\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\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 October $26^{\mathrm{th}}$, 2011}

\vspace{0.3in}

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

\vspace{0.2in}

\noindent {\large 1. (\underline{\hskip 0.35 truein}/20 points)} Cyclic

\begin{enumerate}[(a)]
   \item Let $G = \langle g \rangle$ where $g$ has order 20.

\vspace{0.2in}

            $\langle g^8 \rangle =$ \quad \underline{\hspace*{3in}}

\vspace{0.2in}

            Is $g^{102} \in \langle g^8 \rangle$? Why or why not?

\vfill

   \item Suppose $G$ is a cyclic group with at least one element of order $6$. 

\vspace{0.2in}

            What can you say about the order of $G$? 

\vspace{0.2in}

            How many elements of order 6 can $G$ have? Is there more than one possibility? 

\vfill


 \item List the possible orders of elements in $\mathbb{Z}_{33}$. 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}

\noindent Now fill out a table for $D_{33}$.

\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}

\end{enumerate}

\newpage
\noindent {\large 2. (\underline{\hskip 0.35 truein}/20 points)} The following pairs of groups are {\bf not} isomorphic. Prove this is the case.

\begin{enumerate}[(a)]
   \item $\mathrm{GL}_3(\mathbb{R}) \not\cong A_{500}$

\vfill

   \item $U(8)=\{1,3,5,7\} \not\cong \mathbb{Z}_4$

\vfill

   \item $\mathrm{GL}_2(\mathbb{Z}) \not\cong \mathbb{Q}$

\vfill

   \item $S_4 \not\cong D_{12}$

\vfill
\end{enumerate}

\newpage
\noindent {\large 3. (\underline{\hskip 0.35 truein}/20 points)} Isomorphisms

\begin{enumerate}[(a)]

\item Prove that $U(5) \cong \mathbb{Z}_4$.

\vfill

\item Let $G$ be an {\bf Abelian} group. Define the map $\varphi:G \rightarrow G$ by $\varphi(g)=g^{-1}$. Prove that $\varphi$ is an isomorphism (actually $\varphi$ is an automorphism since its domain and codomain are equal).

\vfill

Is $\varphi$ an automorphism if $G$ is not Abelian? Why or why not?

\end{enumerate}

\newpage
\noindent {\large 4. (\underline{\hskip 0.35 truein}/20 points)} Zombie Apocalypse! Does anyone actually read the directions to these problems?

\begin{enumerate}[(a)]

   \item Let $\sigma = (2453)(1346)(126) \in S_6$

            Write $\sigma$ as a product of disjoint cycles
            
            \vspace{0.2in} 
 
            Find $\sigma^{-1}$
  
            \vspace{0.2in} 

           What is the order of $\sigma$? \underline{\hspace*{2in}}

            \vspace{0.2in} 

           Is $\sigma$ even or odd? \underline{\hspace*{2in}}

            \vspace{0.2in} 

   \item For convenience: $A_4 = \{ (1), (123), (132), (124), (142), (134), (143), (234), (243), (12)(34), (13)(24), (14)(23) \}$
   
   Consider the subgroup $H= \{ (1), (12)(34), (13)(24), (14)(23) \}$. 
    Find the left cosets of $H$ in $A_4$. 

\vfill

   \item Let $\sigma=(14)(23) \in S_4$. What is the order of $\sigma$? \underline{\hspace*{2in}}
 
 \vspace{0.1in} 
 
            Compute $\sigma^{999}$.

\vfill

   \item Find an element of order 15 in $S_8$.

\vspace{0.2in}

\end{enumerate}

\newpage
\noindent {\large 5. (\underline{\hskip 0.35 truein}/20 points)} 

\begin{enumerate}[(a)]

   \item Let $G$ be a group, $H$ be a subgroup of $G$, and $a,b \in G$. Prove $aH=bH$ implies that $a^{-1}b \in H$.\\
            {}[You may {\bf not} use any theorems about cosets.]

\vfill

\vfill

   \item My friend is computing some cosets of $K$ which is a subgroup of $S_4$. He claims has found a left coset $L = \{ (243), (142), (123), (134) \}$.
      
   \vspace{0.2in}
   
             Assuming my friend didn't make a mistake, what is the order of $K$? \underline{\hspace*{2in}}
   
   \vspace{0.2in}
             
             How many cosets will $K$ have in $S_4$? \underline{\hspace*{2in}}
             
   \vspace{0.2in}
   
             My friend then starts computing right cosets and finds a coset
            $R = \{ (1234), (24), (1432), (13), (1324) \}$.        
            
            I know he must have made a mistake. Why? 

\vspace{1in}

   \item $H$ and $K$ are subgroups of $G$ such that $H \subsetneq K \subsetneq G$ (they are proper subsets of each other).
   
   $G$ has order 50. $K$ has order 5. 
   What are the possible orders for $H$?
   
   \vspace{0.1in}
   
   {\it Note:} This is the test as it was given. However, the intended problem is: $G$ has order 50. $H$ has order 5. 
   What are the possible orders for $K$? [switch $H$ and $K$.]

\vfill

\end{enumerate}

\end{document}

