Isomorphisms

An isomorphism is a structure-preserving bijection between two algebraic structures of the same type. If there exists an isomorphism between A and B, the two structures are considered essentially the same, or "the same up to renaming of elements." An isomorphism carries the elements of A to B in such a way that all defining operations and relations are preserved.

Formally, an isomorphism f from A to B is a bijection that preserves the structure. For any operation symbol applied to elements of A, the image under f respects the corresponding operation in B; for any relation, the relation holds for elements of A exactly when it holds for their images in B. In universal algebra this means f preserves all operations, and in relational structures it also preserves all relations. The inverse map f^{-1} is also an isomorphism.

Examples span many areas. In groups, Z and 2Z under addition are isomorphic via the map n ↦ 2n. Finite cyclic groups of order n are all isomorphic to Z/nZ. In linear algebra, any two n-dimensional vector spaces over a field F are isomorphic; once a basis is chosen, coordinates define an isomorphism with F^n. In graph theory, a graph isomorphism is a bijection between vertex sets that preserves adjacency.

Key properties include that isomorphisms are equivalence relations on structures of a fixed language; the composition of isomorphisms is an isomorphism, and every isomorphism has an inverse that is also an isomorphism. An automorphism is an isomorphism from a structure to itself. Isomorphic structures share all invariant properties, such as cardinality, dimension, and adjacency patterns; lack of an isomorphism indicates a genuine structural difference.