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\pagestyle{fancy} \lhead{{\sf Homework 10 $|$ Due June 7 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Summer I 2019 $|$ M.~Macauley}}



\begin{document}

$\;$ \vspace{-4mm}

\noindent Read the following, which can all be found either in the
textbook or on the course website.
\begin{itemize}
\item Chapters 9.2--9.4 of \emph{Visual Group Theory} (VGT). \vspace{-2mm}
\item VGT Exercises 9.1, 9.4, 9.12, 9.14, 9.15, 9.19--9.27. \smallskip
\end{itemize} 


\noindent Write up solutions to the following exercises.

\begin{enumerate}

\item Let $S$ be the following set of $7$ ``binary squares'':
  \[
  \begin{tikzpicture}[scale=.75]
    \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 $S$. Prove 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
    \[
    \phi\colon G\longto\Perm(S)\,,\qquad\phi(g)=\mbox{\small the
      permutation that sends each $x\mapsto g^{-1}xg$.}
    \]
    Prove that $\Fix(\phi)=Z(G)$. 
  \item Prove 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}
  

\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. \bigskip
  \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 A \emph{field} is a set $\F$ containing $0$ and $1$ that is an
  abelian group under addition, and (upon removing $0$) an abelian
  group under multiplication, for which the distributative law
  holds. Common examples of fields are $\Q$, $\R$, and $\C$. \medskip

  There is a unique \emph{finite} field $\F_q$ of order $q=p^k$ for
  every prime $p$ and positive integer $k$. For all other $q\in\N$,
  there is no finite field of order $q$. For each of the fields
  $\F_4$, $\F_5$, and $\F_8$, the Cayley diagrams for addition and
  multiplication are shown below, overlayed on the same set of
  nodes. The solid arrows are the Cayley diagrams for addition and the
  dashed arrows are the Cayley diagrams for multiplication.
  
  \tikzstyle{v} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=5mm] 
  \tikzstyle{g-v} = [circle, draw, fill=darkgreen,inner sep=0pt,
    minimum size=6mm] 
  \tikzstyle{b} = [draw,very thick, Blue]
  \tikzstyle{g} = [draw,very thick, darkgreen]
  \tikzstyle{g-dashed} = [draw,very thick, darkgreen, dashed, -stealth]
  \tikzstyle{r} = [draw, very thick, Red,-stealth,bend left=20]
  \tikzstyle{r2} = [draw, very thick, Red]
  \tikzstyle{p} = [draw, very thick, dashed, darkpurple,-stealth]
  \begin{center}
    \begin{tikzpicture}[scale=1]
      \node (0) at (135:1.5) [v] {{\small $0$}};
      \node (1) at (45:1.5) [v] {\small {$a$}};
      \node (3) at (-135:1.5) [v] {{\small $c$}};
      \node (2) at (-45:1.5) [v] {{\small $b$}};   
      \draw [b] (0) to (1);
      \draw [b] (2) to (3);
      \draw [r2] (0) to (3);
      \draw [r2] (1) to (2);
      \draw [g-dashed,bend left=20] (3) to (1);
      \draw [g-dashed,bend left=40] (1) to (2);
      \draw [g-dashed,bend left=40] (2) to (3);
    \begin{scope}[shift={(5.5,0)}]
      \node (0) at (90:1.5) [v] {{\small $0$}};
      \node (1) at (18:1.5) [v] {\small {$a$}};
      \node (2) at (-54:1.5) [v] {{\small $b$}};
      \node (3) at (234:1.5) [v] {{\small $c$}};
      \node (4) at (162:1.5) [v] {{\small $d$}};
      \draw [r] (0) to (1);
      \draw [r] (1) to (2);
      \draw [r] (2) to (3);
      \draw [r] (3) to (4);
      \draw [r] (4) to (0);
      \draw [g-dashed,bend right=20] (1) to (2);
      \draw [g-dashed,bend right=20] (2) to (4);
      \draw [g-dashed,bend left=20] (4) to (3);
      \draw [g-dashed,bend left=20] (3) to (1);
    \end{scope}
    \begin{scope}[shift={(10,-1.25)}]
      \node (2) at (0,0) [v] {\footnotesize $b$};
      \node (0) at (-0.1,2) [v] {\footnotesize $0$};
      \node (3) at (2,-.4) [v] {\footnotesize $c$};
      \node (1) at (2.1,1.7) [v] {\footnotesize $a$};
      \node (6) at (1,1.1) [v] {\footnotesize $f$};
      \node (4) at (.9,3) [v] {\footnotesize $d$};
      \node (7) at (2.7,.8) [v] {\footnotesize $g$};
      \node (5) at (2.8,2.8) [v] {\footnotesize $e$}; 
      \draw [r2] (0) to (1);
      \draw [r2] (2) to (3);
      \draw [b] (0) to (4);
      \draw [b] (1) to (5);
      \draw [b] (2) to (6);
      \draw [b] (3) to (7);
      \draw [g] (0) to (2);
      \draw [g] (1) to (3);
      \draw [r2] (4) to (5);
      \draw [r2] (6) to (7);
      \draw [g] (4) to (6);
      \draw [g] (5) to (7);
      \draw [p,bend left=20] (1) to (2);
      \draw [p,bend left=10] (2) to (4);
      \draw [p] (4) to (3);
      \draw [p,bend left=35] (3) to (6);
      \draw [p,bend right=40] (6) to (7);
      \draw [p,bend right=40] (7) to (5);
      \draw [p,bend right=55] (5) to (1);
    \end{scope}
    \end{tikzpicture}
  \end{center}
  \begin{enumerate}
  \item For each field above, determine whether or not the addition
    and multiplication operations are in fact, addition and
    multiplication modulo some number. If yes, relabel the vertices
    accordingly. If no, explain why it fails.
  \item Create Cayley diagrams for the finite fields $\F_3$ and
    $\F_7$. \smallskip
  \end{enumerate}




\end{enumerate}


\end{document}



