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\pagestyle{fancy} \lhead{{\sf Homework 9 $|$ Due June 1 (Wednesday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Summer I 2016 (online) $|$ M.~Macauley}}



\begin{document}

$\;$

\noindent Read the following, which can all be found either in the
textbook or on the course website.
\begin{itemize}
\item Chapter 9.1 of \emph{Visual Group Theory} (VGT). \vspace{-2mm}
\item VGT Exercises 8.15--8.18, 9.17.
\end{itemize} 



\noindent Write up solutions to the following exercises.

\begin{enumerate}


\item The commutator subgroup of a group $G$ is the subgroup
  \[
  G'=\<aba^{-1}b^{-1}\mid a,b\in G\>\,.
  \]
  \begin{enumerate}
  \item Prove that $G$ is abelian if and only if $G'=\{e\}$.
  \item Prove that $G'\normal G$. [\emph{Hint}: Take a ``commutator''
    $c=aba^{-1}b^{-1}$ and prove that $gcg^{-1}\in G'$.]
  \item Prove that $G'$ is the intersection of all normal subgroups of $G$
    that contain the set $C:=\{aba^{-1}b^{-1}\mid a,b\in G\}$:
    \[
    G'=\bigcap_{C\subseteq N\normal\,G} N
    \]
  \item If we quotient $G$ by $G'$, then we are in essence,
    ``killing'' all non-abelian parts of the Cayley diagram, as shown
    below:
    
    \colorlet{lightgrey}{black!15}
    \tikzstyle{v} = [circle, draw, fill=lightgrey,inner sep=0pt,
      minimum size=2.5mm] \tikzstyle{r} = [draw,very thick, Red,
      -stealth] \tikzstyle{b} = [draw,very thick, Blue, -stealth]
    \begin{center}
      \begin{tikzpicture}[scale=.8,auto]
        \node at (-1.75,0) {$ab=ba$};
        \node (e) at (0,0) [v] {{\tiny $*$}};
        \node (a) at (1,-1) [v] {};
        \node (b) at (1,1) [v] {};
        \node (ab) at (2,0) [v] {};
        \draw [b] (e) to (a);
        \draw [r] (a) to (ab);
        \draw [r] (e) to (b);
        \draw [b] (b) to (ab);
        \begin{scope}[shift={(8,0)}]
          \node at (-1.75,0) {$ab\neq ba$};
          \node (e) at (0,0) [v] {{\tiny $*$}};
          \node (a) at (1,1) [v] {};
          \node (b) at (2.3,1) [v] {};
          \node (c) at (3.3,0) [v] {};
          \node (d) at (2.3,-1) [v] {};
          \draw [r] (e) to (a);
          \draw [b] (a) to (b);
          \draw [r] (c) to (b);
          \draw [b] (d) to (c);
        \end{scope}
      \end{tikzpicture}
    \end{center}
    Prove algebraically that $G/G'$ is indeed abelian. \medskip
  \end{enumerate}



\item For each of the following groups $G$, compute its commutator
  subgroup $G'$ and its abelianization $G/G'$. Finally, draw the
  subgroup lattice of $G$ and circle every normal subgroup, and circle
  twice the one that is $G'$.
  \begin{enumerate}
    \item $V_4$ 
    \item $D_3$
    \item $Q_8$ \medskip
  \end{enumerate}



\item Find the commutator subgroup of each of the following groups and
  compute its abelianization.
  \begin{enumerate}
    \item An abelian group $A$.
    \item The alternating group $A_n$, for $n\geq 5$. [\emph{Hint}: $A_n$ is a
      \emph{simple group}, which means its only normal subgroups are
      $\<e\>$ and $A_n$.]
    \item The dihedral group $D_n$ for $n$ even.
    \item The dihedral group $D_n$ for $n$ odd. \medskip
  \end{enumerate}


\newpage

$\;$\vspace{-2mm}


\item For each group $G$, find all automorphisms and make a
  multiplication table of $\Aut(G)$. What group is it isomorphic to?
  \begin{enumerate}
    \item $\Z_7$
    \item $\Z_8$
    \item $\Z_{10}$
    \item $V_4$
    \item $D_3$ \medskip
  \end{enumerate}


\item Let $G$ act on a set $S$. Prove that $\Stab(s)$ is a subgroup of
  $G$ for every $s\in S$. \medskip



\item If $C_5$ acts on the set $S=\{A,B,C,D\}$, what will the action
  diagram be? Why? \medskip



\end{enumerate}

\end{document}



