\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 $16^{\mathrm{th}}$, 2020} \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 odd integers $\mathbb{O} = 2\mathbb{Z}+1 = \{ 2k+1 \;|\; k \in \mathbb{Z} \}$ do not form a group under either addition or multiplication. Explain why they fail be a group under both of these operations. \vfill \item On the other hand, the even integers $\mathbb{E}=2\mathbb{Z} = \{ 2k \;|\; k \in\mathbb{Z}\}$ do form a group under one of these operations. Briefly explain why it is a group under that operation and why it isn't a group under the other operation. \vfill \item Why aren't the integers $\mathbb{Z}$ a group under subtraction? \vspace{0.5in} \end{enumerate} \newpage \noindent {\bf\large 2. (20 points)} Some modular arithmetic. \begin{enumerate}[(a)] \item Make a table listing the elements of $\mathbb{Z}_{9}$, their inverses, and their orders. \vfill \item Do the same for $U(9)$. \vfill \item Compute $2^{-1}(3-8)+7$ mod $9$ or explain why this is undefined. \vspace{0.8in} \item Compute $3^{-1}(7-3)-11$ mod $9$ 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 List all of the distinct cyclic subgroups of $\mathbb{Z}_{12}$. Show each subgroup's contents (e.g., $\langle 0 \rangle = \{0\}$). \vfill \item Draw the subgroup lattice for $\mathbb{Z}_{12}$. \vfill \item Find $\displaystyle \left\langle A \right\rangle$ in $\mathrm{GL}_2(\mathbb{Z}_6)$ where $\displaystyle A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$. What is the order of $A$? What is $A^{-1}$? \noindent {\color{red}\bf Embarrassing Error:} I changed $A$ at the last moment to its current definition. However, $\det(A) = 1(1)-2(2) = -3 =3$ (mod $6$) so $\det(A)^{-1}=3^{-1}$ does not exist in $\mathbb{Z}_6$. Thus $A \not\in \mathrm{GL}_2(\mathbb{Z}_6)$ since $A^{-1}$ (working mod 6) does not exist. So this part of the problem is \underline{\bf\color{red}nonsense}! Instead let's change to $A = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}$ and rework this part. \vfill \item Find $10^{-1}$ mod 27 using the extended Euclidean algorithm [Don't just guess and check]. \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 $D_{6}$ to simplify $y^{-2}x^{11}y^3x^{-4}y^{124}x$ \vspace{1.75in} \item Make a table listing the elements of $D_6$, their inverses, and their orders. \vspace{1.25in} \item What is $\langle x^4 \rangle$ in $D_6$? \vspace{0.5in} \item Fill in the following rows of the Cayley table for $D_6$: \hspace*{-0.25in} \begin{tabular}{c||c|c|c|c|c|c|c|c|c|c|c|c|} & $1$ & $x$ & $x^2$ & $x^3$ & $x^4$ & $x^5$ & $y$ & $xy$ & $x^2y$ & $x^3y$ & $x^4y$ & $x^5y$ \\ \hline \hline $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \\ \hline $x^5$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline $y$ & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \\ \hline \end{tabular} \vspace{0.5in} \item Is $H = \{ 1,x^2,y,x^2y \}$ a subgroup of $D_6$? 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 $g^2=1$ for all $g \in G$. Prove that $G$ is abelian. \item For all $g \in G$. Show that $|g|=|g^{-1}|$. \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 $H = \{ 10k+6\ell \;|\; k,\ell \in \mathbb{Z} \}$ is a subgroup of $\mathbb{Z}$. \item Prove that $K = \left\{ \begin{bmatrix} 1 & 0 \\ a & 1 \end{bmatrix} \;\Bigg|\; a \in \mathbb{R} \right\}$ is a subgroup of $\mathrm{GL}_2(\mathbb{R})$. \end{enumerate} \vfill \end{enumerate} \end{document}