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 $\set e$.

$\blacksquare$


Sources

  • 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.1$. Homomorphisms: Example $130$
  • 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 64$
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