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 related issues to classification are the following.

• The isomorphism 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 computable complete set of invariants (together with which invariants are realizable) solves both the classification problem and the isomorphism problem.

There exist many classification theorems in mathematics, as described below.

Geometry

• Classification theorem of surfaces
  • Classification of two-dimensional closed manifolds
  • Enriques-Kodaira classification of algebraic surfaces (complex dimension two, real dimension four)
  • Nielsen-Thurston classification which characterizes homeomorphisms of a compact surface
• Thurston's eight model geometries, and the geometrization conjecture

Algebra

• Classification of finite simple groups
• Artin–Wedderburn theorem — a classification theorem for semisimple rings

Linear algebra

• Finite-dimensional vector spaces (by dimension)
• rank-nullity theorem (by rank and nullity)
• Structure theorem for finitely generated modules over a principal ideal domain
• Jordan normal form
• Sylvester's law of inertia