Group Epimorphism is Isomorphism iff Kernel is Trivial

Theorem
Let $\struct {G, \oplus}$ and $\struct {H, \odot}$ be groups.

Let $\phi: \struct {G, \oplus} \to \struct {H, \odot}$ be a group epimorphism.

Let $e_G$ and $e_H$ be the identities of $G$ and $H$ respectively.

Let $K = \map \ker \phi$ be the kernel of $\phi$.

Then:
 * the epimorphism $\phi$ is an isomorphism


 * $K = \set {e_G}$