\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig,amsmath,latexsym}
\usepackage[all]{xy}
\usepackage{tikz}
\usepackage{enumerate}
\usepackage{multicol}
\usepackage{color}
\colorlet{darkgreen}{green!70!black}
\definecolor{darkblue}{rgb}{0, 0, .6}
\definecolor{midnightblue}{rgb}{.2, .2, .7}
\definecolor{grey}{rgb}{.7, .7, .7}
\definecolor{lightgrey}{rgb}{.85, .85, .85}
\definecolor{Red}{rgb}{.9,0,0}
\definecolor{Blue}{rgb}{0,0,.9}
\definecolor{Green}{rgb}{0,.9,0}
\colorlet{lred}{red!50!white}
\definecolor{lblue}{rgb}{0.60, 0.75, 1}
\colorlet{lgreen}{green!40!white}
\colorlet{lyellow}{yellow!30!white}
\definecolor{lpurple}{rgb}{0.7 0.45 0.9}
\colorlet{lorange}{orange!50!white}
\colorlet{darkpurple}{lpurple!70!black}

\usetikzlibrary{arrows}
\usetikzlibrary{decorations.markings}

\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=0in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in

\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\id}{id}

\def\<{\langle}
\def\>{\rangle}
\def\into{\hookrightarrow}
\def\longto{\longrightarrow}
\def\normal{\lhd}
\def\normaleq{\unlhd}
\DeclareMathOperator\cl{cl}
\DeclareMathOperator\Aut{Aut}
\DeclareMathOperator\Inn{Inn}
\DeclareMathOperator\Char{char}
\DeclareMathOperator\Gal{Gal}
\DeclareMathOperator\Stab{Stab}
\DeclareMathOperator\Fix{Fix}
\DeclareMathOperator\Perm{Perm}
\DeclareMathOperator{\Dic}{Dic}

\renewcommand\labelitemi{\rule[0.12em]{0.4em}{0.4em}}
\renewcommand\labelitemii{\normalfont\bfseries \rule[0.15em]{0.3em}{0.3em}}
\renewcommand\labelitemiii{\textasteriskcentered}
\renewcommand\labelitemiv{\textperiodcentered}

\renewcommand{\footrulewidth}{1pt}
\newenvironment{proof}[1][Proof.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{sol}[1][Solution.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{fminipage}%
{\setlength{\fboxsep}{15pt}\begin{Sbox}\begin{minipage}}%
{\end{minipage}\end{Sbox}\fbox{\TheSbox}}
\newcommand{\qed}{\hfill \ensuremath{\Box}}
\newcommand{\disp}{\displaystyle}


\pagestyle{fancy} \lhead{{\sf Homework 10 $|$ Due November 1 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Fall 2019 $|$ M.~Macauley}}



\begin{document}

$\;$ %\vspace{-4mm}

\noindent Read Chapters 9.2--9.4 of \emph{Visual Group Theory} (VGT),
or Chapters 13--14 of \emph{AATA}. Then write up solutions to the
following exercises. \\

