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 \(G \cong H\)

References