);
      \draw [gg] (e) to (z);
      \draw [gg] (x) to (y);
      \draw [bb] (a) to (c);
      \draw [bb] (d) to (b);
      \draw [gg] (a) to (d);
      \draw [gg] (c) to (b);
      \draw [bb] (aa) to (bb);
      \draw [bb] (cc) to (dd);
      \draw [gg] (aa) to (cc);
      \draw [gg] (bb) to (dd);
    \end{scope}
    %%
    \begin{scope}[shift={(12.5,0)},scale=.6]
      \draw[cosetBlue, fill=cosetBlue] (0,1.625) ellipse (.7 and 1.05);
      \draw[color=cosetRed,fill=cosetRed] (-1.407,-.8125)
      circle [x radius=.7cm, y radius=1.05cm, rotate=-60];
      \draw[color=cosetRed,fill=cosetRed] (1.407,-.8125)
      circle [x radius=.7cm, y radius=1.05cm, rotate=60];
      \draw[cosetBlue, fill=cosetBlue] (0,-2.5) ellipse (3.3 and .85);
      \draw[color=cosetRed,fill=cosetRed] (2.273,1.3125)
      circle [x radius=.9cm, y radius=3.3cm, rotate=30];
      \draw[color=cosetRed,fill=cosetRed] (-2.273,1.3125)
      circle [x radius=.9cm, y radius=3.3cm, rotate=-30];
      \node (l1) at (0,2.25) [v] {$x$};
      \node (l2) at (-.866,3.75) [v] {$b$};
      \node (t1) at (0,1) [v] {$e$};
      \node (t2) at (-.866,-.5) [v] {$a^2$};
      \node (t3) at (.866,-.5) [v] {$a$};
      \node (l3) at (.866,3.75) [v] {$c^2$};
      \node (r1) at (3.68,-1.125) [v] {$d^2$};
      \node (r2) at (2.814,-2.5) [v] {$z$};
      \node (r3) at (1.948,-1.125) [v] {$c$};
      \node (m2) at (-1.948,-1.125) [v] {$b^2$};
      \node (m1) at (-3.68,-1.125) [v] {$d$};
      \node (m3) at (-2.814,-2.5) [v] {$y$};
      \node at (.7,1.625) {\normalsize$\bm{K}$};
      \draw [bb] (l2) to (m1);
      \draw [bb] (m2) to (t2);
      \draw [bb] (r2) to (m3);
      \draw [bb] (l1) to (t1);
      \draw [bb] (l3) to (r1);
      \draw [bb] (r3) to (t3);
      \draw [r] (l1) to (l2);
      \draw [r] (l2) to (l3);
      \draw [r] (l3) to (l1);
      \draw [r] (t1) to (t3);
      \draw [r] (t2) to (t1);
      \draw [r] (t3) to (t2);
      \draw [r] (r1) to (r2);
      \draw [r] (r2) to (r3);
      \draw [r] (r3) to (r1);
      \draw [r] (m1) to (m2);
      \draw [r] (m2) to (m3);
      \draw [r] (m3) to (m1);
    \end{scope}
    %%
    \begin{scope}[shift={(7.25,.3)},scale=.65]
      \draw[cosetBlue, fill=cosetBlue] (-1.2,0) circle (1.2);
      \draw[cosetRed, fill=cosetRed] (1.2,0) circle (1.2);
      \draw[cosetRed, fill=cosetRed] (0,2.7) ellipse (3.4 and 1.05);
      \draw[cosetRed, fill=cosetRed] (0,-2.7) ellipse (3.4 and 1.05);
      \node (a2) at (45:1) [v] {$b$};
      \node (a4) at (135:1) [v] {$e$}; %{$\tiny *$};
      \node (a6) at (225:1) [v] {$a$};
      \node (a8) at (315:1) [v] {$x$};
      \node (b1) at (0:2) [v] {$c^2$};
      \node (b3) at (90:2) [v] {$b^2$};
      \node (b5) at (180:2) [v] {$a^2$};
      \node (b7) at (270:2) [v] {$d^2$};
      \node (c2) at (45:4) [v] {$d$};
      \node (c4) at (135:4) [v] {$y$};
      \node (c6) at (225:4) [v] {$c$};
      \node (c8) at (315:4) [v] {$z$};
      \node at (-1.4,1.2) {\normalsize$\bm{H}$};
      \draw [r] (b1) to (a8); \draw [r] (a8) to (a2); \draw [r] (a2) to (b1);
      \draw [r] (a4) to (a6); \draw [r] (a6) to (b5); \draw [r] (b5) to (a4);
      \draw [r] (c2) to (b3); \draw [r] (b3) to (c4); \draw [r] (c4) to (c2);
      \draw [r] (c6) to (b7); \draw [r] (b7) to (c8); \draw [r] (c8) to (c6);
      \draw [p] (b1) to (c2); \draw [p] (c2) to (c8); \draw [p] (c8) to (b1);
      \draw [p] (b5) to (c6); \draw [p] (c6) to (c4); \draw [p] (c4) to (b5);
      \draw [p] (b3) to (a2); \draw [p] (a2) to (a4); \draw [p] (a4) to (b3);
      \draw [p] (a6) to (a8); \draw [p] (a8) to (b7); \draw [p] (b7) to (a6);
    \end{scope}
  \end{tikzpicture}
  \]
  \begin{enumerate}
  \item For each subgroup shown above, partition $A_4$ into its right
    cosets. Write them as subsets of $A_4$, consisting of permutations
    in cycle notation. Also, highlight them by colors on a fresh copy
    of the Cayley graphs.
  \item For each left coset $gH$, illustrate the construction of the
    conjugate subgroup $gHg^{-1}$ on a fresh copy of the Cayley
    graph. Repeat this for $N$ and $K$. \\
  \end{enumerate}
  
