Definition:Isomorphism (Abstract Algebra)/Group Isomorphism

Definition
Let $\left({G, \circ}\right)$ and $\left({H, *}\right)$ 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).