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\pagestyle{fancy} \lhead{{\sf Homework 7 $|$ Due February 28 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Visual Algebra $|$
    Spring 2025 $|$ M.~Macauley}}

\begin{document}

$\;$

\begin{enumerate}

  %%-------------------------------------------------------------------
  
\item In both the dihedral group $G=D_6$ and dicyclic group
  $G=\Dic_6$, whose Cayley graphs are shown below, the subgroups
  $N=\<r^3\>$ and $H=\<r^2\>$ are normal. For both, construct a
  Cayley table and Cayley graph for the quotients $G/N$ and $G/H$, and
  determine what these are isomorphic to. \vspace{-4mm}
  
  
  %% Dicyclic groups of order 12 and 16.
  \[
  \begin{tikzpicture}[scale=1.4]
    \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt, 
      minimum size=6.5mm]
    \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={(0,0)},scale=1.15]
      \tikzstyle{every node}=[font=\small]
      \node at (0,0) {\normalsize $D_6$};
      \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] {$f$};
      \node (sr5) at (60:1) [v] {$rf$};
      \node (sr4) at (120:1) [v] {$r^2\!f$};
      \node (sr3) at (180:1) [v] {$r^3\!f$};
      \node (sr2) at (240:1) [v] {$r^4\!f$};
      \node (sr) at (300:1) [v] {$r^5\!f$};
      \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={(6,0)},scale=1.15]
      \tikzstyle{every node}=[font=\small]
      \node at (0,0) {\normalsize $\Dic_6$};
      \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);
      %%
    \end{scope}
  \end{tikzpicture}
  \]
  
  %%-------------------------------------------------------------------
  
\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 $C_G(x)$.
  %%
  \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 Fix $x\in G$, and define the map
    \[
    \phi\colon\big\{\text{left cosets of $C_G(x)$}\big\}\longto
    \big\{\text{conjugates of $x$}\big\},\qquad
    \phi\colon gC_G(x)\longmapsto gxg^{-1}.
    \]
    Show that this map is well-defined and a bijection. \medskip
    
  \item Derive the useful formula $|G|=|\cl_G(x)|\cdot|C_G(x)|$, for
    any $x\in G$. \medskip
  \end{enumerate}
  
  %%-------------------------------------------------------------------
  
\item The partition of the groups $Q_8$ and $D_6$ into conjugacy
  classes is shown below.
  
  %% Partition of Q_8 and D_6 into conjugacy classes
  \[
  \begin{tikzpicture}[scale=1.4]
    \tikzstyle{every node}=[font=\normalsize]
    \begin{scope}[shift={(0,0)}]
      \draw[thin,fill=boxPurple] (0,0) rectangle (.7,.7);
      \draw[thin,fill=boxPurple] (0,.7) rectangle (.7,1.4);
      \draw[thin,fill=boxOrange] (.7,0) rectangle (1.4,1.4);
      \draw[thin,fill=boxOrange] (1.4,0) rectangle (2.1,1.4);
      \draw[thin,fill=boxOrange] (2.1,0) rectangle (2.8,1.4);
      \node at (.35,1.05) {$1$};
      \node at (.35,.35) {$-1$};
      \node at (1.05,1.05) {$i$};
      \node at (1.05,.35) {$-i$};
      \node at (1.75,1.05) {$j$};
      \node at (1.75,.35) {$-j$};
      \node at (2.45,1.05) {$k$};
      \node at (2.45,.35) {$-k$};
    \end{scope}
    %%
    \begin{scope}[shift={(5,0)}]
      \tikzstyle{every node}=[font=\normalsize]
      \draw[thin,fill=boxPurple] (0,0) rectangle (.7,.7);
      \draw[thin,fill=boxPurple] (0,.7) rectangle (.7,1.4);
      \draw[thin,fill=boxOrange] (.7,0) rectangle (1.4,1.4);
      \draw[thin,fill=boxOrange] (1.4,0) rectangle (2.1,1.4);
      \draw[thin,fill=boxOrange] (2.1,0) rectangle (4.2,.7);
      \draw[thin,fill=boxOrange] (2.1,.7) rectangle (4.2,1.4);
      \node at (.35,1.05) {$1$};
      \node at (.35,.35) {$r^3$};
      \node at (1.05,1.05) {$r$};
      \node at (1.05,.35) {$r^5$};
      \node at (1.75,1.05) {$r^2$};
      \node at (1.75,.35) {$r^4$};
      \node at (2.45,1.05) {$f$};
      \node at (2.45,.35) {$rf$};
      \node at (3.15,1.05) {$r^2f$};
      \node at (3.15,.35) {$r^3f$};
      \node at (3.85,1.05) {$r^4f$};
      \node at (3.85,.35) {$r^5f$};
    \end{scope}
  \end{tikzpicture}
  \]

