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\noindent
\parbox{1.5in}{\bf Math 3110} 
\hfill {\Large \bf  Test \#2} \hfill
\parbox{1.5in}{\bf \hfill March $4^{\mathrm{th}}$, 2022}

\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)} Random Group Stuff --- Fill out the following table:

\vspace{0.2in}

\mbox{} \hspace{-0.4in}
\begin{tabular}{|c||c|c|c|c|c|c|} 
\hline
$G=$ & \parbox[c][0.5in]{0.85in}{What is the\\ identity of $G$?} & \parbox[c][0.5in]{1in}{What is the\\ order of ...?} & \parbox[c][0.5in]{1.25in}{Does $G$ have an\\ element of order 6?} & \parbox[c][0.5in]{0.85in}{Is $G$ abelian?} & \parbox[c][0.5in]{0.85in}{Is $G$ cyclic?} \\
\hline \hline \hline
$\mathbb{Z}_{55}$ & \varspace{0.75in}{0.75in}  & $|33|=$ & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & \\ \hline
$U(9)$ & \varspace{0.75in}{0.75in}  & $|5|=$ & \varspace{0.75in}{1in} & \varspace{0.75in}{0.75in} & \\ \hline
$D_{10}$ & \varspace{0.75in}{0.75in}  & $|x^4|=$ & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & \\ \hline
$S_7$ & \varspace{0.75in}{0.75in}  & $|(1234)(56)|=$ & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & \\ \hline
\end{tabular} 

\vspace{0.05in}

{\bf Recall:} $D_{10} = \{ 1, x, \dots, x^{9}, y, xy, \dots, x^{9}y \}$ where $x^{10}=1$, $y^2=1$, and $xyxy=1$.

\vspace{0.1in} 

\underline{Scratch Work:}

\newpage
\noindent {\bf\large 2. (20 points)} Cyclic Stuff

   \begin{enumerate}[(a)]

      \item Let $G$ be a finite group and $g \in G$. Suppose that $|g|=120=2^3\cdot 3 \cdot 5$. 
      
      \begin{enumerate}[i.]
      \item What is the order of $g^{100}$? List the distinct elements in $\langle g^{100} \rangle$.
      
      
      \vspace{1.5in}
      
      
      $\langle g^{100} \rangle = \displaystyle \Bigg\{$ \hspace{4.5in} $\displaystyle \Bigg\}$
          
      \vspace{0.3in}
      
      \item  Is $g^{31} \in \langle g^{77} \rangle$? \quad {\large\bf Yes \ \  /  \ \  No}
      
      \vspace{1in}
      
      
      \end{enumerate}

     \item How many elements of order 10 does $\mathbb{Z}_{40}$ have? What are they?
     
\vfill

    \item List the orders of elements in $\mathbb{Z}_{52}$. Then determine the number of elements of each order. 
    
    {\it Note:} It might be helpful to know that $52=2^2\cdot 13$ and its positive divisors are 1, 2, 4, 13, 26, and 52.

\begin{tabular}{c||ccccccc} 
Order = & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} \\ \hline
Number of elements = & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in}
\end{tabular}

\vspace{0.2in}

    \item List the orders of elements in $D_{52}$. Then determine the number of elements of each order.   

\begin{tabular}{c||ccccccc} 
Order = & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} \\ \hline
Number of elements = & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in} & \varspace{0.5in}{0.5in}
\end{tabular}


\end{enumerate}

\newpage

\noindent {\bf\large 3. (22 points)} Permutations \qquad {\it Note:} Please give simplified (``good manners'') answers.
\begin{enumerate}[(a)]
   \item Consider $G = D_3 = \{ 1,x,x^2,y,xy,x^2y \} = \langle x,y \;|\; x^3=1, y^2=1, (xy)^2=1\rangle$. Label $1$ as $1$, $x$ as $2$, $x^2$ as $3$, $y$ as $4$, $xy$ as $5$, and $x^2y$ as $6$. Cayley's theorem says that $G$ is isomorphic to a subgroup of $S_6$. Write down left multiplication by $xy$ does in $D_3$: $L_{xy}:D_3 \to D_3$ is\dots
  
  \vspace{0.2in}
      
   $\;\,1 \quad \mapsto \quad$ \underline{\hspace*{0.75in}}\\ 
   
   $\;\,x \quad \mapsto \quad$ \underline{\hspace*{0.75in}}\\ 
   
   $x^2 \quad \mapsto \quad$ \underline{\hspace*{0.75in}}\\ 
   
   $\;\,y \quad \mapsto \quad$ \underline{\hspace*{0.75in}}\\ 
   
   $\! xy \quad \mapsto \quad$ \underline{\hspace*{0.75in}}\\ 

   $\!\!\!\! x^2y \quad \mapsto \quad$ \underline{\hspace*{0.75in}}\\ 


The corresponding permutation in $S_6$ is \underline{\hspace*{1.5in}}.

%\vspace{0.5in}

   \item Suppose that using Cayley's theorem we found the left multiplication operator of $x$ in $D_4$ corresponds with $(1234)(5678)$ and the left multiplication operator of $y$ corresponds with $(15)(28)(37)(46)$. Find the permutation associated with $x^2y$. [Your answer should be written in terms of disjoint cycles.]

\vspace{1.2in}


   \item Write $\sigma = (124)(1326)(34576)$ as a product of disjoint cycles.\\
   
   $\phantom{{}^{-1}}$$\sigma=$ 
   
   \vspace{0.4in}
   
             $\sigma^{-1}=$ 
             
   \vspace{0.4in}
   
            The order of $\sigma$ is $|\sigma|=$ \underline{\hspace{1in}}.  
            
   \vspace{0.3in}
   
           Write $\sigma$ as a product of transpositions: \underline{\hspace{2.25in}}. \quad $\sigma$ is {\large\bf \  Even \  / \  Odd}

    \vspace{0.4in}
    
         $\sigma^{62}=$

\vspace{0.4in}

\item Does $S_{6}$ have an element of order 8? If not, explain why not. If so, give an example of such an element.

\end{enumerate}

\newpage

\noindent {\bf\large 4. (18 points)} Explain why the following pairs of groups are not isomorphic.
   
\begin{enumerate}[(a)]

\item $Q \not\cong \mathbb{Z}_{8}$ where $Q=\{ \pm 1, \pm i, \pm j, \pm k\}$ is the quaternion group so $i^2=j^2=k^2=-1$, $ij=k$, $ji=-k$, etc.

\vfill

\item $\mathbb{Z}_{9} \not\cong U(9)$

\vfill

\item $S_4 \not\cong D_{12}$ 

\vfill


\end{enumerate}


\newpage

\noindent {\bf\large 5. (20 points)} A few proofs

\begin{enumerate}[(a)]


\item Explain why $A_3 \cong \mathbb{Z}_3$ but $A_n \not\cong \mathbb{Z}_{n}$ for any $n>3$.

\vfill


\item Pick {\bf ONE} of the following\dots

   \begin{enumerate}[I.]
   \item Show that every cyclic group is Abelian. Then explain why the converse is not true.
   
   \item Show that $\mathbb{R} \cong \mathbb{R}_{>0}$. \qquad {\it Hint:} Consider an exponential function for $\varphi$.
   
   {\it Note:} $\mathbb{R}_{>0} = (0,\infty)$ is the group of positive real numbers.
   \end{enumerate}
   
   
\vfill

\end{enumerate}


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