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

\def\bbZ{{\mathbb Z}}
\def\Aut{\operatorname{Aut}}

\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}

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

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\href{http://drorbn.net/?title=14-1100}{\large Math 1100 Core Algebra I}

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{\LARGE Term Test}

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University of Toronto, October 20, 2014

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\noindent{\bf Solve 3 of the 4 problems on the other side of this page. }\\
Each problem is worth 33 points, and you get one point for free.\\You have an hour and fifty minutes to
write this test.

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\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.
\item Advance apology: It may take me some time to grade this exam; sorry.
\item Barring the unforeseen, our final exam will take place on Monday December 15, 12-3PM, at Bahen 6183.
\end{itemize}

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\includegraphics[width=3in]{ElephantProjections.png}
\linebreak
{\scriptsize this image is here for no reason}

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{\bf Good Luck!}

\end{center}

\newpage

\noindent{\bf Solve 3 of the following 4 problems. } Each problem is worth
33 points. You have an hour and fifty minutes. {\bf Neatness counts!
Language counts!}

\vfill \noindent{\bf Problem 1. } Let $n$ be a natural number and let
$F$ be a subset of the set $\{(i,j)\colon 1\leq i\leq j\leq n\}$ that contains the diagonal $\{(i,i)\}$. For each
$(i,j)\in F$ you are given an element $\sigma_{i,j}$ of the permutation
group $S_n$ having the property that $\sigma_{i,j}(\alpha)=\alpha$
if $\alpha<i$, and $\sigma_{i,j}(i)=j$. Assume also that for every $i$, $\sigma_{i,i}=\iota$, the identity permutation. Let $M_1$ be the set of all
``monotone products'':
\[
  M_1:=\left\{
    \sigma_{1,j_1}\sigma_{2,j_2}\cdots\sigma_{n,j_n}
    \colon \forall i\ (i,j_i)\in F
  \right\}.
\]
\begin{enumerate}
\item It is also given that for every $(i,j)\in F$ and every $(k,l)\in F$,
we have $\sigma_{i,j}\sigma_{k,l}\in M_1$. Prove that $M_1$ is a subgroup
of $S_n$.
\item In one or two paragraphs, explain why we cared about this statement
in class. What did it give us that we could not have had without it?
\end{enumerate}

\vfill \noindent{\bf Problem 2. } Let $G$ be a group and let $Z(G)$ denote its centre.
\begin{enumerate}
\item Show that if $G/Z(G)$ is cyclic then $G$ is Abelian.
\item Prove that if the group $\Aut(G)$ of automorphisms of $G$ is cyclic, then $G$ is Abelian.
\end{enumerate}

\vfill \noindent{\bf Problem 3. } Let $G$ be a finite group, let $p$ be
a prime number, let $\alpha$ be the largest natural number such that
$p^\alpha\mid\left|G\right|$, and let $P$ be a subgroup of $G$ whose order is $p^\alpha$.
\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 4. } Let $H$ and $N$ be group, and let $\phi\colon H\to\Aut(N)$ be given. Remember that we may consider $H$ and $N$ as subgroups of the semi-direct product $N\rtimes H\coloneqq N\rtimes_\phi H$.
\begin{enumerate}
\item Prove that $(N\rtimes H)/N=H$.
\item Prove that the centralizer of $N$ within $H$ (that is, the set of elements of $H$ that commute with every element of $N$) is $\ker\phi$.
\item Prove that the centralizer of $H$ within $N$ is equal to the normalizer of $H$ within $N$ (where the latter is the set of $n\in N$ such that $nHn^{-1}=H$).
\end{enumerate}

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\centerline{\bf Good Luck!}

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