  %%-------------------------------------------------------------------
  
\item Show that $A\times\{1_B\}$ is a normal subgroup of $A\times B$,
  where $1_B$ is the identity element of $B$. That is, show first that
  it is a subgroup, and then that it is normal. \\
  
  %%-------------------------------------------------------------------
  
\item Let $G$ be a group, not necessarily finite, and $H\leq G$.
  \begin{enumerate}
  \item Show that the subgroup
    \begin{equation*}
      N:=\bigcap_{x\in G} xHx^{-1}
    \end{equation*}
    is normal in $G$.
  \item Show that \emph{every} normal subgroup $K\normaleq G$ contained
    in $H$ is contained in $N$. In other words, $N$ is the
    \emph{largest normal subgroup} of $G$ contained in $H$.
  \end{enumerate}
  
  %%-------------------------------------------------------------------
  
\item Shown below is a Cayley graph for $G=C_4\rtimes C_4=\<a,b\>$,
  and the construction of the conjugate subgroup $aBa^{-1}$, where
  $B=\<b\>$.
  \[
  \hspace*{-1em}
  \begin{tikzpicture}[scale=1.2]
    \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt, 
      minimum size=7mm]
    \tikzstyle{v-b} = [circle, draw, fill=boxBlue,inner sep=0pt, 
      minimum size=7mm]
    \tikzstyle{v-g} = [circle, draw, fill=gray,inner sep=0pt, 
      minimum size=7mm]
    \tikzstyle{b-faded} = [draw, very thick, fBlue, -stealth]
    \tikzstyle{r-faded} = [draw, very thick, fRed, -stealth]
    %%
    \begin{scope}[shift={(-4.65,0)}]
      \tikzstyle{every node}=[font=\footnotesize]
      \node at (0,3.75) {\normalsize $\Balert{B}$};
      \node (e) at (0,3) [v-b] {$1$};
      \node (b) at (0,2) [v-b] {$b$};
      \node (b2) at (0,1) [v-b] {$b^2$};
      \node (b3) at (0,0) [v-b] {$b^3$};
      \node (a) at (1,3) [v] {$a$};
      \node (ab) at (1,2) [v] {$ab$};
      \node (ab2) at (1,1) [v] {$ab^2$};
      \node (ab3) at (1,0) [v] {$ab^3$};
      \node (a2) at (2,3) [v] {$a^2$};
      \node (a2b) at (2,2) [v] {$a^2b$};
      \node (a2b2) at (2,1) [v] {$a^2\!b^2$};
      \node (a2b3) at (2,0) [v] {$a^2\!b^3$};
      \node (a3) at (3,3) [v] {$a^3$};
      \node (a3b) at (3,2) [v] {$a^3b$};
      \node (a3b2) at (3,1) [v] {$a^3\!b^2$};
      \node (a3b3) at (3,0) [v] {$a^3\!b^3$};
      \draw [r-faded] (e) to (a);
      \draw [r-faded] (a) to (a2);
      \draw [r-faded] (a2) to (a3);
      \draw [r-faded] (a3) to [bend right=25] (e);
      \draw [r-faded] (a3b) to (a2b);
      \draw [r-faded] (a2b) to (ab);
      \draw [r-faded] (ab) to (b);
      \draw [r-faded] (b) to [bend left=25] (a3b);
      \draw [r-faded] (b2) to (ab2);
