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\begin{document}
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{\Large {\bf Homework 4}} \hfill MATH 301/601 \\
{Due Wednesday, February 28, 2024} \\[-2mm]
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\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/groups-exercises.html}{Section~3.5} in the course textbook:
\# 2, 7, 10, 15, 25, 26, \textbf{*27, *31, *32}, 33%, 41, 45, 46, 47, 48
(\# For 25 and 27, use induction.)
\end{Exercise}
\medskip
\begin{Exercise}
Let \( D_4 \) denote the group of symmetries of a square.
\begin{enumerate}[(a)]
\item Describe all the elements of \( D_4 \). (You do not need to prove you have them all, but do your best. We will do an official count in class at a later date.)
\item Describe a permutation of the vertices of the square that cannot be obtained via a symmetry of the square.
(You will need to use the Pythagorean theorem: \( a^2+b^2=c^2 \), where \( a \) and \( b \) are the lengths of the legs of a right triangle and \( c \) is the length of the hypotenuse.)
\end{enumerate}
\end{Exercise}
\medskip
\begin{ExChallenge}
Let \( G \) be a finite group.
Prove that there exists \( N \in \bn \) such that \( g^N = e \) for each \( g \in G \).
\end{ExChallenge}
%\medskip
%\begin{ExGraded}
%Let \( H \) be a subgroup of a group \( G \).
%Define the relation \( \sim \) on \( G \) by \( a\sim b \) if \( b^{-1}a \in H \).
%Prove that \( \sim \) is an equivalence relation on \( G \).
%\end{ExGraded}
%\medskip
%\begin{ExChallenge}
%Suppose \( H \) is a nonempty finite subset of a group \( G \) and that \( H \) is closed under multiplication (that is, \( ab \in H \) for all \( a, b \in H \)).
%Prove that \( H \) is a subgroup of \( G \).
%\end{ExChallenge}
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