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\begin{document}

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\noindent
\parbox{2in}{\bf Math 3110} 
\hfill {\Large \bf Test \#2} \hfill
\parbox{2in}{\bf \hfill October 28$^{th}$, 2009}

\vspace{0.3in}

\noindent {\large\bf Name:} \underline{\hskip 3.0 truein}\\
Don't merely state answers, prove your statements.

\vspace{0.3in}

\begin{description}
   \item[\large 1. (\underline{\hskip 0.35 truein}/16 points)] Either prove $G$ is a group
        or explain why it is not a group.
\begin{enumerate}[(a)]
      \item $G = \{ x \in D_5 \,|\, x \mbox{ is a reflection} \}$ (the operation is composition of symmetries).
                
\vspace{4in}
      
      \item $\mathbb{R}_{>0}$ (positive real numbers) where the operation is multiplication.
\end{enumerate}

\newpage
   \item[\large 2. (\underline{\hskip 0.35 truein}/16 points)] 
\begin{enumerate}[(a)]
      \item Show $\displaystyle{ H = \left\{ \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \,\Bigg|\, a,b \in \mathbb{R} \mbox{ and } a,b \not=0 \right\}}$ is a subgroup of $\mathrm{GL}_2(\mathbb{R})$ 

\vspace{0.1in}

(Recall $\mathrm{GL}_2(\mathbb{R})$ --- the set of $2 \times 2$ invertible matrices ---
 is a group under matrix multiplication).

\vspace{6in}

      \item Quickly (in a few words) why are the even integers a subgroup of the integers?
      
\end{enumerate}

\newpage
   \item[\large 3. (\underline{\hskip 0.35 truein}/20 points)] Workin' mod 5.

\begin{enumerate}[(a)]
\item Fill out the following tables (don't worry about brackets for equivalence classes.)

\begin{center} \hspace*{-0.6in}
\begin{tabular}{c|c|c|c|c|c|}
$(\mathbb{Z}_5,+)$ & 0 & 1 & 2 & 3 & 4 \\ \hline
 0  & \myspace  & \myspace & \myspace & \myspace & \myspace \\ \hline
 1  & \myspace  & \myspace & \myspace & \myspace & \myspace \\ \hline
 2  & \myspace  & \myspace & \myspace & \myspace & \myspace \\ \hline
 3  & \myspace  & \myspace & \myspace & \myspace & \myspace \\ \hline
 4  & \myspace  & \myspace & \myspace & \myspace & \myspace \\ \hline
\multicolumn{6}{c}{$\mathbb{Z}_5$ Addition Table}
\end{tabular} \hspace{0.3in}
\begin{tabular}{c|c|c|c|c|c|}
$(\mathbb{Z}_5,\times)$ & 0 & 1 & 2 & 3 & 4 \\ \hline
 0  & \myspace  & \myspace & \myspace & \myspace & \myspace \\ \hline
 1  & \myspace  & \myspace & \myspace & \myspace & \myspace \\ \hline
 2  & \myspace  & \myspace & \myspace & \myspace & \myspace \\ \hline
 3  & \myspace  & \myspace & \myspace & \myspace & \myspace \\ \hline
 4  & \myspace  & \myspace & \myspace & \myspace & \myspace \\ \hline
\multicolumn{6}{c}{$\mathbb{Z}_5$ Multiplication Table}
\end{tabular}
\end{center}

\item Compute $2^{-1}(4+3)-2$ (mod 5).

\vspace{2in}

\item Find $\langle 3 \rangle$ (the subgroup generated by $3$) in $U(5)$ (NOT $\mathbb{Z}_5$!!!).

\vspace{2in}

\item Find the orders of elements of $U(5)$. Is $U(5)$ cyclic? Why or why not?

\begin{tabular}{r|c}
element = & {\mbox{\parbox[c][0.4in]{4in}{\hfill}}} \\ \hline
order = & {\mbox{\parbox[c][0.4in]{4in}{\hfill}}}
\end{tabular}

\end{enumerate}


\newpage
   \item[\large 4. (\underline{\hskip 0.35 truein}/16 points)] Quick proofs
\begin{enumerate}[(a)]
      \item Let $G$ be a group and suppose that $x=x^{-1}$ for all $x \in G$. Prove that $G$ is Abelian.\\
       Hint: $xy=(xy)^{-1}=\;\;??$

\vspace{4in}

      \item Let $n$ be an integer such that $n \geq 3$. Consider $(12), (13) \in S_n$.
                Compute $(12)(13)$ and $(13)(12)$. Is $S_n$ cyclic? Why or why not?
            
\end{enumerate}

\newpage
   \item[\large 5. (\underline{\hskip 0.35 truein}/16 points)] $G = \{1,a,a^2,b,ab,a^2b \}$ is a group. 
   
   Finish filling out $G$'s Cayley table then answer some questions.
   
 \begin{center}
\begin{tabular}{|c|||c|c|c|c|c|c|}
\hline
$G$  & $1$    & $a$    & $a^2$  & $b$    & $ab$   & $a^2b$ \\ \hline 
\hline \hline
$1$    & $1$    & $a$    & $a^2$  & $b$    & $ab$   & $a^2b$ \\ \hline 
$a$    & $a$    & \myspace & \myspace & $a  b$ & $a^2b$ & $   b$ \\ \hline 
$a^2$  & $a^2$  & $1   $ & \myspace & \myspace & $   b$ & $a  b$ \\ \hline 
$b$    & $b$    & $a^2b$ & $a  b$ & $1   $ & $a^2 $ & \myspace \\ \hline 
$ab$   & $ab$   & $   b$ & $a^2b$ & \myspace & \myspace & \myspace \\ \hline 
$a^2b$ & $a^2b$ & $a  b$ & \myspace & $a^2 $ & \myspace & $1   $ \\ \hline 
\end{tabular}
\end{center}   
   
   
\begin{enumerate}[(a)]
   \item What is the order of $a^2$? Determine $\langle a^2 \rangle$ (the subgroup generated by $a^2$).
   
\vspace{3in}   
   
   \item Is $G$ Abelian? Is $G$ cyclic? Why or why not?

\end{enumerate}   
   
\newpage
   \item[\large 6. (\underline{\hskip 0.35 truein}/16 points)] Permutations! 
\begin{enumerate}[(a)]
      \item Write $\sigma = (125)(35)(24)(264)$ as a product of disjoint cycles.

\vspace{2in}
      
      \item What is the order of $\sigma$? 
      
\vspace{0.5in}

      \item Write $\sigma$ as a product of transpositions. Is $\sigma$ even or odd?
      
\vspace{2in}

      \item Let $\tau = (1452)(367)(89)$. What is the order of $\tau$? 

\vspace{0.5in}

      \item Write $\tau^{26}$ as the product of disjoint cycles.

\vspace{2in}

      \item What is the order of $\tau^6$?\\
       {}[Hint: You shouldn't need to compute any powers of $\tau$ to answer this question.]

   \end{enumerate}
\end{description}

\end{document}