\begin{enumerate}

\item Let $S$ be the following set of $7$ ``binary squares'':
  \[
  \begin{tikzpicture}[scale=1]
    \node at (-1,.5) {$S=\Bigg\{$};
    \node at (13.5,.5) {$\Bigg\}$};
    \node at (1.5,.25) {,};
    \node at (3.5,.25) {,};
    \node at (5.5,.25) {,};
    \node at (7.5,.25) {,};
    \node at (9.5,.25) {,}; 
    \node at (11.5,.25) {,};
    \path[fill=lorange] (0,.5) rectangle ++(.5,.5); 
    \path[fill=lorange] (.5,.5) rectangle ++(.5,.5);
    \path[fill=lorange] (0,0) rectangle ++(.5,.5);
    \path[fill=lorange] (.5,0) rectangle ++(.5,.5);
    \draw (.25,.75) node{\small $0$}; \draw (.75,.75) node{\small $0$};
    \draw (.25,.25) node{\small $0$}; \draw (.75,.25) node{\small $0$};
    \draw (0,0) rectangle (1,1);
    \begin{scope}[shift={(2,0)}]
      \path[fill=lorange] (0,.5) rectangle ++(.5,.5); 
      \path[fill=lpurple] (.5,.5) rectangle ++(.5,.5);
      \path[fill=lpurple] (0,0) rectangle ++(.5,.5);
      \path[fill=lorange] (.5,0) rectangle ++(.5,.5);
      \draw (0,0) rectangle (1,1);
      \draw (.25,.75) node{\small $0$}; \draw (.75,.75) node{\small $1$};
      \draw (.25,.25) node{\small $1$}; \draw (.75,.25) node{\small $0$};
    \end{scope}
    \begin{scope}[shift={(4,0)}]
      \path[fill=lpurple] (0,.5) rectangle ++(.5,.5); 
      \path[fill=lorange] (.5,.5) rectangle ++(.5,.5);
      \path[fill=lorange] (0,0) rectangle ++(.5,.5);
      \path[fill=lpurple] (.5,0) rectangle ++(.5,.5);
      \draw (0,0) rectangle (1,1);
      \draw (.25,.75) node{\small $1$}; \draw (.75,.75) node{\small $0$};
      \draw (.25,.25) node{\small $0$}; \draw (.75,.25) node{\small $1$};
    \end{scope}
    \begin{scope}[shift={(6,0)}]
      \path[fill=lpurple] (0,.5) rectangle ++(.5,.5); 
      \path[fill=lpurple] (.5,.5) rectangle ++(.5,.5);
      \path[fill=lorange] (0,0) rectangle ++(.5,.5);
      \path[fill=lorange] (.5,0) rectangle ++(.5,.5);
      \draw (0,0) rectangle (1,1);
      \draw (.25,.75) node{\small $1$}; \draw (.75,.75) node{\small $1$};
      \draw (.25,.25) node{\small $0$}; \draw (.75,.25) node{\small $0$};
    \end{scope}
    \begin{scope}[shift={(8,0)}]
      \path[fill=lorange] (0,.5) rectangle ++(.5,.5); 
      \path[fill=lpurple] (.5,.5) rectangle ++(.5,.5);
      \path[fill=lorange] (0,0) rectangle ++(.5,.5);
      \path[fill=lpurple] (.5,0) rectangle ++(.5,.5);
      \draw (0,0) rectangle (1,1);
      \draw (.25,.75) node{\small $0$}; \draw (.75,.75) node{\small $1$};
      \draw (.25,.25) node{\small $0$}; \draw (.75,.25) node{\small $1$};
    \end{scope}
    \begin{scope}[shift={(10,0)}]
      \path[fill=lorange] (0,.5) rectangle ++(.5,.5); 
      \path[fill=lorange] (.5,.5) rectangle ++(.5,.5);
      \path[fill=lpurple] (0,0) rectangle ++(.5,.5);
      \path[fill=lpurple] (.5,0) rectangle ++(.5,.5);
      \draw (0,0) rectangle (1,1);
      \draw (.25,.75) node{\small $0$}; \draw (.75,.75) node{\small $0$};
        \draw (.25,.25) node{\small $1$}; \draw (.75,.25) node{\small $1$};
    \end{scope}
    \begin{scope}[shift={(12,0)}]
      \path[fill=lpurple] (0,.5) rectangle ++(.5,.5); 
      \path[fill=lorange] (.5,.5) rectangle ++(.5,.5);
      \path[fill=lpurple] (0,0) rectangle ++(.5,.5);
      \path[fill=lorange] (.5,0) rectangle ++(.5,.5);
      \draw (0,0) rectangle (1,1);
      \draw (.25,.75) node{\small $1$}; \draw (.75,.75) node{\small $0$};
      \draw (.25,.25) node{\small $1$}; \draw (.75,.25) node{\small $0$};
    \end{scope}
  \end{tikzpicture}
  \] \vspace{-12mm}
  \begin{enumerate}
    \item Consider the (right) action of the group $G=V_4=\<v,h\>$ on
      $S$, where $\phi(v)$ reflects each square vertically, and
      $\phi(h)$ reflects each square horizontally. Draw an action
      diagram and compute the stabilizer of each element.
    \item Consider the (right) action of the group $G=C_4=\<r\mid
      r^4=e\>$ on $S$, where $\phi(r)$ rotates each square $90^\circ$
      clockwise. Draw an action diagram and compute the stabilizer of
      each element.
    \item Suppose a group $G$ of order $15$ acts on $S$. Show that
      there must be a fixed point. \bigskip
  \end{enumerate}

  

\item Let $G=S_4$ act on itself by conjugation via the homomorphism 
  \[
  \phi\colon G\longto\Perm(S)\,,\qquad\phi(g)=\mbox{\small the
    permutation that sends each $x\mapsto g^{-1}xg$.}
  \]
  \begin{enumerate}
  \item How many orbits are there? Describe them as specifically as you can.  
  \item Find the orbit and the stabilizer of the following elements: 
    \begin{multicols}{3}
    \item[i.] $e$
    \item[ii.] $(1\;2)$
    \item[iii.] $(1\;2\;3)$ 
    \item[iv.] $(1\;2\;3\;4)$ 
    \item[v.] $(1\;2)\,(3\;4)$ 
    \end{multicols} \smallskip
  \end{enumerate}
  


\item A \emph{$p$-group} is a group of order $p^k$ for some integer
  $k$. Recall that the \emph{center} of a group $G$ is the set of all
  elements that commute with everything:
  \begin{align*}
    Z(G)&=\{z\in G\mid gz=zg,\;\forall g\in G\}
    =\{z\in G\mid g^{-1}zg=z,\;\forall g\in G\}\,.
  \end{align*}
  Finally, a group $G$ is \emph{simple} if its only normal subgroups
  are $G$ and $\<e\>$.
  \begin{enumerate}
  \item Let $G$ act on itself by conjugation via the homomorphism
    \[
    \phi\colon G\longto\Perm(S)\,,\qquad\phi(g)=\mbox{\small the
      permutation that sends each $x\mapsto g^{-1}xg$.}
    \]
    Show that $\Fix(\phi)=Z(G)$. 
  \item Show that if $G$ is a $p$-group, then
    $|Z(G)|>1$. [\emph{Hint}: Revisit the Class Equation.]
  \item Use the result of the previous part to classify all simple
    $p$-groups. \bigskip
  \end{enumerate}
  

