\documentclass[epsfig,12pt, amsmath,amssymb]{article} \usepackage{amssymb} \topmargin -.7in \textheight 9in \textwidth 6.75in \oddsidemargin -.50in \evensidemargin .25in \begin{document} \noindent {\bf Math Studies} \hfill {\bf Exam 1} \hfill {\bf J. Cummings and D. Handron} \vspace{.25in} This is an open book, open notes exam. You may consult any published references you like, provided you cite all sources you cite. You may not consult with your classmates. Any questions you may have should be directed to Dr. Cummings or Dr. Handron. \vspace{.25in} \noindent \begin{enumerate} \item (Limit Comparison Test) Suppose that $\sum a_n$ and $\sum b_n$ are series with positive terms. Show that if $$ \lim_{n\to\infty}\frac{a_n}{b_n}=c $$ where $00$ such that $d(p,q)>\delta$ if $p\in K$ and $q\in F$. \item Show that the conclusion may fail for two disjoint closed sets if neither is compact. \end{enumerate} (cf. Rudin, p. 101 \#20 and 21. You may want to use some of the hints there.) \item Given metric spaces $(X,d_X)$ and $(Y,d_Y)$, a function $d_{X\times Y}:(X\times Y)\times(X\times Y)\to \mathbb{R}$ may be defined by setting $$ d_{X\times Y}((x_1,y_1),(x_2,y_2))=\left[d_X(x_1,x_2)^2+d_Y(y_1,y_2)^2\right]^{1/2} $$ \begin{enumerate} \item Show that $d_{X\times Y}$ is a {\em metric} on $X\times Y$. \item Also, show that in the case where $X$ is compact, $f:X\to Y$ is continuous if and only if the graph of $f$ is a compact subset of $(X\times Y,d_{X\times Y})$. \end{enumerate} \item (The Fundamental Group) If $X$ is a metric space, continuous function $\alpha:[0,1]\to X$ is called a {\em path} in $X$. A path $\alpha$ in $X$ is called a {\em loop} if $\alpha(0)=\alpha(1)$. Two loops $\alpha$ and $\beta$ are said to be {\em path homotopic} if $$ \alpha(0)=\alpha(1)=\beta(0)=\beta(1)=x, $$ and there is a continuous map $H:[0,1]\times[0,1]\to X$ satisfying $$ \begin{array}{c} H(0,t)=\alpha(t)\\ H(1,t)=\beta(t) \\ H(s,0)=H(s,1)=x \end{array} $$ In this case, the function $H$ is called a {\em homotopy} and $x$ is called the {\em basepoint} of the paths $\alpha$ and $\beta$. If $\alpha$ is path homotopic to $\beta$, we write $\alpha\simeq\beta$. \begin{enumerate} \item Show that $\simeq$ is an equivalence relation. Denote the equivalence class of $\alpha$ by $[\alpha]$. \item If $\alpha$ and $\beta$ share the same basepoint, their {\em product} is defined to be $$ (\alpha\ast\beta)(t)=\left\{ \begin{array}{cl} \alpha(2t) & 0\leq t\leq\frac{1}{2} \\ \beta(2t-1) & \frac{1}{2}\leq t\leq1 \end{array} \right. $$ Show that $\alpha*(\beta*\gamma)$ and $(\alpha*\beta)*\gamma$ are different paths in general, but that $$ [\alpha*(\beta*\gamma)]=[(\alpha*\beta)*\gamma]. $$ \item As we discussed in class the set $\{[\alpha]|\alpha\mbox{ is a path in $X$}\}$ with the operation $*$ is the fundamental group of $X$ relative to $x$, denoted $\pi_1(X,x)$. Suppose that $X$ and $Y$ are metric spaces, $x\in X$ and $y\in Y$. Show that a continuous map $f:X\to Y$ satisfying $f(x)=y$, induces a induces a group homomorphism $f_*:\pi_1(X,x)\to\pi_1(Y,y)$. \end{enumerate} \item \begin{enumerate} \item Show that if a group $G$ acts on a set $X$, then this action determines a group action of $G$ on the power set $\mathcal{P}(X)$. Specifically, show that for $g\in G$ and $A\subseteq X$ $$ gA=\{ga|a\in A\} $$ is a group action. The group $\mathbb{Z}^2$ acts on the metric space $\mathbb{R}^2$ by $$ (m,n)(x,y)=(x+m,y+n). $$ \item Suppose $\mathbf{x},\mathbf{y}\in\mathbb{R}^2$. Show that if for some $(m,n)\in \mathbb{Z}^2$ $(m,n)\mathbf{x}\in N_\delta(\mathbf{y})$, then for every $(m',n')\in \mathbb{Z}^2$, there is an $(h,k)\in \mathbb{Z}^2$ such that $$ (m',n')\mathbf{x}\in N_\delta((h,k)\mathbf{y}). $$ \item If $\mathbf{x}\in\mathbb{R}^2$, let $\frak{O}_\mathbf{x}$ denote the orbit of $\mathbf{x}$ under the action of $\mathbb{Z}^2$. The collection of all orbits in $\mathbb{R}^2$ will be denoted $\mathbb{R}^2/\mathbb{Z}^2=\{\frak{O}_\mathbf{x}|\mathbf{x}\in X\}$. Define $$ \Delta(\frak{O}_\mathbf{x},\frak{O}_\mathbf{y})=\inf_{\mathbf{z}\in\frak{O}_\mathbf{x}}d(\mathbf{z},\mathbf{y}) $$ Show this is a metric on $\mathbb{R}^2/\mathbb{Z}^2$. \end{enumerate} \item Let $G$ be a group and suppose that $H \le G$, $N \lhd G$, $G = H N$, $H \cap N = \{ e \}$. For each $h \in H$ define a map $\phi(h)$ with domain $N$ by $\phi(h)(n) = n^h$. \begin{enumerate} \item Show that $\phi(h) \in Aut(N)$. \item Show that $\phi$ is a HM from $H$ to $Aut(N)$. \item Show that every element of $G$ has the form $h n$ for a unique pair $(h, n)$ with $h \in H$, $n \in N$. \item Show that $h_1 n_1 h_2 n_2 = (h_1 \times h_2) (\phi(h_2^{-1})(n_1) \times n_2)$ where $h_i \in H$, $n_i \in N$. \end{enumerate} \item (This is a kind of converse to the preceding question) Let $H$ and $N$ be arbitrary groups, let $\phi: H \rightarrow Aut(N)$ be a HM and let $G = \{ (h, n) : h \in H, n \in N \}$. Define a binary operation $\times_G$ on $G$ by the equation \[ (h_1, n_1) \times_G (h_2, n_2) = (h_1 \times_H h_2, \phi(h_2^{-1})(n_1) \times_N n_2). \] \begin{enumerate} \item Show that $G$ is a group under $\times_G$. \item Let $H^* = \{ (h, e_N) : h \in H \}$ and $N^* = \{ (e_H, n) : n \in N \}$. Show that $H^* \le G$, $N^* \lhd G$, $H^* \cap N^* = \{ e_G \}$, $G = H^* N^*$. \item Compute the conjugate $(e, n)^{ (h, e) }$. \end{enumerate} \item Let $G$ be a cyclic group of order $n$ with a generator $g$. Given $a \in {\Bbb Z}$ let $\phi_a$ be the map from $G$ to $G$ given by $\phi_a(g^i) = g^{i a}$. \begin{enumerate} \item Show that the AMs of $G$ are precisely the maps $\phi_a$ where $0 < a < n$ and $gcd(a, n) = 1$. You may assume the fact from elementary number theory that $gcd(a, n) = 1$ iff $a x + n y =1$ for some integers $x$ and $y$. \item Suppose that $n$ is an odd prime. Show that $G$ has two AMs $\phi$ such that $\phi^2 = id$ and describe them. \end{enumerate} \item Let $p$ be an odd prime and fill in the details in the following analysis of groups of order $2 p$. Suppose $G$ has order $2 p$. \begin{enumerate} \item $G$ has a subgroup $H$ of order $2$ and a normal subgroup $N$ of order $p$. \item $G = H N$, $H \cap N = \{ e \}$. \item $G$ is either cyclic or isomorphic to the group of symmetries of a regular $p$-gon. \end{enumerate} \item Show that for any group $N$ and any $\phi \in Aut(N)$ there is a group $G$ and an element $g \in G$ such that $N \lhd G$, and $n^g = \phi(n)$ for all $n \in N$. \end{enumerate} \end{document} \end