Classification theorem
Describes the objects of a given type, up to some equivalence
title: "Classification theorem" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["mathematical-theorems", "mathematical-classification-systems"] description: "Describes the objects of a given type, up to some equivalence" topic_path: "general/mathematical-theorems" source: "https://en.wikipedia.org/wiki/Classification_theorem" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Describes the objects of a given type, up to some equivalence ::
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 (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)
Analysis
Dynamical systems
Mathematical physics
References
References
- (2006-12-07). "An enormous theorem: the classification of finite simple groups {{!}} plus.maths.org".
::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::