\documentclass[12pt,letterpaper]{hmcpset}
\usepackage[margin=1in]{geometry}
\pagestyle{empty}

\name{}
\class{Math 171 - Abstract Algebra I}
\assignment{HW \# 1}
\duedate{9/6/10}

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

\begin{document}

\problemlist{Dummit \& Foote 1.1 \# 8, 22, 25, 26, 27, 31}

\begin{problem}[1.1.8]
Let $G = \{ z \in \C \ |\ z^n = 1 \text{ for some } n \in \Z^+\}$.
\begin{itemize}
\item[(a)] Prove that $G$ is a group under multiplication (called the group of \emph{roots of unity} in $\C$).
\item[(b)] Prove that $G$ is not a group under addition.
\end{itemize}
\end{problem}
\newpage

\begin{problem}[1.1.22]
If $x$ and $g$ are elements of the group $G$, prove that $|x| = |g^{-1}xg|$.  Deduce that $|ab| = |ba|$ for all $a, b \in G$.
\end{problem}
\newpage

\begin{problem}[1.1.25]
Prove that if $x^2 = 1$ for all $x \in G$ then $G$ is abelian.
\end{problem}
\newpage

\begin{problem}[1.1.26]
Assume $H$ is a nonempty subset of $(G, \star)$ which is closed under the binary operation on $G$ and is closed under inverses, i.e., for all $h$ and $k \in H$, $hk$ and $h^{-1} \in H$.  Prove that $H$ is a group under the operation $\star$ restricted to $H$ (such a subset $H$ is called a \emph{subgroup} of $G$).
\end{problem}
\newpage

\begin{problem}[1.1.27]
Prove that if $x$ is an element of the group $G$ then $\{x^n\ |\ n \in \Z \}$ is a subgroup (cf. the preceding exercise) of $G$ (called the \emph{cyclic subgroup} of $G$ generated by $x$).
\end{problem}
\newpage

\begin{problem}[1.1.31]
Prove that any finite group $G$ of even order contains an element of order 2.  [Let $t(G)$ be the set $\{g \in G \ |\ g \neq g^{-1}\}$.  Show that $t(G)$ has an even number of elements and every nonidentity element of $G - f(G)$ has order 2.]
\end{problem}

\end{document}