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\noindent
\parbox{2in}{\bf Math 3110} 
\hfill {\Large \bf Test \#1} \hfill
\parbox{2in}{\bf \hfill September $16^{\mathrm{th}}$, 2015}

\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? Give details.

\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 $G = \mathbb{Z}_{>0}$ be the set of positive integers. Define a binary operation as follows: $x \star y = x^y$ (example: $2 \star 3 = 2^3=8$). $G$ paired with this operation is {\bf not} a group. Why? [Use concrete counterexample(s).]

\vfill

   \item Let $G = \left\{ n\sqrt{2} \;|\; n \in \mathbb{Q}_{\not=0} \right\}$. (The non-zero rational multiples of $\sqrt{2}$. For example: $\sqrt{2}$ and $-\dfrac{3\sqrt{2}}{4}$ both belong to $G$.) $G$ is {\bf not} a group under \underline{multiplication}. Why? [Use concrete counterexample(s).]

\vfill

\end{enumerate}

\newpage

\noindent {\bf\large 2. (20 points)} Some modular arithmetic.

   \begin{enumerate}[(a)]

      \item What is the inverse of $80$ in the group $\mathbb{Z}_{100}$? What operation makes $\mathbb{Z}_{100}$ a group?

\vspace{0.8in}

      \item Is $80$ an element of $U(100)$? If not, \underline{why not}? If so, \underline{why} so and what is its \underline{inverse}?

\vfill

      \item Is $11$ an element of $U(100)$? If not, \underline{why not}? If so, \underline{why} so and what is its \underline{inverse}?

\vfill

      \item What is the order of $80$ in $\mathbb{Z}_{100}$?

\vspace{0.8in}

      \item Draw the subgroup lattice for $\mathbb{Z}_{100}$.

\vfill

\end{enumerate}

\newpage

\noindent {\bf\large 3. (25 points)} More Modular Arithmetic.

\begin{enumerate}[(a)]

   \item List the elements of $U(10)$. 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}

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$\langle x \rangle=$ & $\displaystyle\left\{\mbox{\myspacea}\right\}$ &  $\displaystyle\left\{\mbox{\myspacea}\right\}$ &  $\displaystyle\left\{\mbox{\myspacea}\right\}$ &  $\displaystyle\left\{\mbox{\myspacea}\right\}$ &  $\displaystyle\left\{\mbox{\myspacea}\right\}$ &  $\displaystyle\left\{\mbox{\myspacea}\right\}$  \\ \hline
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\vfill

\item Is $U(10)$ cyclic? \qquad {\bf\Large Yes \quad / \quad No} \qquad (Circle the correct answer.)

\vspace{0.15in}

   \item Find $\displaystyle \left\langle A \right\rangle$  in $\mathrm{GL}_2(\mathbb{Z}_4)$ where $\displaystyle A = \begin{bmatrix} 3 & 0 \\ 1 & 1 \end{bmatrix}$. What is the order of $A$? What is $A^{-1}$?

\vfill

   \item Find $7^{-1}$ mod 25 using the extended Euclidean algorithm [Don't just guess and check].

\vfill

\end{enumerate}

\newpage
\noindent {\bf\large 4. (15 points)} Recall $D_n = \{1,x,\dots,x^{n-1},y,xy,\dots,x^{n-1}y \} = \langle x,y \;|\; x^n=1, y^2=1, (xy)^2=1 \rangle$.

\begin{enumerate}[(a)]
   \item Use the relations for $D_{8}$ to simplify $x^{-2}y^3x^{11}yx^2y^{124}$
      
   \vspace{1.75in}
   
   \item What is the inverse of $x^6$ in $D_8$? What is its order?
   
   \vspace{1.25in}
   
   \item Fill in the Cayley table for $D_3$:
   
   \begin{tabular}{c||c|c|c|c|c|c|} 
              &  $1$ & $x$ & $x^2$ & $y$ & $xy$ & $x^2y$ \\ \hline \hline  
       $1$ & $1$ & $x$ & $x^2$ & $y$ & $xy$ & $x^2y$ {\Large\strut} \\ \hline 
       $x$ & $x$ & \myspace &  \myspace & \myspace & \myspace & \myspace \\ \hline
       $x^2$ & $x^2$ & \myspace &  \myspace & \myspace & \myspace & \myspace \\ \hline
       $y$ & $y$ & \myspace &  \myspace & \myspace & \myspace & \myspace \\ \hline
       $xy$ & $xy$ & \myspace &  \myspace & \myspace & \myspace & \myspace \\ \hline
       $x^2y$ & $x^2y$ & \myspace &  \myspace & \myspace & \myspace & \myspace \\ \hline
   \end{tabular}
  
  \vspace{0.2in}
  
 \item The {\bf rotations} form a subgroup of $D_3$ Explain why. Is this a cyclic subgroup?
 
\end{enumerate}

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

\begin{enumerate}[(a)]
   \item Choose one of the following:
   \begin{enumerate}[I.] 
   \item Suppose that $(ab)^{2}=a^{2}b^{2}$ for all $a,b \in G$. Prove that $G$ is abelian.
   \item Let $G = \langle g \rangle$ be a cyclic group. Prove that $G$ is abelian.
   \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 $H = \{ x \in \mathbb{Z} \;|\; \mbox{there exists }k,\ell \in \mathbb{Z}\mbox{ such that }x=4k+6\ell \}$ is a subgroup of $\mathbb{Z}$.
   \item Prove that $K = \left\{ \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \;\Bigg|\; a,b \in \mathbb{R} \mbox{ and } a,b \not=0 \right\}$ is a subgroup of $\mathrm{GL}_2(\mathbb{R})$.
   \end{enumerate}
   
\vfill
   
   


\end{enumerate}


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