# Morphisms

## Homomorphism

- A homomorphism is the mathematical tool for succinctly expressing precise structural correspondences
- It is a function between groups satisfying a few “natural” properties

- A homomorphism is a function \(h: G \rightarrow H\) between two groups satisfying

$$ h(ab) = h(a)h(b), \quad \forall a,b \in G $$

- Note that \(a \cdot b\) is occurring in the
*domain*while \(h(a) \cdot h(b)\) occurs in the*codomain* - Not all functions from one group to another are homomorphisms
- The condition \(h(ab) = h(a)h(b)\) means that the map \(h\)
*preserves the structure*of \(G\)

- The condition \(h(ab) = h(a)h(b)\) means that the map \(h\)

### Example

- Consider the function \(h\) that reduces an integer to integer modulo 5:

$$ h: \mathbb{Z} \rightarrow \mathbb{Z}_5, \, h(n) = n \quad (\text{mod } 5) $$

- Since the group operation is
*additive*, the “homomorphism property” becomes

$$ h(a + b) = h(a) + h(b) $$

- This means “first add, then reduce modulo 5” OR “first reduce modulo 5, then add”

- A homomorphism that is injective is called
*embedding*: the group \(G\) “embeds” into \(H\) as a*subgroup*- If \(h\) is not injective, is called
*quotient* - If \(h(G) = H\), then \(h\) is surjective

- If \(h\) is not injective, is called
- A homomorphism that is both
**injective and surjective**is an*isomorphism*

## Isomorphism

- Two isomorphic groups may name their elements differently and may look different
- But the isomorphism between them guarantees that they have the same structure

- When two groups \(G\) and \(H\) have an isomorphism between them, we say that “G and H are isomorphic”, written \(G \cong H\)