\section{Groups}

Group theory gives us a set tools for understanding complex systems. We can use groups to solve the Rubik's cube, to solve problems in physics and chemistry, and to understand complex geometric symmetries. It's also worth noting that all modern crypography relies heavily on group theory.

\definition{}
A \textit{group} $(G, \ast)$ consists of a set $G$ and an operator $\ast$. \\
A group must have the following properties: \\

\begin{enumerate}
	\item $G$ is closed under $\ast$. In other words, $a, b \in G \implies a \ast b \in G$.
	\item $\ast$ is associative: $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a,b,c \in G$
	\item There is an \textit{identity} $e \in G$, so that $a \ast e = a \ast e = a$ for all $a \in G$.
	\item For any $a \in G$, there exists a $b \in G$ so that $a \ast b = b \ast a = e$. $b$ is called the \textit{inverse} of $a$. \\
	This element is written as $-a$ if our operator is addition and $a^{-1}$ otherwise.
\end{enumerate}

Any pair $(G, \ast)$ that satisfies these properties is a group.

\problem{}
Is $(\mathbb{Z}_5, +)$ a group? \\
Is $(\mathbb{Z}_5, -)$ a group? \\
\hint{$+$ and $-$ refer to our usual definition of modular arithmetic.}
\vfill


\problem{}
Show that $(\mathbb{R}, \times)$ is not a group, then make it one by modifying $\mathbb{R}$. \\

\begin{solution}
	$(\mathbb{R}, \times)$ is not a group because $0$ has no inverse. \\
	The solution is simple: remove the problem.

	\vspace{3mm}

	$(\mathbb{R} - \{0\}, \times)$ is a group.
\end{solution}

\vfill
\pagebreak

\problem{}
Show that a group has exactly one identity element.
\vfill

\problem{}
Show that each element in a group has exactly one inverse.
\vfill

\problem{}
Show that $(\mathbb{Z}_n^\times, \times)$ is a group for any $n \in \mathbb{Z}^+$.
\vfill

\problem{}
Let $(G, \ast)$ be a group and $a, b, c \in G$. Show that...
\begin{itemize}
	\item $a \ast b = a \ast c \implies b = c$
	\item $b \ast a = c \ast a \implies b = c$
\end{itemize}
This means that we can \say{cancel} operations in groups, much like we do in algebra.
\vfill
\pagebreak


\problem{}
What is the smallest group we can create?

\begin{solution}
	Let $(G, \circledcirc)$ be our group, where $G = \{\star\}$ and $\circledcirc$ is defined by the identity $\star \circledcirc \star = \star$

	Verifying that the trivial group is a group is trivial.
\end{solution}
\vfill

\problem{}
Let $G$ be the set of all bijections $A \to A$. \\
Let $\circ$ be the usual composition operator. \\
Is $(G, \circ)$ a group?

\vfill

\definition{}
Note that our definition of a group does \textbf{not} state that $a \ast b = b \ast a$. \\
Many interesting groups do not have this property.
Those that do are called \textit{abelian} groups. \\

\vspace{2mm}

One example of a non-abelian group is the set of invertible 2x2 matrices under matrix multiplication. In this handout, all groups are abelian.\\



\problem{}
Show that if $G$ has four elements, $(G, \ast)$ is abelian.

\vfill
\pagebreak

\problem{}
Let $(G, \ast)$ be a finite group (i.e, $G$ has finitely many elements), and let $g \in G$. \\
Show that $\exists~n \in \mathbb{Z}^+$ so that $g^n = e$ \\
\hint{$g^n = g \ast g \ast ... \ast g$ $n$ times.}

\vspace{2mm}

The smallest such $n$ defines the \textit{order} of $g$.

\vfill

\problem{}
What is the order of 5 in $(\mathbb{Z}_{25}, +)$? \\
What is the order of 2 in $(\mathbb{Z}_{17}^\times, \times)$? \\

\vfill
\pagebreak

% \problem{}
% Let $e, a, b, c$ be counterclockwise rotations of a square by $0, \frac{\pi}{2}, \pi,$ and $\frac{3\pi}{2}$. \\
% Create a multiplication table for this group.
% \vfill
%
% \problem{}
% Let $d, f, g, h$ correspond to reflections of the square along the following axis. \\
% Create a multiplication table for this group.
%
% \begin{center}
% \begin{tikzpicture}[scale=2]
% 	\draw (0,0) -- (1,0) -- (1,1) -- (0,1) -- (0,0);
%
% 	\draw[gray] (1.25,1.25) -- (-0.25,-0.25) node[below left]{$d$};
% 	\draw[gray] (1.25,-0.25) -- (-0.25,1.25) node[above left]{$f$};
% 	\draw[gray] (0.5,-0.25) -- (0.5,1.25) node[above]{$g$};
% 	\draw[gray] (-0.25, 0.5) -- (1.25,0.5) node[right]{$h$};
%
% \end{tikzpicture}
% \end{center}
% \vfill
%
% \problem{}
% Create a multiplication table for all symmetries of a square.
% \vfill
% \pagebreak
%
% \problem{}
% Create a multiplication table for all symmetries of a rhombus.
% \vfill
% \pagebreak
%
% \problem{}
% Find the order of each element in...
% \begin{itemize}
% 	\item The group of symmetries of a square
% 	\item The group of symmetries of a rhombus
% \end{itemize}
%
%
% \vfill
% \pagebreak

