Isomorphieinvariant

Isomorphieinvariant, or isomorphism invariant, is a term used in mathematics to denote a property or quantity of an object that is preserved under isomorphisms. More precisely, if two objects A and B are isomorphic, then A has the invariant if and only if B has it. This makes invariants useful for distinguishing objects up to isomorphism and for organizing objects into equivalence classes.

In practice, invariants can be either qualitative properties or quantitative measurements. Common examples across areas include the number of connected components in a graph; planarity and bipartiteness in graph theory; in algebra, the order of a finite group and whether the group is cyclic or abelian; in linear algebra, the dimension of a vector space and the eigenvalues or characteristic polynomial of a linear operator; in topology, invariants such as the genus of a surface or homology groups. These quantities do not change under a relabeling or re-description of the object, provided the new description respects the same structure.

A key use of isomorphism invariants is to classify objects up to isomorphism and to identify when two objects are not isomorphic. Invariants can be partial, meaning different non-isomorphic objects may share the same invariant. A complete invariant determines the entire isomorphism class, such as the invariant factors or elementary divisors that classify finite abelian groups. The term is common in English and German mathematical literature, with the German form isomorphieinvariant.