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\pagestyle{fancy} \lhead{{\sf Homework 6 $|$ Due May 30 (Monday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Summer I 2016 (online) $|$ 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 7 of \emph{Visual Group Theory} (VGT). \vspace{-2mm}
\item VGT Exercises 7.3, 7.7--7.10, 7.12, 7.13, 7.17--7.20,
  7.24--7.27, 7.30, 7.32--7.35.

\end{itemize} 


\noindent Write up solutions to the following exercises.

\begin{enumerate}


\item Let $G$ be the group whose Cayley diagram is shown below, and
  suppose $e$ is the identity element. Consider the subgroups
  $H=\<a\>=\{a,b,c,d,e\}$ and $K=\<j\>=\{e,j,o,t\}$.
  \begin{center}
    \colorlet{lightgrey}{black!15}
    \tikzstyle{v} = [circle, draw, fill=lightgrey,inner sep=0pt, 
      minimum size=5.5mm]
    \tikzstyle{r} = [draw, very thick, Red, -stealth]
    \tikzstyle{b} = [draw, very thick, Blue, -stealth]
    \begin{tikzpicture}[scale=.95,auto]
      \node (a) at (0,4.95) [v] {{\footnotesize $a$}};
      \node (b) at (1.5,4.95) [v] {{\footnotesize $b$}};
      \node (c) at (3,4.95) [v] {{\footnotesize $c$}};
      \node (d) at (4.5,4.95) [v] {{\footnotesize $d$}};
      \node (e) at (6,4.95) [v] {{\footnotesize $e$}};
      \node (f) at (0,3.3) [v] {{\footnotesize $f$}};
      \node (g) at (1.5,3.3) [v] {{\footnotesize $g$}};
      \node (h) at (3,3.3) [v] {{\footnotesize $h$}};
      \node (i) at (4.5,3.3) [v] {{\footnotesize $i$}};
      \node (j) at (6,3.3) [v] {{\footnotesize $j$}};
      \node (k) at (0,1.65) [v] {{\footnotesize $k$}};
      \node (l) at (1.5,1.65) [v] {{\footnotesize $l$}};
      \node (m) at (3,1.65) [v] {{\footnotesize $m$}};
      \node (n) at (4.5,1.65) [v] {{\footnotesize $n$}};
      \node (o) at (6,1.65) [v] {{\footnotesize $o$}};
      \node (p) at (0,0) [v] {{\footnotesize $p$}};
      \node (q) at (1.5,0) [v] {{\footnotesize $q$}};
      \node (r) at (3,0) [v] {{\footnotesize $r$}};
      \node (s) at (4.5,0) [v] {{\footnotesize $s$}};
      \node (t) at (6,0) [v] {{\footnotesize $t$}};
      \draw [b] (a) to (b);
      \draw [b] (b) to (c);
      \draw [b] (c) to (d);
      \draw [b] (d) to (e);
      \draw [b] (e) to [bend right=25] (a);
      \draw [b] (f) to [bend left=40] (h);
      \draw [b] (g) to [bend left=40] (i);
      \draw [b] (h) to [bend left=40] (j);
      \draw [b] (i) to [bend left=40] (f);
      \draw [b] (j) to [bend left=40] (g);
      \draw [b] (o) to (n);
      \draw [b] (n) to (m);
      \draw [b] (m) to (l);
      \draw [b] (l) to (k);
      \draw [b] (k) to [bend right=20] (o);
      \draw [b] (p) to [bend left] (s);
      \draw [b] (q) to [bend left] (t);
      \draw [b] (r) to [bend left=40] (p);
      \draw [b] (s) to [bend left=40] (q);
      \draw [b] (t) to [bend left=40] (r);
      \draw [r] (a) to (f);
      \draw [r] (b) to (g);
      \draw [r] (c) to (h);
      \draw [r] (d) to (i);      
      \draw [r] (e) to (j);
      \draw [r] (f) to (k);
      \draw [r] (g) to (l);
      \draw [r] (h) to (m);
      \draw [r] (i) to (n);
      \draw [r] (j) to (o);
      \draw [r] (k) to (p);
      \draw [r] (l) to (q);
      \draw [r] (m) to (r);
      \draw [r] (n) to (s);
      \draw [r] (o) to (t);
      \draw [r] (p) to [bend left=25] (a);
      \draw [r] (q) to [bend left=25] (b);
      \draw [r] (r) to [bend left=25] (c);
      \draw [r] (s) to [bend right=25] (d);
      \draw [r] (t) to [bend right=25] (e);
    \end{tikzpicture}
  \end{center}
  Carry out the following steps for both of these subgroups.
  \begin{enumerate}
    \item Write $G$ as a disjoint union of the subgroup's left cosets.
    \item Write $G$ as a disjoint union of the subgroup's right cosets.
    \item Compute the normalizer of the subgroup. 
    \item If the subgroup (call it $N$) is indeed normal, then compute
      the quotient $G/N$, draw its Cayley diagram, and label the nodes
      appropriately. \smallskip
  \end{enumerate}



\item All of the following statements are \emph{false}. For each one,
  exhibit an explicit counter-example, and justify your
  reasoning. Assume that each $H_i\lhd G_i$ for $i=1,2$.
  \begin{enumerate}
  \item If every proper subgroup $H$ of a group $G$ is cyclic, then
    $G$ is cyclic.
  \item If $K\lhd H\lhd G$, then $K\lhd G$.
  \item If $G_1\cong G_2$ and $H_1 \cong H_2$, then $G_1/H_1\cong
    G_2/H_2$.
  \item If $G_1\cong G_2$ and $G_1/H_1\cong G_2/H_2$, then $H_1\cong
    H_2$.
  \item If $H_1\cong H_2$ and $G_1/H_1\cong G_2/H_2$, then $G_1\cong
    G_2$. \smallskip
  \end{enumerate}



\item Prove the following ``subgroup criterion'', which can be very
  useful when trying to show that a subset is indeed a subgroup: A
  nonempty subset $H$ of a group $G$ is a subgroup if and only if
  $xy^{-1}\in H$ holds for all $x,y\in H$. \smallskip


\newpage

$\;$


\item Let $A$ be a subset of a group $G$. The \emph{centralizer} of
  $A$, denoted $C_G(A)$, is the set of all elements that commute with
  everything in $A$:
  \[
  C_G(A)=\{g\in G\mid ga=ag,\;\forall a\in A\}\,.
  \]
  \begin{enumerate}
    \item Prove that $C_G(A)$ is a subgroup of $G$. 
    \item If $A$ is additionally a subgroup of $G$, prove that
      $C_G(A)\normal N_G(A)$. \medskip
  \end{enumerate}


\item Let $H$ be a subgroup of an abelian group $G$. Prove that $H$
  and $G/H$ are both abelian. \medskip


\item Partition the following groups into conjugacy classes:
  \begin{enumerate}
  \item $\Z_4$;
  \item $Q_8$;
  \item $D_5$;
  \item $A_4$;
  \item $S_4$. \smallskip
  \end{enumerate}

 
\end{enumerate}

\end{document}


