# 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 $$G$$

### 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
• 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 $$G \cong H$$