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\pagestyle{fancy} \lhead{{\sf Homework 7 $|$ Due October 8 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Fall 2021 $|$ M.~Macauley}}

\begin{document}

%$\;$

\begin{enumerate}

\item In both the dihedral group $G=D_6$ and dicyclic group
  $G=\Dic_6$, whose Cayley diagrams are shown below, the subgroups
  $\<r^3\>$ and $\<r^2\>$ are normal. For both, construct a
  Cayley table for the quotients $G/\<r^2\>$ and $G/\<r^3\>$, and
  determine what these are isomorphic to. \vspace{-4mm}

  
  %% Generalized quaternion group (dicyclic) of order 12 and 16.
\tikzstyle{R6-out} = [draw, very thick, Red,-stealth,bend right=18]
\tikzstyle{R6-in} = [draw, very thick, Red,-stealth,bend left=12]
\tikzstyle{R-out} = [draw, very thick, Red,-stealth,bend right=15]
\tikzstyle{R-in} = [draw, very thick, Red,-stealth,bend left=12]
\tikzstyle{B} = [draw, very thick, Blue,-stealth,bend right=25]
\[
\tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=6mm] 
  \begin{tikzpicture}[scale=1.25]
    \begin{scope}[shift={(0,0)},scale=1.15]
      \tikzstyle{every node}=[font=\footnotesize]
      \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 (s) at (0:1) [v-small] {$s$};
      \node (sr5) at (60:1) [v-small] {$rs$};
      \node (sr4) at (120:1) [v-small] {$r^2s$};
      \node (sr3) at (180:1) [v-small] {$r^3s$};
      \node (sr2) at (240:1) [v-small] {$r^4s$};
      \node (sr) at (300:1) [v-small] {$r^5s$};
      \path[bb] (1) to (s);
      \path[bb] (r) to (sr5);
      \path[bb] (r2) to (sr4);
      \path[bb] (r3) to (sr3);
      \path[bb] (r4) to (sr2);
      \path[bb] (r5) to (sr);
      \path[R-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);
    \end{scope}
    %%
    \begin{scope}[shift={(7,0)},scale=1.15]
      \tikzstyle{every node}=[font=\footnotesize]
      \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 (s) at (0:1) [v-small] {$s$};
      \node (sr5) at (60:1) [v-small] {$rs$};
      \node (sr4) at (120:1) [v-small] {$r^2s$};
      \node (sr3) at (180:1) [v-small] {$r^3s$};
      \node (sr2) at (240:1) [v-small] {$r^4s$};
      \node (sr) at (300:1) [v-small] {$r^5s$};
      %%
      \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);
      %%
    \end{scope}
  \end{tikzpicture}
  \]
  %\begin{enumerate}
  %\item Construct a Cayley table of the 
  %\end{enumerate}


\item Let $H$ be a subgroup of $G$.
  \begin{enumerate}
  \item Show that if $G$ is abelian, then $H$ and $G/H$ are abelian. \smallskip
  \item Show that if $G/Z(G)$ is cyclic, then $G$ is abelian. \smallskip
  \item What cyclic groups can arise as a quotient $G/Z(G)$? Justify
    your answer. \bigskip
  \end{enumerate}


  
\item Let $X$ be a subset of a group $G$. The \emph{centralizer} of
  $X$, denoted $C_G(X)$, is the set of all elements that commute with
  everything in $X$:
  \[
  C_G(X)=\big\{g\in G\mid gx=xg,\;\forall x\in X\big\}.
  \]
  If $X=\{x\}$, then we denote the centralizer as $C_G(x)$. You may
  use the fact, without proof, that $|G|=|\cl_G(x)|\cdot|C_G(x)|$ for
  all $x\in G$.
  \begin{enumerate}
  \item Show that $C_G(X)$ is a subgroup of $G$. \medskip

  \item If $H$ is a subgroup of $G$, show that $C_G(H)\normaleq
    N_G(H)$. \medskip
    
