Homomorphism
Previous: Category theory
A homomorphism is not an Isomorphism when it is not bijective, meaning it either:
- Fails to be injective (one-to-one): This happens when different elements in the domain are mapped to the same element in the codomain. For example, the kernel of the homomorphism (the set of elements that map to the identity element in the codomain) is not trivial, indicating that the homomorphism is not injective.
- Fails to be surjective (onto): This occurs when not every element in the codomain has a preimage in the domain. In this case, some elements of the codomain are not “reached” by the homomorphism.