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\pagestyle{fancy} \lhead{{\sf Homework 2 $|$ Due May 17 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Summer I 2019 $|$ M.~Macauley}}


\begin{document}

$\;$ \vspace{-3mm}

\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}, or Chapters 5.1--5.3 of \emph{IBL Abstract Algebra} \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 Consider the frieze pattern shown below at left. 
  \tikzstyle{up}=[shape=diamond, orange,
    aspect=.5,anchor=south,fill=orange,draw]
  \tikzstyle{down}=[shape=diamond, orange,
    aspect=.5,anchor=north,fill=orange,draw]
  \colorlet{lightgrey}{black!15}
  \tikzstyle{vert} = [circle=2pt, draw, fill=lightgrey]
  \tikzstyle{B} = [draw,very thick, blue]
  \tikzstyle{R} = [draw, very thick, Red,-stealth']

  \begin{tikzpicture}[scale=1.05,baseline=(current bounding box.center)]
    \begin{scope}[scale=.65,shift={(0,0)}] % glide and horizontal flip
    \draw (0,0) node [up] {};
    \draw (1,0) node [down] {};
    \draw (2,0) node [up] {};
    \draw (3,0) node [down] {};
    \draw (4,0) node [up] {};
    \draw (5,0) node [down] {};
    \draw (6,0) node [up] {};
    \draw (7,0) node [down] {};
    \draw (8,0) node [up] {};
    \draw (-1,0) -- (9,0);
    \draw (-1.5,0) node {$\cdots$};
    \draw (9.5,0) node {$\cdots$};
    \draw[dotted] (2,-1.75)--(2,1.75);
    \draw[dotted] (3,-1.75)--(3,1.75);
    \draw[dashed,Red,thick] (4,-2)--(4,2);
    \draw[dotted] (5,-1.75)--(5,1.75);
    \draw[dotted] (6,-1.75)--(6,1.75);
    \end{scope}
    \begin{scope}[scale=.9,shift={(9.5,-.7)}]
      \foreach \x in {0,1.5,...,5} {
        \node[vert] at (\x,0) {};
        \node[vert] at (\x,1.5) {};
        \path[B] (\x, .15) -- (\x, 1.35);
      }
      \foreach \x in {0,1.5,...,4} {
        \path[R] (\x+1.35, 0) -- (\x+.15, 0);
        \path[R] (\x+.15, 1.5) -- (\x+1.35, 1.5);
      }
      \path[R] (-.15, 0) -- (-.85, 0);
      \path[R] (-.85, 1.5) -- (-.15, 1.5);
      \path[R] (5.4, 0) -- (4.7, 0);
      \path[R] (4.7, 1.5) -- (5.4, 1.5);
      \draw (-1.1,0) node[red] {$\cdots$};  
      \draw (-1.1,1.5) node[red] {$\cdots$};
      \draw (5.75,0) node[red] {$\cdots$};
      \draw (5.75,1.5) node[red] {$\cdots$};
    \end{scope}
  \end{tikzpicture}

  Let $g$ be a glide-reflection to the right, and $h$ a horizontal
  flip about the dashed red line. These actions generate the symmetry group of this frieze, also called its \emph{frieze group}, and its Cayley diagram is shown on the right. For each of the four dotted lines shown in the freize, express the reflection about that line in terms of $g$ and $h$. 

\medskip
  
\item There are 7 freize groups. In addition to the one in the previous problem, the other 6 are symmetry groups of the friezes shown below. \vspace{-4mm}

  \tikzstyle{up}=[shape=diamond, orange,
    aspect=.5,anchor=south,fill=orange,draw]
  \tikzstyle{down}=[shape=diamond, orange,
    aspect=.5,anchor=north,fill=orange,draw]

