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\pagestyle{fancy} \lhead{{\sf Homework 2 $|$ Due May 19 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Summer I 2017 (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 3 \& 4 of \emph{Visual Group Theory} (VGT). \vspace{-2mm}
\item VGT Exercises 3.5--3.10, 3.12. 4.3--4.5, 4.10--4.14. 

\end{itemize} 



\noindent Write up solutions to the following exercises.

\begin{enumerate}

\item Draw a Cayley diagram for each of the $7$ frieze groups using a
  minimal generating set. Make it clear what your generators are and
  write out a group presentation for each.



\item Shown below are the Cayley graphs of two groups: $D_4$ is on the
  left, and the right is called an ``alternating group,'' denoted $A_4$.
  \begin{center}
    \colorlet{lightgrey}{black!15}
    \tikzstyle{vert} = [circle, draw, fill=lightgrey,inner sep=0pt, 
      minimum size=5.5mm]
    \tikzstyle{vert1} = [circle, draw, fill=lightgrey,inner sep=0pt, 
      minimum size=7mm]
    \tikzstyle{r} = [draw, very thick, Red, -stealth]
    \tikzstyle{b} = [draw, very thick, Blue]
    \begin{tikzpicture}[scale=.5,auto] 
      \node (1) at (135:2.5) [vert1] {$e$};
      \node (i) at (45:2.5) [vert1] {$r$};
      \node (-1) at (315:2.5) [vert1] {$r^2$};
      \node (-i) at (225:2.5) [vert1] {$r^3$};
      \node (k) at (135:5.5) [vert1] {$f$};
      \node (-j) at (45:5.5) [vert1] {$rf$};
      \node (-k) at (315:5.5) [vert1] {$r^2\!f$};
      \node (j) at (225:5.5) [vert1] {$r^3\!f$};
      \draw [r] (1) to (i);
      \draw [r] (-k) to (-j);
      \draw [r] (-1) to (-i);
      \draw [r] (k) to (j);
      \draw [r] (j) to (-k);
      \draw [r] (-i) to (1);
      \draw [r] (-j) to (k);
      \draw [r] (i) to (-1);
      \draw [b] (1) to (k);
      \draw [b] (i) to (-j);
      \draw [b] (-i) to (j);
      \draw [b] (-k) to (-1);
      %\node at (0,0) {$D_4$};
    \end{tikzpicture}
    \hspace{30mm}
    \begin{tikzpicture}[scale=.95,auto]
      \node (e) at (0,4) [vert] {{\small $e$}};
      \node (x) at (1.5,4) [vert] {{\small $x$}};
      \node (c) at (3,4) [vert] {{\small $c$}};
      \node (d) at (4.5,4) [vert] {{\small $d$}};
      \node (a) at (0,2) [vert] {{\small $a$}};
      \node (b) at (1.5,2) [vert] {{\small $b$}};
      \node (dd) at (3,2) [vert] {{\small $d^2$}};
      \node (bb) at (4.5,2) [vert] {{\small $b^2$}};
      \node (aa) at (0,0) [vert] {{\small $a^2$}};
      \node (cc) at (1.5,0) [vert] {{\small $c^2$}};
      \node (z) at (3,0) [vert] {{\small $z$}};
      \node (y) at (4.5,0) [vert] {{\small $y$}};
      \draw [b] (e) to (x);
      \draw [b] (a) to (c);
      \draw [b] (b) to (d);
      \draw [b] (aa) to (bb);
      \draw [b] (cc) to (dd);
      \draw [b] (z) to (y);
      \draw [r] (e) to (a);
      \draw [r] (a) to (aa);
      \draw [r] (aa) to [bend left] (e);
      \draw [r] (x) to (b);
      \draw [r] (b) to (cc);
      \draw [r] (cc) to [bend left] (x);
      \draw [r] (c) to (dd);
      \draw [r] (dd) to (z);
      \draw [r] (z) to [bend right] (c);
      \draw [r] (d) to (bb);
      \draw [r] (bb) to (y);
      \draw [r] (y) to [bend right] (d);
    \end{tikzpicture}
  \end{center}
  \begin{enumerate}
  \item Create a multiplication table for each group. For consistency,
    order the elements in $D_4$ by $(e,r,r^2,r^3,f,rf,r^2f,r^3f)$ and
    by $(e,x,y,z,a,a^2,b,b^2,c,c^2,d,d^2)$ in $A_4$.
  \item Find the inverse of each element of each group.
  \item Write out a group presentation for each group using the
    generators shown in the Cayley graph.
  \end{enumerate}



