# Homomorphism set is disjoint union of injective homomorphism sets

From Groupprops

## Statement

### For groups

Suppose and are groups. Then, the set of homomorphisms can be identified with the disjoint union of the sets of injective homomorphisms from to for every normal subgroup , i.e.,:

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 .

### For algebras in an arbitrary variety

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