      \draw [r-faded] (ab2) to (a2b2);
      \draw [r-faded] (a2b2) to (a3b2);
      \draw [r-faded] (a3b2) to [bend left=25] (b2);
      \draw [r-faded] (a3b3) to (a2b3);
      \draw [r-faded] (a2b3) to (ab3);
      \draw [r-faded] (ab3) to (b3);
      \draw [r-faded] (b3) to [bend right=25] (a3b3);
      \draw [b] (e) to (b);
      \draw [b] (b) to (b2);
      \draw [b] (b2) to (b3);
      \draw [b] (b3) to [bend left=25] (e);
      \draw [b-faded] (a) to (ab);
      \draw [b-faded] (ab) to (ab2);
      \draw [b-faded] (ab2) to (ab3);
      \draw [b-faded] (ab3) to [bend left=25] (a);
      \draw [b-faded] (a2) to (a2b);
      \draw [b-faded] (a2b) to (a2b2);
      \draw [b-faded] (a2b2) to (a2b3);
      \draw [b-faded] (a2b3) to [bend right=25] (a2);
      \draw [b-faded] (a3) to (a3b);
      \draw [b-faded] (a3b) to (a3b2);
      \draw [b-faded] (a3b2) to (a3b3);
      \draw [b-faded] (a3b3) to [bend right=25] (a3);
    \end{scope}
    %%
    \begin{scope}[shift={(0,0)}]
      \tikzstyle{every node}=[font=\footnotesize]
      \node at (1,3.75) {\normalsize $\color{xBlue}aB$};
      \node (e) at (0,3) [v-g] {$1$};
      \node (b) at (0,2) [v] {$b$};
      \node (b2) at (0,1) [v] {$b^2$};
      \node (b3) at (0,0) [v] {$b^3$};
      \node (a) at (1,3) [v-b] {$a$};
      \node (ab) at (1,2) [v-b] {$ab$};
      \node (ab2) at (1,1) [v-b] {$ab^2$};
      \node (ab3) at (1,0) [v-b] {$ab^3$};
      \node (a2) at (2,3) [v] {$a^2$};
      \node (a2b) at (2,2) [v] {$a^2b$};
      \node (a2b2) at (2,1) [v] {$a^2\!b^2$};
      \node (a2b3) at (2,0) [v] {$a^2\!b^3$};
      \node (a3) at (3,3) [v] {$a^3$};
      \node (a3b) at (3,2) [v] {$a^3b$};
      \node (a3b2) at (3,1) [v] {$a^3\!b^2$};
      \node (a3b3) at (3,0) [v] {$a^3\!b^3$};
      \draw [r] (e) to (a);
      \draw [r-faded] (a) to (a2);
      \draw [r-faded] (a2) to (a3);
      \draw [r-faded] (a3) to [bend right=25] (e);
      \draw [r-faded] (a3b) to (a2b);
      \draw [r-faded] (a2b) to (ab);
      \draw [r-faded] (ab) to (b);
      \draw [r-faded] (b) to [bend left=25] (a3b);
      \draw [r-faded] (b2) to (ab2);
      \draw [r-faded] (ab2) to (a2b2);
      \draw [r-faded] (a2b2) to (a3b2);
      \draw [r-faded] (a3b2) to [bend left=25] (b2);
      \draw [r-faded] (a3b3) to (a2b3);
      \draw [r-faded] (a2b3) to (ab3);
      \draw [r-faded] (ab3) to (b3);
      \draw [r-faded] (b3) to [bend right=25] (a3b3);
      \draw [b-faded] (e) to (b);
      \draw [b-faded] (b) to (b2);
      \draw [b-faded] (b2) to (b3);
      \draw [b-faded] (b3) to [bend left=25] (e);
      \draw [b] (a) to (ab);
      \draw [b] (ab) to (ab2);
