\definition{Group} A group $\langle G, *\rangle$ is a set $G$, closed under a binary operation $*$, such that the following axioms are satisfied:
\begin{itemize}
\item For all $a, b, c \in G$, we have
\begin{equation*}
(a * b) * c = a * (b * c).\quad\text{{\bf associativity} of $*$}
\end{equation*}
\item There is an element $e$ in $G$ such that for all $x \in G$,
\begin{equation*}
e * x = x * e = x.\quad\text{{\bf identity} $e$ for $*$}
\end{equation*}
\item Corresponding to each $a \in G$, there is an element $a'$ in G such that
\begin{equation*}
a * a' = a' * a = e.\quad\text{{\bf inverse} $a'$ of $a$}
\end{equation*}
\end{itemize}