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\newcommand{\bq}{\mathbb Q} % rationals
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% This is for the title
{\Large {\bf Homework 11}} \hfill MATH 301/601 \\
{Due Wednesday, May 8, 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/normal-exercises.html}{Section~10.4} in the course textbook:

\# 1, 2, 3, 4, 5, 8, 9, \textbf{*11}

\end{Exercise}

\medskip
\begin{Exercise}
Complete the following exercises from \href{http://abstract.ups.edu/aata/homomorph-exercises.html}{Section~11.4} in the course textbook:

\# 9, 10, 13

\end{Exercise}

\medskip
\begin{ExGraded}
Let \( \varphi \co G_1 \to G_2 \) be a homomorphism, let \( H_2 \) be a subgroup of \( G_2 \), and let \( H_1 = \varphi^{-1}(H_2) = \{ g \in G_1 : \varphi(g) \in H_2 \} \).
\begin{enumerate}[(a)]
\item Prove that \( H_1 \) is a subgroup of \( G_1 \).
\item Prove that if \( H_2 \) is normal in \( G_2 \), then \( H_1 \) is normal in \( G_1 \). 
\end{enumerate}
(Note: since the trivial subgroup is always normal, it follows that \( \ker \varphi \) is a normal subgroup of \( G_1 \).)
\end{ExGraded}

\medskip
\begin{ExGraded}
Let \( G \) be a cyclic group, let \( a \) be a generator of \( G \), and let \( \varphi, \psi \co G \to H \) be homomorphisms. Prove that if \( \varphi(a) = \psi(a) \), then \( \varphi = \psi \). 
(This says that a homomorphism defined on a cyclic group is completely determined by its action on a generator of the group.)
\end{ExGraded}
%
%\medskip
%\begin{Exercise}
%Use the previous exercise to answer to solve the following problems.
%\begin{enumerate}[(a)]
%\item Find all homomorphisms from \( \bz \) to \( \bz_6 \).
%\item Show that \( \varphi \co \bz_6 \to \bz_4 \) defined by \( \varphi(\bar 1) = \bar 1 \) is not a homomorphism (there are multiple ways to see this, but one is to realize that it is not even well defined). 
%\item Find all homomorphisms \( \bz_{24} \) to \( \bz_{18} \). 
%\end{enumerate}
%\end{Exercise}

\medskip
\begin{Exercise}
Let \( \varphi \co G \to H \) be a homomorphism. 
Prove that \( \varphi \) is injective if and only if \( \ker \varphi \) is trivial. 
\end{Exercise}

\medskip
\begin{ExChallenge}
The subgroup of a group \( G \) generated by the set \( \{ xyx^{-1}y^{-1} : x,y \in G \} \) is called the \emph{commutator subgroup of \( G \)} and is denoted \( G' \) (or \( [G,G] \)). 
\begin{enumerate}[(a)]
\item Prove that \( G' \) is normal in \( G \).
\item Prove that \( G/G' \) is abelian.
\item Let \( N \) be a normal subgroup of \( G \). Prove that \( G/N \) is abelian if and only if \( G' \subset N \). 
\end{enumerate}
(The group \( G/G' \) is called the \emph{abelianization} of \( G \).)
\end{ExChallenge}

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