Identity Mapping is Group Endomorphism
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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: $\S 60$