homomorphism

A homomorphism is a map between algebraic structures of the same type that preserves the structure's operation(s). For groups, a group homomorphism f: G → H satisfies f(xy) = f(x) f(y) for all x, y in G; equivalently, f maps the identity to the identity and preserves inverses. For additive structures like abelian groups or modules, the operation is addition, so f(x + y) = f(x) + f(y). For rings, a homomorphism preserves both addition and multiplication: f(a + b) = f(a) + f(b) and f(ab) = f(a) f(b). For vector spaces and modules, a homomorphism is a linear map, preserving addition and scalar multiplication.

The kernel Ker f = {g in G : f(g) = e_H} measures how far f is from being injective; the image Im f = {f(g) : g in G} is a substructure of H. The first isomorphism theorem states that G/Ker f ≅ Im f; in particular, f is injective iff Ker f is trivial, and surjective iff Im f = H. Endomorphisms are homomorphisms from a structure to itself, and automorphisms are bijective endomorphisms.

In category theory, the concept of a homomorphism generalizes to morphisms between objects, and the set of all homomorphisms from A to B is denoted Hom(A, B). Homomorphisms thus capture the idea of preserving algebraic structure across mappings and form a foundational tool across many areas of algebra.