      \draw [b] (ab2) to (ab3);
      \draw [b] (ab3) to [bend left=25] (a);
      \draw [b-faded] (a2) to (a2b);
      \draw [b-faded] (a2b) to (a2b2);
      \draw [b-faded] (a2b2) to (a2b3);
      \draw [b-faded] (a2b3) to [bend right=25] (a2);
      \draw [b-faded] (a3) to (a3b);
      \draw [b-faded] (a3b) to (a3b2);
      \draw [b-faded] (a3b2) to (a3b3);
      \draw [b-faded] (a3b3) to [bend right=25] (a3);
    \end{scope}
    %%
    \begin{scope}[shift={(4.65,0)}]
      \tikzstyle{every node}=[font=\footnotesize]
      \node at (0,3.75) {\normalsize ${\color{xBlue}aB}{\color{xBlue}a^{-1}}$};
      \node (e) at (0,3) [v-b] {$1$};
      \node (b) at (0,2) [v] {$b$};
      \node (b2) at (0,1) [v-b] {$b^2$};
      \node (b3) at (0,0) [v] {$b^3$};
      \node (a) at (1,3) [v-g] {$a$};
      \node (ab) at (1,2) [v-g] {$ab$};
      \node (ab2) at (1,1) [v-g] {$ab^2$};
      \node (ab3) at (1,0) [v-g] {$ab^3$};
      \node (a2) at (2,3) [v] {$a^2$};
      \node (a2b) at (2,2) [v-b] {$a^2b$};
      \node (a2b2) at (2,1) [v] {$a^2\!b^2$};
      \node (a2b3) at (2,0) [v-b] {$a^2\!b^3$};
      \node (a3) at (3,3) [v] {$a^3$};
      \node (a3b) at (3,2) [v] {$a^3b$};
      \node (a3b2) at (3,1) [v] {$a^3\!b^2$};
      \node (a3b3) at (3,0) [v] {$a^3\!b^3$};
      \draw [r] (e) to (a);
      \draw [r-faded] (a) to (a2);
      \draw [r-faded] (a2) to (a3);
      \draw [r-faded] (a3) to [bend right=25] (e);
      \draw [r-faded] (a3b) to (a2b);
      \draw [r] (a2b) to (ab);
      \draw [r-faded] (ab) to (b);
      \draw [r-faded] (b) to [bend left=25] (a3b);
      \draw [r] (b2) to (ab2);
      \draw [r-faded] (ab2) to (a2b2);
      \draw [r-faded] (a2b2) to (a3b2);
      \draw [r-faded] (a3b2) to [bend left=25] (b2);
      \draw [r-faded] (a3b3) to (a2b3);
      \draw [r] (a2b3) to (ab3);
      \draw [r-faded] (ab3) to (b3);
      \draw [r-faded] (b3) to [bend right=25] (a3b3);
      \draw [b-faded] (e) to (b);
      \draw [b-faded] (b) to (b2);
      \draw [b-faded] (b2) to (b3);
      \draw [b-faded] (b3) to [bend left=25] (e);
      \draw [b] (a) to (ab);
      \draw [b] (ab) to (ab2);
      \draw [b] (ab2) to (ab3);
      \draw [b] (ab3) to [bend left=25] (a);
      \draw [b-faded] (a2) to (a2b);
      \draw [b-faded] (a2b) to (a2b2);
      \draw [b-faded] (a2b2) to (a2b3);
      \draw [b-faded] (a2b3) to [bend right=25] (a2);
      \draw [b-faded] (a3) to (a3b);
      \draw [b-faded] (a3b) to (a3b2);
      \draw [b-faded] (a3b2) to (a3b3);
      \draw [b-faded] (a3b3) to [bend right=25] (a3);
    \end{scope}
  \end{tikzpicture}
  \]  
  \begin{enumerate}
    
