Classification theorem explained

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 (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 Platonic solids
• Classification theorems of surfaces
  • • • of algebraic surfaces (complex dimension two, real dimension four)
  • which characterizes homeomorphisms of a compact surface
• Thurston's eight model geometries, and the

Algebra

• • • • • • • • • • - a classification theorem for semisimple rings
    • Classification of Simple Lie algebras and groups

Linear algebra

• s (by dimension)
• (by rank and nullity)
  • • (rational canonical form)

Dynamical systems

• • Ratner classification theorem

See also

• • • • • List of theorems

Notes and References

1. Web site: 2006-12-07 . An enormous theorem: the classification of finite simple groups plus.maths.org . 2025-09-16 . plus.maths.org . en.