Homomorphism set is disjoint union of injective homomorphism sets
where denotes the set of injective homomorphisms. Here denotes a canonical bijection of the sets.
The correspondence is as follows:
- For any homomorphism from to , we denote its kernel by . By the first isomorphism theorem, there is an isomorphism to the image of the homomorphism, which when composed with the inclusion map to , gives an injective homomorphism from to .
- Conversely, given an injective homomorphism to , we compose with the quotient map to get a homomoorphism from to with kernel precisely .