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 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 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

Complex analysis

• Classification of Fatou components