\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 420-620\\Fall 2012\\Homework 5}
\date{}
\maketitle
\thispagestyle{empty}
\pagestyle{empty}



\centerline{\it Due Monday October 1, 2012.}

\centerline{}

\noindent 1. Consider the additive group of the rationals $\mathbb{Q}$.
\begin{itemize}
\item[a)] (5 pt) Show that any finitely generated subgroup of $\mathbb{Q}$ is cyclic.
\item[b)] (5 pt) Show that $\mathbb{Q}$ is not finitely generated.
\end{itemize}


\centerline{}

\noindent 2. (5 pt) Let $H$ and $K$ normal subgroups of $G$ such that $H\bigcap K=1$. Show that $hk=kh$ for all $h\in H$ and $k\in K$.





\centerline{}

\noindent 3. (5 pt) Classify all groups of order $2p$ where $p$ is an odd prime.


\centerline{}

\noindent 4. (5 pt) Show that if $G$ is a finite abelian group of order greater than $2$, then $\text{Aut}(G)$ is a finite group of even order.

\centerline{}

\noindent 5. Suppose that $G$ is a finite group and $N\unlhd G$.
\begin{itemize}
\item[a)] (5 pt) Show that if $H$ is a subgroup of $G$ such that $\text{gcd}(\vert H\vert,[G:N])=1$ then $H$ is a subgroup of $N$.
\item[b)] (5 pt) Show that if $\text{gcd}(\vert N\vert,[G:N])=1$ then $N$ is the unique subgroup of $G$ of order $\vert N\vert$.
\end{itemize}



\end{document}