# Finite group

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A finite group has a finite carrier and complete closed Cayley table, satisfying associativity, a unique identity and inverses for every element.

mko-finite-group · definition · v0.10.0



## Mathematical statement

Let a finite set G carry a binary operation *. If it is closed on G, associative for every triple, has a two-sided identity and every element has a two-sided inverse, then (G,*) is a finite group. A complete Cayley table allows exhaustive checking of the specified finite model.





## Representative formula (exact TeX)

G(S)





## Assumptions


- Elements are labeled from 0 to n−1, and the question gives a complete n×n operation table.
- Sampled rows or a few triples do not prove a group. Associativity must be checked over every one of the n³ ordered triples.





## Symbol scope


- G — finite-group candidate
- S — finite carrier





## Intuition, definition, and misconceptions

Closure means that every table output is still an element of G.

Associativity compares (a*b)*c with a*(b*c) for all triples and is not commutativity.

An identity must work on both sides. An inverse must also multiply to the same identity on both sides.

Do not infer a group from table symmetry or a few examples, or replace two-sided identities and inverses with one-sided conditions.





## Adopted derivation guide

[Finite group: auditing the group axioms from a complete Cayley table](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-group-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Set](https://ocme.evemisslab.com/en/mko/mko-set/) — A group has a fixed carrier set.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — A binary operation must be total and single-valued.
- [Binary relations and set projections](https://ocme.evemisslab.com/en/mko/mko-relation/) — A Cayley table assigns an output to each input pair.
- [Mathematical proof](https://ocme.evemisslab.com/en/mko/mko-proof/) — Group qualification must close every axiom obligation.





## Verification boundary

A complete finite table can exactly audit its specified candidate; no infinite group or unlisted operation is inferred.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add subgroups, homomorphisms, cosets, normal subgroups and quotient groups.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/finite-group.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-finite-group/index.md)