kb/data/en.wikipedia.org/wiki/Classification_theorem-0.md

title	chunk	source	category	tags	date_saved	instance
Classification theorem	1/1	https://en.wikipedia.org/wiki/Classification_theorem	reference	science, encyclopedia	2026-05-05T09:08:23.500509+00:00	kb-cron

In mathematics, a classification theorem answers the classification problem: "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class. A few issues related to classification are the following.

The equivalence problem is "given two objects, determine if they are equivalent". A complete set of invariants, together with which invariants are realizable, solves the classification problem, and is often a step in solving it. (A combination of invariant values is realizable if there in fact exists an object whose invariants take on the specified set of values) A computable complete set of invariants (together with which invariants are realizable) solves both the classification problem and the equivalence problem. A canonical form solves the classification problem, and is more data: it not only classifies every class, but provides a distinguished (canonical) element of each class. There exist many classification theorems in mathematics, as described below.

== Geometry == Classification of Euclidean plane isometries – Isometry of the Eluclidean plane Classification of Platonic solids Classification theorems of surfaces Classification of two-dimensional closed manifolds – Two-dimensional manifoldPages displaying short descriptions of redirect targets Enriques–Kodaira classification – Mathematical classification of surfaces of algebraic surfaces (complex dimension two, real dimension four) Nielsen–Thurston classification – Characterizes homeomorphisms of a compact orientable surface which characterizes homeomorphisms of a compact surface Thurston's eight model geometries, and the geometrization conjecture – Three dimensional analogue of uniformization conjecture Berger classification – Concept in differential geometry Classification of Riemannian symmetric spaces – (pseudo-)Riemannian manifold whose geodesics are reversible Classification of 3-dimensional lens spaces – Class of topological space Classification of manifolds – Basic question in geometry and topology

== Algebra == Classification of finite simple groups – Theorem classifying finite simple groups Classification of Abelian groups – Commutative group (mathematics) Classification of Finitely generated abelian group – Commutative group where every element is the sum of elements from one finite subset Classification of Rank 3 permutation group – Five sporadic simple groupsPages displaying short descriptions of redirect targets Classification of 2-transitive permutation groups Artin–Wedderburn theorem – Classification of semi-simple rings and algebrasPages displaying short descriptions of redirect targets — a classification theorem for semisimple rings Classification of Clifford algebras – Classification in abstract algebra Classification of low-dimensional real Lie algebras Classification of Simple Lie algebras and groups Classification of simple complex Lie algebras – Direct sum of simple Lie algebras Classification of simple real Lie algebras – Term in mathematics Classification of centerless simple Lie groups – Connected non-abelian Lie group lacking nontrivial connected normal subgroups Classification of simple Lie groups – Connected non-abelian Lie group lacking nontrivial connected normal subgroupsPages displaying short descriptions of redirect targets Bianchi classification – Lie algebra classification ADE classification – Mathematical classification Langlands classification – Mathematical theory

== Linear algebra == Finite-dimensional vector space – Number of vectors in any basis of the vector spacePages displaying short descriptions of redirect targetss (by dimension) Rank–nullity theorem – In linear algebra, relation between 3 dimensions (by rank and nullity) Structure theorem for finitely generated modules over a principal ideal domain – Statement in abstract algebra Jordan normal form – Form of a matrix indicating its eigenvalues and their algebraic multiplicities Frobenius normal form – Canonical form of matrices over a field (rational canonical form) Sylvester's law of inertia – Theorem of matrix algebra of invariance properties under basis transformations

== Analysis == Classification of discontinuities – Mathematical analysis of discontinuous points

== Dynamical systems == Classification of Fatou components – Components of the Fatou set Ratner classification theorem

== Mathematical physics == Classification of electromagnetic fields Petrov classification – Classification used in differential geometry and general relativity Segre classification – Algebraic classification of rank two symmetric tensors Wigner's classification – Classification of irreducible representations of the Poincaré group

== See also == Representation theorem – Proof that every structure with certain properties is isomorphic to another structure Comparison theorem Moduli space – Geometric space whose points represent algebro-geometric objects of some fixed kind List of manifolds List of theorems

== References ==