  \item For $Q_8$ and $D_6$, compute the centralizers of each element
    $x\in G$, as well as $N_G(\<x\>)$. The partition of these
    groups by conjugacy classes is shown below. %\smallskip
    
    \[
    \begin{tikzpicture}[scale=1.4]
      \tikzstyle{every node}=[font=\normalsize]
      \begin{scope}[shift={(0,0)}]
        \draw[thin,fill=conj-purple] (0,0) rectangle (.7,.7);
        \draw[thin,fill=conj-purple] (0,.7) rectangle (.7,1.4);
        \draw[thin,fill=conj-orange] (.7,0) rectangle (1.4,1.4);
        \draw[thin,fill=conj-orange] (1.4,0) rectangle (2.1,1.4);
        \draw[thin,fill=conj-orange] (2.1,0) rectangle (2.8,1.4);
        \node at (.35,1.05) {$1$};
        \node at (.35,.35) {$-1$};
        \node at (1.05,1.05) {$i$};
        \node at (1.05,.35) {$-i$};
        \node at (1.75,1.05) {$j$};
        \node at (1.75,.35) {$-j$};
        \node at (2.45,1.05) {$k$};
        \node at (2.45,.35) {$-k$};
      \end{scope}
      %%
      \begin{scope}[shift={(5,0)}]
        \tikzstyle{every node}=[font=\normalsize]
        \draw[thin,fill=conj-purple] (0,0) rectangle (.7,.7);
        \draw[thin,fill=conj-purple] (0,.7) rectangle (.7,1.4);
        \draw[thin,fill=conj-orange] (.7,0) rectangle (1.4,1.4);
        \draw[thin,fill=conj-orange] (1.4,0) rectangle (2.1,1.4);
        \draw[thin,fill=conj-orange] (2.1,0) rectangle (4.2,.7);
        \draw[thin,fill=conj-orange] (2.1,.7) rectangle (4.2,1.4);
        \node at (.35,1.05) {$1$};
        \node at (.35,.35) {$r^3$};
        \node at (1.05,1.05) {$r$};
        \node at (1.05,.35) {$r^5$};
        \node at (1.75,1.05) {$r^2$};
        \node at (1.75,.35) {$r^4$};
        \node at (2.45,1.05) {$f$};
        \node at (2.45,.35) {$rf$};
        \node at (3.15,1.05) {$r^2f$};
        \node at (3.15,.35) {$r^3f$};
        \node at (3.85,1.05) {$r^4f$};
        \node at (3.85,.35) {$r^5f$};
      \end{scope}
    \end{tikzpicture}
    \]

    %\medskip
    
  \item Partition the elements of the groups $D_5$ and $D_8$ by
    conjugacy classes, and arrange them in a table, as above. Then repeat the
    previous part for these groups. The Cayley diagram and subgroup
    lattice for $D_8$ is shown below, for convenience.

