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\begin{document}

% This is for the title
{\Large {\bf Homework 6}} \hfill MATH 301/601 \\
{Due Wednesday, March 20, 2024} \\[-2mm]
\line(1,0){470}

\medskip

{\bf Instructions.} \textit{Read the appropriate homework guide (\href{http://qc.edu/~nvlamis/301S24/Homework_Guide_301.pdf}{Homework Guide for 301} or \href{http://qc.edu/~nvlamis/301S24/Homework_Guide_601.pdf}{Homework Guide for 601}) to make sure you understand how to successfully complete the assignment. 
All claims must be sufficiently justified. }


\medskip
\begin{Exercise}
Complete the following exercises from  \#1 from \href{http://abstract.ups.edu/aata/cyclic-exercises.html}{Section~4.5} in the course textbook:
\# 24, 25, 29, \textbf{*34}, 36
\end{Exercise}

\medskip
\begin{Exercise}
Prove that the order of \( S_n \) is \( n! \).
\end{Exercise}

\medskip
\begin{Exercise}
\begin{enumerate}[(a)]
\item Write down the elements in the cyclic subgroups generated by the cycles \( (1 \; 4 \; 3) \) and \( (1 \; 3 \; 5 \; 2 \; 4 \; 6 ) \). 
\item
Prove the order of a \( k \)-cycle is \( k \). 
\end{enumerate}
\end{Exercise}

\medskip
\begin{Exercise}
Complete the following exercises from \href{http://abstract.ups.edu/aata/permute-exercises.html}{Section~5.4} in the course textbook:

\#1, 2(a,b,c,d), 4, 5 (ignore the first sentence, and just find each of the sets in a, b, and c, and decide whether they are subgroups or not), 17,  23, \textbf{*33}

\end{Exercise}

\medskip
\begin{ExGraded}
Prove that a \( k \)-cycle can be expressed as the product of \( k-1 \) transpositions. 
(Hint: You should use induction.)
\end{ExGraded}

%\medskip
%\begin{ExGraded}
%From Exercise 1, \( |S_4| = 4! = 24 \). 
%Show that for any divisor \( d \) of 24 there exists a subgroup \( H \) such that \( |H| = d \). 
%\end{ExGraded}

\medskip
\begin{ExChallenge}
Prove that any two \( k \)-cycles in \( S_n \) are conjugate, that is, if \( \sigma, \tau \in S_n \) are \( k \)-cycles, then there exists \( \mu \in S_n \) such that \( \mu \sigma \mu^{-1} = \tau \). 
\end{ExChallenge}

\end{document}

