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\begin{document}

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\noindent
\parbox{2in}{\bf Math 3110} 
\hfill {\Large \bf  Big Quiz \#1} \hfill
\parbox{2in}{\bf \hfill September $23^{\mathrm{rd}}$, 2013}

\vspace{0.3in}

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

\vspace{0.2in}

\noindent {\bf\large 1. (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 Abelian} group?

\vspace{0.2in}

           \item \mbox{}

\vspace{0.2in}
          \end{enumerate}   
      

   \item Let $G = \mathbb{Q}_{\leq 0}$ be the non-positive rational numbers. Prove $G$ is {\bf not} a group under addition.

\vfill

   \item Let $G = \mathbb{R}$ (the real numbers). Prove $G$ is {\bf not} a group under subtraction.

\vfill

\end{enumerate}

\newpage
\noindent {\bf\large 2. (20 points)} Working mod $20$.

   \begin{enumerate}[(a)]

      \item What is the inverse of $3$ in the group $\mathbb{Z}_{20}$? What operation makes $\mathbb{Z}_{20}$ a group?

\vspace{0.8in}

      \item Is $3$ an element of $U(20)$? If not, why not? If so, why so \& what is its inverse?

\vspace{0.8in}

      \item List all of the {\it distinct} cyclic subgroups, $\langle x \rangle$, of $\mathbb{Z}_{20}$. 

\vfill

      \item What is the order of $15$ in $\mathbb{Z}_{20}$?

\vspace{0.8in}

      \item Draw the subgroup lattice of $\mathbb{Z}_{20}$ [Note: $20=2^2\cdot 5$].

\vfill

\end{enumerate}

\newpage

\noindent {\bf\large 3. (20 points)} More Modular Arithmetic.

\begin{enumerate}[(a)]

%   \item Draw a Cayley table for $U(8)$.

%\vspace{1.5in}

   \item List the elements of $U(8)$. Then find their {\bf order}s and the list the elements in {\bf cyclic subgroup} generated by that element. [{\it Note:} There may be more spaces than you need.]

\vspace{0.1in}

\force \hspace{-0.45in}
\begin{tabular}{c||c|c|c|c|c|c|} 
$x=$ & \myspacea & \myspacea & \myspacea & \myspacea & \myspacea & \myspacea \\ \hline\hline
$|x|=$ & \myspacea & \myspacea & \myspacea & \myspacea & \myspacea & \myspacea \\ \hline
$\langle x \rangle=$ & $\displaystyle\left\{\mbox{\myspacea}\right\}$ &  $\displaystyle\left\{\mbox{\myspacea}\right\}$ &  $\displaystyle\left\{\mbox{\myspacea}\right\}$ &  $\displaystyle\left\{\mbox{\myspacea}\right\}$ &  $\displaystyle\left\{\mbox{\myspacea}\right\}$ &  $\displaystyle\left\{\mbox{\myspacea}\right\}$  \\ \hline
\end{tabular}

\vspace{0.15in}

   \item Is $U(8)$ cyclic? \qquad {\bf\Large Yes \quad / \quad No} \qquad (Circle the correct answer.)

\vspace{0.15in}

   \item Let $\displaystyle A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$. Is $A \in \mathrm{GL}_2(\mathbb{Z}_{7})$? 
             If so, find $A^{-1}$. If not, explain why not.
             
%\vspace{1in}             

%   \item Find $\displaystyle \left\langle B \right\rangle = \left\langle \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \right\rangle$ in $\mathrm{GL}_2(\mathbb{Z}_6)$. What is the order of $B$?

\vfill

   \item Find $36^{-1}$ mod 151 using the extended Euclidean algorithm.

\vfill

\end{enumerate}

\newpage
\noindent {\bf\large 4. (20 points)} Recall that $D_4 = \{1,x,x^2,x^3,y,xy,x^2y,x^3y \} = \langle x,y \;|\; x^4=1, y^2=1, (xy)^2=1 \rangle$.

\begin{enumerate}[(a)]
   \item Use the relations for $D_4$ to derive the relation: $yx = x^{-1}y$.
   
   \vspace{1.75in}
   
   \item Fill in the Cayley table for $D_4$:
   
   \begin{tabular}{c||c|c|c|c|c|c|c|c|} 
              &  $1$ & $x$ & $x^2$ & $x^3$ & $y$ & $xy$ & $x^2y$ & $x^3y$ \\ \hline \hline  
       $1$ & $1$ & $x$ & $x^2$ & $x^3$ & $y$ & $xy$ & $x^2y$ & $x^3y$ {\Large\strut} \\ \hline 
       $x$ & $x$ & \myspace &  \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline
       $x^2$ & $x^2$ & \myspace &  \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline
       $x^3$ & $x^3$ & \myspace &  \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline
       $y$ & $y$ & \myspace &  \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline
       $xy$ & $xy$ & \myspace &  \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline
       $x^2y$ & $x^2y$ & \myspace &  \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline
       $x^3y$ & $x^3y$ & \myspace &  \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline
   \end{tabular}
  
  \vspace{0.2in}
  
  \item Find the inverse and order of each element in $D_4$.

\begin{tabular}{c||c|c|c|c|c|c|c|c|} \hline
$g=$ & $1$ & $x$ & $x^2$ & $x^3$ & $y$ & $xy$ & $x^2y$ &  $x^3y$ \\ \hline \hline
$g^{-1}=$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline
$|g|=$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline
\end{tabular} 
 
 \vspace{0.5in}
 
 \item Do the rotations form a subgroup of $D_4$? If so, why? If not, why not?
 
\end{enumerate}

\newpage
\noindent {\bf\large 5. (20 points)} Proofs! 

\begin{enumerate}[(a)]
   \item Choose one of the following:
   \begin{enumerate}[I.] 
   \item Prove that $f:\mathbb{Z} \to \mathbb{Z}$ defined by $f(x)=5x$ is 1-1 but not onto.
   \item Let $G$ be a group and $g \in G$. Prove that $|g|=|g^{-1}|$ \quad (i.e. $g$ and its inverse have the same order).
   \end{enumerate}   
   
   \vfill

   \item Choose one of the following:  \qquad (You {\bf must} use a subgroup test in your proof.)
   \begin{enumerate}[I.] 
   \item Prove that $\mathrm{SL}_2(\mathbb{Z}) = \left\{ A \in \mathbb{Z}^{2 \times 2} \;|\; \mathrm{det}(A)=1 \right\}$ is a subgroup of $\mathrm{GL}_2(\mathbb{Z})$.
   \item Prove that $H = \{ n \in \mathbb{Z} \;|\; n = 10x+6y \mbox{ for some } x,y \in \mathbb{Z} \}$ is a subgroup of $\mathbb{Z}$.
   \end{enumerate}
   
\vfill
   
   


\end{enumerate}


\end{document}


