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\noindent
\parbox{2in}{\bf Math 3110} 
\hfill {\Large \bf Test \#1} \hfill
\parbox{2in}{\bf \hfill February $17^{\mathrm{th}}$, 2021}

\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 The positive real numbers $\mathbb{R}_{>0} = \{ r \in \mathbb{R} \;|\; r>0 \}$ do not form a group under division. Why not?
 
\vfill
   
   \item On the other hand, the positive real numbers $\mathbb{R}_{>0} = \{ r \in \mathbb{R} \;|\; r>0 \}$ do form a group if we select the right operation. Which operation turns this collection of numbers into a group: Addition or Multiplication? Then explain why the other operation does not yield a group.

\vfill

   \item The non-zero rational numbers $\mathbb{Q}_{\not=0}$ form a group under multiplication. On the other hand, the (non-zero) irrational numbers $\mathbb{I}=\mathbb{R}-\mathbb{Q} = \{ x \in \mathbb{R} \;|\; x \not\in \mathbb{Q}\}$ do not. Why?

\vspace{0.5in}

\end{enumerate}

\newpage

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

   \begin{enumerate}[(a)]
      \item Make a list of all of the cyclic subgroups of $\mathbb{Z}_{10}$ along with their contents (for example: $\langle 0 \rangle = \{0\}$).
      
      \vfill

      \item Fill out the following table referring to the operations of addition and multiplication modulo 10:
      
      {\it Note:} Just put an {\large\bf X} if something is undefined / does not exist.

      \hspace{-0.5in}      
      \begin{tabular}{|r|c|c|c|c|c|c|c|c|c|c|} \hline
                     Element $x=$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \large\strut \\ \hline
      Additive Inverse $-x=$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline
      Order (in $\mathbb{Z}_{10}$) $|x|=$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline
      Multiplicative Inverse $x^{-1}=$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline
      Order (in $U(10)$) $|x|=$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline
      \end{tabular}

\vspace{2in}
      
      \item Compute $2^{-1}(3-8)+7$ mod $10$ or explain why this is undefined.

\vspace{0.8in}

      \item Compute $3^{-1}(7-3)-11$ mod $10$ or explain why this is undefined.

\vspace{0.8in}

\end{enumerate}

\newpage

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

\begin{enumerate}[(a)]


      \item Write down a Cayley table for $U(8)$. Is $U(8)$ cyclic (circle the correct answer)? \quad {\Large Yes \quad / \quad No}
      
\vfill

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

\vfill


\vfill

   \item Find $10^{-1}$ mod 77 using the extended Euclidean algorithm [Don't just guess and check].

\vfill

\vfill

\end{enumerate}

\newpage
\noindent {\bf\large 4. (20 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 $x^{13}y^3x^{-2}y^{888}x$
      
   \vspace{1.75in}
   
   \item Make a table listing the elements of $D_8$, their inverses, and their orders.
   
   \vspace{1.25in}
   
   
   \item What is $\langle x^6 \rangle$ in $D_8$?
   
   
   \vspace{0.5in}
  
  
   \item Fill in the following rows of the Cayley table for $D_4$:
  
  \hspace*{-0.25in} 
   \begin{tabular}{c||c|c|c|c|c|c|c|c|c|c|c|c|} 
              &  $1$ & $x$ & $x^2$ & $x^3$ & $y$ & $xy$ & $x^2y$ & $x^3y$ \\ \hline \hline  
$\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \\ \hline
$x^3$ &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace \\ \hline
$y$ &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace \\ \hline
$\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \\ \hline
   \end{tabular}
  
  \vspace{0.5in}
  
 \item Is $H = \{ 1, y, x^2y \}$ a subgroup of $D_4$? Why or why not?
 
 \vspace{0.75in}
 
\end{enumerate}

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

\begin{enumerate}[(a)]
   \item Choose one of the following:  \qquad \underline{Assume $G$ is a group under multiplication with identity $1$.}
   \begin{enumerate}[I.] 
   \item Suppose that $g=g^{-1}$ for all $g \in G$. Prove that $G$ is abelian.  
   \item Suppose that $G=\langle g \rangle$ is 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 $H = 8\mathbb{Z} = \{ 8k \;|\; k \in \mathbb{Z} \}$ is a subgroup of $\mathbb{Z}$.
   \item Prove that $K = \left\{ \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \;\Bigg|\; a,b \in \mathbb{R}_{\not=0} \right\}$ is a subgroup of $\mathrm{GL}_2(\mathbb{R})$.
   \end{enumerate}
   
\vfill
   
   


\end{enumerate}


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