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\begin{document}
\title{Math 120: Homework 1}
\date{}
\maketitle
\begin{itemize}
	\item Section 1.1 \# 6,12,15,22,25 (Dummit and Foote)
	\item Section 1.2 \# 1,2,3
	\item Section 7.1 \# 3
\end{itemize}

\bigbreak\noindent
{\textbf{Problem 1.1 \# 6}}. Determine which of the following sets are
groups under addition.

(a) Rational numbers in lowest terms including $0 = 0 / 1$ whose denominators
are odd.

(b) Rational numbers in lowest terms whose denominators are even, together
with $0$.

(c) The set of rational numbers of absolute value $< 1$.

(d) The set of rational numbers of absolute value $\geqslant 1$ together with
$0$.

(e) The set of rational numbers with denominators equal to $1$ or $2$.

(f) The set of rational numbers with denominators equal to $1$, $2$ or 3.

\bigbreak\noindent
{\textbf{Problem 1.1 \# 12}}. Find the orders of the following elements of
$(\mathbb{Z}/ 12\mathbb{Z})^{\times}$: $\overline{1}$, $\overline{- 1}$,
$\overline{5}$, $\overline{7}$, $\overline{- 7}$ and $\overline{13}$.

\bigbreak\noindent
{\textbf{Problem 1.1 \# 15}}. Prove that $(a_1 \cdots a_n)^{- 1} = a_n^{-
1} \cdots a_1^{- 1}$ in a group $G$.

\bigbreak\noindent
{\textbf{Problem 1.1 \# 22}}. If $x$ and $g$ are elements of a group $G$
prove that $| x | = | g^{- 1} x g |$. Deduce that $| a b | = | b a |$ for $a,
b \in G$.

\bigbreak\noindent
{\textbf{Problem 1.1 \# 25}}. Prove that if $x^2 = 1$ for all $x \in G$
then $G$ is abelian.

\bigbreak\noindent
{\textbf{Problem 1.2 \# 1}}.\quad Compute the order of each of the elements
in the following groups:

(a) $D_6$; (b) $D_8$; (c) $D_{10}$.

\bigbreak\noindent
{\textbf{Problem 1.2 \# 2}}. Use the generators and relations above to show
that if $x$ is any element of $D_{2 n}$ which is not a power of $r$ then $r x
= x r^{- 1}$.

\bigbreak\noindent
{\textbf{Problem 1.2 \# 3}} Use the generators and relations above to show
that every element of $D_{2 n}$ which is not a power of $r$ has order $2$.
Deduce that $D_{2 n}$ is generated by the two elements $s$ and $s r$, both of
which have order 2.

\bigbreak\noindent
{\textbf{Problem 7.1 \# 3}}. Let $R$ be a ring with $1$. Let $S$ be a
subring of $R$ containing $1$. Prove that if $u$ is a unit in $S$ then $u$ is
a unit in $R$. Give an example to show that the converse is not true.

\vfill

\end{document}
