\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amssymb}
\pagestyle{empty}

\def\bbZ{{\mathbb Z}}

\begin{document}

\begin{center}

\vfill

\noindent{\bf Do not turn this page until instructed.}

\vfill

{\large Math 1100 Core Algebra I}

\vfill

{\LARGE Final Examination}

\vfill

University of Toronto, December 9, 2011

\vfill

\noindent{\bf Solve the 5 of the 6 problems on the other side of this page. }\\
Each problem is worth 20 points.\\
You have three hours to write this test.

\vfill

\noindent{\bf Notes.}

\begin{itemize}
\item No outside material other than stationary is allowed.
\item {\bf Neatness counts! Language counts!} The {\em ideal} written
solution to a problem looks like a page from a textbook; neat and clean
and made of complete and grammatical sentences. Definitely phrases like
``there exists'' or ``for every'' cannot be skipped. Lectures are mostly
made of spoken words, and so the blackboard part of proofs given
during lectures often omits or shortens key phrases. The ideal written
solution to a problem does not do that.
\end{itemize}

\end{center}

\vfill

\centerline{\bf Good Luck!}

\vfill

\newpage

\noindent{\bf Solve 5 of the following 6 problems. } Each problem is worth
20 points. You have three hours. {\bf Neatness counts!  Language counts!}

\vfill \noindent{\bf Problem 1. } Let $G$ be a group, $K$ a subgroup of
$G$, and $H$ a subgroup of the normalizer $N_G(K)$ of $K$ in $G$. Prove
that $KH$ is a subgroup of $G$, that $K$ is normal in $KH$, that $H\cap K$
is normal in $H$, and that $KH/K$ is isomorphic to $H/(H\cap K)$.

\vfill \noindent{\bf Problem 2. } Let $S_n$ denote the symmetric group on
$n$ elements.
\begin{enumerate}
\item How many Sylow-$2$ subgroups does $S_4$ have? Describe one of those
explicitly.
\item Let $p$ be a prime. How many Sylow-$p$ subgroups does $S_p$ have?
Describe one of those explicitly.
\end{enumerate}

\vfill \noindent{\bf Problem 3. } If $S$ is a set, denote by $F(S)$ the
$\bbZ$-module of integer-valued functions on $S$, with pointwise
operations. Let $X$ and $Y$ be two sets.
\begin{enumerate}
\item Construct an injection $\iota:F(X)\otimes_\bbZ F(Y)\to F(X\times Y)$.
\item Show that if $X$ and $Y$ are finite then $\iota$ is an isomorphism.
\item By means of an example, show that $\iota$ need not be an isomorphism
if $X$ and $Y$ are both infinite.
\end{enumerate}

\vfill \noindent{\bf Problem 4. } Define a ``Principal Ideal Domain
(PID)'' and a ``Unique Factorization Domain (UFD)'' and show that every
PID is a UFD. If you need to use the lemma that an increasing chain of
ideals in a PID must become constant at some point (i.e., that a PID is
``Noetherian''), prove it.

\vfill \noindent{\bf Problem 5. } Prove the following simplified version
of the structure theorem for finitely generated modules over a PID:

Let $R$ be a PID and let $M$ be the $R$-module $R^n/\langle
r_1,\ldots,r_m\rangle$, where $n$ and $m$ are natural numbers and
$r_1,\ldots,r_m\in R^n$. Then there exists a natural number $k$ and elements
$a_1,\ldots,a_l$ of $R$ so that $M\cong R^k\oplus\bigoplus_{i=1}^l R/\langle
a_i\rangle$.

\vfill \noindent{\bf Problem 6. } Let $p$ and $q$ be primes in a PID
$R$ such that $p\not\sim q$, let $\hat{p}$ denote the operation of
``multiplication by $p$'', acting on any $R$-module $M$, and let $s$
and $t$ be positive integers. For each of the $R$-modules $R$, $R/\langle
q^t\rangle$, and $R/\langle p^t\rangle$, determine $\ker\hat{p}^s$ and
$(R/\langle p\rangle)\otimes\ker\hat{p}^s$.

\vfill

\centerline{\bf Good Luck!}

\vfill

\end{document}