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\noindent
\parbox{1.5in}{\bf Math 3110} 
\hfill {\Large \bf  Test \#2} \hfill
\parbox{1.5in}{\bf \hfill March $6^{\mathrm{th}}$, 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{-0.6in}
\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]{0.85in}{What is the\\ order of ...?} & \parbox[c][0.5in]{1.25in}{Does $G$ have an\\ element of order 10?} \\
\hline \hline \hline
$\mathbb{Z}_{100}$ & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in}  & $|15|=$ & \\ \hline
$U(4)$ & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in}  & $|3|=$ \varspace{0.75in}{1in} & \\ \hline
$D_5$ & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in}  & $|x^2y|=$ & \\ \hline
$S_7$ & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in}  & $|(12)(3456)|=$ & \\ \hline
\end{tabular} 

{\bf Recall:} $D_5 = \{ 1, x, \dots, x^4, y, xy, \dots, x^4y \}$ where $x^5=1$, $y^2=1$, 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 $|g|=35$. 
      
      \begin{enumerate}[i.]
      \item What is the order of $g^{10}$? List the distinct elements in $\langle g^{10} \rangle$.
      
      
      \vspace{1.5in}
      
      
      $\langle g^{10} \rangle = \displaystyle \Bigg\{$ \hspace{4in} $\displaystyle \Bigg\}$
          
      \vspace{0.3in}
      
      \item  Is $g^{100} \in \langle g^{10} \rangle$? \quad {\large\bf Yes \ \  /  \ \  No}
      
      \vspace{1in}
      
      
      \end{enumerate}

     \item How many elements of order 8 does $\mathbb{Z}_{40}$ have? What are they?
     
\vfill

    \item List the orders of elements in $\mathbb{Z}_{35}$. 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 $D_{35}$. 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

Recall that $Q = \{ 1,-1,i,-i,j,-j,k,-k \}$ where $ij=k, jk=i, ki=j, ji=-k,i^2=j^2=k^2=-1,$ etc.
\begin{enumerate}[(a)]
   \item Write down what the left multiplication operator of $i$ does in $Q$. Then write down the corresponding
            permutation if we label $1$ as $1$, $-1$ as $2$, $i$ as $3$, $-i$ as $4$, $j$ as $5$, $-j$ as $6$, $k$ as $7$, and $-k$ as $8$.
            
            \vspace{0.2in}
            
            \parbox[c][2in]{2in}{
            $L_i : Q  \rightarrow  Q$
            
            \vspace{-0.1in}

            \begin{eqnarray*}
            1 & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\
            -1 & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\
            i & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\
            -i & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\
            j & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\
            -j & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\
            k & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\
            -k & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\
            \end{eqnarray*}
            }
            
            The corresponding permutation is...?

\vspace{0.5in}

   \item Suppose that using Cayley's theorem we found the left multiplication operator of $x$ in $D_3$ corresponds to
             $(123)(456)$ and $y$ corresponds to $(14)(26)(35)$. What would the left multiplication operator of $x^2y$ correspond to?
             [Your answer should be a permutation written as a product of disjoint cycles.]
             
 \vspace{1.25in}
 

   \item Write $\sigma = (152)(3467)(25)(37)$ 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 $\sigma^{65}$.

\end{enumerate}

\newpage

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

\begin{enumerate}[(a)]

\item $U(1000) \not\cong \mathbb{R}$

\vfill

\item $A_5 \not\cong \mathbb{Z}_{60}$

\vfill

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

\vfill

\end{enumerate}


\newpage

\noindent {\bf\large 5. (16 points)} A few proofs

\begin{enumerate}[(a)]

\item Prove that $U(10) \cong \mathbb{Z}_4$.

\vfill

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

   \begin{enumerate}[I.]
   \item Let $g \in G$ ($G$ a group). Show that $\varphi:G \to G$ defined by $\varphi(x)=gxg^{-1}$ is an automorphism of $G$.
   \item Show that if $G$ is a cyclic group, then $G$ is abelian. Is the converse true?
   \end{enumerate}
   
   
\vfill

\end{enumerate}


\end{document}


