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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 25, 2011
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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 $G$ be a finite group, let $p$ be
a prime number, and let $\alpha$ be the largest natural number such that
$p^\alpha\mid\left|G\right|$.
\begin{enumerate}
\item Prove that there is a subgroup $P$ of $G$ whose order is $p^\alpha$.
(You are not allowed to use the Sylow theorems, of course).
\item Suppose that $x\in G$ is an element whose order is a power of $p$,
and suppose that $x$ normalizes $P$. Show that $x\in P$.
\end{enumerate}
\vfill \noindent{\bf Problem 2. } A group $G$ is said to be ``torsion free''
if every non-trivial element thereof has infinite order.
\begin{enumerate}
\item Prove that a semi-direct of two torsion free groups is again torsion
free.
\item Let $\beta$ be a pure braid on $n$ strands. Prove that if $\beta^7=e$
then $\beta=e$.
\end{enumerate}
\vfill \noindent{\bf Problem 3. } Let $H_1$ and $H_2$ be subgroups of some
group $G$. Prove that the left $G$-sets $G/H_1$ and $G/H_2$ are isomorphic
(as left $G$-sets) iff the subgroups $H_1$ and $H_2$ are conjugate.
\vfill \noindent{\bf Problem 4. }
\begin{enumerate}
\item Let $G$ be a subgroup of $S_n$ that contains both the transposition
$(12)$ and the $n$-cycle $(123\ldots n)$. Prove that $G=S_n$. (Hint: Conjugate
your way up, do not use non commutative Gaussian elimination).
\item Let $n$ be odd and let $G$ be a subgroup of $S_n$ that contains both the
3-cycle $(123)$ and the $n$-cycle $(123\ldots n)$. Prove that
$G=A_n$. (Hint: For the lower bound, conjugate your way up, do not use
non commutative Gaussian elimination).
\item In the previous part, what if $n$ is even?
\end{enumerate}
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\centerline{\bf Good Luck!}
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