Isomorphism

An isomorphism is a bijective map between structured objects that preserves their structure in both directions. It shows that the objects are equivalent in the relevant mathematical sense.

Bijection: A function that is both injective and surjective. Isomorphisms must pair every element of one structure with exactly one element of the other.
Group isomorphism: A bijective group homomorphism whose inverse is also a group homomorphism. It preserves group multiplication and therefore also the identity and inverses.
Monomorphism: A morphism that is left-cancellable under composition. Unlike an isomorphism, a monomorphism need not have an inverse or be surjective.
Cayley's theorem: The theorem that every group is isomorphic to a subgroup of a permutation group. It represents abstract group structure concretely through permutations.
Category theory: The study of objects and composable structure-preserving maps between them. Its abstract language defines isomorphisms uniformly across different kinds of mathematical objects.
Homomorphism: A map between algebraic structures that preserves specified operations. An isomorphism is a bijective homomorphism whose inverse is also a homomorphism.
Graph isomorphism: A bijection between graph vertices that preserves adjacency in both directions. It captures the same structural idea using edges rather than algebraic operations.
Epimorphism: A morphism that is right-cancellable under composition. An epimorphism need not be invertible, even when it captures a categorical form of surjectivity.
Classification of finite abelian groups: The theorem describing every finite abelian group as a product of cyclic groups of prime-power order. Its statement identifies groups up to isomorphism rather than by their element names.
Structuralism in mathematics: The view that mathematical objects are characterized by their relations within structures. Isomorphic objects exemplify sameness of structure despite different underlying elements.

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