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\section*{Math 3140: Homework 4}




\begin{enumerate}
\item[A.] 
\item[6.3.]  Show that the elements $w\in S_9$ such that $\{w(2),w(5).w(7)\}=\{2,5,7\}$ form a subgroup of $S_9$.  What is the order of this subgroup?
\item[6.7+.]  
\begin{enumerate}
\item[(a)] Describe/characterize the elements of order 2 of $S_n$.
\item[(b)] Show that if $n\geq 4$, then every permutation can be written as a product of two permutations of order 2.  Hint: Answer the question first for cycles.
\item[(c)] What goes wrong in (b) if $n<4$?
\end{enumerate}
\item[B.] Recall the definitions of in and out shuffles from Homework 3.  We may view them as elements of $S_{52}$.  
\begin{enumerate}
\item[(a)] Explain how to determine whether a permutation $w\in S_n$ is even based on the lengths of the cycles in its cycle decomposition.
\item[(b)] Are the in and out shuffles in $A_{52}$?   
\end{enumerate}
\item[C.]  6.6, 6.9
\end{enumerate}

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