\documentclass[12pt]{amsart}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage[all]{xy}
\voffset=-1in
\hoffset=-1in
\textheight=10in
\textwidth=6in
\begin{document}
\author{}
\title{Math 720\\Fall 2010\\Homework 3}
\date{}
\maketitle
\thispagestyle{empty}
\pagestyle{empty}



\centerline{\it Due Friday October 1, 2010.}

\centerline{}

\noindent 1. (5 pt) Give an example to show that the notions of direct product and direct sum are distinct (that is, give a family of groups such that $\oplus G_i\ncong\prod G_i$).

\centerline{}

\noindent 2. (5 pt) Suppose that $G$ and $H$ are groups and $\phi:G\longrightarrow H$ and $\psi:H\longrightarrow G$ are both injective homomorphisms. Does this imply that $G\cong H$?

\centerline{}

\noindent 3. We define the following generalizations of $S_n$. Let $S_{\infty}=\{f:\mathbb{N}\longrightarrow\mathbb{N}\vert f\text{ is bijective}\}$ and $\overline{S}=\bigcup_{n=1}^\infty S_n$.
\begin{itemize}
\item[a)] (5 pt) Show that $S_{\infty}$ is a group with subgroup $\overline{S}$.
\item[b)] (5 pt) Show that $S_{\infty}$ and $\overline{S}$ are not isomorphic.
\end{itemize} 
 

\centerline{}

\noindent 4. (5 pt) Show that $S_n$ can be generated by the following sets.
\begin{itemize}
\item[a)] (5 pt) $\{(1\ 2), (1\ 2\cdots n)\}$.
\item[b)] (5 pt) $\{(1\ 2), (1\ 3),\cdots, (1\ n)\}$.
\end{itemize}

\centerline{}

\noindent 5. For this problem, we consider the groups $A_n\trianglelefteq S_n$.
\begin{itemize}
\item[a)] (5 pt) Show that $A_n$ is the unique subgroup of $S_n$ of index 2.
\item[b)] (5 pt) Show that for all $n\neq 4$, $A_n$ is the unique normal subgroup of $S_n$.
\end{itemize}




\end{document}