\documentclass{article} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{enumerate} \usepackage{ifthen} \usepackage{graphicx} \setlength{\unitlength}{0.1in} \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][0.35in]{0.35in}{\hfill}}} \newcommand{\myspacea}{\mbox{\parbox[c][0.35in]{0.65in}{\hfill}}} \newcommand{\varspace}[2]{\mbox{\parbox[c][#1]{#2}{\hfill}}} \newcommand{\force}{{$\mbox{\;}$}} \begin{document} \pagestyle{empty} \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} \end{document}