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\begin{center}
{\large MATH 150A Winter 2020 \ - \ {\bf Problem Set 6}}
% Put your name and section here.
{\large due February 24}
\end{center}
\begin{enumerate}
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\item Let $G$ be a group of order $n$ that acts operates non-trivially on a set of size $r$. Prove that if $n > r!$, then $G$ has a proper normal subgroup. (A {\em proper} subgroup of $G$ is a subgroup that is neither trivial nor equal to $G$.)
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\item \begin{enumerate}
\item Prove that the transpositions $(1\;2),(2\;3),\ldots,(n-1\;n)$ generate the symmetric group $S_n$.
\item How many transpositions are needed to write the cycle $(1\;2\;3\cdots n)$?
\item Prove that the cycle $(1\;2\;3\cdots n)$ and $(1\;2)$ generate the symmetric group $S_n$.
\end{enumerate}
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\item Let $\sigma$ be the 5-cycle $(1\:2\;3\;4\;5)$ in $S_5$. Find the element $\tau \in S_5$ which accomplishes the specified conjugation:
\begin{enumerate}
\item $\tau \sigma \tau^{-1} = \sigma^2$,
\item $\tau \sigma \tau^{-1} = \sigma^{-1}$,
\item $\tau \sigma \tau^{-1} = \sigma^{-2}$.
\end{enumerate}
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\item Let $C$ be the conjugacy class of an even permutation $p$ in $S_n$. Show that $C$ is either a conjugacy class in $A_n$, or else the union of two conjugacy classes in $A_n$ of equal size. Explain how to decide which case occurs in terms of the centralizer of $p$.
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\item Find the class equation for $S_6$ and give a representative for each conjugacy class.
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\item Let $G$ be a group of order 200. Prove that $G$ has a normal Sylow 5-subgroup.
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\item Let $G$ be a group of order 105. Prove that $G$ has a proper normal subgroup.
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\end{enumerate}
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