\begin{definition}[Definition:Isomorphism (Abstract Algebra)/Group Isomorphism]
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.
Let $\phi: G \to H$ be a (group) homomorphism.
Then $\phi$ is a group isomorphism {{iff}} $\phi$ is a bijection.
That is, $\phi$ is a group isomorphism {{iff}} $\phi$ is both a monomorphism and an epimorphism.
If $G$ is isomorphic to $H$, then the notation $G \cong H$ can be used (although notation varies).
\end{definition}