Group Epimorphism is Isomorphism iff Kernel is Trivial

Theorem
Let $\left({G, \oplus}\right)$ and $\left({H, \odot}\right)$ be groups.

Let $\phi: \left({G, \oplus}\right) \to \left({H, \odot}\right)$ be a group epimorphism.

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

Let $K = \ker \left({\phi}\right)$ be the kernel of $\phi$.

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


 * $K = \left\{{e_G}\right\}$