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. ^[1]

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^{[ clarify ]} (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  – 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 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
• List of manifolds
• List of theorems

References

1. "An enormous theorem: the classification of finite simple groups | plus.maths.org". plus.maths.org. 2006-12-07. Retrieved 2025-09-16.