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\noindent
\parbox{2in}{\bf Math 3110} 
\hfill {\Large \bf Test \#1} \hfill
\parbox{2in}{\bf \hfill September $16^{\mathrm{th}}$, 2020}

\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 odd integers $\mathbb{O} = 2\mathbb{Z}+1 = \{ 2k+1 \;|\; k \in \mathbb{Z} \}$ do not form a group under either addition or multiplication. Explain why they fail be a group under both of these operations. 
   
\vfill
   
   \item On the other hand, the even integers $\mathbb{E}=2\mathbb{Z} = \{ 2k \;|\; k \in\mathbb{Z}\}$ do form a group under one of these operations. Briefly explain why it is a group under that operation and why it isn't a group under the other operation.

\vfill

   \item Why aren't the integers $\mathbb{Z}$ a group under subtraction?

\vspace{0.5in}

\end{enumerate}

\newpage

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

   \begin{enumerate}[(a)]

      \item Make a table listing the elements of $\mathbb{Z}_{9}$, their inverses, and their orders.

\vfill

      \item Do the same for $U(9)$.

\vfill

      \item Compute $2^{-1}(3-8)+7$ mod $9$ or explain why this is undefined.

\vspace{0.8in}

      \item Compute $3^{-1}(7-3)-11$ mod $9$ 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 List all of the distinct cyclic subgroups of $\mathbb{Z}_{12}$. Show each subgroup's contents (e.g., $\langle 0 \rangle = \{0\}$).
      
\vfill

      \item Draw the subgroup lattice for $\mathbb{Z}_{12}$.

\vfill

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

\noindent
{\color{red}\bf Embarrassing Error:} I changed $A$ at the last moment to its current definition. However, $\det(A) = 1(1)-2(2) = -3 =3$ (mod $6$) so $\det(A)^{-1}=3^{-1}$ does not exist in $\mathbb{Z}_6$. Thus $A \not\in \mathrm{GL}_2(\mathbb{Z}_6)$ since $A^{-1}$ (working mod 6) does not exist. So this part of the problem is \underline{\bf\color{red}nonsense}! 

Instead let's change to $A = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}$ and rework this part.


\vfill

   \item Find $10^{-1}$ mod 27 using the extended Euclidean algorithm [Don't just guess and check].

\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 $D_{6}$ to simplify $y^{-2}x^{11}y^3x^{-4}y^{124}x$
      
   \vspace{1.75in}
   
   \item Make a table listing the elements of $D_6$, their inverses, and their orders.
   
   \vspace{1.25in}
   
   
   \item What is $\langle x^4 \rangle$ in $D_6$?
   
   
   \vspace{0.5in}
  
  
   \item Fill in the following rows of the Cayley table for $D_6$:
  
  \hspace*{-0.25in} 
   \begin{tabular}{c||c|c|c|c|c|c|c|c|c|c|c|c|} 
              &  $1$ & $x$ & $x^2$ & $x^3$ & $x^4$ & $x^5$ & $y$ & $xy$ & $x^2y$ & $x^3y$ & $x^4y$ & $x^5y$ \\ \hline \hline  
$\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \\ \hline
$x^5$ & \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace \\ \hline
$y$ & \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace &  \myspace \\ \hline
$\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \\ \hline
   \end{tabular}
  
  \vspace{0.5in}
  
 \item Is $H = \{ 1,x^2,y,x^2y \}$ a subgroup of $D_6$? 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 $g^2=1$ for all $g \in G$. Prove that $G$ is abelian.  
   \item For all $g \in G$. Show that $|g|=|g^{-1}|$.
   \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 $H = \{ 10k+6\ell \;|\; k,\ell \in \mathbb{Z} \}$ is a subgroup of $\mathbb{Z}$.
   \item Prove that $K = \left\{ \begin{bmatrix} 1 & 0 \\ a & 1 \end{bmatrix} \;\Bigg|\; a \in \mathbb{R} \right\}$ is a subgroup of $\mathrm{GL}_2(\mathbb{R})$.
   \end{enumerate}
   
\vfill
   
   


\end{enumerate}


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