\documentclass[12pt]{article}

\usepackage{latexsym}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage[all]{xy}


\xyoption{dvips}

\newtheorem*{goal}{Goal}

\newcommand{\CC}{\mathbb{C}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\dd}{\displaystyle}
\newcommand{\scs}{\scriptstyle}

\setlength{\evensidemargin}{1in}
\addtolength{\evensidemargin}{-1in}
\setlength{\oddsidemargin}{1.5in}
\addtolength{\oddsidemargin}{-1.5in}
\setlength{\topmargin}{1in}
\addtolength{\topmargin}{-1.5in}

\setlength{\textwidth}{16cm}
\setlength{\textheight}{23cm}


\pagestyle{empty}


\begin{document}



\section*{Math 3140: Homework 1}
\subsection*{Due: Wednesday, September 3}



\begin{enumerate}
\item[A.] Figure 1.4 essentially describes two symmetries of the tetrahedron:
$$ \xy<0cm,.9cm>\xymatrix@R=.3cm@C=.5cm{*={} & *+{\scs 1} \\ \\ *+{\scs 2} \ar @{--}[rr] \ar @{-} [uur] & *={} & *+{\scs 4}\ar @{-} [uul]\\ *={} & *+{\scs 3}\ar @{-} [ul]\ar @{-} [uuu] \ar @{-} [ur]}\endxy\overset{r}{\longrightarrow}  \xy<0cm,.9cm>\xymatrix@R=.3cm@C=.5cm{*={} & *+{\scs 1} \\ \\ *+{\scs 4} \ar @{--}[rr] \ar @{-} [uur] & *={} & *+{\scs 3}\ar @{-} [uul]\\ *={} & *+{\scs 2}\ar @{-} [ul]\ar @{-} [uuu] \ar @{-} [ur]}\endxy\overset{s}{\longrightarrow}  \xy<0cm,.9cm>\xymatrix@R=.3cm@C=.5cm{*={} & *+{\scs 3} \\ \\ *+{\scs 2} \ar @{--}[rr] \ar @{-} [uur] & *={} & *+{\scs 1}\ar @{-} [uul]\\ *={} & *+{\scs 4}\ar @{-} [ul]\ar @{-} [uuu] \ar @{-} [ur]}\endxy$$
\begin{enumerate}
\item[1.3] Adopt the notation of Figure 1.4.  Show that the axis of the composite rotation $srs$ passes through the vertex 4, and that the axis of $rsrr$ is determined by the midpoints of edges $12$ and $34$. \item[1.4]  Having completed 1.3, Express each of the twelve rotational symmetries of the tetrahedron in terms of $r$ and $s$.
\item[1.5]  Again with the notation of Figure 1.4, check that $r^{-1}=rr$, $s^{-1}=s$, $(rs)^{-1}=srr$, and $(sr)^{-1}=rrs$.
\item[1.9] Find all plane symmetries (rotations and reflections) of a regular pentagon.
\end{enumerate}
\item[B.] 
\begin{enumerate}
 \item[2.5] A function from the plane to itself which preserves the distance between any two points is called and \emph{isometry}.  Prove that an isometry must be a bijection and check that the collection of all isometries of the plane forms a group under composition of functions.
 \item[2.6]  Show that the collection of all rotations of the plane about a fixed point $P$ forms a group under composition of functions.  Is the same true of the set of all reflections in lines which pass through $P$? What happens if we take all the rotations and all the reflections?
 \item[2.7]  Let $x$ and $y$ be elements of a groups $G$.  Prove that $G$ contains elements $w$, $z$ which satisfy $wx=y$ and $xz=y$, and show that these elements are unique.
 \item[2.8] If $x$ and $y$ are elements of a group, prove that $(xy)^{-1}=y^{-1}x^{-1}$.
 \end{enumerate}
\item[C.]  What are all the group structures on the set $\{1,a,b,c\}$, where you may assume $1$ is the identity?
\end{enumerate}

\end{document} 