  \item For both order-$4$ subgroups, $\<ab\>$ and $\<a^2,b^2\>$,
    illustrate the construction of its conjugate subgroups on the Cayley
    graph, in the same $3$-step process that was done above for
    $B=\<b\>$. Carry this out for each of its three distinct left cosets
    (excluding the subgroup itself).
    
  \item The subgroup lattice of $G=\<a,b\>$ is shown below. Redraw this
    with the subgroups written by generator(s). Then partition
    the subgroups into conjugacy classes, and fully justify your
    answer. \vspace{-2mm}
    \[
    \hspace*{-3em}
    \begin{tikzpicture}[shorten >= -2pt, shorten <= -2pt,xscale=1]
      \tikzstyle{every node}=[font=\normalsize]
      \begin{scope}[shift={(0,0)},scale=1.5]
        \node (G) at (0,4) {$C_4\rtimes C_4$};
        \node (C4xC2-1) at (-1.5,3) {$C_4\times C_2$};
        \node (C4xC2-2) at (0,3) {$C_4\times C_2$};
        \node (C4xC2-3) at (1.5,3) {$C_4\times C_2$};
        \node (C4-1) at (-3,2) {$C_4$};
        \node (C4-2) at (-2,2) {$C_4$};
        \node (C4-3) at (-1.33,2) {$C_4$};
        \node (C4-4) at (-.67,2) {$C_4$};
        \node (C4-5) at (1.5,2) {$C_4$};
        \node (C4-6) at (3,2) {$C_4$};
        \node (V4) at (0,2) {$V_4$};
        \node (C2-1) at (-1.5,1) {$C_2$};
        \node (C2-2) at (0,1) {$C_2$};
        \node (C2-3) at (1.5,1) {$C_2$};
        \node (1) at (0,0) {$\<1\>$};
        \draw[f] (G) to (C4xC2-1); \draw[f] (G) to (C4xC2-2);
        \draw[f] (G) to (C4xC2-3);
        \draw[f] (C4xC2-1) to (C4-1);
        \draw[f] (C4xC2-1) to (C4-2);
        \draw[f] (C4xC2-1) to (V4);
        \draw[f] (C4xC2-2) to (C4-3);
        \draw[f] (C4xC2-2) to (C4-4);
        \draw[f] (C4xC2-2) to (V4); 
        \draw[f] (C4xC2-3) to (C4-5);
        \draw[f] (C4xC2-3) to (C4-6);
        \draw[f] (C4xC2-3) to (V4);
        \draw[f] (C4-1) to (C2-1); \draw[f] (C4-2) to (C2-1);
        \draw[f] (C4-3) to (C2-1); \draw[f] (C4-4) to (C2-1);
        \draw[f] (V4) to (C2-1); \draw[f] (V4) to (C2-2);
        \draw[f] (V4) to (C2-3);
        \draw[f] (C4-5) to (C2-3); \draw[f] (C4-6) to (C2-3);
        \draw[f] (C2-1) to (1); \draw[f] (C2-2) to (1); \draw[f] (C2-3) to (1);
      \end{scope}
    \end{tikzpicture}
    \]

  \item Without computing left or right cosets, i.e., only from the
    knowledge of conjugacy classes, find the normalizer of each
    subgroup. Justify your answer.

  \item Construct a (labeled) cycle graph. Which subgroup is the
    center, $Z(G)$, and why?

