\documentclass[12pt,letterpaper]{hmcpset}
\usepackage[margin=1in]{geometry}
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\name{}
\class{Math 171 - Abstract Algebra I}
\assignment{HW \# 2}
\duedate{9/9/10}

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

\begin{document}

\problemlist{Dummit \& Foote (1.1) 2, 3. (2.1) 5,6}

\begin{problem}[1.1.2]
Decide which of the following binary operations are commutative:
\begin{itemize}
\item[(a)] the operation $\star$ on $\Z$ defined by $a \star b = a - b$
\item[(b)] the operation $\star$ on $\R$ defined by $a \star b = a + b + ab$
\item[(c)] the operation $\star$ on $\Q$ defined by $a \star b = \frac{a+b}{5}$
\item[(d)] the operation $\star$ on $\Z \times \Z$ defined by $(a,b) \star (c,d) = (ad+bc, bd)$
\item[(e)] the operation $\star$ on $\Q - \{0\}$ defined by $a \star b = \frac{a}{b}$.
\end{itemize}
\end{problem}
\newpage

\begin{problem}[1.1.3]
Prove that addition of residue classes in $\Z/n\Z$ is associative (you may assume it is well defined).
\end{problem}
\newpage

\begin{problem}[2.1.5]
Prove that $G$ cannot have a subroup $H$ with $|H| = n - 1$, where $n = |G| > 2$.
\end{problem}
\newpage

\begin{problem}[2.1.6]
Let $G$ be an abelian group.  Prove that $\{g \in G\ |\ |g| < \infty\}$ is a subgroup of $G$ (called the \emph{torsion subgroup} of $G$). Give an explicit example where this set is not a subgroup when $G$ is non-abelian.
\end{problem}

\end{document}