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\pagestyle{fancy} \lhead{{\sf Homework 10 $|$ Due October 29 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Fall 2021 $|$ M.~Macauley}}

\begin{document}

%$\;$

\begin{enumerate}

  %%--------------------------------------------------------------------
  
\item Consider the right action of $G=D_6=\<r,f\>$ on following set of $31$
  ``binary hexagons,'' where $r$ rotates each one $60^\circ$ counterclockwise, and $f$ flips each one horizontally (i.e., across a vertical axis).
  \[
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        \node at (0,0) {$S=\Bigg\{$};
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      \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$1$}; \node at (150:.6) {$1$};
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      \node at (30:.6) {$1$}; \node at (90:.6) {$1$}; \node at (150:.6) {$1$};
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      \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
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      \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    \begin{scope}[shift={(7.5,-6)}]
      \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    \begin{scope}[shift={(10.5,-6)}]
      \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    \begin{scope}[shift={(13.5,-6)}]
      \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    \begin{scope}[shift={(16.5,-6)}]
      \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    \begin{scope}[shift={(0,-9)}]
      \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    \begin{scope}[shift={(3,-9)}]
      \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
     \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    \begin{scope}[shift={(6,-9)}]
      \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    \begin{scope}[shift={(9,-9)}]
      \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    \begin{scope}[shift={(12,-9)}]
      \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    \begin{scope}[shift={(15,-9)}]
      \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    \begin{scope}[shift={(1.5,-12)}]
      \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    \begin{scope}[shift={(4.5,-12)}]
      \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    \begin{scope}[shift={(7.5,-12)}]
      \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$1$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    \begin{scope}[shift={(10.5,-12)}]
      \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$1$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$0$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    \begin{scope}[shift={(13.5,-12)}]
      \path[fill=lorange] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$0$}; \node at (90:.6) {$0$}; \node at (150:.6) {$1$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$0$}; \node at (330:.6) {$1$};
      \node at (1.5,-.2) {$,$};
    \end{scope}
    \begin{scope}[shift={(16.5,-12)}]
      \path[fill=lpurple] (0,0) -- (0:1) -- (60:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (60:1) -- (120:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (120:1) -- (180:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (180:1) -- (240:1) -- (0,0);
      \path[fill=lpurple] (0,0) -- (240:1) -- (300:1) -- (0,0);
      \path[fill=lorange] (0,0) -- (300:1) -- (0:1) -- (0,0);
      \node at (30:.6) {$1$}; \node at (90:.6) {$0$}; \node at (150:.6) {$0$};
      \node at (210:.6) {$1$}; \node at (270:.6) {$1$}; \node at (330:.6) {$0$};
    \end{scope}
    \begin{scope}[shift={(18.25,-12)}]
      \node at (0,0) {$\Bigg\}$};
    \end{scope}
  \end{tikzpicture}
  \]
  \begin{enumerate}
  \item Draw the action diagram.
  \item Construct the ``\emph{fixed point table}'', which has a
    checkmark in row $g$ and column $s$ if $\phi(g).s=s$.
  \item Next to each $s\in S$ on your action diagram, write
    $\stab(s)$, the stabilizer subgroup, using its generators. Which
    subgroups of $D_6$ don't appear, and why?
  \item The fixed point set $\fix(g)$ of each $g\in D_6$ can be read
    off of the the fixed point table. What is the average size $|\fix(g)|$? 
  \item Find $\Ker(\phi)$ and $\Fix(\phi)$.
  \item Repeat the previous parts of this problem, but for the action
    of $G=D_3=\<r,f\>$ on $S$, where $r$ rotates each hexagon $120^\circ$
    counterclockwise. %\bigskip
  \end{enumerate}

\item Suppose that $G$ acts on $S$ via the homomorphism $\phi\colon
  G\to\Perm(S)$.
  \begin{enumerate}
  \item Show that $\stab(s)$ is a subgroup for all $s\in S$. Use the
    notational conventions that we have been using in lecture.
  \item Show that the stablizers of any two elements in the same orbit
    are conjugate -- specifically, that
    $\stab(s\Dot\phi(g))=g^{-1}\stab(s)g$ for all $g\in G$ and $s\in
    S$. This relationship is summarized by the following commutative
    diagram.
    \[
      \begin{tikzcd}[column sep=5em,row sep=3.0em]
s \arrow[r,mapsto,"\phi(x)"] \arrow[d,mapsto,swap,"\phi(g)"]
& s \arrow[d,mapsto,"\phi(g)"] \\   
s' \arrow[r,mapsto,dashed,"\phi(g^{-1}xg)"] & s'
      \end{tikzcd}
      \]
  \end{enumerate}

