# Classification theorem

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
* 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 – 3-manifold that is a quotient of S³ by ℤ/p actions: (z,w) ↦ (exp(2πi/p)z, exp(2πiq/p)w)
* Classification of manifolds – basic questionPages displaying wikidata descriptions as a fallback

## Algebra

* Classification of finite simple groups – Massive theorem assigning all but 26 finite simple groups to a few infinite families
** 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 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 – graph encoding the structure of a reductive group over a non-algebraically-closed field, in which vertices are colored black or white according to whether they vanish on a maximal split torus, and the white vertices are acted upon by the Galois groupPages displaying wikidata descriptions as a fallback
** 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
* Langlands classification

## 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
* 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