\documentclass{article} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{enumerate} \usepackage{ifthen} \usepackage{graphicx} \usepackage{color} % for \sout to strike through text \usepackage{ulem} \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{2in}{\bf Math 3110} \hfill {\Large \bf Test \#1} \hfill \parbox{2in}{\bf \hfill February $17^{\mathrm{th}}$, 2021} \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)} Definition and Basics \begin{enumerate}[(a)] \item Suppose that $G$ is a non-empty set equipped an operation. What 4 things do I need to check to see if $G$ is a group? Give details. \vspace{0.2in} \begin{enumerate}[1:] \item \mbox{} \vspace{0.2in} \item \mbox{} \vspace{0.2in} \item \mbox{} \vspace{0.2in} \item \mbox{} \vspace{0.2in} \mbox{}\hspace{-0.35in} What additional property needs to hold for $G$ to be an {\bf Abelian} group? \vspace{0.2in} \item \mbox{} \vspace{0.2in} \end{enumerate} \item The positive real numbers $\mathbb{R}_{>0} = \{ r \in \mathbb{R} \;|\; r>0 \}$ do not form a group under division. Why not? \vfill \item On the other hand, the positive real numbers $\mathbb{R}_{>0} = \{ r \in \mathbb{R} \;|\; r>0 \}$ do form a group if we select the right operation. Which operation turns this collection of numbers into a group: Addition or Multiplication? Then explain why the other operation does not yield a group. \vfill \item The non-zero rational numbers $\mathbb{Q}_{\not=0}$ form a group under multiplication. On the other hand, the (non-zero) irrational numbers $\mathbb{I}=\mathbb{R}-\mathbb{Q} = \{ x \in \mathbb{R} \;|\; x \not\in \mathbb{Q}\}$ do not. Why? \vspace{0.5in} \end{enumerate} \newpage \noindent {\bf\large 2. (20 points)} Some modular arithmetic. \begin{enumerate}[(a)] \item Make a list of all of the cyclic subgroups of $\mathbb{Z}_{10}$ along with their contents (for example: $\langle 0 \rangle = \{0\}$). \vfill \item Fill out the following table referring to the operations of addition and multiplication modulo 10: {\it Note:} Just put an {\large\bf X} if something is undefined / does not exist. \hspace{-0.5in} \begin{tabular}{|r|c|c|c|c|c|c|c|c|c|c|} \hline Element $x=$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \large\strut \\ \hline Additive Inverse $-x=$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline Order (in $\mathbb{Z}_{10}$) $|x|=$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline Multiplicative Inverse $x^{-1}=$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline Order (in $U(10)$) $|x|=$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline \end{tabular} \vspace{2in} \item Compute $2^{-1}(3-8)+7$ mod $10$ or explain why this is undefined. \vspace{0.8in} \item Compute $3^{-1}(7-3)-11$ mod $10$ or explain why this is undefined. \vspace{0.8in} \end{enumerate} \newpage \noindent {\bf\large 3. (20 points)} More Modular Arithmetic. \begin{enumerate}[(a)] \item Write down a Cayley table for $U(8)$. Is $U(8)$ cyclic (circle the correct answer)? \quad {\Large Yes \quad / \quad No} \vfill \item Draw the subgroup lattice for $\mathbb{Z}_{20}$. [$20 = 2^2 \cdot 5$] \vfill \vfill \item Find $10^{-1}$ mod 77 using the extended Euclidean algorithm [Don't just guess and check]. \vfill \vfill \end{enumerate} \newpage \noindent {\bf\large 4. (20 points)} Recall $D_n = \{1,x,\dots,x^{n-1},y,xy,\dots,x^{n-1}y \} = \langle x,y \;|\; x^n=1, y^2=1, (xy)^2=1 \rangle$. \begin{enumerate}[(a)] \item Use the relations for $D_{8}$ to simplify $x^{13}y^3x^{-2}y^{888}x$ \vspace{1.75in} \item Make a table listing the elements of $D_8$, their inverses, and their orders. \vspace{1.25in} \item What is $\langle x^6 \rangle$ in $D_8$? \vspace{0.5in} \item Fill in the following rows of the Cayley table for $D_4$: \hspace*{-0.25in} \begin{tabular}{c||c|c|c|c|c|c|c|c|c|c|c|c|} & $1$ & $x$ & $x^2$ & $x^3$ & $y$ & $xy$ & $x^2y$ & $x^3y$ \\ \hline \hline $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \\ \hline $x^3$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline $y$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \\ \hline \end{tabular} \vspace{0.5in} \item Is $H = \{ 1, y, x^2y \}$ a subgroup of $D_4$? Why or why not? \vspace{0.75in} \end{enumerate} \newpage \noindent {\bf\large 5. (20 points)} Proofs! \begin{enumerate}[(a)] \item Choose one of the following: \qquad \underline{Assume $G$ is a group under multiplication with identity $1$.} \begin{enumerate}[I.] \item Suppose that $g=g^{-1}$ for all $g \in G$. Prove that $G$ is abelian. \item Suppose that $G=\langle g \rangle$ is a cyclic group. Prove that $G$ is abelian. \end{enumerate} \vfill \item Choose one of the following: \qquad (You {\bf must} use a subgroup test in your proof.) \begin{enumerate}[I.] \item Prove that $H = 8\mathbb{Z} = \{ 8k \;|\; k \in \mathbb{Z} \}$ is a subgroup of $\mathbb{Z}$. \item Prove that $K = \left\{ \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \;\Bigg|\; a,b \in \mathbb{R}_{\not=0} \right\}$ is a subgroup of $\mathrm{GL}_2(\mathbb{R})$. \end{enumerate} \vfill \end{enumerate} \end{document}