Nonisomorphic

Nonisomorphic is an adjective used to describe two mathematical objects that do not have an isomorphism between them. An isomorphism is a bijective map that preserves the relevant structure, so isomorphic objects are considered the same from a structural standpoint. Nonisomorphic objects, in contrast, cannot be matched by any such structure-preserving bijection.

Isomorphisms are defined within specific contexts, such as graphs, groups, rings, vector spaces, or manifolds. In graphs, an isomorphism is a relabeling of vertices that preserves adjacency. In groups, it is a bijective homomorphism that preserves the group operation. In vector spaces, it is a linear bijection that preserves addition and scalar multiplication. In rings, it is a bijective ring homomorphism that preserves addition and multiplication, and typically the identity element.

Two objects are nonisomorphic when they belong to different isomorphism classes, meaning no structure-preserving bijection exists between them. Invariants—properties preserved under isomorphism—are used to distinguish nonisomorphic objects. Examples include the order and degree sequence of graphs, or the group structure in groups of a given order. If invariants differ, the objects cannot be isomorphic.

Common examples illustrate the concept. On two vertices, the empty graph and the single-edge graph are nonisomorphic. Among groups of order four, the cyclic group and the Klein four group are nonisomorphic. Recognizing nonisomorphism helps mathematicians classify objects up to structural equivalence and understand the diversity of mathematical structures within a given category. Proving nonisomorphism typically relies on invariants or explicit obstruction arguments.