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{\large Math 1100 Core Algebra I}

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{\LARGE Term Test}

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University of Toronto, October 26, 2010

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\noindent{\bf Solve the 4 problems on the other side of this page. }\\
Each problem is worth 25 points.\\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 us a while to grade this exam; sorry.
\end{itemize}

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\centerline{\bf Good Luck!}

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\noindent{\bf Solve the following 4 problems. } Each problem is worth
25 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 $\{(i,j)\colon 1\leq i<j\leq n\}$. 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 $\sigma_{i,j}(i)=j$. Let $M_1$ be the set of all
``monotone products'':
\[
  M_1:=\left\{
    \sigma_{i_1,j_1}\sigma_{i_2,j_2}\cdots\sigma_{i_t,j_t}
    \colon i_1<i_2<\dots<i_t,
      \text{ and }\forall\alpha\ (i_\alpha,j_\alpha)\in F
  \right\}.
\]
\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 of odd order. Show
that $x\in G$ is not conjugate to $x^{-1}$ unless $x=e$.

\vfill \noindent{\bf Problem 3. } 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 4. } Let $A_4$ be the alternating subgroup
of $S_4$, the permutation group of $\{1,2,3,4\}$, let $C=\langle x\colon
x^3=e\rangle$ be the cyclic group of order 3, and let $P=(\bbZ/2)\times
(\bbZ/2)$ be
the direct product of the cyclic group of order 2 with itself.
\begin{enumerate}
\item Explain how $C$ and $P$ can be viewed as subgroups of $A_4$. 
\item Is $C$ normal in $A_4$? Is $P$ normal in $A_4$?
\item Find an action $\phi\colon P\times C\to P$ so that
$A_4\cong C\ltimes_\phi P$.
\end{enumerate}

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\centerline{\bf Good Luck!}

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