# Finite abelian group

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

A finite group is abelian when its complete Cayley table satisfies commutativity for every pair of elements.

mko-finite-abelian-group · definition · v0.10.0



## Mathematical statement

A finite abelian group must first qualify as a group using its complete Cayley table: closure, associativity, a unique identity and inverses. Then check a*b=b*a for all a,b. This slice classifies only explicitly supplied finite tables.





## Representative formula (exact TeX)

A(G)





## Assumptions


- Carrier labels run from 0 to n−1 and a complete n-by-n table is supplied.
- A few commuting samples or apparently commuting generators are not used to infer commutativity of the whole group.





## Symbol scope


- A — abelian-group record
- G — finite group





## Intuition, definition, and misconceptions

Full group qualification precedes the commutativity classification.

Element orders and integer multiples derive from the same group operation and identity.

The cyclic subgroup generated by one element belongs only to the specified finite group.

Do not declare a whole group abelian because a few elements commute or the diagonal of its table appears symmetric.





## Adopted derivation guide

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





## Prerequisites


- [Finite group](https://ocme.evemisslab.com/en/mko/mko-finite-group/) — Commutativity is an additional obligation on an already qualified group structure.
- [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/mko/mko-quantifier/) — The abelian property quantifies over all element pairs.





## Verification boundary

A complete finite table establishes facts only about the specified finite group; it does not cover arbitrary groups.

not_formalized

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





## Next obligation

Add the additive abelian group of a finite unital ring and its second operation.





## 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-abelian-group.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-finite-abelian-group/index.md)