\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 September $16^{\mathrm{th}}$, 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)} 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 Let $G = \mathbb{Z}_{>0}$ be the set of positive integers. Define a binary operation as follows: $x \star y = x^y$ (example: $2 \star 3 = 2^3=8$). $G$ paired with this operation is {\bf not} a group. Why? [Use concrete counterexample(s).] \vfill \item Let $G = \left\{ n\sqrt{2} \;|\; n \in \mathbb{Q}_{\not=0} \right\}$. (The non-zero rational multiples of $\sqrt{2}$. For example: $\sqrt{2}$ and $-\dfrac{3\sqrt{2}}{4}$ both belong to $G$.) $G$ is {\bf not} a group under \underline{multiplication}. Why? [Use concrete counterexample(s).] \vfill \end{enumerate} \newpage \noindent {\bf\large 2. (20 points)} Some modular arithmetic. \begin{enumerate}[(a)] \item What is the inverse of $80$ in the group $\mathbb{Z}_{100}$? What operation makes $\mathbb{Z}_{100}$ a group? \vspace{0.8in} \item Is $80$ an element of $U(100)$? If not, \underline{why not}? If so, \underline{why} so and what is its \underline{inverse}? \vfill \item Is $11$ an element of $U(100)$? If not, \underline{why not}? If so, \underline{why} so and what is its \underline{inverse}? \vfill \item What is the order of $80$ in $\mathbb{Z}_{100}$? \vspace{0.8in} \item Draw the subgroup lattice for $\mathbb{Z}_{100}$. \vfill \end{enumerate} \newpage \noindent {\bf\large 3. (25 points)} More Modular Arithmetic. \begin{enumerate}[(a)] \item List the elements of $U(10)$. Then find their {\bf order}s and the list the elements in {\bf cyclic subgroup} generated by that element. [{\it Note:} There may be more spaces than you need.] \vspace{0.1in} \force \hspace{-0.45in} \begin{tabular}{c||c|c|c|c|c|c|} $x=$ & \myspacea & \myspacea & \myspacea & \myspacea & \myspacea & \myspacea \\ \hline\hline $|x|=$ & \myspacea & \myspacea & \myspacea & \myspacea & \myspacea & \myspacea \\ \hline $\langle x \rangle=$ & $\displaystyle\left\{\mbox{\myspacea}\right\}$ & $\displaystyle\left\{\mbox{\myspacea}\right\}$ & $\displaystyle\left\{\mbox{\myspacea}\right\}$ & $\displaystyle\left\{\mbox{\myspacea}\right\}$ & $\displaystyle\left\{\mbox{\myspacea}\right\}$ & $\displaystyle\left\{\mbox{\myspacea}\right\}$ \\ \hline \end{tabular} \vfill \item Is $U(10)$ cyclic? \qquad {\bf\Large Yes \quad / \quad No} \qquad (Circle the correct answer.) \vspace{0.15in} \item Find $\displaystyle \left\langle A \right\rangle$ in $\mathrm{GL}_2(\mathbb{Z}_4)$ where $\displaystyle A = \begin{bmatrix} 3 & 0 \\ 1 & 1 \end{bmatrix}$. What is the order of $A$? What is $A^{-1}$? \vfill \item Find $7^{-1}$ mod 25 using the extended Euclidean algorithm [Don't just guess and check]. \vfill \end{enumerate} \newpage \noindent {\bf\large 4. (15 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 $x^{-2}y^3x^{11}yx^2y^{124}$ \vspace{1.75in} \item What is the inverse of $x^6$ in $D_8$? What is its order? \vspace{1.25in} \item Fill in the Cayley table for $D_3$: \begin{tabular}{c||c|c|c|c|c|c|} & $1$ & $x$ & $x^2$ & $y$ & $xy$ & $x^2y$ \\ \hline \hline $1$ & $1$ & $x$ & $x^2$ & $y$ & $xy$ & $x^2y$ {\Large\strut} \\ \hline $x$ & $x$ & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline $x^2$ & $x^2$ & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline $y$ & $y$ & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline $xy$ & $xy$ & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline $x^2y$ & $x^2y$ & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline \end{tabular} \vspace{0.2in} \item The {\bf rotations} form a subgroup of $D_3$ Explain why. Is this a cyclic subgroup? \end{enumerate} \newpage \noindent {\bf\large 5. (20 points)} Proofs! \begin{enumerate}[(a)] \item Choose one of the following: \begin{enumerate}[I.] \item Suppose that $(ab)^{2}=a^{2}b^{2}$ for all $a,b \in G$. Prove that $G$ is abelian. \item Let $G = \langle g \rangle$ be 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 $H = \{ x \in \mathbb{Z} \;|\; \mbox{there exists }k,\ell \in \mathbb{Z}\mbox{ such that }x=4k+6\ell \}$ is a subgroup of $\mathbb{Z}$. \item Prove that $K = \left\{ \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \;\Bigg|\; a,b \in \mathbb{R} \mbox{ and } a,b \not=0 \right\}$ is a subgroup of $\mathrm{GL}_2(\mathbb{R})$. \end{enumerate} \vfill \end{enumerate} \end{document}