# Identity Mapping is Group Endomorphism

Jump to navigation
Jump to search

## 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 endomorphism.

## Proof

The Identity Mapping is Group Automorphism.

By definition, a group endomorphism is a group homomorphism from $G$ to itself.

A group automorphism is a group isomorphism from $G$ to itself.

As a group isomorphism is a group homomorphism which is also a bijection, the result follows by definition.

$\blacksquare$

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 60$