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\pagestyle{fancy} \lhead{{\sf Homework 2 $|$ Due September 12 (Wednesday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra I $|$
    Fall 2018 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-6mm}


\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}
  \] 
  For the following actions, draw an action diagram
  and find the stabilizer of each element:
  \begin{enumerate}
    \item The action of $G=V_4=\<v,h\>$ on
      $S$, where $\phi(v)$ reflects each square vertically, and
      $\phi(h)$ reflects each square horizontally. 
    \item The action of $G=C_4=\<r\>$ on $S$, where $\phi(r)$ rotates each square $90^\circ$
      clockwise. \medskip
  \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}{5}
    \item[i.] $id$
    \item[ii.] $(1\;2)$
    \item[iii.] $(1\;2\;3)$ 
    \item[iv.] $(1\;2\;3\;4)$ 
    \item[v.] $(1\;2)\,(3\;4)$. \medskip 
    \end{multicols} 
  \end{enumerate}


\item Prove that if $H\leq S_n$ contains an odd permutation, then
   exactly half of its permutations are odd.~ \medskip 



\item Suppose $H,K\leq G$ both have finite index.
  \begin{enumerate}
    \item Show that there is some $N\normaleq G$ with $N\leq H$ such
      that $[G:N]<\infty$.
    \item Show that $[G:H\cap K]\leq[G:H][G:K]$. \medskip
  \end{enumerate}


\item Let $G$ be a finite group acting on a set $S$.
\begin{enumerate}
\item Prove that if $G$ has no subgroup of index $2$, then any
  subgroup of index $3$ is normal.
\item Prove that if $[G:H]=p$, the smallest prime dividing $|G|$, then
  $H\normaleq G$. \medskip
\end{enumerate}


% Grove 12.11
\item Given a left action, show that
    $\Stab_G(xs)=x\Stab_G(s)x^{-1}$ for all $x\in G$, $s\in S$.~ \medskip


% Grove 12.14
\item Permutation groups $G_1$ and $G_2$ acting on sets $S_1$ and
$S_2$ are called \emph{permutation isomorphic} if there exists an
isomorphism $\theta\colon G_1\to G_2$ and a bijection $\phi\colon
S_1\to S_2$ such that $(\theta x)(\phi s)=\phi(xs)$ for all $x\in G_1$
and $s\in S_1$. In other words, the following diagram commutes:
\[
\xymatrix{S_1\ar@{->}[r]^x\ar@{->}[d]_\phi & S_1 \ar@{->}[d]^\phi \\ 
  S_2\ar@{->}[r]_{\theta x} & S_2}
\]
Show that the following two actions of $G$ on itself are permutation isomorphic:
\begin{itemize}
  \item[(i)] the action of $x\in G$ is left multiplication by $x$;
  \item[(ii)] the action of $x\in G$ is right muliplication by $x^{-1}$. 
\end{itemize}




\end{enumerate}

\end{document}

