Homomorphism Injectivity and Surjectivity
A homomorphism is surjective exactly when , and injective exactly when .
Proposition 2.9
Let be a homomorphism.
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is surjective if and only if .
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is injective if and only if .
Proof
Surjectivity is equivalent to by the definition of image. If is injective, then has at most one element, and it contains . Conversely, if and , then , so and .
Related
Stated in
- Proposition 2.9§2.1 Introduction on Homomorphisms and Isomorphisms