Homomorphism

A homomorphism is a map between groups that respects the group operation. Homomorphisms are the morphisms of group theory — everything we do with groups, from comparing them to decomposing them, goes through homomorphisms.

Definition

Let G and H be groups. A map φ : G → H is a homomorphism if

φ(ab) = φ(a) φ(b)   for all a, b ∈ G.

• Injective φ is a monomorphism (or embedding).
• Surjective φ is an epimorphism.
• Bijective φ is an isomorphism. G ≅ H means such a φ exists.
• A homomorphism G → G is an endomorphism; a bijective one is an automorphism. The automorphisms of G form a group Aut(G) under composition.

For a homomorphism φ : G → H:

• The kernel is ker φ = {a ∈ G : φ(a) = eH}.
• The image is im φ = φ(G) = {φ(a) : a ∈ G}.

First isomorphism theorem

For any homomorphism φ : G → H,

G / ker φ  ≅  im φ.

Every homomorphism factors as a surjection onto a quotient followed by an injection. Proof deferred to the quotient page, where we have the machinery to state it cleanly.

Examples

• Sign. sgn : Sₙ → {±1} sends each permutation to +1 (even) or −1 (odd). Kernel is the alternating group Aₙ.
• Determinant. det : GL(n,F) → F* on invertible matrices. Kernel is SL(n,F).
• Reduction mod n. ℤ → Cn, k ↦ k mod n. Kernel nℤ.
• Inclusion. H ↪ G for H ≤ G. Kernel trivial, image H.
• Quotient map. π : G → G/N for N ⊴ G (see quotient). Kernel N.
• Trivial hom. G → H, a ↦ eH. Kernel G, image {eH}.

Properties

• A homomorphism φ: G → H preserves the identity and inverses: φ(eG) = eH and φ(a⁻¹) = φ(a)⁻¹.
• A homomorphism φ: G → H has subgroups for its kernel and image: ker φ is even normal in G, and im φ ≤ H.
• A homomorphism φ: G → H is injective exactly when its kernel is trivial: ker φ = {eG}.
• A homomorphism φ: G → H carries subgroups to subgroups both ways: if K ≤ G then φ(K) ≤ H, and if K ≤ H then φ⁻¹(K) ≤ G.
• A composition of homomorphisms is again a homomorphism.

See also

• Normal subgroup — kernels are normal.
• Quotient group — the target of the canonical surjection.
• Characteristic subgroup — subgroups fixed by every automorphism; opens with Aut(G). M₁₁ is a concrete example with a rich Out(G).
• Wikipedia: Group homomorphism