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% This is for the title
{\Large {\bf Homework 9}} \hfill MATH 301/601 \\
{Due Wednesday, April 17, 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}
Let \( \varphi \co G \to H \) be an isomorphism.
\begin{enumerate}[(a)]
\item Prove that \( \varphi(e_G) = e_H \). (Hint: use the fact that \( e_Ge_G = e_G \).)
\item Prove that \( \varphi(g)^{-1} = \varphi(g^{-1}) \) for all \( g \in G \).
\item Prove that \( \varphi(g^n) = \varphi(g)^n \) for all \( g \in G \) and for all \( n \in \bz \).
\end{enumerate}
\end{Exercise}
%\medskip
%\begin{Exercise}
%Let \( \varphi \co G \to H \) be an isomorphism.
%Prove that \( \varphi^{-1} \co H \to G \) is an isomorphism.
%\end{Exercise}
\bigskip
\begin{Def}
An \emph{automorphism} of a group \( G \) is an isomorphism \( G \to G \).
\end{Def}
\medskip
\begin{ExGraded}
Let \( G \) be a finite abelian group of order \( n \).
Suppose \( m \in \bn \) is relatively prime to \( n \).
Prove that \( \varphi \co G \to G \) given by \( \varphi(g) = g^m \) is an automorphism of \( G \).
(This says that every element of \( G \) has an \( m^{\text{th}} \)-root.)
\end{ExGraded}
\medskip
\begin{Exercise}
Let \( G \) be a group.
Prove that the set of automorphisms of \( G \), denoted \( \mathrm{Aut}(G) \), is a group with respect to function composition (this group is called \emph{the automorphism group of \( G \)}).
\end{Exercise}
\medskip
\begin{ExGraded}
Let \( G \) be a cyclic group, and let \( \varphi, \psi \in \mathrm{Aut}(G) \).
Prove that if \( a \in G \) is a generator of \( G \) and \( \varphi(a) = \psi(a) \), then \( \varphi = \psi \).
\end{ExGraded}
\medskip
\begin{Exercise}
Complete the following exercises from \href{http://abstract.ups.edu/aata/isomorph-exercises.html}{Section~9.4} in the course textbook:
\# 2, 8, 11, 12, 14, \textbf{*31}, 38, 39, 41, 46
(Hint for \#38 and \#39: use Exercise 4.)
\end{Exercise}
\medskip
\begin{ExChallenge}
Let \( \bq \) denote the group \( (\bq, +) \), and let \( \bq^\times \) denote the group \( (\bq\smallsetminus\{0\}, \cdot) \).
\begin{enumerate}[(a)]
\item
Let \( \varphi \co \bq \to \bq \) be an isomorphism.
Prove that \( \varphi(x) = x\cdot \varphi(1) \) for all \( x \in \bq \).
(This is saying that every automorphism of \( \bq \) is \( \bq \)-linear.)
\item
Use part (a) to prove that if \( \varphi \co \bq \to \bq \) is an isomorphism, then there exists \( q \in \bq \smallsetminus \{0\} \) such that \( \varphi(x) = qx \) for all \( x \in \bq \).
\item
Use part (b) to prove that \( \mathrm{Aut}(\bq) \cong \bq^\times \).
\end{enumerate}
\end{ExChallenge}
\end{document}