  \newpage

  
\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=25] (e);
        \draw [a] (b) to (ba);
        \draw [a] (ba) to (baa);
        \draw [a] (baa) to (baaa);
        \draw [a] (baaa) to [bend left=25] (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=25] (e);
          \draw [a] (b) to (ba);
          \draw [a] (ba) to (baa);
          \draw [a] (baa) to (baaa);
          \draw [a] (baaa) to [bend left=25] (b);
          \draw [b] (e) to (b);
        \end{scope}
      \end{tikzpicture}
    \end{center}

    \vspace{-1mm}
    
    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. \medskip
  \end{enumerate}
 

\item In this problem, we will find the Sylow subgroups of all
  $3$ nonabelian groups of order $12$.
  \begin{enumerate}
  \item Find all Sylow $2$-subgroups and Sylow $3$-subgroups of the
    groups whose subgroup lattices are shown below. Determine which
    are normal \vspace{-5mm}
    \begin{center}
      \begin{tikzpicture}[node distance=1cm,shorten >= -2pt, shorten <= -2pt]
        \begin{scope}[shift={(0,0)},scale=.8]
        \node(1) at (0,0) {\footnotesize $\left\{1\right\}$};
        \node (ghost) [above of=1,node distance=3cm] {};
        \node(C21) at (.75,2) {\footnotesize $\<(12)(34))\>$};
        \node(C22) at (3,2) {\footnotesize $\<(13)(24)\>$};
        \node(C23) at (5.25,2) {\footnotesize $\<(14)(23))\>$};
        \node(C34) at (-.75,3) {\footnotesize $\<(234)\>$};
        \node(C33) at (-2.25,3){\footnotesize $\<(134)\>$};
        \node(C32) at (-3.75,3){\footnotesize $\<(124)\>$};
        \node(C31) at (-5.25,3) {\footnotesize $\<(123)\>$};
        \node(V4) at (2.25,4) {\footnotesize $\<(12)(34),(13)(24)\>$};
        \node(A4) at (0,6) {\small $A_4$};
        \draw(A4)--(V4);
        \draw(A4)--(C31);
        \draw(A4)--(C32);
        \draw(A4)--(C33);
        \draw(A4)--(C34);
        \draw(C31)--(1);
        \draw(C32)--(1);
        \draw(C33)--(1);
        \draw(C34)--(1);
        \draw(V4)--(C21);
        \draw(V4)--(C22);
        \draw(V4)--(C23);
        \draw(C21)--(1);
        \draw(C22)--(1);
        \draw(C23)--(1);
    \end{scope}
        %%
    \begin{scope}[shift={(7.5,0)},scale=.8]
      \node(G) at (0,6) {\small $\Dic_{12}$};
      \node(b) at (-1.5,4.5) {\small $\<b\>$};
      \node(aa) at (0,1.5) {\small $\<a^2\>$};
      \node(bb) at (-2,2.25) {\small $\<b^2\>$};
      \node(a) at (.5,24/7) {\small $\<a\>$};
      \node(ab) at (1.75,24/7) { $\<ab\>$};
      \node(ba) at (3,24/7) {\small $\<ba\>$};
      \node(1) at (0,0) {\small $\left\<1\right\>$};
      \draw(1)--(aa); \draw(1)--(bb);
      \draw(b)--(aa); \draw(b)--(bb);
      \draw(G)--(a); \draw(G)--(ab); \draw(G)--(ba); \draw(G)--(b);
      \draw(aa)--(a); \draw(aa)--(ab); \draw(aa)--(ba);
      %\node at (0,-1) {\small $\Dic_{12}=\<a,b\mid a^4=b^3,\;bab=a\>$};
    \end{scope}
    \end{tikzpicture}
      \end{center}
    
    \item Find all Sylow subgroups of $D_6=\<r,f\mid
      r^6=f^2=e,\,rfr=f\>$, and determine which are normal. \medskip
  \end{enumerate}

    
\item Recall that a group $G$ is called \emph{simple} if its only
  normal subgroups are $G$ and $\{e\}$. Use group actions and/or the
  Sylow theorems to show the following.
  \begin{enumerate}
  \item There is no simple group of order $45=3^2\cdot 5$.
  \item There is no simple group of order $pq$, where $p<q$ and are
    both prime.
  \item There is no simple group of order $56=2^3\cdot 7$.
  \item There is no simple group of order $108=2^2\cdot 3^3$. 
  \item If $G$ has a subgroup $H$ with $[G:H]=p$, the smallest
    prime dividing $|G|$, then $H\normaleq G$, and hence $G$ cannot be
    simple. [\emph{Hint}: Let $G$ act on the right cosets of $H$.]
  \end{enumerate}
  
\end{enumerate}

\end{document}
