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\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 $0<c<\infty$, then either both series converge or both series
diverge.


\item If $E$ is a nonempty subset of a metric space $X$, define the
  distance from $x\in X$ to $E$ by
$$
\rho_E(x)=\inf_{z\in E} d(x,z).
$$

\begin{enumerate}

\item Show that $\rho_E(x)=0$ if and only if $x\in\overline{E}$.  

\item Also show
that $\rho_E$ is a uniformly continuous function on $X$ , by showing
that 
$$
|\rho_E(x)-\rho_E(y)|\leq d(x,y)
$$
for all $x\in X$ and $y\in X$.

\item Now let $K$ and $F$ be disjoint sets in a metric space $X$.  Suppose
that $K$ is compact and $F$ is closed.  
Prove that there exists a $\delta>0$ 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}






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