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\pagestyle{fancy} \lhead{{\sf Homework 6 $|$ Due Wednesday, October 2
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Fall 2019 $|$ M.~Macauley}}



\begin{document}

$\;$

\noindent Read Chapter 7 of \emph{Visual Group Theory}, or Chapter 8
of \emph{IBL Abstract Algebra}, and then 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
  $A=\<a\>=\{a,b,c,d,e\}$ and $J=\<j\>=\{e,j,o,t\}$.
  \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=1.05,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. List the
  cosets element-wise.
  \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 Attempt the quotient process, by shrinking the left cosets
      into individual nodes. The the resulting diagram and determine
      whether it is a valid Cayley diagram, and if so, which familiar
      group the quotient is isomorphic to.
    \item Find all conjugates of the subgroup. \bigskip
  \end{enumerate}



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



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


  $\;$ \vspace{-2mm}

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



\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 $A$:
  \[
  C_G(A)=\{g\in G\mid ga=ag,\;\forall a\in A\}\,.
  \]
  If $A=\{a\}$, then we denote the centralizer as $C_G(a)$. 
  \begin{enumerate}
  \item Prove that $C_G(A)$ is a subgroup of $G$. 
  \item For $D_4$, and $Q_8$, compute the centralizers of each
    element.
  \item Compute the centralizers of the following elements in $S_4$:
    $e$, $(12)$, $(123)$, $(1234)$, and $(12)(34)$. 
  \item If $A$ is a subgroup of $G$, prove that $C_G(A)\normaleq
    N_G(A)$. \bigskip 
  \end{enumerate}

  

\item Recall that for any group $G$, conjugacy is an equivalence
  relation on $G$, and the \emph{conjugacy class} of an element $x\in
  G$ is $\cl_G(x)=\{gxg^{-1}\mid g\in G\}$.
\begin{enumerate}
\item Partition the following groups into conjugacy classes:
  \begin{multicols}{2}
    \begin{enumerate}[(i)]
    \item $\Z_4$; 
    \item $D_5$; 
    \item $D_8$; 
    \item $Q_8$; 
    \item $S_4$;
    \item $A_4$. \bigskip
    \end{enumerate}
  \end{multicols}
\item For each of the following elements $\sigma\in S_4$: $e$, $(12)$,
  $(123)$, $(1234)$, and $(12)(34)$, compare the size of its
  centralizer $C_{S_4}(\sigma)$ to the size of its conjugacy class,
  $\cl_{S_4}(\sigma)$. What do you notice about the product of these
  two numbers?
\end{enumerate}

\end{enumerate}

\end{document}


