\documentclass{article} \usepackage{graphicx} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{enumerate} \usepackage{ifthen} \setlength{\oddsidemargin}{0.0in} \setlength{\evensidemargin}{0.0in} \setlength{\topmargin}{0.0in} \setlength{\headheight}{0.0in} \setlength{\headsep}{0.0in} \setlength{\topskip}{0.0in} \setlength{\textheight}{8.9in} \setlength{\textwidth}{6.5in} \newcommand{\comp}{ \,{\scriptstyle \stackrel{\circ}{}}\, } \newcommand{\nullset}{\mathrm{O}\!\!\!\!\big/\,} \newcommand{\divides}{\,\Big|\,} \newcommand{\myspace}{\mbox{\parbox[c][0.3in]{0.3in}{\hfill}}} \begin{document} \pagestyle{empty} \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}