\documentclass[10pt]{article}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amsmath}
\addtolength{\textwidth}{100pt}
\addtolength{\evensidemargin}{-50pt}
\addtolength{\oddsidemargin}{-50pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Begin user defined commands
\newcommand{\map}[1]{\xrightarrow{#1}}
\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
% End user defined commands
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% These establish different environments for stating Theorems, Lemmas, Remarks, etc.
\newtheorem{Thm}{Theorem}
\newtheorem{Prop}[Thm]{Proposition}
\newtheorem{Lem}[Thm]{Lemma}
\newtheorem{Cor}[Thm]{Corollary}
\theoremstyle{definition}
\newtheorem{Def}[Thm]{Definition}
\theoremstyle{remark}
\newtheorem{Rem}[Thm]{Remark}
\newtheorem{Ex}[Thm]{Example}
\theoremstyle{definition}
\newtheorem{Exercise}{Exercise}
\newenvironment{Solution}{\noindent\textbf{Solution.}}{\qed}
\renewcommand{\labelenumi}{(\alph{enumi})}
% End environments
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Now we're ready to start
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\author{Your Name}
\title{MT310 Homework 1}
\date{January 20, 2010}
\maketitle
\thispagestyle{empty}
\noindent{\bf Definition.\ } A {\it group} consists of a set $G$ and a function
$G\times G\map\circ G$, denoted $(a,b)\mapsto a\circ b$,
satisfying the following three axioms:
\begin{itemize}
\item (associativity) For all $a,b,c\in G$, we have $a\circ (b\circ c)=(a\circ b)\circ c$.
\item (identity) There exists an element $e\in G$ such that $e\circ a=a\circ e=a$ for all $a\in G$.
\item (inverses) For every $a\in G$ there exists an element $a^{-1}\in G$ such that
$a\circ a^{-1}=a^{-1}\circ a=e$.
\end{itemize}
\noindent{\bf Remark.\ } The function $\circ$ is called the {\bf group operation} or {\bf group law}.
It is usually omitted, and we write $ab$ instead of $a\circ b$.
The exception is when $G$ has a natural group operation coming from addition;
in this case we write $a+b$ instead of $a\circ b$.
\medskip
\noindent{\bf Example.\ } The group $\Z_3=\{0,1,2\}$ of integers modulo $3$ has group law as shown in the following {\it Cayley table}:
$$
\begin{array}{c|ccc}
+ & 0 & 1 & 2 \\
\hline
0 & 0 & 1 &2\\
1 & 1& 2 & 0\\
2 & 2 & 0& 1
\end{array}
$$
\begin{Exercise}
Let $G=\{e,x,y\}$ be any group with three elements. Without knowing the group law, fill in the Cayley table:
$$
\begin{array}{c|ccc}
\circ & e &x & y \\
\hline
e & & & \\
x & & & \\
y & & &
\end{array}
$$
\end{Exercise}
\begin{Exercise}
Let $G=\{e,x,y,z\}$ be a group with four elements. Again, you are not told the group law. Show that there are exactly two possibilities for the Cayley table:
$$
\begin{array}{c|cccc}
\circ & e &x & y &z\\
\hline
e & & & &\\
x & & & &\\
y & & & & \\
z & & & &
\end{array}\quad\quad\quad
\begin{array}{c|cccc}
\circ & e &x & y &z\\
\hline
e & & & &\\
x & & & &\\
y & & & & \\
z & & & &
\end{array}
$$
\end{Exercise}
\begin{Exercise}
Let $G$ be a group and let $g_1, g_2,\dots, g_n$ be elements of $G$.
Prove that
$$(g_1 g_2\cdots g_n)^{-1}=g_n^{-1}g_{n-1}^{-1}\cdots g_2\cdot g_1.$$
\end{Exercise}
\begin{proof}
Write your proof here.
\end{proof}
\begin{Exercise} Let $\Z_n^\times$ be the group of units of $\Z_n$ and assume that $n\geq 3$. Prove that there is an element $a\in \Z_n^\times$ such that $a^2=1$, but $a\neq 1$.
\end{Exercise}
\begin{proof}
Write your proof here.
\end{proof}
\begin{Exercise} Let $G$ be a group for which $g^2=e$ for all $g\in G$.
Prove that $G$ is abelian.
\end{Exercise}
\begin{proof}
Write your proof here.
\end{proof}
\begin{Exercise} Let $G$ be the symmetry group of an equilateral triangle,
and let $a,b\in G$ be two reflections. Write the remaining three non-identity elements of $G$ in terms of $a$ and $b$.
\end{Exercise}
\begin{Solution}
Write your solution here.
\end{Solution}
\end{document}