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\noindent
\parbox{1.5in}{\bf Math 3110} 
\hfill {\Large \bf  Test \#2} \hfill
\parbox{1.5in}{\bf \hfill October $9^{\mathrm{th}}$, 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)} Random Group Stuff --- Fill out the following table:

\vspace{0.2in}

\mbox{} \hspace{-0.4in}
\begin{tabular}{|c|||c|c|c|l|c|} 
\hline
$G=$ & \parbox[c][0.5in]{0.85in}{What is the\\ identity of $G$?} & \parbox[c][0.5in]{0.85in}{Is $G$ abelian?} & \parbox[c][0.5in]{0.85in}{Is $G$ cyclic?} & \parbox[c][0.5in]{1.75in}{What is the\\ order of ...?} & \parbox[c][0.5in]{1.25in}{Does $G$ have an\\ element of order 5?} \\
\hline \hline \hline
$\mathbb{Z}_{99}$ & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in}  & $|15|=$ & \\ \hline
$U(10)$ & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in}  & $|3|=$ \varspace{0.75in}{1in} & \\ \hline
$D_{20}$ & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in}  & $|x^6|=$ & \\ \hline
$S_9$ & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in}  & $|(123)(4567)(89)|=$ & \\ \hline
\end{tabular} 

{\bf Recall:} $D_{20} = \{ 1, x, \dots, x^{19}, y, xy, \dots, x^{19}y \}$ where $x^{20}=1$, $y^2=1$, and $xyxy=1$.

\vspace{0.1in} 

\underline{Scratch Work:}

\newpage
\noindent {\bf\large 2. (24 points)} Cyclic Stuff

   \begin{enumerate}[(a)]

      \item Let $G$ be a finite group and $g \in G$. Suppose that $|g|=66$. 
      
      \begin{enumerate}[i.]
      \item What is the order of $g^{55}$? List the distinct elements in $\langle g^{55} \rangle$.
      
      
      \vspace{1.5in}
      
      
      $\langle g^{55} \rangle = \displaystyle \Bigg\{$ \hspace{4in} $\displaystyle \Bigg\}$
          
      \vspace{0.3in}
      
      \item  Is $g^{22} \in \langle g^{22} \rangle$? \quad {\large\bf Yes \ \  /  \ \  No}
      
      \vspace{1in}
      
      
      \end{enumerate}

     \item How many elements of order 6 does $\mathbb{Z}_{120}$ have? What are they?
     
\vfill

    \item List the orders of elements in $\mathbb{Z}_{55}$. Then determine the number of elements of each order.   

\begin{tabular}{c||c|c|c|c|c|c} 
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} \\ \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}
\end{tabular}

\vspace{0.2in}

    \item List the orders of elements in $D_{55}$. Then determine the number of elements of each order.   

\begin{tabular}{c||c|c|c|c|c|c} 
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} \\ \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}
\end{tabular}


\end{enumerate}

\newpage

\noindent {\bf\large 3. (22 points)} Permutations
\begin{enumerate}[(a)]
   \item Let $G = \langle i \rangle = \{ 1,i,-1,-i \}$ where $i=\sqrt{-1}$.  \qquad [$G$ is a subgroup of $\mathbb{C}_{\not=0}$ (nonzero complex numbers).] 
   
   Label $1$ as $1$, $i$ as $2$, $-1$ as $3$, and $-i$ as $4$. Cayley's theorem says that $G$ is isomorphic to a subgroup of $S_4$. Find this subgroup [using left multiplication maps and the labels provided].
   
\vspace{1.5in}

 $G \cong \displaystyle \Bigg\{$ \hspace{4in} $\displaystyle \Bigg\}$

   \item Write $\sigma = (237)(1724)(27563)$ as a product of disjoint cycles.
   
   \vspace{0.75in}
   
             $\sigma^{-1}=$ 
             
   \vspace{0.5in}
   
            The order of $\sigma$ is $|\sigma|=$ \underline{\hspace{1in}}.  
            
   \vspace{0.3in}
   
           Write $\sigma$ as a product of transpositions. \qquad $\sigma$ is \quad {\large\bf Even \ \  / \ \  Odd}

    \vspace{0.75in}
    
         Compute $\sigma^{30}$.

\end{enumerate}

\newpage

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

\begin{enumerate}[(a)]

\item $\mathbb{Q} \not\cong \mathbb{Z}_{123}$ \quad [$\mathbb{Q}$ is the rational numbers.]

\vfill

\item $Q \not\cong \mathrm{Aut}(\mathbb{Z}_{8})$  \quad [$Q = \{\pm 1, \pm i, \pm j, \pm k\}$ is the group of quaternions.]

\vfill

\item $A_4 \not\cong D_{6}$

\vfill

\end{enumerate}


\newpage

\noindent {\bf\large 5. (16 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 $n>3$.

\vfill

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

   \begin{enumerate}[I.]
   \item Let $G$ be an abelian group. Show that $\varphi:G \to G$ defined by $\varphi(x)=x^{-1}$ is an automorphism of $G$.
   \item Let $\varphi, \psi$ be automorphisms of $G$. Prove that $H = \{ x \in G \;|\; \varphi(x)=\psi(x) \}$ is a subgroup of $G$.
   \end{enumerate}
   
   
\vfill

\end{enumerate}


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