\item In each of the following multiplication tables, let $e$ denote
  the identity element. Complete each table so its depicts a
  group. There may be more than one way to complete a table, in which
  case you need to give all possibilities.

  \begin{tikzpicture}[scale=0.75]
    \foreach \i in {1,...,3} {
      \draw [very thin] (\i,1) -- (\i,3); 
      \draw [very thin] (\i,3+.1) -- (\i,3+1); 
      \draw [very thin] (1,\i) -- (3,\i); 
      \draw [very thin] (0,\i) -- (.9,\i); 
    } 
    \draw [very thin] (1,3+.1) rectangle (3,3+1);
    \draw [very thin] (0,1) rectangle (.9,3); 
    \node at (0.5,2.5) {$e$}; 
    \node at (0.5,1.5) {$a$};
    %
    \node at (1.5,3.5) {$e$}; 
    \node at (2.5,3.5) {$a$};
    %%
  \end{tikzpicture}
  \hspace{20mm}
  \begin{tikzpicture}[scale=0.75]
    \foreach \i in {1,...,4} {
      \draw [very thin] (\i,1) -- (\i,4); 
      \draw [very thin] (\i,4+.1) -- (\i,4+1); 
      \draw [very thin] (1,\i) -- (4,\i); 
      \draw [very thin] (0,\i) -- (.9,\i); 
    } 
    \draw [very thin] (1,4+.1) rectangle (4,4+1);
    \draw [very thin] (0,1) rectangle (.9,4); 
    \node at (0.5,3.5) {$e$}; 
    \node at (0.5,2.5) {$a$};
    \node at (0.5,1.58) {$b$};
    %
    \node at (1.5,4.5) {$e$}; 
    \node at (2.5,4.5) {$a$};
    \node at (3.5,4.58) {$b$};
  \end{tikzpicture}
  \hspace{20mm}
  \begin{tikzpicture}[scale=0.75]
    \foreach \i in {1,...,5} {
      \draw [very thin] (\i,1) -- (\i,5); 
      \draw [very thin] (\i,5+.1) -- (\i,5+1); 
      \draw [very thin] (1,\i) -- (5,\i); 
      \draw [very thin] (0,\i) -- (.9,\i); 
    } 
    \draw [very thin] (1,5+.1) rectangle (5,5+1);
    \draw [very thin] (0,1) rectangle (.9,5); 
    \node at (0.5,4.5) {$e$}; 
    \node at (0.5,3.5) {$a$};
    \node at (0.5,2.58) {$b$};
    \node at (0.4,1.5) {$c$};
    %
    \node at (1.5,5.5) {$e$}; 
    \node at (2.5,5.5) {$a$};
    \node at (3.5,5.58) {$b$};
    \node at (4.5,5.5) {$c$};
    %%
    \node at (2.5,3.5) {$e$};
  \end{tikzpicture}



\item Prove that an element cannot appear twice in the same column of
  a multiplication table.



\item Using our ``unofficial'' definition of a group, prove that every
  group has a unique identity action $e$, satisfying $ge=g=eg$ for
  every action $g$ in $G$. [\emph{Hint}: You need to prove both
    existence and uniqueness. For the latter, assume that $e$ and $f$
    are both identity actions. Can you prove that $e=f$?]



