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 cryptography relies heavily on group theory.
\definition{}
A \textit{group}$(G, \ast)$ consists of a set $G$ and an operator $\ast$. \par
Groups always 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 $a$. \par
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? \par
Is $(\mathbb{Z}_5, -)$ a group? \par
\hint{$+$ and $-$ refer to the usual operations in modular arithmetic.}
\vfill
\problem{}
Show that $(\mathbb{R}, \times)$ is not a group, then make it one by modifying $\mathbb{R}$. \par
\begin{solution}
$(\mathbb{R}, \times)$ is not a group because $0$ has no inverse. \par
The solution is simple: remove the problem.
\vspace{3mm}
$(\mathbb{R}-\{0\}, \times)$ is a group.
\end{solution}
\vfill
\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.
