\section{Definition and Examples}

\begin{definition}[Group Isomorphism]
	An \textit{isomorphism} $\phi$ from a group $G$ to a group $\overline{G}$ is a one-to-one mapping (or function) from $G$ onto $\overline{G}$ that preserves the group operation. That is,
	\[ \phi(ab) = \phi(a)\phi(b),\ \forall a,b \in G \]
	If there is an isomorphism from $G$ onto $\overline{G}$, we say that $G$ and $\overline{G}$ are \textit{isomorphic} and write $G \approx \overline{G}$.
\end{definition}