\documentclass[11pt]{article}
\usepackage{amsmath, amsthm, amssymb}
\usepackage[mathscr]{eucal}
\usepackage{fullpage}
\usepackage{enumitem}
\usepackage[all]{xy}
\usepackage{boxedminipage}
\usepackage{tikz}

% Math commands.
\newcommand{\R}{{\mathbb R}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\N}{{\mathbb N}}
\newcommand{\T}{\mathbb{T}}
\newcommand{\A}{\mathscr{A}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\norm}[1]{{\left\| #1 \right\|}}
\newcommand{\abs}[1]{{\left| #1 \right|}}
\newcommand{\script}[1]{{\mathscr{#1}}}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\conv}{\mathrm{conv}}
\newcommand{\Hil}{\mathcal{H}}
\newcommand{\supp}{\operatorname{supp}}
\newcommand{\B}{\mathcal{B}}
\newcommand{\id}{{\rm{id}}}
\DeclareMathOperator*{\mn}{\text{M}_n}
\DeclareMathOperator*{\gln}{\text{GL}_n}


% Theorems.
\newtheorem{defn}{Definition}
\newtheorem{prop}{Proposition}
\newtheorem{thm}[prop]{Theorem}
\newtheorem{cor}[prop]{Corollary}
\newtheorem{lem}[prop]{Lemma}
\theoremstyle{definition}
\newtheorem{aside}[defn]{Aside}
\newtheorem{exmp}[defn]{Example}
\newtheorem{rem}[defn]{Remark}

% New List.
\newlist{thmenum}{enumerate}{10}
\setlist[thmenum,1]{label=\textnormal{(\alph*)}}
\setlist[thmenum,2]{label=\textnormal{(\roman*)}}

\begin{document}

\begin{center}
	\Large{Math 31 - Homework 2}
	
	\vspace{2mm}
	\large{Due Wednesday, July 3}
\end{center}

\subsection*{Easy}

\paragraph{1.} [Saracino, Section 2, \#1 (a), (b), (h), (i)] Which of the following are groups? Why? (That is, either verify that the axioms hold, or explain why one of them
fails.)
\begin{thmenum}
	\item $\R^+$ under addition. (Here $\R^+$ denotes the set of all \emph{positive} real numbers.)
	\item The set $3\Z$ of integers that are multiples of 3, under addition.
	\item $\R-\{1\}$ under the operation $a*b = a+b-ab$.
	\item $\Z$ under the operation $a*b = a+b-1$.
\end{thmenum}

\paragraph{2.} [Saracino, Section 2, \#5] The following table defines a binary operation on the set $S = \{a, b, c\}$.
\begin{center}
\begin{tabular}{c||c|c|c}
	$*$ & $a$ & $b$ & $c$ \\
	\hline
	\hline
	$a$ & $a$ & $b$ & $c$ \\
	\hline
	$b$ & $b$ & $b$ & $c$ \\
	\hline
	$c$ & $c$ & $c$ & $c$
\end{tabular}
\end{center}
Is $\la S, * \ra$ a group?

%\paragraph{3.} Let $D_4$ be the 4th dihedral group, which consists of symmetries of the square. Let $r \in D_4$ denote counterclockwise 
%rotation by $90^\circ$, and let $m$ denote reflection across the vertical axis.
%\begin{center}
%\begin{tikzpicture}
%	\draw [black,line width=1pt] (0,0) node [left]{$1$} -- (2,0) node [right]{$2$} -- (2,2) node [right]{$3$} -- (0,2) node [left]{$4$} -- cycle;
%	\draw [->,thick] (3,1) -- (5, 1);
%	\draw [black,line width=1pt] (6,0) node [left]{$4$} -- (8,0) node [right]{$1$} -- (8,2) node [right]{$2$} -- (6,2) node [left]{$3$} -- cycle; 
%	\node [above] at (4,1.1) {$r$};
%\end{tikzpicture}
%\end{center}
%\vspace{2mm}
%\begin{center}
%\begin{tikzpicture}
%	\draw [black,line width=1pt] (0,0) node [left]{$1$} -- (2,0) node [right]{$2$} -- (2,2) node [right]{$3$} -- (0,2) node [left]{$4$} -- cycle;
%	\draw [->,thick] (3,1) -- (5, 1);
%	\draw [black,line width=1pt] (6,0) node [left]{$2$} -- (8,0) node [right]{$1$} -- (8,2) node [right]{$4$} -- (6,2) node [left]{$3$} -- cycle; 
%	\node [above] at (4,1.1) {$m$};
%\end{tikzpicture}
%\end{center}
%Check that
%\[
%	rm = mr^{-1}.
%\]
%Conclude that $D_4$ is a nonabelian group of order 8.
%
%\paragraph{4.} We mentioned in class that elements of $D_n$ can be thought of as permutations of the vertices of the regular $n$-gon. For example, the rotation 
%$r$ of the square mentioned in the last problem can be identified with the permutation
%\[
%	\rho = \left( \begin{array}{cccc}
%		1 & 2 & 3 & 4 \\
%		2 & 3 & 4 & 1
%	\end{array} \right).
%\]
%Write the reflection $m$ as a permutation $\mu \in S_4$, and compute the product $\rho \mu$ in $S_4$. Then compute $rm \in D_4$, and write it as a permutation 
%$\sigma$. Check that $\sigma = \rho \mu$. (In other words, this identification of symmetries of the square with permutations respects the group operations.)

\subsection*{Medium}
\paragraph{3.} [Saracino, Section 2, \#8] Let $G$ be the set of all real-valued functions $f$ on the real line which have the property that $f(x) \neq 0$ for all $x \in \R$.
In other words,
\[
	G = \{ f : \R \to \R : f(x) \neq 0 \text{ for all } x \in \R \}.
\]
Define the product $f \times g$ of two functions $f, g \in G$ by
\[
	(f \times g)(x) = f(x) g(x) \text{ for all } x \in \R.
\]
With this operation, does $G$ form a group? Prove or disprove.

\paragraph{4.} If $G$ is a group in which $a*a=e$ for all $a \in G$, show that $G$ is abelian.

\paragraph{Extra credit:} We saw in class that any group of order 1, 2, or 3 is abelian. Show that any group of order 4 must be abelian. [Hint: Try to write down all the possible group tables in this case. Up to reordering the elements of the group, there will only be two possibilities.]

\paragraph{Extra extra credit:} Try to extend the first extra credit problem by showing that any group of order 5 must also be abelian.













\end{document}