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{\Large {\bf Homework 7}} \hfill MATH 301/601 \\
{Due Wednesday, March 27, 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 \href{http://abstract.ups.edu/aata/permute-exercises.html}{Section~5.4} in the course textbook:
\# 7, 8, 9, 11, 14, 18, 23, \textbf{*25}, 37(a,b)
\end{Exercise}
\medskip
\begin{Exercise}
Let \( n \in \bn \).
\begin{enumerate}[(a)]
\item Prove that \( A_n \) is a subgroup of \( S_n \).
\item Explain why the subset of odd permutations in \( S_n \) is not a subgroup of \( S_n \).
\end{enumerate}
\end{Exercise}
\medskip
\begin{ExGraded}
From a previous homework exercise, \( |S_4| = 4! = 24 \).
Show that for any divisor \( d \) of 24 there exists a subgroup \( H \) such that \( |H| = d \).
\end{ExGraded}
\medskip
\begin{ExGraded}
Let \( \Gamma = (V,E) \) be the graph with \( V = \bz \) and \( (m,n) \in \bz \) if and only if \( |m-n| = 1 \).
So, \( \Gamma \) is just the number line (a portion of which is drawn here):
\begin{figure}[h]
\centering
\includegraphics{numberline}
\end{figure}
The \emph{infinite dihedral group}, denoted \( D_\infty \), is the automorphism group of the graph \( \Gamma \).
Let \( \tau, \rho \in D_\infty \) be given by \( \tau(n) = n+1 \) and \( \rho(n) = -n \) for \( n \in \bz \).
\begin{enumerate}[(a)]
\item For \( k \in \bz \), write down a formula for \( \tau^k \).
\item Prove that if \( f \in D_\infty \) such that \( f(0) = 0 \) and \( f(1) = 1 \), then \( f \) is the identity.
(Hint: Let's first focus on the natural numbers. Use strong induction: Let \( k \in \bn \smallsetminus \{1\} \). Suppose that \( f(j) = j \) for all \( 0\leq j < k \) and prove that \( f(k) = k \). A similar argument works for the negative integers.)
\item Prove that every element of \( D_\infty \) can be written as either \( \tau^k \) or \( \tau^k \rho \) for some \( k \in \bz \). (Hint: let \( f \in D_\infty \). Use a power of \( \tau \) to get \( f(0) \) back to \( 0 \), and then use \( \rho \) to get 1 back to itself if necessary.)
\end{enumerate}
\end{ExGraded}
\medskip
\begin{ExChallenge}
Let \( \sigma = (1 \,\, 2 \,\, 3 \,\, 4 \,\, 5) \in A_9 \) and \( \tau = ( 5 \,\, 6 \,\, 7 \,\, 8 \,\, 9 ) \in A_9 \).
Prove that the permutation \( ( 1 \, \, 2 \, \, 3 ) \) can be written as a word in \( \{\sigma, \tau\} \), i.e., there exists \( r \in \bn \) and \( n_1, \ldots, n_r, m_1, \ldots m_r \in \bz \) such that \( ( 1 \, \, 2 \, \, 3 ) = \sigma^{n_1}\tau^{m_1}\sigma^{n_2}\tau^{m_2} \cdots \sigma^{n_r}\tau^{m_r} \).
(Hint: It will help to visualize the problem.
Look at the figure on the next page.
Imagine \( \sigma \) as a counterclockwise rotation of the left circle and \( \tau \) as a counterclockwise rotation of the right circle.
Imagine playing a game where you are rotating the two circles to get the points in the desired position.
Record the moves you take.)
(From Exercise 5 you should feel confident in the fact that every 3-cycle can be expressed as a word in \( \{\sigma, \tau\} \), and hence by \S5.4 \#25, every permutation in \( A_9 \) can be expressed as a word in \( \sigma \) and \( \tau \)).
\end{ExChallenge}
\begin{figure}[t]
\centering
\includegraphics{5cycles}
\caption{Visualizing Exercise 5}
\end{figure}
\end{document}