\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{enumerate}
\usepackage{ifthen}
\usepackage{graphicx}

\setlength{\unitlength}{0.1in}

\setlength{\oddsidemargin}{0.0in}
\setlength{\evensidemargin}{0.0in}
\setlength{\topmargin}{0.0in}
\setlength{\headheight}{0.0in}
\setlength{\headsep}{0.0in}
\setlength{\topskip}{0.0in}
\setlength{\textheight}{8.9in}
\setlength{\textwidth}{6.5in}

\newcommand{\comp}{ \,{\scriptstyle \stackrel{\circ}{}}\, } 
\newcommand{\nullset}{\mathrm{O}\!\!\!\!\big/\,}
\newcommand{\divides}{\,\Big|\,}
\newcommand{\myspace}{\mbox{\parbox[c][0.35in]{0.35in}{\hfill}}}
\newcommand{\varspace}[2]{\mbox{\parbox[c][#1]{#2}{\hfill}}}

\begin{document}

\pagestyle{empty}

\noindent
\parbox{2in}{\bf Math 3110} 
\hfill {\Large \bf Test \#2} \hfill
\parbox{2in}{\bf \hfill March $31^{\mathrm{st}}$, 2010}

\vspace{0.3in}

\noindent {\large\bf Name:} \underline{\hskip 3.0 truein} \hfill {\bf Be sure to show your work!}

\begin{description}
\item[\large 1. (\underline{\hskip 0.35 truein}/20 points)] Random Group Stuff --- Fill out the following table:

%\mbox{} \hspace{-0.6in}
\begin{tabular}{|c|||c|c|c|l|c|} 
\hline
$G=$ & \parbox[c][0.5in]{0.85in}{What is the\\ identity of $G$?} & \parbox[c][0.5in]{0.85in}{Is $G$ abelian?} & \parbox[c][0.5in]{0.85in}{Is $G$ cyclic?} & \parbox[c][0.5in]{0.85in}{What is the\\ order of ...?} & \parbox[c][0.5in]{1.25in}{Does $G$ have an\\ element of order 4?} \\
\hline \hline \hline
$\mathbb{Z}_{40}$ & \varspace{1in}{0.75in} & \varspace{1in}{0.75in} & \varspace{1in}{0.75in}  & $|30|=$ & \\ \hline
$U(8)$ & \varspace{1in}{0.75in} & \varspace{1in}{0.75in} & \varspace{1in}{0.75in}  & $|5|=$ \varspace{1.25in}{0.75in} & \\ \hline
$D_7$ & \varspace{1in}{0.75in} & \varspace{1in}{0.75in} & \varspace{1in}{0.75in}  & $|x^4y|=$ & \\ \hline
$S_4$ & \varspace{1in}{0.75in} & \varspace{1in}{0.75in} & \varspace{1in}{0.75in}  & $|(12)(34)|=$ & \\ \hline
\end{tabular} 

{\bf Recall:} $D_7 = \{ 1, x, \dots, x^6, y, xy, \dots, x^6y \}$ where $x^7=1$, $y^2=1$, and $xyxy=1$.

\vspace{0.1in} 

\underline{Scratch Work:}

\newpage
\item[\large 2. (\underline{\hskip 0.35 truein}/20 points)] Group or not? Are the following sets with operations groups or not? 
If $G$ is a group, prove it --- you may use a subgroup test if it applies. If $G$ fails to be a group, explain what property fails.
\begin{enumerate}[(a)]
   \item Let $G = [-1,1] = \{ r \in \mathbb{R} \,|\, -1 \leq r \leq 1 \}$ with the operation ``$+$'' (addition).

\vspace{4in}

   \item Let $\displaystyle{ G = \left\{ \begin{bmatrix} 1 & 0 \\ r & 1 \end{bmatrix} \,\Big|\, r \in \mathbb{R} \right\} }$ with the operation
             of matrix multiplication.

\end{enumerate}

\newpage
\item[\large 3. (\underline{\hskip 0.35 truein}/15 points)] Cayley's Theorem and Permutations.\\
Recall that $D_n = \{ 1,x,x^2,\dots,x^{n-1},y,xy,\dots,x^{n-1}y \}$ where $x^n=1$, $y^2=1$, and $xyxy=1$.
\begin{enumerate}[(a)]
   \item Write down what the left multiplication operator of $y$ does in $D_3$. Then write down the corresponding
            permutation if we label $1$ as $1$, $x$ as $2$, $x^2$ as $3$, $y$ as $4$, $xy$ as $5$, $x^2y$ as $6$.
            
            \vspace{0.2in}
            
            \parbox[c][2in]{2in}{
            $L_y : D_3  \rightarrow  D_3$
            
            \vspace{-0.1in}

            \begin{eqnarray*}
            1 & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\
            x & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\
            x^2 & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\
            y & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\
            xy & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\
            x^2y & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\
            \end{eqnarray*}
            }
            
            The corresponding permutation is...?

\vspace{1in}

   \item Suppose that using Cayley's theorem we found the left multiplication operator of $x$ in $D_4$ corresponds to
             $(1234)(5678)$ and $y$ corresponds to $(15)(28)(37)(46)$. What would the left multiplication operator of $x^2y$ correspond to?
             [Your answer should be a permutation written as a product of disjoint cycles.]
             
 \vspace{2.5in}
 
             Now write your answer as a product of transpositions. Is this permutation even or odd?

\end{enumerate}

\newpage
\item[\large 4. (\underline{\hskip 0.35 truein}/25 points)] Mod stuff.
\begin{enumerate}[(a)]
   \item Draw the subgroup lattice of $\mathbb{Z}_{44}$. {\it Note:} $44 = 2^2 \cdot 11$. 

\vspace{2.5in}

   \item List the possible orders of elements in $\mathbb{Z}_{44}$. Then determine the number of elements of each order.   

\begin{tabular}{c||c|c|c|c|c|c} 
Order = & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} \\ \hline
Number of elements = & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in}
\end{tabular}

\vspace{0.2in}

   \item Show that $f : \mathbb{Z}_{6} \rightarrow \mathbb{Z}_{4}$ defined by $f(x)=2x$ is a {\bf well-defined} homomorphism.
   
   \vspace{2in}
   
   \item $\mathrm{Ker}(f) =$ \underline{\hspace{3in}} \vspace{0.5in} \\
             $f(\mathbb{Z}_6) =$ \underline{\hspace{3in}} \vspace{0.5in} \\
             Is $f$ 1-1? \vspace{0.2in}\\
             Is $f$ onto? \vspace{0.2in}\\
             Is $f$ an isomorphism?           
\end{enumerate}

\newpage
\item[\large 5. (\underline{\hskip 0.35 truein}/20 points)] POOF! ...I mean... PROOFS! [No magic please.]
\begin{enumerate}[(a)]
   \item Let $G$ be a group and let $a,b \in G$ such that $(ab)^2=a^2b^2$. Show that $ab=ba$.
   
   \vspace{1in}

   \item Let $G$ be a group and let $g \in G$. Define the map $\varphi:G \rightarrow G$ by $\varphi(x)=gxg^{-1}$.\\
             Prove that $\varphi$ is an isomorphism.

\vspace{3.5in}

   \item Let $G$ and $G'$ be groups and let $\psi:G \rightarrow G'$ be a homomorphism. Suppose that $G$ is cyclic. Show
             that $\psi(G)$ is abelian. [Extra Credit: Show that $\psi(G)$ is cyclic.]

\end{enumerate}

\end{description}

\end{document}
