\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{1.5in}{\bf Math 3110} \hfill {\Large \bf Test \#2} \hfill \parbox{1.5in}{\bf \hfill March $6^{\mathrm{th}}$, 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)} Random Group Stuff --- Fill out the following table: \vspace{0.2in} %\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 10?} \\ \hline \hline \hline $\mathbb{Z}_{100}$ & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & $|15|=$ & \\ \hline $U(4)$ & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & $|3|=$ \varspace{0.75in}{1in} & \\ \hline $D_5$ & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & $|x^2y|=$ & \\ \hline $S_7$ & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & \varspace{0.75in}{0.75in} & $|(12)(3456)|=$ & \\ \hline \end{tabular} {\bf Recall:} $D_5 = \{ 1, x, \dots, x^4, y, xy, \dots, x^4y \}$ where $x^5=1$, $y^2=1$, and $xyxy=1$. \vspace{0.1in} \underline{Scratch Work:} \newpage \noindent {\bf\large 2. (24 points)} Cyclic Stuff \begin{enumerate}[(a)] \item Let $G$ be a finite group and $g \in G$. Suppose that $|g|=35$. \begin{enumerate}[i.] \item What is the order of $g^{10}$? List the distinct elements in $\langle g^{10} \rangle$. \vspace{1.5in} $\langle g^{10} \rangle = \displaystyle \Bigg\{$ \hspace{4in} $\displaystyle \Bigg\}$ \vspace{0.3in} \item Is $g^{100} \in \langle g^{10} \rangle$? \quad {\large\bf Yes \ \ / \ \ No} \vspace{1in} \end{enumerate} \item How many elements of order 8 does $\mathbb{Z}_{40}$ have? What are they? \vfill \item List the orders of elements in $\mathbb{Z}_{35}$. 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 List the orders of elements in $D_{35}$. 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} \end{enumerate} \newpage \noindent {\bf\large 3. (22 points)} Permutations Recall that $Q = \{ 1,-1,i,-i,j,-j,k,-k \}$ where $ij=k, jk=i, ki=j, ji=-k,i^2=j^2=k^2=-1,$ etc. \begin{enumerate}[(a)] \item Write down what the left multiplication operator of $i$ does in $Q$. Then write down the corresponding permutation if we label $1$ as $1$, $-1$ as $2$, $i$ as $3$, $-i$ as $4$, $j$ as $5$, $-j$ as $6$, $k$ as $7$, and $-k$ as $8$. \vspace{0.2in} \parbox[c][2in]{2in}{ $L_i : Q \rightarrow Q$ \vspace{-0.1in} \begin{eqnarray*} 1 & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\ -1 & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\ i & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\ -i & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\ j & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\ -j & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\ k & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\ -k & \mapsto & \underline{\hspace{0.75in}} \vspace{0.25in} \\ \end{eqnarray*} } The corresponding permutation is...? \vspace{0.5in} \item Suppose that using Cayley's theorem we found the left multiplication operator of $x$ in $D_3$ corresponds to $(123)(456)$ and $y$ corresponds to $(14)(26)(35)$. 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{1.25in} \item Write $\sigma = (152)(3467)(25)(37)$ as a product of disjoint cycles. \vspace{0.75in} $\sigma^{-1}=$ \vspace{0.5in} The order of $\sigma$ is $|\sigma|=$ \underline{\hspace{1in}}. \vspace{0.3in} Write $\sigma$ as a product of transpositions. \qquad $\sigma$ is \quad {\large\bf Even \ \ / \ \ Odd} \vspace{0.75in} Compute $\sigma^{65}$. \end{enumerate} \newpage \noindent {\bf\large 4. (18 points)} Explain why the following pairs of groups are not isomorphic. \begin{enumerate}[(a)] \item $U(1000) \not\cong \mathbb{R}$ \vfill \item $A_5 \not\cong \mathbb{Z}_{60}$ \vfill \item $S_4 \not\cong D_{12}$ \vfill \end{enumerate} \newpage \noindent {\bf\large 5. (16 points)} A few proofs \begin{enumerate}[(a)] \item Prove that $U(10) \cong \mathbb{Z}_4$. \vfill \item Pick {\bf ONE} of the following\dots \begin{enumerate}[I.] \item Let $g \in G$ ($G$ a group). Show that $\varphi:G \to G$ defined by $\varphi(x)=gxg^{-1}$ is an automorphism of $G$. \item Show that if $G$ is a cyclic group, then $G$ is abelian. Is the converse true? \end{enumerate} \vfill \end{enumerate} \end{document}