Group Epimorphism is Isomorphism iff Kernel is Trivial/Proof 2

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

if and only if

$K = \set {e_G}$

Proof

From Kernel is Trivial iff Group Monomorphism, $\phi$ is a monomorphism if and only if $K = \set {e_G}$.

By definition, a group $G$ is an epimorphism is an isomorphism if and only if $G$ is also a monomorphism.

Hence the result.

$\blacksquare$