  \begin{enumerate} 
  \item For $Q_8$ and $D_6$, compute the centralizers of each element
    $x\in G$, and the normalizer $N_G(\<x\>)$ as well.
    
  \item Partition the elements of the group $D_8$ by conjugacy
    classes, and arrange them in a table, as above. Then repeat the
    previous part for this group. The Cayley graph and subgroup
    lattice for $D_8$ is shown below, for convenience.
    
    %% Cayley graph and subgroup lattice of D_8
    \[
    \hspace*{-15mm}
    \begin{tikzpicture}
      \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt, 
        minimum size=6.25mm]  
      \tikzstyle{R-out} = [draw, very thick, eRed,-stealth,bend right=15]
      \tikzstyle{R-in} = [draw, very thick, eRed,-stealth,bend left=12]
      %%
      \begin{scope}[shift={(0,0)},scale=1.45]
        \tikzstyle{every node}=[font=\small]
        \node (s) at (0:1.1) [v] {$f$};
        \node (rs) at (45:1.1) [v] {$rf$};
        \node (r2s) at (90:1.1) [v] {$r^2\!f$};
        \node (r3s) at (135:1.1) [v] {$r^3\!f$};
        \node (r4s) at (180:1.1) [v] {$r^4\!f$};
        \node (r5s) at (225:1.1) [v] {$r^5\!f$};
        \node (r6s) at (270:1.1) [v] {$r^6\!f$};
        \node (r7s) at (315:1.1) [v] {$r^7\!f$};
        %%
        \node (1) at (0:2) [v] {$1$};
        \node (r) at (45:2) [v] {$r$};
        \node (r2) at (90:2) [v] {$r^2$};
        \node (r3) at (135:2) [v] {$r^3$};
        \node (r4) at (180:2) [v] {$r^4$};
        \node (r5) at (225:2) [v] {$r^5$};
        \node (r6) at (270:2) [v] {$r^6$};
        \node (r7) at (315:2) [v] {$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 (1.5,1) {$\<r^7f\>$};
        \node (r3f) at (.75,1) {$\<r^3f\>$};
        \node (r5f) at (3,1) {$\<r^5f\>$};
        \node (rf) at (2.25,1) {$\<rf\>$};    
        %%
        \node (1) at (0,0) {$\<1\>$};
        %%
        \draw[f] (D8) to (r2-f); \draw[f] (D8) to (r); \draw[f] (D8) to (r2-rf);
        \draw[f] (r2-f) to (r4-f); \draw[f] (r2-f) to (r4-r2f);
        \draw[f] (r2-f) to (r2);
        \draw[f] (r2-rf) to (r4-r5f); \draw[f] (r2-rf) to (r4-r3f);
        \draw[f] (r2-rf) to (r2);
        \draw[f] (r2) to (r); \draw[f] (r2) to (r4);
        \draw[f] (r4-f) to (f); \draw[f] (r4-f) to (r4f);
        \draw[f] (r4-f) to (r4); \draw[f] (r4-r2f) to (r2f);
        \draw[f] (r4-r2f) to (r6f); \draw[f] (r4-r2f) to (r4);
        \draw[f] (r4-r3f) to (r7f); \draw[f] (r4-r3f) to (r3f);
        \draw[f] (r4-r3f) to (r4);
        \draw[f] (r4-r5f) to (r5f); \draw[f] (r4-r5f) to (rf);
        \draw[f] (r4-r5f) to (r4);
        \draw[f] (f) to (1); \draw[f] (r4f) to (1); \draw[f] (r2f) to (1);
        \draw[f] (r6f) to (1);
        \draw[f] (r7f) to (1); \draw[f] (r3f) to (1); \draw[f] (r5f) to (1);
        \draw[f] (rf) to (1); \draw[f] (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}    
    
  
 