\[
  \begin{tikzpicture}[scale=.7]%,baseline=(current bounding box.center)]
    \begin{scope}[shift={(0,0)}] % glide reflections only
      \filldraw[orange] (0,0) arc (90:-90:10pt);
      \filldraw[white] (-.15,0) arc (90:-90:10pt);
      \filldraw[orange] (1,0) arc (90:-90:10pt);
      \filldraw[white] (0.85,0) arc (90:-90:10pt);
      \filldraw[orange] (2,0) arc (90:-90:10pt);
      \filldraw[white] (1.85,0) arc (90:-90:10pt);
      \filldraw[orange] (3,0) arc (90:-90:10pt);
      \filldraw[white] (2.85,0) arc (90:-90:10pt);
      \filldraw[orange] (4,0) arc (90:-90:10pt);
      \filldraw[white] (3.85,0) arc (90:-90:10pt);
      \draw (-1,0) -- (5,0);
      \draw (-1.5,0) node {$\cdots$};
      \draw (5.5,0) node {$\cdots$};
    \end{scope}
    \begin{scope}[shift={(10,0)}]  % translations only
      \filldraw[orange] (1,0) arc (90:-90:10pt);
      \filldraw[white] (0.85,0) arc (90:-90:10pt);
      \filldraw[orange] (3,0) arc (90:-90:10pt);
      \filldraw[white] (2.85,0) arc (90:-90:10pt);
      \filldraw[orange] (0,.75) arc (90:-90:10pt);
      \filldraw[white] (-.15,.75) arc (90:-90:10pt);
      \filldraw[orange] (2,.75) arc (90:-90:10pt);
      \filldraw[white] (1.85,.75) arc (90:-90:10pt);
      \filldraw[orange] (4,.75) arc (90:-90:10pt);
      \filldraw[white] (3.85,.75) arc (90:-90:10pt);
      \draw (-1,0) -- (5,0);
      \draw (-1.5,0) node {$\cdots$};
      \draw (5.5,0) node {$\cdots$};
    \end{scope}
    %%
  \begin{scope}[shift={(0,-2.5)}]
    \filldraw[orange] (0,0) arc (90:-90:10pt);
    \filldraw[white] (-.15,0) arc (90:-90:10pt);
    \filldraw[orange] (1,0) arc (90:-90:10pt);
    \filldraw[white] (0.85,0) arc (90:-90:10pt);
    \filldraw[orange] (2,0) arc (90:-90:10pt);
    \filldraw[white] (1.85,0) arc (90:-90:10pt);
    \filldraw[orange] (3,0) arc (90:-90:10pt);
    \filldraw[white] (2.85,0) arc (90:-90:10pt);
    \filldraw[orange] (4,0) arc (90:-90:10pt);
    \filldraw[white] (3.85,0) arc (90:-90:10pt);
    \filldraw[orange] (0,.73) arc (90:-90:10pt);
    \filldraw[white] (-.15,.73) arc (90:-90:10pt);
    \filldraw[orange] (1,.73) arc (90:-90:10pt);
    \filldraw[white] (0.85,.73) arc (90:-90:10pt);
    \filldraw[orange] (2,.73) arc (90:-90:10pt);
    \filldraw[white] (1.85,.73) arc (90:-90:10pt);
    \filldraw[orange] (3,.73) arc (90:-90:10pt);
    \filldraw[white] (2.85,.73) arc (90:-90:10pt);
    \filldraw[orange] (4,.73) arc (90:-90:10pt);
    \filldraw[white] (3.85,.73) arc (90:-90:10pt);
    \draw (-1,0) -- (5,0);
    \draw (-1.5,0) node {$\cdots$};
    \draw (5.5,0) node {$\cdots$};
  \end{scope}
  \begin{scope}[shift={(10,-2.5)}]
    \filldraw[orange] (0,0) arc (90:-90:10pt);
    \filldraw[white] (-.15,0) arc (90:-90:10pt);
    \filldraw[orange] (1,0) arc (90:-90:10pt);
    \filldraw[white] (0.85,0) arc (90:-90:10pt);
    \filldraw[orange] (2,0) arc (90:-90:10pt);
    \filldraw[white] (1.85,0) arc (90:-90:10pt);
    \filldraw[orange] (3,0) arc (90:-90:10pt);
    \filldraw[white] (2.85,0) arc (90:-90:10pt);
    \filldraw[orange] (4,0) arc (90:-90:10pt);
    \filldraw[white] (3.85,0) arc (90:-90:10pt);
    \filldraw[orange] (0,0) arc (270:90:10pt);
    \filldraw[white] (.15,0) arc (270:90:10pt);
    \filldraw[orange] (1,0) arc (270:90:10pt);
    \filldraw[white] (1.15,0) arc (270:90:10pt);
    \filldraw[orange] (2,0) arc (270:90:10pt);
    \filldraw[white] (2.15,0) arc (270:90:10pt);
    \filldraw[orange] (3,0) arc (270:90:10pt);
    \filldraw[white] (3.15,0) arc (270:90:10pt);
    \filldraw[orange] (4,0) arc (270:90:10pt);
    \filldraw[white] (4.15,0) arc (270:90:10pt);
    \draw (-1,0) -- (5,0);
    \draw (-1.5,0) node {$\cdots$};
    \draw (5.5,0) node {$\cdots$};
  \end{scope}
  %%
  \begin{scope}[shift={(0,-5)}]  % translation and horizontal flip
    \draw (0,0) node [down] {};
    \draw (1,0) node [down] {};
    \draw (2,0) node [down] {};
    \draw (3,0) node [down] {};
    \draw (4,0) node [down] {};
    \draw (-1,0) -- (5,0);
    \draw (-1.5,0) node {$\cdots$};
    \draw (5.5,0) node {$\cdots$};
    \draw (0,-1.5) node {}; 
    \end{scope}
    \begin{scope}[shift={(10,-5)}]  % 3 generators
    \draw (0,0) node [up] {}; \draw (0,0) node [down] {};
    \draw (1,0) node [up] {}; \draw (1,0) node [down] {};
    \draw (2,0) node [up] {}; \draw (2,0) node [down] {};
    \draw (3,0) node [up] {}; \draw (3,0) node [down] {};
    \draw (4,0) node [up] {}; \draw (4,0) node [down] {};
    \draw (-1,0) -- (5,0);
    \draw (-1.5,0) node {$\cdots$};
    \draw (5.5,0) node {$\cdots$};
  \end{scope} 
  \end{tikzpicture}  
\]
For each of these friezes, draw a Cayley diagram for its frieze group
using a minimal generating set. Make it clear what the generators are
and write out a group presentation. 

\medskip



\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}

  \medskip


\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. Draw a Cayley diagram for each.

  \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}

\medskip


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

  \medskip



\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 $e=f$?] \medskip



\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, give a minimal generating set if there is one.
    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, 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 $n\in\N$ 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}


