\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 March $4^{\mathrm{th}}$, 2022} \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|c|c|c|} \hline $G=$ & \parbox[c][0.5in]{0.85in}{What is the\\ identity of $G$?} & \parbox[c][0.5in]{1in}{What is the\\ order of ...?} & \parbox[c][0.5in]{1.25in}{Does $G$ have an\\ element of order 6?} & \parbox[c][0.5in]{0.85in}{Is $G$ abelian?} & \parbox[c][0.5in]{0.85in}{Is $G$ cyclic?} \\ \hline \hline \hline $\mathbb{Z}_{55}$ & \varspace{0.75in}{0.75in} & $|33|=$ & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & \\ \hline $U(9)$ & \varspace{0.75in}{0.75in} & $|5|=$ & \varspace{0.75in}{1in} & \varspace{0.75in}{0.75in} & \\ \hline $D_{10}$ & \varspace{0.75in}{0.75in} & $|x^4|=$ & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & \\ \hline $S_7$ & \varspace{0.75in}{0.75in} & $|(1234)(56)|=$ & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & \\ \hline \end{tabular} \vspace{0.05in} {\bf Recall:} $D_{10} = \{ 1, x, \dots, x^{9}, y, xy, \dots, x^{9}y \}$ where $x^{10}=1$, $y^2=1$, and $xyxy=1$. \vspace{0.1in} \underline{Scratch Work:} \newpage \noindent {\bf\large 2. (20 points)} Cyclic Stuff \begin{enumerate}[(a)] \item Let $G$ be a finite group and $g \in G$. Suppose that $|g|=120=2^3\cdot 3 \cdot 5$. \begin{enumerate}[i.] \item What is the order of $g^{100}$? List the distinct elements in $\langle g^{100} \rangle$. \vspace{1.5in} $\langle g^{100} \rangle = \displaystyle \Bigg\{$ \hspace{4.5in} $\displaystyle \Bigg\}$ \vspace{0.3in} \item Is $g^{31} \in \langle g^{77} \rangle$? \quad {\large\bf Yes \ \ / \ \ No} \vspace{1in} \end{enumerate} \item How many elements of order 10 does $\mathbb{Z}_{40}$ have? What are they? \vfill \item List the orders of elements in $\mathbb{Z}_{52}$. Then determine the number of elements of each order. {\it Note:} It might be helpful to know that $52=2^2\cdot 13$ and its positive divisors are 1, 2, 4, 13, 26, and 52. \begin{tabular}{c||ccccccc} 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} & \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} & \varspace{0.5in}{0.5in} \end{tabular} \vspace{0.2in} \item List the orders of elements in $D_{52}$. Then determine the number of elements of each order. \begin{tabular}{c||ccccccc} 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} & \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} & \varspace{0.5in}{0.5in} \end{tabular} \end{enumerate} \newpage \noindent {\bf\large 3. (22 points)} Permutations \qquad {\it Note:} Please give simplified (``good manners'') answers. \begin{enumerate}[(a)] \item Consider $G = D_3 = \{ 1,x,x^2,y,xy,x^2y \} = \langle x,y \;|\; x^3=1, y^2=1, (xy)^2=1\rangle$. Label $1$ as $1$, $x$ as $2$, $x^2$ as $3$, $y$ as $4$, $xy$ as $5$, and $x^2y$ as $6$. Cayley's theorem says that $G$ is isomorphic to a subgroup of $S_6$. Write down left multiplication by $xy$ does in $D_3$: $L_{xy}:D_3 \to D_3$ is\dots \vspace{0.2in} $\;\,1 \quad \mapsto \quad$ \underline{\hspace*{0.75in}}\\ $\;\,x \quad \mapsto \quad$ \underline{\hspace*{0.75in}}\\ $x^2 \quad \mapsto \quad$ \underline{\hspace*{0.75in}}\\ $\;\,y \quad \mapsto \quad$ \underline{\hspace*{0.75in}}\\ $\! xy \quad \mapsto \quad$ \underline{\hspace*{0.75in}}\\ $\!\!\!\! x^2y \quad \mapsto \quad$ \underline{\hspace*{0.75in}}\\ The corresponding permutation in $S_6$ is \underline{\hspace*{1.5in}}. %\vspace{0.5in} \item Suppose that using Cayley's theorem we found the left multiplication operator of $x$ in $D_4$ corresponds with $(1234)(5678)$ and the left multiplication operator of $y$ corresponds with $(15)(28)(37)(46)$. Find the permutation associated with $x^2y$. [Your answer should be written in terms of disjoint cycles.] \vspace{1.2in} \item Write $\sigma = (124)(1326)(34576)$ as a product of disjoint cycles.\\ $\phantom{{}^{-1}}$$\sigma=$ \vspace{0.4in} $\sigma^{-1}=$ \vspace{0.4in} The order of $\sigma$ is $|\sigma|=$ \underline{\hspace{1in}}. \vspace{0.3in} Write $\sigma$ as a product of transpositions: \underline{\hspace{2.25in}}. \quad $\sigma$ is {\large\bf \ Even \ / \ Odd} \vspace{0.4in} $\sigma^{62}=$ \vspace{0.4in} \item Does $S_{6}$ have an element of order 8? If not, explain why not. If so, give an example of such an element. \end{enumerate} \newpage \noindent {\bf\large 4. (18 points)} Explain why the following pairs of groups are not isomorphic. \begin{enumerate}[(a)] \item $Q \not\cong \mathbb{Z}_{8}$ where $Q=\{ \pm 1, \pm i, \pm j, \pm k\}$ is the quaternion group so $i^2=j^2=k^2=-1$, $ij=k$, $ji=-k$, etc. \vfill \item $\mathbb{Z}_{9} \not\cong U(9)$ \vfill \item $S_4 \not\cong D_{12}$ \vfill \end{enumerate} \newpage \noindent {\bf\large 5. (20 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 any $n>3$. \vfill \item Pick {\bf ONE} of the following\dots \begin{enumerate}[I.] \item Show that every cyclic group is Abelian. Then explain why the converse is not true. \item Show that $\mathbb{R} \cong \mathbb{R}_{>0}$. \qquad {\it Hint:} Consider an exponential function for $\varphi$. {\it Note:} $\mathbb{R}_{>0} = (0,\infty)$ is the group of positive real numbers. \end{enumerate} \vfill \end{enumerate} \end{document}