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\pagestyle{fancy} \lhead{{\sf Homework 3 $|$ Due September 11 (Wednesday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Fall 2019 $|$ M.~Macauley}}



\begin{document}

$\;$ \vspace{-3mm}

\noindent Read Chapter 5 of \emph{Visual Group Theory}, or Chapter 6 of \emph{IBL Abstract Algebra}. Then write up solutions to the following exercises.

\begin{enumerate}

\item Carry out the following steps for the groups $A_4$ and $Q_8$,
  whose Cayley graphs are shown below.
%%
  \begin{center}
    \colorlet{lightgrey}{black!15}
    \tikzstyle{vert} = [circle, draw, fill=lightgrey,inner sep=0pt, 
      minimum size=5.5mm]
    \tikzstyle{vert1} = [circle, draw, fill=lightgrey,inner sep=0pt, 
      minimum size=7mm]
    \tikzstyle{r} = [draw, very thick, Red, -stealth]
    \tikzstyle{b2} = [draw, very thick, Blue]
    \tikzstyle{b} = [draw, very thick, Blue, -stealth]
    \begin{tikzpicture}[scale=.95,auto]
      \node (e) at (0,4) [vert] {{\small $e$}};
      \node (x) at (1.5,4) [vert] {{\small $x$}};
      \node (c) at (3,4) [vert] {{\small $c$}};
      \node (d) at (4.5,4) [vert] {{\small $d$}};
      \node (a) at (0,2) [vert] {{\small $a$}};
      \node (b) at (1.5,2) [vert] {{\small $b$}};
      \node (dd) at (3,2) [vert] {{\small $d^2$}};
      \node (bb) at (4.5,2) [vert] {{\small $b^2$}};
      \node (aa) at (0,0) [vert] {{\small $a^2$}};
      \node (cc) at (1.5,0) [vert] {{\small $c^2$}};
      \node (z) at (3,0) [vert] {{\small $z$}};
      \node (y) at (4.5,0) [vert] {{\small $y$}};
      \draw [b2] (e) to (x);
      \draw [b2] (a) to (c);
      \draw [b2] (b) to (d);
      \draw [b2] (aa) to (bb);
      \draw [b2] (cc) to (dd);
      \draw [b2] (z) to (y);
      \draw [r] (e) to (a);
      \draw [r] (a) to (aa);
      \draw [r] (aa) to [bend left] (e);
      \draw [r] (x) to (b);
      \draw [r] (b) to (cc);
      \draw [r] (cc) to [bend left] (x);
      \draw [r] (c) to (dd);
      \draw [r] (dd) to (z);
      \draw [r] (z) to [bend right] (c);
      \draw [r] (d) to (bb);
      \draw [r] (bb) to (y);
      \draw [r] (y) to [bend right] (d);
    \end{tikzpicture}
    \hspace{30mm}
    \begin{tikzpicture}[scale=.48,auto] 
      \node (1) at (135:2.5) [vert] {{\small $1$}};
      \node (i) at (45:2.5) [vert] {{\small $i$}};
      \node (-1) at (315:2.5) [vert] {{\small $-1$}};
      \node (-i) at (225:2.5) [vert] {{\small $-i$}};
      \node (k) at (135:5.5) [vert] {{\small $k$}};
      \node (-j) at (45:5.5) [vert] {{\small $-j$}};
      \node (-k) at (315:5.5) [vert] {{\small $-k$}};
      \node (j) at (225:5.5) [vert] {{\small $j$}};
      \draw [r] (1) to (i);
      \draw [r] (-k) to (-j);
      \draw [r] (-1) to (-i);
      \draw [r] (k) to (j);
      \draw [r] (j) to (-k);
      \draw [r] (-i) to (1);
      \draw [r] (-j) to (k);
      \draw [r] (i) to (-1);
      \draw [b] (1) to (j);
      \draw [b] (k) to (-i);
      \draw [b] (-1) to (-j);
      \draw [b] (-k) to (i);
      \draw [b] (j) to (-1);
      \draw [b] (-i) to (-k);
      \draw [b] (-j) to (1);
      \draw [b] (i) to (k);
    \end{tikzpicture}
  \end{center}
  %%
  \begin{enumerate}
  \item Find the orbit of each element.
  \item Draw the orbit graph of the group. \smallskip
  \end{enumerate}


  % VGT Exercise 5.38 (modified) 
\item Show algebraically that if $g^2=e$ for every element of a
  group $G$, then $G$ must be abelian.~ \medskip



\item Compute the product of the following permutations. Your answer
  for each should be a single permutation written in cycle notation as
  a product of disjoint cycles.
  \begin{enumerate}
  \item $(1\;3\;2)\,(1\;2\;5\;4)(1\;5\;3)$ in $S_5$;
  \item $(1\;5)\,(1\;2\;4\;6)\,(1\;5\;4\;2\;6\;3)$ in $S_6$. \medskip
  \end{enumerate}



\item Write out all $4!=24$ permutations in $S_4$ in cycle notation as
  a product of disjoint cycles. Additionally, write each as a product
  of transpositions, and decide if they are even or odd. Which of
  these permutations are also in $A_4$? \medskip


  
  % VGT Exercise 5.34 (modified)
\item 
  \begin{enumerate}
    \item The group $S_3$ can be generated by the transpositions $(1\;2)$
    and $(2\;3)$. In fact, it has the following presentation
    \[
    S_3=\<a,b\mid a^2=e,\,b^2=e\,,(ab)^3=e\>,
    \]
    where one can take $a=(1\;2)$ and $b=(2\;3)$. Make a Cayley
    diagram for $S_3$ using this generating set.
  \item Make a Cayley diagram for the group generated by the
    permutations $a=(1\;2)$ and $c=(3\;4)$, and write down a group
    presentation for this.
  \item The group $S_4$ can be generated by the transpositions
    $(1\;2)$, $(2\;3)$, and $(3\;4)$. Make a Cayley diagram for $S_4$
    using this generating set. This can be laid out on a polytope
    called a \emph{permutahedron}, which is a truncated octahedron
    shown below. Make a Cayley graph by labeling the vertices on the
    unlabeled permutahedron with the $24$ permutations of $S_4$
    \emph{in cycle notation}, and color the edges appropriately. Let
    the vertex at the top denote the identity
    permutation. \hspace{-8mm}

    \begin{center}
      \includegraphics[width=0.5\textwidth]{permutahedron-unlabeled}
    \end{center}


  \item Write down a group presentation for $S_4$ using the generators
    $a,b,c$ as defined in Parts (a) and (b). \bigskip
  \end{enumerate}





\item The Cayley diagram for $A_4$ shown above labels the elements
  with letters instead of permutations:
  \[
  A_4=\{e,a,a^2,b,b^2,c,c^2,d,d^2,x,y,z\}.
  \]
  Redraw this Cayley diagram but label the nodes with the $12$ even
  permutations from the previous problem. That is, you need to
  determine which permutation corresponds to $a$, which to $b$, and so
  on. [\emph{Hint}: There are many possible ways to do this. If you
    let $a$ be any one of the permutations of order 3, and let $x$ be any one
    of the permutations of order 2, then you will be able to
    determine the remaining elements.]
    
\end{enumerate}

\end{document}


