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\noindent
\parbox{2in}{\bf Math 3110} 
\hfill {\Large \bf Test \#1} \hfill
\parbox{2in}{\bf \hfill September $23^{\mathrm{rd}}$, 2011}

\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)] Definition and Basics

\begin{enumerate}[(a)]
   \item Suppose that $G$ is a non-empty set equipped an operation. What 4 things do I need to check to see if $G$ is a group?

\vspace{0.2in}

          \begin{enumerate}[1:]
             \item \mbox{}

\vspace{0.2in}

             \item \mbox{}

\vspace{0.2in}

             \item \mbox{}

\vspace{0.2in}

             \item \mbox{}

\vspace{0.2in}

           \mbox{}\hspace{-0.35in} What additional property needs to hold for $G$ to be an {\bf Abelain} group?

\vspace{0.2in}

           \item \mbox{}

\vspace{0.2in}
          \end{enumerate}


         

   \item Let $G = \mathbb{R}_{\not=0}$ be the non-zero real numbers. Also, for each $x,y \in G$ let $\displaystyle x \div y = \frac{x}{y}$.
            Prove $G$ (with this operation) is {\bf not}  a group.

\vfill

   \item Let $G = \{ \dots, -16, -9, -4, -1, 0, 1, 4, 9, 16, \dots \}$ (the set of perfect squares and their negatives). Prove $G$ fails to be a group under the operation of addition.

\vfill

\end{enumerate}

\newpage
\item[\large 2. (\underline{\hskip 0.35 truein}/20 points)] The group $\mathbb{Z}_{n}$.

   \begin{enumerate}[(a)]

      \item What is the inverse of $4$ in $\mathbb{Z}_{12}$?

\vspace{0.8in}

      \item List all of the cyclic subgroups, $\langle x \rangle$, of $\mathbb{Z}_6$. 

\vfill

      \item What is the order of $8$ in $\mathbb{Z}_{12}$?

\vspace{0.8in}

      \item Draw the subgroup lattice of $\mathbb{Z}_{50}$ [Note: $50=2\cdot 5^2$].

\vfill

\end{enumerate}

\newpage
\item[\large 3. (\underline{\hskip 0.35 truein}/23 points)] More Modular Arithmetic.

\begin{enumerate}[(a)]

   \item Draw a Cayley table for $U(10)$.

\vspace{2in}

   \item Find $\langle 3 \rangle$ in $U(10)$.

\vspace{0.4in}

   \item Is $U(10)$ cyclic?

\vspace{0.4in}

   \item Let $\displaystyle A = \begin{bmatrix} 2 & 3 \\ 4 & 2 \end{bmatrix}$. Is $A \in \mathrm{GL}_2(\mathbb{Z}_{10})$? 
             If so, find $A^{-1}$. If not, explain why not.
             
\vspace{1in}             

   \item Does $33^{-1}$ exist in $\mathbb{Z}_{300}$? If so find $33^{-1}$. Otherwise explain why it does not exist.

\vfill

   \item Does $59^{-1}$ exist in $\mathbb{Z}_{120}$? If so find $59^{-1}$. Otherwise explain why it does not exist.

\vfill

\end{enumerate}

\newpage
\item[\large 4. (\underline{\hskip 0.35 truein}/15 points)] Dihedral Groups.

\includegraphics[width=4in]{math3110-fall2011-test1-figure01.jpg}


\begin{enumerate}[(a)]
   \item Compute $D' V$. Draw a few squares to illustrate your computation.
   
   \vspace{1.5in}
   
   \item Complete the Cayley table for $D_4$:
   
   \begin{tabular}{c||c|c|c|c|c|c|c|c|} 
      $D_4$ & $R_0$ & $R_{90}$ & $R_{180}$ & $R_{270}$ & $H$ & $D'$ & $V$ & $D$ \\ \hline \hline
      $R_0$ & $R_0$ & $R_{90}$ & $R_{180}$ & $R_{270}$ & $H$ & $D'$ & $V$ & $D$ \\ \hline
 $R_{90}$ & $R_{90}$ & \myspace & \myspace & \myspace & $D'$ & $V$ & \myspace & \myspace \\ \hline
 $R_{180}$ & $R_{180}$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline
 $R_{270}$ & $R_{270}$ & \myspace & \myspace & \myspace & $D$ & \myspace & \myspace & \myspace \\ \hline
 $H$ & $H$ & $D$ & \myspace & \myspace & \myspace & \myspace & \myspace & $R_{90}$ \\ \hline
 $D'$ & $D'$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline
 $V$ & $V$ & \myspace & \myspace & $D$ & $R_{180}$ & \myspace & \myspace & \myspace \\ \hline
 $D$ & $D$ & \myspace & $D'$ & $H$ & \myspace & \myspace & \myspace & \myspace \\ \hline
   \end{tabular}
  
  \vspace{0.1in}
  
  \item In a sentence or two explain why the set of rotations is a subgroup of $D_n$, but the set of reflections is not.
  
\end{enumerate}

\newpage
\item[\large 5. (\underline{\hskip 0.35 truein}/22 points)] Proofs!

\begin{enumerate}[(a)]
   \item Let $G$ be a group and suppose that for all $a,b \in G$ we have $(ab)^{-1}=a^{-1}b^{-1}$. Prove $G$ is Abelian.
   
   \vfill


   \item Let $g \in G$ where $G$ is a group. Prove $\langle g \rangle = \{ g^k \,|\, k \in \mathbb{Z} \}$ is a subgroup
             of $G$. Then show $\langle g \rangle$ is Abelian.

\vfill

   \item Show $H = \{0,3,6,9\}$ is a subgroup of $\mathbb{Z}_{12}$.
   
   \vspace{1in}
   
   


\end{enumerate}

\end{description}

\end{document}