\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{2in}{\bf Math 3110} \hfill {\Large \bf Big Quiz \#1} \hfill \parbox{2in}{\bf \hfill September $23^{\mathrm{rd}}$, 2013} \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? \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 $G = \mathbb{Q}_{\leq 0}$ be the non-positive rational numbers. Prove $G$ is {\bf not} a group under addition. \vfill \item Let $G = \mathbb{R}$ (the real numbers). Prove $G$ is {\bf not} a group under subtraction. \vfill \end{enumerate} \newpage \noindent {\bf\large 2. (20 points)} Working mod $20$. \begin{enumerate}[(a)] \item What is the inverse of $3$ in the group $\mathbb{Z}_{20}$? What operation makes $\mathbb{Z}_{20}$ a group? \vspace{0.8in} \item Is $3$ an element of $U(20)$? If not, why not? If so, why so \& what is its inverse? \vspace{0.8in} \item List all of the {\it distinct} cyclic subgroups, $\langle x \rangle$, of $\mathbb{Z}_{20}$. \vfill \item What is the order of $15$ in $\mathbb{Z}_{20}$? \vspace{0.8in} \item Draw the subgroup lattice of $\mathbb{Z}_{20}$ [Note: $20=2^2\cdot 5$]. \vfill \end{enumerate} \newpage \noindent {\bf\large 3. (20 points)} More Modular Arithmetic. \begin{enumerate}[(a)] % \item Draw a Cayley table for $U(8)$. %\vspace{1.5in} \item List the elements of $U(8)$. 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} \vspace{0.15in} \item Is $U(8)$ cyclic? \qquad {\bf\Large Yes \quad / \quad No} \qquad (Circle the correct answer.) \vspace{0.15in} \item Let $\displaystyle A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$. Is $A \in \mathrm{GL}_2(\mathbb{Z}_{7})$? If so, find $A^{-1}$. If not, explain why not. %\vspace{1in} % \item Find $\displaystyle \left\langle B \right\rangle = \left\langle \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \right\rangle$ in $\mathrm{GL}_2(\mathbb{Z}_6)$. What is the order of $B$? \vfill \item Find $36^{-1}$ mod 151 using the extended Euclidean algorithm. \vfill \end{enumerate} \newpage \noindent {\bf\large 4. (20 points)} Recall that $D_4 = \{1,x,x^2,x^3,y,xy,x^2y,x^3y \} = \langle x,y \;|\; x^4=1, y^2=1, (xy)^2=1 \rangle$. \begin{enumerate}[(a)] \item Use the relations for $D_4$ to derive the relation: $yx = x^{-1}y$. \vspace{1.75in} \item Fill in the Cayley table for $D_4$: \begin{tabular}{c||c|c|c|c|c|c|c|c|} & $1$ & $x$ & $x^2$ & $x^3$ & $y$ & $xy$ & $x^2y$ & $x^3y$ \\ \hline \hline $1$ & $1$ & $x$ & $x^2$ & $x^3$ & $y$ & $xy$ & $x^2y$ & $x^3y$ {\Large\strut} \\ \hline $x$ & $x$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline $x^2$ & $x^2$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline $x^3$ & $x^3$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline $y$ & $y$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline $xy$ & $xy$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline $x^2y$ & $x^2y$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline $x^3y$ & $x^3y$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline \end{tabular} \vspace{0.2in} \item Find the inverse and order of each element in $D_4$. \begin{tabular}{c||c|c|c|c|c|c|c|c|} \hline $g=$ & $1$ & $x$ & $x^2$ & $x^3$ & $y$ & $xy$ & $x^2y$ & $x^3y$ \\ \hline \hline $g^{-1}=$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline $|g|=$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline \end{tabular} \vspace{0.5in} \item Do the rotations form a subgroup of $D_4$? If so, why? If not, why not? \end{enumerate} \newpage \noindent {\bf\large 5. (20 points)} Proofs! \begin{enumerate}[(a)] \item Choose one of the following: \begin{enumerate}[I.] \item Prove that $f:\mathbb{Z} \to \mathbb{Z}$ defined by $f(x)=5x$ is 1-1 but not onto. \item Let $G$ be a group and $g \in G$. Prove that $|g|=|g^{-1}|$ \quad (i.e. $g$ and its inverse have the same order). \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 $\mathrm{SL}_2(\mathbb{Z}) = \left\{ A \in \mathbb{Z}^{2 \times 2} \;|\; \mathrm{det}(A)=1 \right\}$ is a subgroup of $\mathrm{GL}_2(\mathbb{Z})$. \item Prove that $H = \{ n \in \mathbb{Z} \;|\; n = 10x+6y \mbox{ for some } x,y \in \mathbb{Z} \}$ is a subgroup of $\mathbb{Z}$. \end{enumerate} \vfill \end{enumerate} \end{document}