  % VGT Exercise 4.16 (modified)
\item In this exercise, you will see one reason why our unofficial
  definition of a group doesn't quite match our formal
  definition. Consider the two diagrams below, neither of which is a
  valid Cayley diagram.
  %%
  \vspace{-5mm}
  \begin{center}
    \tikzstyle{vert} = [circle=3pt, draw, fill=grey]
    \tikzstyle{b} = [draw,very  thick, blue]
    \tikzstyle{r} = [draw,very  thick, Red,-stealth]
    \tikzstyle{rr} = [draw,very  thick, Red]
    \begin{tikzpicture}[scale=2]
      \node (e) at (0,0) [vert] {};
      \node (a) at (1,0) [vert] {};
      \node (b) at (2,0) [vert] {};
      \node (c) at (0,1) [vert] {};
      \node (d) at (1,1) [vert] {};
      \node (f) at (2,1) [vert] {};
      \node (g) at (-1,0) [vert] {};
      \node (h) at (-1,1) [vert] {};
      \path[b] (e) to (c);
      \path[b] (a) to (f);
      \path[b] (b) to (d);
      \path[b] (g) to (h);
      \path[r] (e) to (a);
      \path[r] (a) to (b);
      \path[r] (b) to [bend left=20] (g);
      \path[r] (c) to (d);
      \path[r] (d) to (f);
      \path[r] (h) to (c);
      \path[r] (g) to (e);
      \path[r] (f) to [bend right=20] (h);
    \end{tikzpicture}
    \hspace{25mm}
    \begin{tikzpicture}[scale=2]
      \node (e) at (0,0) [vert] {};
      \node (a) at (1,0) [vert] {};
      \node (b) at (2,0) [vert] {};
      \node (c) at (0,1) [vert] {};
      \node (d) at (1,1) [vert] {};
      \node (f) at (2,1) [vert] {};
      \node (g) at (-1,0) [vert] {};
      \node (h) at (-1,1) [vert] {};
      \path[b] (e) to (c);
      \path[b] (a) to (d);
      \path[b] (b) to (f);
      \path[b] (g) to (h);
      \path[rr] (a) to (b);
      \path[r] (c) to (d);
      \path[r] (d) to (f);
      \path[r] (h) to (c);
      \path[rr] (g) to (e);
      \path[r] (f) to [bend right=25] (h);
      \node at (0,-.36) {};
    \end{tikzpicture}
  \end{center}
  \vspace{-5mm}
  \begin{enumerate}
    \item Assuming that the arrows represent actions in a group, are
      the four rules in our unofficial definition of a group
      satisfied? For each rule, give a brief explanation why or why
      not.
    \item Try to convert each of these diagrams into a multiplication
      table. What problem arises in each case?
    \item Try to write a group presentation describing each ``group.''
      What problem arises in each case?
    \item Explain what is wrong with these diagrams that caused the
      problems you encountered in Parts~(b) and (c)?
    \item Create another diagram that is not a valid Cayley diagram
      for the same reason.
  \end{enumerate}



\item Let $\Z$, $\Q$, and $\R$ denote the set of integers, rational
  numbers, and real numbers, respectively. Let $\Z^+$, $\Q^+$, and
  $\R^+$ denote the positive integers, rationals, and reals. Let
  $\Z^*$, $\Q^*$, and $\R^*$ denote the nonzero integers, rationals,
  and reals.
  \begin{enumerate}
  \item Which the above sets are groups under addition? For each one
    that is a group (excluding $\R$, $\R^+$, and $\R^*$), give a
    minimal generating set.  For each one that is not, give an
    explicit reason for why it fails.
  \item Which of the above sets are groups under multiplication? For
    each one that is a group (excluding $\R$, $\R^+$, and $\R^*$),
    give a minimal generating set if there is one.  For each one that
    is not, give an explicit reason for why it fails.
  \item Let $n\Z$ denote the set of all integers that are multiples of
    $n$. For what $n$ is the set $n\Z$ a group under addition? Give a
    minimal generating set for each one that is a group.
\end{enumerate}


\end{enumerate}

\end{document}