%% Cayley diagram and subgroup lattice of D_8
\tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt, 
  minimum size=6mm]
\tikzstyle{R-out} = [draw, very thick, Red,-stealth,bend right=15]
\tikzstyle{R-in} = [draw, very thick, Red,-stealth,bend left=12]
\[
\hspace*{-15mm}
  \begin{tikzpicture}
    \begin{scope}[shift={(0,0)},scale=1.45]
    \tikzstyle{every node}=[font=\footnotesize]
      \node (s) at (0:1.1) [v-small] {$s$};
      \node (rs) at (45:1.1) [v-small] {$rs$};
      \node (r2s) at (90:1.1) [v-small] {$r^2\!s$};
      \node (r3s) at (135:1.1) [v-small] {$r^3\!s$};
      \node (r4s) at (180:1.1) [v-small] {$r^4\!s$};
      \node (r5s) at (225:1.1) [v-small] {$r^5\!s$};
      \node (r6s) at (270:1.1) [v-small] {$r^6\!s$};
      \node (r7s) at (315:1.1) [v-small] {$r^7\!s$};
      %%
      \node (1) at (0:2) [v-small] {$1$};
      \node (r) at (45:2) [v-small] {$r$};
      \node (r2) at (90:2) [v-small] {$r^2$};
      \node (r3) at (135:2) [v-small] {$r^3$};
      \node (r4) at (180:2) [v-small] {$r^4$};
      \node (r5) at (225:2) [v-small] {$r^5$};
      \node (r6) at (270:2) [v-small] {$r^6$};
      \node (r7) at (315:2) [v-small] {$r^7$};
      %%
      \draw [R-out] (1) to (r);
      \draw [R-out] (r) to (r2);
      \draw [R-out] (r2) to (r3);
      \draw [R-out] (r3) to (r4);
      \draw [R-out] (r4) to (r5);
      \draw [R-out] (r5) to (r6);
      \draw [R-out] (r6) to (r7);
      \draw [R-out] (r7) to (1);
      %%
      \draw [R-in] (s) to (r7s);
      \draw [R-in] (r7s) to (r6s);
      \draw [R-in] (r6s) to (r5s);
      \draw [R-in] (r5s) to (r4s);
      \draw [R-in] (r4s) to (r3s);
      \draw [R-in] (r3s) to (r2s);
      \draw [R-in] (r2s) to (rs);
      \draw [R-in] (rs) to (s);
      %%
      \draw [bb] (1) to (s); \draw [bb] (r) to (rs);
      \draw [bb] (r2) to (r2s); \draw [bb] (r3) to (r3s);
      \draw [bb] (r4) to (r4s); \draw [bb] (r5) to (r5s);
      \draw [bb] (r6) to (r6s); \draw [bb] (r7) to (r7s);
      \node at (0,0) {\normalsize $D_8$};
    \end{scope}
    %%
    \begin{scope}[shift={(8.5,-3)},shorten >= -3pt, shorten <= -3pt,scale=1.45]
      \node (D8) at (0,4) {$\<r,f\>$};
      \node (r2-f) at (-1.25,3) {$\<r^2,f\>$};
      \node (r) at (0,3) {$\<r\>$};
      \node (r2-rf) at (1.25,3) {$\<r^2,rf\>$};
      %%
      \node (r4-f) at (-2.5,2) {$\<r^4,f\>$};
      \node (r4-r2f) at (-1.25,2) {$\<r^4,r^2f\>$};
      \node (r2) at (0,2) {$\<r^2\>$};
      \node (r4-r3f) at (1.25,2) {$\<r^4,r^3f\>$};
      \node (r4-r5f) at (2.5,2) {$\<r^4,r^5f\>$};
      %%
      \node (f) at (-3,1) {$\<f\>$};
      \node (r4f) at (-2.25,1) {$\<r^4f\>$};
      \node (r2f) at (-1.5,1) {$\<r^2f\>$};
      \node (r6f) at (-.75,1) {$\<r^6f\>$};
      \node (r4) at (0,1) {$\<r^4\>$};
      \node (r7f) at (.75,1) {$\<r^7f\>$};
      \node (r3f) at (1.5,1) {$\<r^3f\>$};
      \node (r5f) at (2.25,1) {$\<r^5f\>$};
      \node (rf) at (3,1) {$\<rf\>$};    
      %%
      \node (1) at (0,0) {$\<1\>$};
      %%
      \draw (D8) to (r2-f); \draw (D8) to (r); \draw (D8) to (r2-rf);
      \draw (r2-f) to (r4-f); \draw (r2-f) to (r4-r2f); \draw (r2-f) to (r2);
      \draw (r2-rf) to (r4-r5f); \draw (r2-rf) to (r4-r3f); \draw (r2-rf) to (r2);
      \draw (r2) to (r); \draw (r2) to (r4);
      \draw (r4-f) to (f); \draw (r4-f) to (r4f); \draw (r4-f) to (r4);
      \draw (r4-r2f) to (r2f); \draw (r4-r2f) to (r6f); \draw (r4-r2f) to (r4);
      \draw (r4-r3f) to (r7f); \draw (r4-r3f) to (r3f); \draw (r4-r3f) to (r4);
      \draw (r4-r5f) to (r5f); \draw (r4-r5f) to (rf); \draw (r4-r5f) to (r4);
      \draw (f) to (1); \draw (r4f) to (1); \draw (r2f) to (1); \draw (r6f) to (1);
      \draw (r7f) to (1); \draw (r3f) to (1); \draw (r5f) to (1); \draw (rf) to (1);
      \draw (r4) to (1);
    \end{scope}
  \end{tikzpicture}
  \]
    
  \end{enumerate}


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

  