  \item Shown below are three copies of the Cayley graph for $G$,
    each with a peculiar labeling of the nodes. In each graph, a pair
    of nodes that differ by a sign represent a left coset of one of the
    three order-$2$ subgroups, all of which are normal. \vspace{-4mm}
    \[
    \hspace*{-15mm}
    \begin{tikzpicture}[scale=1.2,auto]
      \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt, 
        minimum size=7mm]
      \tikzstyle{v-b} = [circle, draw, fill=boxBlue,inner sep=0pt, 
        minimum size=7mm]
      \tikzstyle{v-g} = [circle, draw, fill=gray,inner sep=0pt, 
        minimum size=7mm]
      \tikzstyle{b-faded} = [draw, very thick, lightblue, -stealth]
      \tikzstyle{r-faded} = [draw, very thick, lightred, -stealth]
      %%
      \begin{scope}[shift={(-4.65,0)}]
        \tikzstyle{every node}=[font=\scriptsize]
        \node (e) at (0,3) [v] {$1$};
        \node (b) at (0,2) [v] {$b$};
        \node (b2) at (0,1) [v] {$b^2$};
        \node (b3) at (0,0) [v] {$b^3$};
        \node (a) at (1,3) [v] {$a$};
        \node (ab) at (1,2) [v] {$ab$};
        \node (ab2) at (1,1) [v] {$ab^2$};
        \node (ab3) at (1,0) [v] {$ab^3$};
        \node (a2) at (2,3) [v] {$-1$};
        \node (a2b) at (2,2) [v] {$-b$};
        \node (a2b2) at (2,1) [v] {$-b^2$};
        \node (a2b3) at (2,0) [v] {$-b^3$};
        \node (a3) at (3,3) [v] {$-a$};
        \node (a3b) at (3,2) [v] {$-ab$};
        \node (a3b2) at (3,1) [v] {$-ab^2$};
        \node (a3b3) at (3,0) [v] {$-ab^3$};
        \draw [r] (e) to (a);
        \draw [r] (a) to (a2);
        \draw [r] (a2) to (a3);
        \draw [r] (a3) to [bend right=25] (e);
        \draw [r] (a3b) to (a2b);
        \draw [r] (a2b) to (ab);
        \draw [r] (ab) to (b);
        \draw [r] (b) to [bend left=25] (a3b);
        \draw [r] (b2) to (ab2);
        \draw [r] (ab2) to (a2b2);
        \draw [r] (a2b2) to (a3b2);
        \draw [r] (a3b2) to [bend left=25] (b2);
        \draw [r] (a3b3) to (a2b3);
        \draw [r] (a2b3) to (ab3);
        \draw [r] (ab3) to (b3);
        \draw [r] (b3) to [bend right=25] (a3b3);
        \draw [b] (e) to (b);
        \draw [b] (b) to (b2);
        \draw [b] (b2) to (b3);
        \draw [b] (b3) to [bend left=25] (e);
        \draw [b] (a) to (ab);
        \draw [b] (ab) to (ab2);
        \draw [b] (ab2) to (ab3);
        \draw [b] (ab3) to [bend left=25] (a);
        \draw [b] (a2) to (a2b);
        \draw [b] (a2b) to (a2b2);
        \draw [b] (a2b2) to (a2b3);
        \draw [b] (a2b3) to [bend right=25] (a2);
        \draw [b] (a3) to (a3b);
        \draw [b] (a3b) to (a3b2);
        \draw [b] (a3b2) to (a3b3);
        \draw [b] (a3b3) to [bend right=25] (a3);
      \end{scope}
      %%
      \begin{scope}[shift={(0,0)}]
        \tikzstyle{every node}=[font=\scriptsize]
        \node at (0,3.75) {};
        \node at (1,3.75) {};
        \node (e) at (0,3) [v] {$1$};
        \node (b) at (0,2) [v] {$b$};
        \node (b2) at (0,1) [v] {$-1$};
        \node (b3) at (0,0) [v] {$-b$};
        \node (a) at (1,3) [v] {$a$};
        \node (ab) at (1,2) [v] {$ab$};
        \node (ab2) at (1,1) [v] {$-a$};
        \node (ab3) at (1,0) [v] {$-ab$};
        \node (a2) at (2,3) [v] {$a^2$};
        \node (a2b) at (2,2) [v] {$a^2b$};
        \node (a2b2) at (2,1) [v] {$-a^2$};
        \node (a2b3) at (2,0) [v] {$-a^2\!b$};
        \node (a3) at (3,3) [v] {$a^3$};
        \node (a3b) at (3,2) [v] {$a^3b$};
        \node (a3b2) at (3,1) [v] {$-a^3$};
        \node (a3b3) at (3,0) [v] {$-a^3\!b$};
        \draw [r] (e) to (a);
        \draw [r] (a) to (a2);
        \draw [r] (a2) to (a3);
        \draw [r] (a3) to [bend right=25] (e);
        \draw [r] (a3b) to (a2b);
