\documentclass[12pt]{article}
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\usepackage[setpagesize=false]{hyperref}
\usepackage[usenames,dvipsnames]{xcolor}
% Following http://tex.stackexchange.com/a/847/22475:
\usepackage[setpagesize=false]{hyperref}\hypersetup{colorlinks,
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}

\def\myurl{http://www.math.toronto.edu/~drorbn/}

\def\red{\color{red}}

\def\bbC{{\mathbb C}}
\def\bbF{{\mathbb F}}
\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\bbV{{\mathbb V}}
\def\bbZ{{\mathbb Z}}
\def\calS{{\mathcal S}}
\def\Aut{\operatorname{Aut}}

\def\problempage{{
  \newpage\noindent\parbox{0.5\linewidth}{
    \footnotesize Solve all 5 problems. Write your solutions only where indicated, or write explicitly, ``continued on page $k$''.
    \newline{\bf Neatness counts! Language counts!}
  }
  \vskip 2mm
  }\parshape 4 0in 0.5\linewidth 0in 0.5\linewidth 0in 0.5\linewidth 0in \linewidth}
\def\yoursolution#1{{
  \vskip 2mm \noindent{\bf Your solution of Problem #1.}
  \newpage\noindent\hfill{\bf Your solution of Problem #1, continued.}
}}
\def\scratchright{{
  \newpage\noindent\hfill\parbox{0.35\linewidth}{\bf
    Scratch work --- this page will not be read unless you explicitly request it.
  }
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\def\scratchleft{{
  \newpage\noindent\parbox{0.35\linewidth}{\bf
    Scratch work --- this page will not be read unless you explicitly request it.
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}}
\def\scratchboth{{\scratchleft\scratchright}}

% Comment out the following line for the Crowdmark version:
\pagestyle{empty} \def\problempage{} \def\yoursolution#1{\vfill} \def\scratchright{} \def\scratchleft{} \def\scratchboth{}

\begin{document}
\noindent{\small
  \url{http://drorbn.net/25-347}
  \newline\href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#2526}{2025-26}:
}

\vskip 4mm
\noindent{\bf Do not open this notebook until instructed.}
\vskip 4mm
\begin{center}
  \href{http://drorbn.net/24-327}{\large MAT 347 Groups, Rings, Fields}
  \vskip 4mm
  {\LARGE Term Test 1}
  \vskip 4mm
  University of Toronto, November 4, 2025
  \vskip 5mm{\bf Solve all 5 problems in this booklet. }\\
  The problems are of equal weight.\\
  You have an hour and fifty minutes to write this test.
\end{center}

\vskip 9cm

\noindent{\bf Notes}
\begin{itemize}
\item No outside material allowed other than stationery, minimal hydration and snacks, and stuffed animals.
\item Write your solution to each problem on its page and on the back of that page. If you run out of space you may continue into the scratch pages, but you {\bf must} indicate this on the problem page or else the scratch pages will not be read.
\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 consisting 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.
\item {\red {\bf In Red.} Added after the test took place.}
\item {\red Please return this booklet intact, without tearing out any pages. If you use some of the pages as scratch, mark them SCRATCH and return them too.}
\end{itemize}

\vfill\begin{center} {\bf Good Luck!} \end{center}\vfill\eject

\scratchright

\problempage
\noindent{\bf Problem 1. } In this question, all permutations are written using cycles notation.

Last Thursday I wanted to analyze the group $G=\langle(12345),\,(2354)\rangle<S_5$, so I ran the NCGE algorithm on it. The output was the following table:
\[
  \def\s{\hspace{12mm}}
  \begin{array}{|c|c|c|c|c|c|}
    \hline 1 & I       &          & \s & \s & \s \\
    \hline 2 & (12345) & I        &    &    &    \\
    \hline 3 & (13524) & (2354)   & I  &    &    \\
    \hline 4 & (14253) & (2453)   & -  & I  &    \\
    \hline 5 & (15432) & (25)(34) & -  & -  & I  \\
    \hline   & 1       & 2        & 3  & 4  & 5  \\
    \hline
  \end{array}
\]
\begin{enumerate}
\item What is $|G|$? Show your calculation.
\item Is the permutation $(354)$ a member of $G$? Justify your answer.
\end{enumerate}

\par\noindent{\small {\bf Tip. } Don't start working! Read the whole exam first. You may wish to start with the questions that are easiest for you.}

\par\noindent{\small {\bf Tip. } Neatness, cleanliness and organization count, here and everywhere else!}

\yoursolution{1}

\problempage
\noindent{\bf Problem 2. } Let $G$ be a group.
\begin{enumerate}
\item Prove that if $a,b\in G$ are conjugate, then their orders are the same.
\item Prove that if $x,y\in G$, then the order of $xy$ is equal to the order of $yx$.
\end{enumerate}

\yoursolution{2}

\problempage
\noindent{\bf Problem 3. } Let $G$ be a group and $N\lhd G$ be a normal subgroup thereof. We know from class that $G/N$ is a group.

Explain why it was necessary to assume that $N$ was normal for $G/N$ to be a group. What exactly breaks if $N$ is not normal? How does the normality condition fix that?

\vskip 2mm
\par\noindent{\small {\bf Tip. } Often in mathematics, writing the right things is more valued than writing a lot of things.}

\yoursolution{3}

\problempage
\noindent{\bf Problem 4. } Prove that if a group $G$ is Abelian and simple then it is isomorphic to $\bbZ/p$ for some prime $p$. Note that we are not assuming that $G$ is finite.

\yoursolution{4}

\problempage
\noindent{\bf Problem 5. } Assuming that a group $G$ has a Sylow-$p$ subgroup, prove that the number of Sylow-$p$ subgroups of $G$ is congruent to $1$ modulo $p$.

\vskip 2mm
\par\noindent{\small {\bf Tip. } As always in math exams, when proving a theorem you may freely assume anything that preceded it but you may not assume anything that followed it.}

\par\noindent{\small {\bf Tip. } Once you have finished writing an exam, if you have time left, it is always a good idea to go back and re-read and improve everything you have written, and perhaps even completely rewrite any parts that came out messy.}

\yoursolution{5}

\scratchboth
\scratchboth
\scratchboth
\scratchboth
\scratchboth

\end{document}

