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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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