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% This is for the title
{\Large {\bf Homework 5}} \hfill MATH 301/601 \\
{Due Wednesday, March 13, 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/groups-exercises.html}{Section~3.5} in the course textbook:
\# 39, 41, 43, 45, 46, 47, 48

\end{Exercise}

\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}

\medskip
\begin{Exercise}
Complete the following exercises from \href{http://abstract.ups.edu/aata/cyclic-exercises.html}{Section~4.5} in the course textbook:

\#1(a,b,c,d), 2(a,e,f), 3(b,c,e), 4(a,b,c), 9, 11, \textbf{*23}, 30, 31, 39

\end{Exercise}


\medskip
\begin{Def}
The \emph{center} of a group \( G \), denoted \( Z(G) \), is the subgroup \[ Z(G) = \{ a \in G : ag = ga \text{ for all } g \in G \}, \] i.e., \( Z(G) \) is  the subgroup consisting of group elements that commute with every element of \( G \). (You proved that \( Z(G) \) is a subgroup in \#48 in \S3.5.)
\end{Def}


\medskip
\begin{Exercise}
\begin{enumerate}[(a)]
\item Compute the center of \( \mathrm{GL}(2, \mathbb R) \).  (Hint: use the following test matrices \( \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \) and \( \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \).)
\item Compute the center of \( \mathrm{SL}(2, \mathbb R) \). 
\end{enumerate}
\end{Exercise}

%\medskip
%\begin{ExGraded}
%Let \( G \) be a group.
%Let \( a \in G \smallsetminus \{e\} \) such that the order of \( a \), denoted \( |a| \), is \( m \) (recall, this means the cyclic subgroup \( \langle a \rangle \) is order \( m \)).
%The goal of this exercise is to prove that \( |a| = \min\{ n \in \bn : a^n = e \} \). 
%
%\begin{enumerate}[(a)]
%\item Show that the set \( S= \{ n \in \bn : a^n = e \} \) is not empty.
%(Hint: argue that there exists \( j,k \in \{1, 2, \ldots, m+1 \} \) such that \( a^{j} = a^k \).)
%Then, by the well-ordering principle, \( S \) has a least element, call it \( \ell \). 
%\item Show that \( a^k \neq a^j \) if \( 0 \leq j < k < \ell \). 
%\item Given \( k \in \bz \), use the division algorithm to show that \( a^k \in \{ e, a, a^2, \ldots, a^{\ell-1} \} \). 
%\item Conclude that \( \langle a \rangle = \{ e, a, a^2, \ldots, a^{\ell-1} \} \), and hence \( |a| = \ell \). 
%\end{enumerate}
%\end{ExGraded}
%
%\medskip
%\begin{Exercise}
%Let \( a, b \in \bz \). 
%\begin{enumerate}[(a)]
%\item Let \( \langle a,b \rangle = \{ as + bt : s,t \in \bz \} \). 
%Prove that \( \langle a,b \rangle \) is a subgroup of \( \bz \).
%\item Show that if \( H \) is a subgroup such that \( a, b \in H \), then \( \langle a,b \rangle < H \) (this says that \( \langle a,b \rangle\) is the subgroup generated by \( a \) and \( b \)).
%\item Find \( n \in \bn \cup \{0\} \) such that \( \langle a,b \rangle = n \bz \). Justify your answer. (It might help to try some concrete values for \( a \) and \( b \) if you are not sure what \( n \) should be.)
%\end{enumerate}
%\end{Exercise}

\medskip
\begin{ExGraded}
Suppose \( G \) is a nontrivial group in which the only two subgroups of \( G \) are itself and the trivial subgroup.
\begin{enumerate}[(a)]
\item Prove that \( G \) is cyclic.
\item Using part (a), prove that \( G \) is a finite group of prime order.
\end{enumerate}
\end{ExGraded}


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