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\pagestyle{fancy} \lhead{{\sf Homework 7 $|$ Due March 3 (Monday)
}} \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 8.1, 8.2 of \emph{Visual Group Theory} (VGT). \vspace{-2mm}
\item VGT Exercises 8.2, 8.5, 8.6, 8.9, 8.11, 8.13, 8.14.

\end{itemize} 



\noindent Write up solutions to the following exercises.

\begin{enumerate}


\item Do the following steps for each mapping $\phi\colon G\to H$
  listed below.
  \begin{enumerate}[(i)]
    \item Determine whether $\phi$ is a homomorphism. 
    \item Find $\ker\phi:=\{g\in G\mid \phi(G)=e\}$ and $\im\phi=\phi(G)$. 
    \item Draw Cayley diagrams of the domain and codomain, and
      arrange them so one can ``visually see'' the cosets of
      $\ker\phi$ in $G$. Draw dotted lines around these cosets.
    \item Is the quotient $G/\ker\phi$ a group? If so, what is it
      isomorphic to?
  \end{enumerate}
  Here is an example of Step~(iii) for the map
  $\phi\colon\Z_6\to\Z_3$, defined by $\phi(n)=n\pmod{3}$.

  \tikzstyle{v} = [circle, draw, fill=grey,inner sep=0pt, minimum size=4mm] 
  \tikzstyle{r} = [draw,very thick, Red,-stealth]
  \tikzstyle{b} = [draw,very thick, Blue]
  \tikzstyle{g} = [draw, very thick, darkgreen,-stealth]
  \tikzstyle{to} = [draw, thick,-stealth]  

  \vspace{-3mm}
  \begin{center}
    \begin{tikzpicture}[scale=.4,auto]
      \node (3) at (90:2) [v] {{\tiny $3$}};
      \node (5) at (-30:2) [v] {{\tiny $5$}};
      \node (1) at (210:2) [v] {{\tiny $1$}};
      \node (0) at (90:4.5) [v] {{\tiny $0$}};
      \node (4) at (210:4.5) [v] {{\tiny $4$}};
      \node (2) at (-30:4.5) [v] {{\tiny $2$}};
      \draw [r] (3) to [bend left] (5);
      \draw [r] (5) to [bend left] (1);
      \draw [r] (1) to [bend left] (3);
      \draw [r] (0) to [bend left] (2);
      \draw [r] (2) to [bend left] (4);
      \draw [r] (4) to [bend left] (0);
      \draw [b] (3) to (0);
      \draw [b] (5) to (2);
      \draw [b] (1) to (4);
      \draw [to] (5,0) to (12,0);
      \node at (8.5,.75) {$\phi$};
      \draw [dashed] (0,3.25) ellipse (1.25 and 2.5);
      \draw [dashed, rotate=120] (0,3.25) ellipse (1.35 and 2.7);
      \draw [dashed, rotate=240] (0,3.25) ellipse (1.35 and 2.7);
    \begin{scope}[shift={(15,0)}]
      \node (1) at (-30:2.5) [v] {{\tiny $1$}};
      \node (0) at (90:2.5) [v] {{\tiny $0$}};
      \node (2) at (210:2.5) [v] {{\tiny $2$}};
      \draw [g] (0) to [bend left] (1);
      \draw [g] (1) to [bend left] (2);
      \draw [g] (2) to [bend left] (0);
    \end{scope}
    \end{tikzpicture}
  \end{center}
  \vspace{-2mm}

  Now, do steps (i)--(iv) for the following maps.
  \begin{enumerate}[(a)]
  \item The map $\phi\colon\Z\to\Z$ defined by $\phi(n)=4n$.
  \item The map $\phi\colon D_4\to\Z_4$ defined by $\phi(r^kf)=k$.
  \item The map $\phi\colon D_4\to\Z_4$ defined by $\phi(r^kf)=2k$.
  \item The map $\phi\colon \Z_4\to D_4$ defined by $\phi(k)=r^k$.
  \item The map $\phi\colon D_4\to V_4$ defined by $\phi(r)=h$ and
    $\phi(f)=v$. \medskip
  \end{enumerate}



\item Prove that $A\times B\cong B\times A$. \medskip



\item For each part below, list \emph{all} homomorphisms with the given
  domain and codomain.
  \begin{enumerate}
  \item Domain $\Z_{15}$ and codomain $\Z_4$.
  \item Domain $\Z_{412}$ and codomain $\Z_{450}$.
  \item Domain and codomain both $\Z_4$.
  \item Domain $C_4$ and codomain $V_4$.
  \item Domain and codomain both $V_4$. 
  \end{enumerate}


\newpage

$\;$

\item Prove that there is no embedding $\phi\colon\Z_n\into\Z$. \medskip



\item Let $H\leq G$, and fix $x\in G$. Recall that we showed in class
  that $xHx^{-1}$ is always a subgroup of $G$. 
  \begin{enumerate}
  \item Prove additionally that $xHx^{-1}\cong H$. [\emph{Hint}:
    Define a mapping from $H$ to $xHx^{-1}$ and prove that it is a
    homomorphism, one-to-one, and onto.]
  \item Use Part~(a) to show that $|xy|=|yx|$ for any $x,y\in G$. \medskip
  \end{enumerate}



\item Let $\phi\colon G\to H$ be a homomorphism, and $N\normal H$. 
  \begin{enumerate}[(i)]
  \item Show that the set $\phi^{-1}(N):=\{g\in G\mid\phi(g)\in N\}$
    is a subgroup of $G$. 
  \item Show that $\phi^{-1}(N)$ is a \emph{normal} subgroup of $G$.
  \item Show by example that if $M\normal G$, then $\phi(M)$ need not
    be a normal subgroup of $H$.
  \end{enumerate}


\end{enumerate}

\end{document}


