\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amssymb}
\usepackage[usenames,dvipsnames]{xcolor}
% Following http://tex.stackexchange.com/a/847/22475:
\usepackage[setpagesize=false]{hyperref}\hypersetup{colorlinks,
linkcolor={blue!50!black},
citecolor={blue!50!black},
urlcolor=blue
}
\pagestyle{empty}
\def\myurl{http://www.math.toronto.edu/~drorbn/}
\def\bbZ{{\mathbb Z}}
\begin{document}
{\small
\href{\myurl}{Dror Bar-Natan}:
\href{\myurl/classes/}{Classes}:
\href{\myurl/classes/\#1415}{2014-15}:
\hfill\url{http://drorbn.net/?title=14-1100}
}
\begin{center}
\vfill
\noindent{\bf Do not turn this page until instructed.}
\vfill
\href{http://drorbn.net/?title=14-1100}{\large Math 1100 Core Algebra I}
\vfill
{\LARGE Final Examination}
\vfill
University of Toronto, December 15, 2014
\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. } Prove that a ring $R$ is a PID iff it is a UFD in which $\gcd(a,b)\in\langle a, b\rangle$ for every non-zero $a,b\in R$.
{\color{red} Correction. You may use the result ``PID$\Rightarrow$UFD'' without proof.}
\vfill \noindent{\bf Problem 2. } Let $G$ be a finite group, let $p$ be
a prime number, and let $P$ be a Sylow-$p$ subgroup of $G$.
\begin{enumerate}
\item Suppose that $x\in G$ is an element whose order is a power of $p$,
and suppose that $x$ normalizes $P$. Show that $x\in P$.
\item Prove that the number of conjugates of $P$ in $G$ is 1 modulo $p$.
(You are not allowed to use the Sylow theorems, of course).
\end{enumerate}
\vfill \noindent{\bf Problem 3. } Let $M$ and $N$ be some modules over a ring $R$.
\begin{enumerate}
\item Define $M\otimes N$ as the solution of some universal property.
\item Prove that if such a solution exists, it is unique up to an isomorphism.
\item Using only this universal-property definition, prove that $M\otimes R\cong M$.
\end{enumerate}
\vfill
\noindent{\bf Problem 4. } 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 5. } Let $V=F^n$ be an $n$-dimensional
vector space over some field $F$, let $T\colon V\to V$ be a linear
transformation, let $R=F[x]$ denote the ring of polynomials in a variable
$x$ with coefficients in $F$, and consider $V$ as an $R$-module by setting
$xv=Tv$ for any $v\in V$. Let $\pi\colon R^n\to F^n$ be the morphism
of $R$-modules defined by mapping the standard basis elements $e_i$
of $R^n$ to their obvious counterparts in $F^n$. Propose a set of $n$
generators $r_i$ of $\ker\pi$ and prove in detail that your proposed $r_i$
indeed generate $\ker\pi$.
\vfill \noindent{\bf Problem 6. } Let $H$ and $K$ be subgroups of some
group $G$. Prove that the left $G$-sets $G/H$ and $G/K$ are isomorphic
(as left $G$-sets) iff the subgroups $H$ and $K$ are conjugate.
\vskip 2mm{\small
\noindent{\bf Hint.} The map $G/H\to G/K$ is induced by multiplication on the right by some element $y\in G$. What could $y$ be, if $H$ and $K$ are conjugate? How would you recover such a $y$, given an isomorphism of left $G$-sets $\phi\colon G/H\to G/K$?
}
\vfill
\centerline{\bf Good Luck!}
\vfill
\end{document}