        \draw [r] (a2b) to (ab);
        \draw [r] (ab) to (b);
        \draw [r] (b) to [bend left=25] (a3b);
        \draw [r] (b2) to (ab2);
        \draw [r] (ab2) to (a2b2);
        \draw [r] (a2b2) to (a3b2);
        \draw [r] (a3b2) to [bend left=25] (b2);
        \draw [r] (a3b3) to (a2b3);
        \draw [r] (a2b3) to (ab3);
        \draw [r] (ab3) to (b3);
        \draw [r] (b3) to [bend right=25] (a3b3);
        \draw [b] (e) to (b);
        \draw [b] (b) to (b2);
        \draw [b] (b2) to (b3);
        \draw [b] (b3) to [bend left=25] (e);
        \draw [b] (a) to (ab);
        \draw [b] (ab) to (ab2);
        \draw [b] (ab2) to (ab3);
        \draw [b] (ab3) to [bend left=25] (a);
        \draw [b] (a2) to (a2b);
        \draw [b] (a2b) to (a2b2);
        \draw [b] (a2b2) to (a2b3);
        \draw [b] (a2b3) to [bend right=25] (a2);
        \draw [b] (a3) to (a3b);
        \draw [b] (a3b) to (a3b2);
        \draw [b] (a3b2) to (a3b3);
        \draw [b] (a3b3) to [bend right=25] (a3);
      \end{scope}
      %%
      \begin{scope}[shift={(4.65,0)}]
        \tikzstyle{every node}=[font=\scriptsize]
        \node (e) at (0,3) [v] {$1$};
        \node (b) at (0,2) [v] {$b$};
        \node (b2) at (0,1) [v] {$-a^2$};
        \node (b3) at (0,0) [v] {$-a^2\!b$};
        \node (a) at (1,3) [v] {$a$};
        \node (ab) at (1,2) [v] {$ab$};
        \node (ab2) at (1,1) [v] {$-a^3$};
        \node (ab3) at (1,0) [v] {$-a^3\!b$};
        \node (a2) at (2,3) [v] {$a^2$};
        \node (a2b) at (2,2) [v] {$a^2b$};
        \node (a2b2) at (2,1) [v] {$-1$};
        \node (a2b3) at (2,0) [v] {$-b$};
        \node (a3) at (3,3) [v] {$a^3$};
        \node (a3b) at (3,2) [v] {$a^3b$};
        \node (a3b2) at (3,1) [v] {$-a$};
        \node (a3b3) at (3,0) [v] {$-ab$};
        \draw [r] (e) to (a);
        \draw [r] (a) to (a2);
        \draw [r] (a2) to (a3);
        \draw [r] (a3) to [bend right=25] (e);
        \draw [r] (a3b) to (a2b);
        \draw [r] (a2b) to (ab);
        \draw [r] (ab) to (b);
        \draw [r] (b) to [bend left=25] (a3b);
        \draw [r] (b2) to (ab2);
        \draw [r] (ab2) to (a2b2);
        \draw [r] (a2b2) to (a3b2);
        \draw [r] (a3b2) to [bend left=25] (b2);
        \draw [r] (a3b3) to (a2b3);
        \draw [r] (a2b3) to (ab3);
        \draw [r] (ab3) to (b3);
        \draw [r] (b3) to [bend right=25] (a3b3);
        \draw [b] (e) to (b);
        \draw [b] (b) to (b2);
        \draw [b] (b2) to (b3);
        \draw [b] (b3) to [bend left=25] (e);
        \draw [b] (a) to (ab);
        \draw [b] (ab) to (ab2);
        \draw [b] (ab2) to (ab3);
        \draw [b] (ab3) to [bend left=25] (a);
        \draw [b] (a2) to (a2b);
        \draw [b] (a2b) to (a2b2);
        \draw [b] (a2b2) to (a2b3);
        \draw [b] (a2b3) to [bend right=25] (a2);
        \draw [b] (a3) to (a3b);
        \draw [b] (a3b) to (a3b2);
        \draw [b] (a3b2) to (a3b3);
        \draw [b] (a3b3) to [bend right=25] (a3);
      \end{scope}
    \end{tikzpicture}
    \]
    For each graph, construct a Cayley table and Cayley graph
    consisting of these eight ``cosets''. Determine what the resulting
    \emph{quotient group} is isomorphic to, and describe where that
    subgroup lattice appears, hiding in the lattice for $G$. \medskip
\end{enumerate}

  %%-------------------------------------------------------------------
  
\item Let $H$ be a subgroup of $G$. Given two fixed elements $a,b\in
  G$, define the sets
  \[
  aHbH=\{ah_1bh_2\mid h_1,h_2\in H\}\,\qquad\mbox{ and }\qquad
  abH=\{abh\mid h\in H\}\,.
  \]
  Show that if $H\normaleq G$, then $aHbH=abH$. \medskip
  
  %%-------------------------------------------------------------------
  
\end{enumerate}

\end{document}

