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\pagestyle{fancy} \lhead{{\sf Homework 11 $|$ Due April 4 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Spring 2014 $|$ 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 Chapters 9.3, 9.4 of \emph{Visual Group Theory} (VGT). \vspace{-2mm}
\item VGT Exercises 9.1, 9.4, 9.12, 9.14, 9.15, 9.21--9.27. \smallskip

\end{itemize} 


\noindent Write up solutions to the following exercises.

\begin{enumerate}


\item Let $G$ be an unknown group of order $8$. By the First Sylow
  Theorem, $G$ must contain a subgroup $H$ of order $4$.
  
  \begin{enumerate}
  \item If all subgroups of $G$ of order 4 are isomorphic to $V_4$,
    then what group must $G$ be? Completely justify your answer. 
  \item Next, suppose that $G$ has a subgroup $H\cong C_4$. Then $G$
    has a Cayley diagram like one of the following:
    \colorlet{lightgrey}{black!15}
    \tikzstyle{v} = [circle, draw, fill=lightgrey,inner sep=0pt, 
      minimum width=4.5mm]
    \tikzstyle{a} = [draw, very thick, Red, -stealth]
    \tikzstyle{b} = [draw, very thick, Blue, -stealth]
    \tikzstyle{b2} = [draw, very thick, Blue]
    
    \begin{center}
      \begin{tikzpicture}[scale=1.4,auto]
        \node (b) at (0,0) [v] {{\tiny $b$}};
        \node (e) at (0,1) [v] {{\tiny $e$}};
        \node (ba) at (1,0) [v] {{\tiny $ba$}};
        \node (a) at (1,1) [v] {{\tiny $a$}};
        \node (baa) at (2,0) [v] {{\tiny $ba^{\!2}$}};
        \node (aa) at (2,1) [v] {{\tiny $a^{\!2}$}};
        \node (baaa) at (3,0) [v] {{\tiny $ba^{\!3}$}};
        \node (aaa) at (3,1) [v] {{\tiny $a^{\!3}$}};
        \draw [a] (e) to (a);
        \draw [a] (a) to (aa);
        \draw [a] (aa) to (aaa);
        \draw [a] (aaa) to [bend right] (e);
        \draw [a] (b) to (ba);
        \draw [a] (ba) to (baa);
        \draw [a] (baa) to (baaa);
        \draw [a] (baaa) to [bend left] (b);
        \draw [b2] (e) to (b);
        %%
        \begin{scope}[shift={(6,0)}]
          \node (b) at (0,0) [v] {{\tiny $b$}};
          \node (e) at (0,1) [v] {{\tiny $e$}};
          \node (ba) at (1,0) [v] {{\tiny $ba$}};
          \node (a) at (1,1) [v] {{\tiny $a$}};
          \node (baa) at (2,0) [v] {{\tiny $ba^{\!2}$}};
          \node (aa) at (2,1) [v] {{\tiny $a^{\!2}$}};
          \node (baaa) at (3,0) [v] {{\tiny $ba^{\!3}$}};
          \node (aaa) at (3,1) [v] {{\tiny $a^{\!3}$}};
          \draw [a] (e) to (a);
          \draw [a] (a) to (aa);
          \draw [a] (aa) to (aaa);
          \draw [a] (aaa) to [bend right] (e);
          \draw [a] (b) to (ba);
          \draw [a] (ba) to (baa);
          \draw [a] (baa) to (baaa);
          \draw [a] (baaa) to [bend left] (b);
          \draw [b] (e) to (b);
        \end{scope}
      \end{tikzpicture}
    \end{center}
    
    Find all possibilities for finishing the Cayley
    diagram. \smallskip
  
  \item Label each completed Cayley diagram by isomorphism
    type. Justify your answer.
  \item Make a complete list of all groups of order $8$, up to
    isomorphism.
  \end{enumerate}
 


\item Prove that a Sylow $p$-subgroup of $G$ is normal if and only
  if it is the unique Sylow $p$-subgroup of $G$. \smallskip


  
\item Recall that a group $G$ is called \emph{simple} if its only
  normal subgroups are $G$ and $\{e\}$.
  \begin{enumerate}
  \item Show that there is no simple group of order $45=3^2\cdot
    5$. 
  \item Show that there is no simple group of order $pq$, where
    $p<q$ and are both prime.
  \item Show that there is no simple group of order $12=2^2\cdot 3$.
  \item Show that there is no simple group of order $56=2^3\cdot
    7$. \smallskip
  \end{enumerate}



\item Suppose that $H\lneq G$, and let $S=G/H$, the set of right
  cosets of $H$ in $G$. 
  \begin{enumerate}
  \item Show that if $|G|$ does not divide $[G:H]!$, then $G$ cannot
    be simple. [\emph{Hint}: Consider the action of $G$ on $S$, where
      $\phi(g)\colon Hx\mapsto Hxg$. Prove that
      $\{e\}\lneq\ker\phi\lneq G$.]
  \item Use Part~(a), together with the Sylow theorems, to show that
    any group of order $108$ cannot be simple.
  \end{enumerate}


    
  %\item Let $G$ be a simple group of order $168$. Show that $G$ is
  %    a subgroup of $A_8$, the alternating group. (Hint: look at the
  %    Sylow subgroups of $G$.

  

\end{enumerate}


\end{document}

