Isomorphie

Isomorphie, in English often rendered as isomorphism, is a core concept in mathematics describing a strong form of structural sameness between two objects of the same kind. Two objects A and B are isomorphic if there exists a bijective map f from A to B that preserves the essential operations or relations that define the structure. In other words, A and B have the same shape, just with elements relabeled.

In different areas, the precise notion of preservation varies:

- In groups, a bijective group homomorphism f satisfies f(a · b) = f(a) · f(b) for all elements a, b.

- In vector spaces, a linear isomorphism is a bijective linear map that preserves addition and scalar multiplication.

- In rings or algebras, an isomorphism preserves both addition and multiplication (and typically the identity element, when required).

- In graphs, a graph isomorphism is a bijection between vertex sets that preserves adjacency.

- In order theory, poset isomorphisms preserve the order relation.

- In topology, a homeomorphism preserves open sets and continuity.

Isomorphism is an equivalence relation: it is reflexive, symmetric, and transitive. Two isomorphic structures are considered essentially the same from the perspective of the structure’s defining operations and relations. Consequently, many properties are invariants of isomorphism; for example, isomorphic groups have the same order, isomorphic vector spaces have the same dimension, and isomorphic graphs have the same number of vertices and edges and the same connectivity patterns.

An automorphism is an isomorphism from a structure to itself, describing the structure’s internal symmetries. Isomorphie thus provides a unifying way to classify mathematical objects up to structural equivalence.