Isomorfism

Isomorfizm, or isomorphism, is a concept in mathematics describing a structure-preserving bijection between two objects of the same kind. If A and B are structures such as groups, rings, vector spaces, graphs, or topological spaces, an isomorphism is a function f from A to B that is bijective and preserves the defining operations or relations: for groups, f(xy) = f(x)f(y); for vector spaces, f(a x + b y) = a f(x) + b f(y); for graphs, f preserves adjacency.

Because f is bijective and its inverse is also structure-preserving, A and B are said to be isomorphic, meaning they have the same structure up to renaming of elements. Isomorphism is an equivalence relation on the class of such structures, partitioning objects into isomorphism classes. The existence of an isomorphism implies that many invariants are preserved, such as order for finite groups, dimension for vector spaces, or Euler characteristic for certain topological spaces, depending on the context.

Examples help illustrate the idea. In finite group theory, all cyclic groups of the same order n are isomorphic to the cyclic group Z/nZ. In linear algebra, any two n-dimensional vector spaces over the same field are isomorphic. In graph theory, two graphs are isomorphic if there exists a bijection between their vertex sets that preserves adjacency. In ring theory, ring isomorphisms preserve both addition and multiplication.

Automorphism refers to an isomorphism from a structure to itself. In category theory, an isomorphism is a morphism with a two-sided inverse, generalizing the notion of structural sameness across different mathematical settings.