\documentclass[11pt]{article}
\usepackage{amsmath, amsthm, amssymb}
\usepackage[mathscr]{eucal}
\usepackage{fullpage}
\usepackage{enumitem}
\usepackage[all]{xy}
\usepackage{boxedminipage}
\usepackage{tikz}

% Math commands.
\newcommand{\R}{{\mathbb R}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\N}{{\mathbb N}}
\newcommand{\T}{\mathbb{T}}
\newcommand{\abs}[1]{{\left| #1 \right|}}
\newcommand{\script}[1]{{\mathscr{#1}}}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\id}{{\rm{id}}}
\DeclareMathOperator*{\Aut}{\text{Aut}}
\DeclareMathOperator*{\mn}{\text{M}_n}
\DeclareMathOperator*{\gln}{\text{GL}_n}


% Theorems.
\newtheorem{defn}{Definition}
\newtheorem{prop}{Proposition}
\newtheorem{thm}[prop]{Theorem}
\newtheorem{cor}[prop]{Corollary}
\newtheorem{lem}[prop]{Lemma}
\theoremstyle{definition}
\newtheorem{aside}[defn]{Aside}
\newtheorem{exmp}[defn]{Example}
\newtheorem{rem}[defn]{Remark}

% New List.
\newlist{thmenum}{enumerate}{10}
\setlist[thmenum,1]{label=\textnormal{(\alph*)}}
\setlist[thmenum,2]{label=\textnormal{(\roman*)}}

\begin{document}

\begin{center}
	\LARGE{Math 31 - Homework 4}
	
	\vspace{2mm}
	\large{Due Friday, July 20}
\end{center}

\subsection*{Easy}
\paragraph{1.} Determine if each mapping is a homomorphism. State why or why not. If it is a homomorphism, find its kernel, and determine whether it is one-to-one and onto.
\begin{thmenum}
	\item Define $\varphi : \Z \to \R$ by $\varphi(n) = n$. (Both are groups under addition here.)
	\item Let $G$ be a group, and define $\varphi : G \to G$ by $\varphi(a)=a^{-1}$ for all $a \in G$.
	\item Let $G$ be an \emph{abelian} group, and define $\varphi: G \to G$ by $\varphi(a)=a^{-1}$ for all $a \in G$.
	\item Define $\varphi : \gln(\R) \to \R^\times$ by $\varphi(A) = \det(A)$. (Recall that $\R^\times$ denotes the nonzero real numbers under 				multiplication.)
	\item Let $G$ be a group, and define $\varphi : G \to G$ by $\varphi(a) = a^2$ for all $a \in G$.
\end{thmenum}

\paragraph{2.} Consider the subgroup $H = \{i, m_1\}$ of the dihedral group $D_3$. Find all the left cosets of $H$, and then find all of the right cosets of $H$. Observe that the left and right cosets do not coincide.

\paragraph{3.} Let $G$ and $G'$ be groups, and suppose that $\abs{G} = p$ for some prime number $p$. Show that any group homomorphism $\varphi : G \to G'$ must either be the trivial homomorphism or a one-to-one homomorphism.

\subsection*{Medium}
\paragraph{4.} Let $G$ be a group and $H$ a subgroup of $G$. Show that there are the same number of left cosets of $H$ as there are right cosets of $H$. That is, exhibit a one-to-one map from the set of all left cosets of $H$ onto the set of all right cosets of $H$. (Note that this can be accomplished for finite groups by simply counting. Your proof must work for \emph{all} groups.)

\paragraph{5.} [Herstein, Section 2.5 \#2] Recall that $G_1 \cong G_2$ means that $G_1$ is isomorphic to $G_2$. Prove the following statements.
\begin{thmenum}
	\item For any group $G$, we have $G \cong G$.
	\item If $G_1$ and $G_2$ are groups and $G_1 \cong G_2$, then $G_2 \cong G_1$.
	\item If $G_1$, $G_2$, and $G_3$ are groups, and $G_1 \cong G_2$ and $G_2 \cong G_3$, then $G_1 \cong G_3$.
\end{thmenum}
[Note that you are essentially proving that isomorphism is an equivalence relation on the class of all groups.]

\paragraph{6.} [Herstein, Section 2.5 \#14] If $G$ is abelian and $\varphi : G \to G'$ is a homomorphism from $G$ \emph{onto} $G'$, prove that $G'$ is abelian.

\subsection*{Hard}
\paragraph{7.} [Herstein, Section 2.5 \#28] Let $G$ be a group, and let $\Aut(G)$ denote the set of all automorphisms of $G$. We can define a binary operation on $\Aut(G)$ by:
\[
	\theta \psi = \theta \circ \psi
\]
for $\theta, \psi \in \Aut(G)$.
\begin{thmenum}
	\item Prove that if $\theta, \psi \in \Aut(G)$, then $\theta \psi \in \Aut(G)$. (That is, show that we have indeed defined a binary operation by checking 		that $\Aut(G)$ is closed.
	
	\item If $\theta \in \Aut(G)$, then $\theta$ is in particular a bijection, so it has an inverse $\theta^{-1}$. Prove that $\theta^{-1}$ is a homomorphism, 		so that $\theta^{-1} \in \Aut(G)$ for all $\theta \in \Aut(G)$.
	
	\item Use parts (a) and (b) to show that $\Aut(G)$ is itself a group under composition.
\end{thmenum}


























\end{document}