\item Recall that two elements in $S_n$ are conjugate if and only if
  they have the same cycle type.
  \begin{enumerate}
    \item Determine how many elements there are of each cycle type in
      $S_4$, and in $S_5$. Note that the sum of your answers should add
      up to $|S_4|=4!=24$ and $|S_5|=5!=120$, respectively.
    \item Partition the elements of $S_4$ by conjugacy class.
    \item Compute the centralizers of $e$, $(12)$, $(123)$, $(1234)$,
      and $(12)(34)$ in $S_4$.
    \item Partition the elements of $A_4$ by conjugacy class. Then
      pick one element from each class, and find its
      centralizer. [\emph{Hint:} two-thirds of the elements in $A_4$
        are 3-cycles, so they cannot all be in the same conjugacy
        class.]
    \item Find the centralizer of each of the elements $e$, $(12)$,
      $(123)$, $(1234)$, $(12345)$, $(12)(34)$, and $(123)(45)$ in
      $S_5$.      
  \end{enumerate}


  
  
\end{enumerate}


\end{document}    
    
  
  %%-------------------------------------------------------------------
  
\item Shown below is a Cayley diagram for $G=C_4\rtimes C_4=\<a,b\>$,
  and the construction of the conjugate subgroup $aBa^{-1}$, where
  $B=\<b\>$.
  \[
  \hspace*{-1em}
  \tikzstyle{v} = [circle, draw, fill=lightgrey,inner sep=0pt, 
    minimum size=7mm]
  \tikzstyle{v-r} = [circle, draw, fill=lightred,inner sep=0pt, 
    minimum size=7mm]
  \tikzstyle{v-b} = [circle, draw, fill=lightblue,inner sep=0pt, 
    minimum size=7mm]
  \tikzstyle{v-g} = [circle, draw, fill=grey,inner sep=0pt, 
    minimum size=7mm]
  \tikzstyle{b-faded} = [draw, very thick, lightblue, -stealth]
  \tikzstyle{r-faded} = [draw, very thick, lightred, -stealth]
  \begin{tikzpicture}[scale=1.2]
    %%
    \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);
      %\node at (-1,3) {$\color{Blue}aB$};
    \end{scope}
    %%
    \begin{scope}[shift={(0,0)}]
      \tikzstyle{every node}=[font=\footnotesize]
      \node at (1,3.75) {\normalsize $\color{Blue}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);
      %\node at (-1,3) {$\color{Blue}aB$};
        \end{scope}
        %%
    \begin{scope}[shift={(4.65,0)}]
      \tikzstyle{every node}=[font=\footnotesize]
      \node at (0,3.75) {\normalsize ${\color{Blue}aB}{\color{Blue}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);
      %\node at (-1,3) {${\color{Blue}aB}{\color{Red}a^{-1}}$};
      %\node at (-1,2) {$\color{black}aB$};
\end{scope}
  \end{tikzpicture}
%\caption{The subgroup $B=\<b\>$ is not normal. On the left is the right coset $Ba$, in the middle is the left coset $aB$, and the blue nodes on the right are the conjugate $aBa^{-1}\neq B$.}\label{fig3:aBa^{-1}}
\]  
\begin{enumerate}[(a)]

\item Find all conjugates of the subgroups $\<ab\>$ and $\<a^2,b^2\>$,
  and illustrate the construction of each on the Cayley diagram, as
  was done above for $B=\<b\>$. 
  