  %\newpage
    
  
\item Suppose a group $G$ of order $55$ acts on a set $S$ of size
  $14$. Let $s\in S$ be an arbitrary element.
  \begin{enumerate}
  \item What are the possible sizes of the orbit of $s$?
  \item What are the possible sizes of the stabilizer of $s$?
  \item Show that this action must have a fixed point.
  \item What is the fewest number of fixed points that this action can
    have? Justify your answer. \bigskip
  \end{enumerate}
  
  
\item Let $G=D_6=\<r,f\>$ act on its set $S=\{H\leq D_6\}$ of
  subgroups by conjugation, i.e.,
  \[
  \phi\colon G\longto\Perm(S)\,,\qquad \phi(g)=\mbox{\footnotesize the
    permutation that sends each $H\mapsto g^{-1}Hg$.}
  \]
  A Cayley diagram for $D_6$ and its subgroup lattice are shown below.
  
  \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{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]
  \[
  \hspace*{-5mm}
  \tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=6mm] 
  \begin{tikzpicture}
    \begin{scope}[shift={(0,4.5)},scale=1.5]
      \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] {$f$};
      \node (sr5) at (60:1) [v-small] {$rf$};
      \node (sr4) at (120:1) [v-small] {$r^2\!f$};
      \node (sr3) at (180:1) [v-small] {$r^3\!f$};
      \node (sr2) at (240:1) [v-small] {$r^4\!f$};
      \node (sr) at (300:1) [v-small] {$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[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}
    %%
    \begin{scope}[shift={(8,0)},scale=1.5]
      \tikzstyle{every node}=[font=\footnotesize]
      \node(D6) at (0,\Hf) {$D_6$};
      \node(r2-f) at (-1,\He) {$\<r^2,f\>$};
      \node(r2-rf) at (0,\He) {$\<r^2,rf\>$}; 
      \node(r) at (-2,\He) {$\<r\>$}; 
      \node(r3-f) at (1.75,\Hd) {$\<r^3,f\>$};
      \node(r3-rf) at (2.6,\Hd) {$\<r^3,rf\>$};
      \node(r3-r2f) at (3.5,\Hd) {$\<r^3,r^2\!f\>$};
      \node(r2) at (-2,\Hc) {$\<r^2\>$};
      \node(r3) at (0,\Hb) {$\<r^3\>$};
      \node(f) at (1,\Hb) {$\<f\>$};
      \node(r2f) at (1.5,\Hb) {$\<r^2\!f\>$};      
      \node(r4f) at (2,\Hb) {$\<r^4\!f\>$};      
      \node(rf) at (2.5,\Hb) {$\<rf\>$};
      \node(r3f) at (3,\Hb) {$\<r^3\!f\>$};
      \node(r5f) at (3.5,\Hb) {$\<r^5\!f\>$};
      \node (1) at (0,\Ha) {$\<1\>$};
      \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-f) to (f); 
      \draw (r2-f) -- (r2f);
      \draw (r2-f) to (r4f);
      \draw (r2-f) to (r2);
      \draw (r) -- (r2);
      \draw (r) -- (r3);
      \draw (r2-rf) -- (r2);
      \draw (r2-rf) to (rf);
      \draw (r2-rf) to (r3f);
      \draw (r2-rf) 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}
  \end{tikzpicture}
  \]
  \begin{enumerate}
  \item Construct the action diagram, and superimpose it on the
    subgroup lattice.
  \item Construct the fixed point table.
  \item Find $\stab(H)$ for each subgroup $H\leq D_6$, and $\fix(g)$
    for each $g\in D_6$.
  \item Find $\Ker(\phi)$ and $\Fix(\phi)$.
  \item Interpret $\orb(H)$, $\stab(H)$, $[G:\stab(H)]$, $\Fix(\phi)$,
    $\Ker(\phi)$, $\fix(g)$, and the average size of $|\fix(g)|$ in terms of
    familiar algebraic objects. 
  \end{enumerate}

    
\end{enumerate}

\end{document}



