Identity Mapping is Automorphism/Groups

Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.

Then $I_G: \struct {G, \circ} \to \struct {G, \circ}$ is a group automorphism.

Its kernel is $\set e$.

Proof
The main result Identity Mapping is Automorphism holds directly.

As $I_G$ is a bijection, the only element that maps to $e$ is $e$ itself.

Thus the kernel is $\set e$.