\item The subgroup lattice of $G=\<a,b\>$ is shown below. Re-draw this
  with the subgroups written by generator(s). Then partition
    the subgroups into conjugacy classes, and fully justify your
    answer.
    \[
    \hspace*{-3em}
    \begin{tikzpicture}[shorten >= -2pt, shorten <= -2pt,xscale=1.6]
      \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 (-2,2) {$C_4$};
        \node (C4-2) at (-1.5,2) {$C_4$};
        \node (C4-3) at (-1,2) {$C_4$};
        \node (C4-4) at (-.5,2) {$C_4$};
        \node (C4-5) at (1,2) {$C_4$};
        \node (C4-6) at (2,2) {$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 (G) to (C4xC2-1); \draw (G) to (C4xC2-2); \draw (G) to (C4xC2-3);
        \draw (C4xC2-1) to (C4-1);
        \draw (C4xC2-1) to (C4-2);
        \draw (C4xC2-1) to (V4);
        \draw (C4xC2-2) to (C4-3);
        \draw (C4xC2-2) to (C4-4);
        \draw (C4xC2-2) to (V4); 
        \draw (C4xC2-3) to (C4-5);
        \draw (C4xC2-3) to (C4-6);
        \draw (C4xC2-3) to (V4);
        \draw (C4-1) to (C2-1); \draw (C4-2) to (C2-1);
        \draw (C4-3) to (C2-1); \draw (C4-4) to (C2-1);
        \draw (V4) to (C2-1); \draw (V4) to (C2-2); \draw (V4) to (C2-3);
        \draw (C4-5) to (C2-3); \draw (C4-6) to (C2-3);
        \draw (C2-1) to (1); \draw (C2-2) to (1); \draw (C2-3) to (1);
        %\draw[bend right=5] (G) to (s); 
        %\draw (G) to (st); \draw (G) to (s3t);
        %\draw (G) to (ts); \draw (G) to (ts3); \draw (G) to (s2-t);
        %%
        %\draw (s2-t) to (t); \draw (t) to (1);
        %\draw (s2-t) to (s2); \draw (s2-t) to (ts2);
        %\draw (s2-t) to (sts);
        %\draw (s2-t) to (s2t2); 
        %\draw[bend left=10] (s2-t) to (s2t);
        %%
        %\draw (s) to (s2); \draw (st) to (ts2); \draw (s3t) to (sts);
        %\draw (ts) to (s2t2); \draw (ts3) to (s2t); 
        %%
        %\draw (1) to (s2); \draw (1) to (ts2); \draw (1) to (sts);
        %\draw (1) to (s2t2); \draw (1) to (s2t);
      \end{scope}
      %%
    \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 Find the center $Z(G)$ of this group. 

  \item Shown below are three copies of the Cayley diagram for $G$,
    each with a peculiar labeling of the nodes. In each diagram, pair
    of nodes that differ by a sign are a left coset of one of the
    three order-$2$ subgroups, all of which are normal. \vspace{-4mm}
    \[
    \hspace*{-15mm}
  \tikzstyle{v} = [circle, draw, fill=lightgrey,inner sep=0pt, 
      minimum size=7mm]
    \tikzstyle{v-r} = [circle, draw, fill=lightred,inner sep=0pt, 
      minimum size=7mm]
    \tikzstyle{v-b} = [circle, draw, fill=lightblue,inner sep=0pt, 
      minimum size=7mm]
    \tikzstyle{v-g} = [circle, draw, fill=grey,inner sep=0pt, 
      minimum size=7mm]
    \tikzstyle{b-faded} = [draw, very thick, lightblue, -stealth]
  \tikzstyle{r-faded} = [draw, very thick, lightred, -stealth]
  \begin{tikzpicture}[scale=1.2,auto]
    \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 diagram, construct a multiplication table and Cayley
  diagram 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 $G$. 

    
  %\item Find the normalizer of each subgroup. 
\end{enumerate}


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


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

    

    
\end{enumerate}

\end{document}












  \[
  \begin{tikzpicture}[scale=1]
  \tikzstyle{every node}=[font=\small,anchor=south]
  \begin{scope}[shift={(0,0)}]
      \draw[thin,fill=faded] (0,0) rectangle (8,.8);
      \draw[thin] (0,1.6) to (8,1.6); 
      \draw[thin] (0,0) to (0,1.6);
      \draw[thin] (2,0) to (2,1.6);
      \draw[thin] (4,0) to (4,1.6);
      \draw[thin] (6,0) to (6,1.6);
      \draw[thin] (8,0) to (8,1.6);
      \node at (.5,.15) {$1$};
      \node at (1.5,.15) {$r^4$};
      \node at (2.5,.15) {$r$};
      \node at (3.5,.15) {$r^5$};
      \node at (4.5,.15) {$r^2$};
      \node at (5.5,.15) {$r^6$};
      \node at (6.5,.15) {$r^3$};
      \node at (7.5,.15) {$r^7$};
      %%
      \node at (.5,.95) {$s$};
      \node at (1.5,.95) {$r^4s$};
      \node at (2.5,.95) {$rs$};
      \node at (3.5,.95) {$r^5s$};
      \node at (4.5,.95) {$r^2s$};
      \node at (5.5,.95) {$r^6s$};
      \node at (6.5,.95) {$r^3s$};
      \node at (7.5,.95) {$r^7s$};
      %\node at (2,-.8) {$\<r\>/\<r^4\>\leq G/\<r^4\>$};
    \end{scope}
  \end{tikzpicture}
  \]

