homomorfia

Homomorfia is a concept in abstract algebra describing a structure-preserving map between two algebraic structures of the same type. In general, a homomorphism preserves the defining operation(s) of the structures involved.

For groups, a function f: G → H is a group homomorphism if f(xy) = f(x)f(y) for all x, y in G. Consequently, f(e_G) = e_H and f(x⁻¹) = f(x)⁻¹. For rings, a ring homomorphism preserves both addition and multiplication (f(a + b) = f(a) + f(b) and f(ab) = f(a)f(b)); some definitions also require f(1_R) = 1_S. For vector spaces, a linear transformation preserves vector addition and scalar multiplication: f(av + bw) = af(v) + bf(w).

Key concepts associated with homomorphisms include the kernel and the image. The kernel ker(f) = {x in G | f(x) = e_H} measures the failure of injectivity and is a normal subgroup of G; the image im(f) is a subgroup of the codomain. The first isomorphism theorem states that G/ker(f) is isomorphic to im(f). If a homomorphism is bijective, it is an isomorphism, with an inverse that is also a homomorphism. Endomorphisms map a structure to itself; automorphisms are bijective endomorphisms.

Homomorfia is fundamental for comparing and classifying algebraic structures and appears across many areas of mathematics, including symmetry, modules, and topology, where it provides a rigorous way to study how structures relate to one another.