\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{\varspace}[2]{\mbox{\parbox[c][#1]{#2}{\hfill}}} \begin{document} \pagestyle{empty} \noindent \parbox{2in}{\bf Math 3110} \hfill {\Large \bf Test \#2} \hfill \parbox{2in}{\bf \hfill March $31^{\mathrm{st}}$, 2010} \vspace{0.3in} \noindent {\large\bf Name:} \underline{\hskip 3.0 truein} \hfill {\bf Be sure to show your work!} \begin{description} \item[\large 1. (\underline{\hskip 0.35 truein}/20 points)] Random Group Stuff --- Fill out the following table: %\mbox{} \hspace{-0.6in} \begin{tabular}{|c|||c|c|c|l|c|} \hline $G=$ & \parbox[c][0.5in]{0.85in}{What is the\\ identity of $G$?} & \parbox[c][0.5in]{0.85in}{Is $G$ abelian?} & \parbox[c][0.5in]{0.85in}{Is $G$ cyclic?} & \parbox[c][0.5in]{0.85in}{What is the\\ order of ...?} & \parbox[c][0.5in]{1.25in}{Does $G$ have an\\ element of order 4?} \\ \hline \hline \hline $\mathbb{Z}_{40}$ & \varspace{1in}{0.75in} & \varspace{1in}{0.75in} & \varspace{1in}{0.75in} & $|30|=$ & \\ \hline $U(8)$ & \varspace{1in}{0.75in} & \varspace{1in}{0.75in} & \varspace{1in}{0.75in} & $|5|=$ \varspace{1.25in}{0.75in} & \\ \hline $D_7$ & \varspace{1in}{0.75in} & \varspace{1in}{0.75in} & \varspace{1in}{0.75in} & $|x^4y|=$ & \\ \hline $S_4$ & \varspace{1in}{0.75in} & \varspace{1in}{0.75in} & \varspace{1in}{0.75in} & $|(12)(34)|=$ & \\ \hline \end{tabular} {\bf Recall:} $D_7 = \{ 1, x, \dots, x^6, y, xy, \dots, x^6y \}$ where $x^7=1$, $y^2=1$, and $xyxy=1$. \vspace{0.1in} \underline{Scratch Work:} \newpage \item[\large 2. (\underline{\hskip 0.35 truein}/20 points)] Group or not? Are the following sets with operations groups or not? If $G$ is a group, prove it --- you may use a subgroup test if it applies. If $G$ fails to be a group, explain what property fails. \begin{enumerate}[(a)] \item Let $G = [-1,1] = \{ r \in \mathbb{R} \,|\, -1 \leq r \leq 1 \}$ with the operation ``$+$'' (addition). \vspace{4in} \item Let $\displaystyle{ G = \left\{ \begin{bmatrix} 1 & 0 \\ r & 1 \end{bmatrix} \,\Big|\, r \in \mathbb{R} \right\} }$ with the operation of matrix multiplication. \end{enumerate} \newpage \item[\large 3. (\underline{\hskip 0.35 truein}/15 points)] Cayley's Theorem and Permutations.\\ Recall that $D_n = \{ 1,x,x^2,\dots,x^{n-1},y,xy,\dots,x^{n-1}y \}$ where $x^n=1$, $y^2=1$, and $xyxy=1$. \begin{enumerate}[(a)] \item Write down what the left multiplication operator of $y$ does in $D_3$. Then write down the corresponding permutation if we label $1$ as $1$, $x$ as $2$, $x^2$ as $3$, $y$ as $4$, $xy$ as $5$, $x^2y$ as $6$. \vspace{0.2in} \parbox[c][2in]{2in}{ $L_y : D_3 \rightarrow D_3$ \vspace{-0.1in} \begin{eqnarray*} 1 & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\ x & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\ x^2 & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\ y & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\ xy & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\ x^2y & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\ \end{eqnarray*} } The corresponding permutation is...? \vspace{1in} \item Suppose that using Cayley's theorem we found the left multiplication operator of $x$ in $D_4$ corresponds to $(1234)(5678)$ and $y$ corresponds to $(15)(28)(37)(46)$. What would the left multiplication operator of $x^2y$ correspond to? [Your answer should be a permutation written as a product of disjoint cycles.] \vspace{2.5in} Now write your answer as a product of transpositions. Is this permutation even or odd? \end{enumerate} \newpage \item[\large 4. (\underline{\hskip 0.35 truein}/25 points)] Mod stuff. \begin{enumerate}[(a)] \item Draw the subgroup lattice of $\mathbb{Z}_{44}$. {\it Note:} $44 = 2^2 \cdot 11$. \vspace{2.5in} \item List the possible orders of elements in $\mathbb{Z}_{44}$. Then determine the number of elements of each order. \begin{tabular}{c||c|c|c|c|c|c} 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} \\ \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} \end{tabular} \vspace{0.2in} \item Show that $f : \mathbb{Z}_{6} \rightarrow \mathbb{Z}_{4}$ defined by $f(x)=2x$ is a {\bf well-defined} homomorphism. \vspace{2in} \item $\mathrm{Ker}(f) =$ \underline{\hspace{3in}} \vspace{0.5in} \\ $f(\mathbb{Z}_6) =$ \underline{\hspace{3in}} \vspace{0.5in} \\ Is $f$ 1-1? \vspace{0.2in}\\ Is $f$ onto? \vspace{0.2in}\\ Is $f$ an isomorphism? \end{enumerate} \newpage \item[\large 5. (\underline{\hskip 0.35 truein}/20 points)] POOF! ...I mean... PROOFS! [No magic please.] \begin{enumerate}[(a)] \item Let $G$ be a group and let $a,b \in G$ such that $(ab)^2=a^2b^2$. Show that $ab=ba$. \vspace{1in} \item Let $G$ be a group and let $g \in G$. Define the map $\varphi:G \rightarrow G$ by $\varphi(x)=gxg^{-1}$.\\ Prove that $\varphi$ is an isomorphism. \vspace{3.5in} \item Let $G$ and $G'$ be groups and let $\psi:G \rightarrow G'$ be a homomorphism. Suppose that $G$ is cyclic. Show that $\psi(G)$ is abelian. [Extra Credit: Show that $\psi(G)$ is cyclic.] \end{enumerate} \end{description} \end{document}