\documentclass[12pt,letterpaper]{hmcpset}
\usepackage[margin=1in]{geometry}
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\name{}
\class{Math 171 - Abstract Algebra I}
\assignment{HW \#6}
\duedate{9/23/10}

\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}

\begin{document}

\problemlist{Dummit \& Foote 3.1 \# 14, 24, 40, 41}

\begin{problem}[3.1.14]
Consider the additive quotient group $\Q/\Z$.
\begin{itemize}
\item[(a)] Show that every coset of $\Z$ in $\Q$ contains exactly one representative $q \in \Q$ in the range $0 \leq q  < 1$.
\item[(b)] Show that every element of $\Q/\Z$ has finite order but that there are elements of arbitrarily large order.
\item[(c)] Show that $\Q/\Z$ is the torsion subgroup of $\R/\Z$ (cf. Exercise 6, Section 2.1).
\item[(d)] Prove that $\Q/\Z$ is isomorphic to the multiplicative group of root of unity in $\C^\times$.
\end{itemize}	
\end{problem}
\newpage

\begin{problem}[3.1.24]
Prove that if $N \unlhd G$ and $H$ is any subgroup of $G$ then $N \cap H \unlhd H$.
\end{problem}
\newpage

\begin{problem}[3.1.40]
Let $G$ be a group, let $N$ be a normal subgroup of $G$ and let $\overline G = G / N$. Prove that $\overline x$ and $\overline y$ commute in $\overline G$ if and only if $x^{-1}y^{-1}xy \in N$. (The element $x^{-1}y^{-1}xy$ is called the \emph{commutator} of $x$ and $y$ and is denoted by $[x,y]$.)
\end{problem}
\newpage

\begin{problem}[3.1.41]
Let $G$ be a group. Prove that $N=\langle x^{-1}y^{-1}xy \vert x,y \in G \rangle$ is a normal subgroup of $G$ and $G/N$ is abelian ($N$ is called the \emph{commutator subgroup} of $G$).
\end{problem}
\newpage

\end{document}