• Source: 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




See also


Representation theorem – Proof that every structure with certain properties is isomorphic to another structure
Comparison theorem
List of manifolds
List of theorems




References

Kata Kunci Pencarian:

• The Zero Theorem
• Klasifikasi grup sederhana hingga
• Jaringan saraf tiruan
• RSA
• Statistika
• Permukaan (topologi)
• Segitiga sama kaki
• Ilmu aktuaria
• Statistika matematika
• Geometri kompleks
• Classification theorem
• Classification of finite simple groups
• Surface (topology)
• Classification
• Nielsen–Thurston classification
• List of theorems called fundamental
• Measure-preserving dynamical system
• Feit–Thompson theorem
• Petrov classification
• Wedderburn–Artin theorem