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\pagestyle{fancy} \lhead{{\sf Homework 1 $|$ Due May 13 (Wednesday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Summer 2020 $|$ 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 1--2 of \emph{Visual Group Theory} (VGT), or Chapters
  1--3 of \emph{IBL Abstract Algebra}. \vspace{-2mm}
\item VGT Exercises 1.1--1.4, 1.8--1.12, 2.13--2.17. \vspace{-2mm}
\item The article \emph{Group Think} by Steven Strogatz, which
  appeared in the NY Times in 2010. \vspace{-2mm}
\item Francis Su's essay on good mathematical writing.
\end{itemize} 



\noindent Write up solutions to the following exercises.

\begin{enumerate}

\item Answer the following questions after reading Francis Su's essay
  on good mathematical writing.
  \begin{enumerate}
  \item What is a good rule of thumb for what you should assume of your
    audience as you write your homework sets?  
  \item Is chalkboard writing formal or informal writing?  
  \item Why is the proof by contradiction on page 3 not really a
    proof by contradiction?
  \item Name three things a lazy writer would do that a good writer
    would not.  
  \item What's the difference in meaning between these
    three phrases? 
    \[
    \mbox{``Let $A=12$.''}\qquad\qquad\mbox{``So $A=12$.''}
    \qquad\qquad\mbox{``$A=12$.''}
    \]
  \end{enumerate}    



\item Given a regular $n$-gon, let $r$ be a rotation of it by $2\pi/n$
  radians. This time, assume that we are not allowed to flip over the
  $n$-gon. These $n$ actions form a group denoted
  $C_n=\<r\>=\{e,r,r^2,\dots,r^{n-1}\}$.
  \begin{enumerate}
  \item Draw a Cayley diagram for $C_n$ for $n=4$, $n=5$, and $n=6$.
  \item For $n=4,5,6$, find all minimal generating sets of
    $C_n$. [\emph{Note:} There are minimal generating sets of $C_6$
    of size $2$.]
  \item Make a conjecture of what integers $k$ does $C_n=\<r^k\>$ for
    a general fixed integer $n$. \\
  \end{enumerate}  



\item As we saw in lecture, the six symmetries of an equilaterial
  triangle \hspace{0mm}
  %%
  \begin{tikzpicture}[scale=.8,baseline=1ex]
    \draw [black, fill=lightgray] (0,0)--(.5,.86)--(1,0)--cycle;
    \draw (.5,.55) node {\small $1$}; 
    \draw (.3,.2) node {\small $3$};\draw (.7,.2) node {\small $2$};
  \end{tikzpicture}  \hspace{.6mm}
  form a group denoted $D_3=\{e,r,r^2,f,rf,r^2f\}$, where $r$ is a
  $120^\circ$ clockwise rotation and $f$ is a flip about a vertical
  axis (which fixes the top corner). Since $r$ and $f$ suffice to
  generate all six of these symmetries, we write $D_3=\<r,f\>$.
  \begin{enumerate}
    \item Let $g$ be the reflection of the triangle that fixes the
      lower-left corner. Which of the six actions in $D_3$ is $g$
      equal to? Which action is $fg$?
    \item Write all 6 actions of $D_3$ using only $f$ and $g$. Draw a
      Cayley diagram using $f$ and $g$ as generators.
    \item To generate $D_3$, we need at least 2 actions. It is not
      difficult to show that if we have 3 generators, then one of them
      is unnecessary. Find all \emph{minimal} generating sets of
      $D_3=\{e,r,r^2,f,rf,r^2f\}$; note that all of them should have
      exactly two actions. Do not use $g$ in this list. \\
  \end{enumerate}



\item The eight symmetries of a square \hspace{0mm}
  %%
  \begin{tikzpicture}[scale=.8,baseline=1ex]
    \draw [black, fill=lightgray] (0,0) rectangle (1,1);
    \draw (.25,.75) node{\small $1$}; \draw (.75,.75) node{\small $2$};
    \draw (.25,.25) node{\small $4$}; \draw (.75,.25) node{\small $3$}; 
  \end{tikzpicture}  \hspace{.6mm}
  form a group denoted $D_4$. Let $r$ be a $90^\circ$ clockwise
  rotation and $f$ a horizontal flip (that is, about a vertical
  axis). It is not difficult to show that $D_4=\<r,f\>$.
  \begin{enumerate}
    \item Write all $8$ actions of $D_4$ using $r$ and $f$ and draw a
      Cayley diagram using these two actions as generators. 
    \item Let $g$ be the reflection of the square that fixes the
      lower-left and upper-right corner. Which of the eight actions in
      $D_4$ is $g$ equal to? Which action is $fg$?
    \item Draw a Cayley diagram of $D_4$ using $f$ and $g$ as
      generators.
    \item Find all \emph{minimal} generating sets of
      $D_4$. [\emph{Hint}: There are 12.]
    \item Find a minimal generating set of
      $D_6$ (the 12 symmetries of a regular hexagon) that has three actions. \\
  \end{enumerate}


%% VGT 1.13, modified
%\item Pick any integer and consider this set of actions: adding any
%  integer to the one you choose. This is an infinite set of actions;
%  we might name them like ``add $1$'' and ``add $-4210$,'' etc.
%  This is a group. Find all minimal generating sets. Sketch a Cayley
%  graph for this group using one of these minimal generating sets. \\


% VGT 1.13, modified
\item Pick any integer and consider this set of actions: adding any
  integer to the one you choose. This is an infinite set of actions;
  we might name them like ``add $1$'' and ``add $-4120$,'' etc.
  This is a group.
  \begin{enumerate}
  \item Find all generating sets of size $1$ and sketch a Cayley diagram
    for each. 
  \item Find a minimal generating set of size $2$ and sketch the Cayley diagram.
  \item Find a minimal generating set of size $3$.
  \item Explain how to construct a minimal generating set of size $n$,
    for any $n\in\N$.
  \end{enumerate}


\end{enumerate}

\end{document}
