# Finite group action

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

An action of a finite group on a finite set is a total action table satisfying the identity and multiplication-compatibility laws.

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



## Mathematical statement

Use one complete finite Cayley table, complete point carrier and total action table. Check e*x=x and (g*h)*x=g*(h*x) for all relevant indices. Each group element then induces a permutation, but validity alone does not certify faithful, free or transitive action. Sampled pairs are insufficient.





## Representative formula (exact TeX)

A(g,x)





## Assumptions


- The full group table passes exact closure, associativity, identity and inverse checks.
- For every group element and point the action table supplies exactly one value in the point carrier.





## Symbol scope


- A — finite group action
- g — group element





## Intuition, definition, and misconceptions

Identity action and multiplication compatibility are separate obligations.

Complete finite tables allow exact checks of every relevant triple.

Faithful, free and transitive properties need their own definitions and certificates.

A few tested pairs or permutation rows alone do not establish a group action.





## Adopted derivation guide

[A complete six-element action on four points](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-group-action-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/) — The group operation is complete.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — The action is a total map.
- [Set](https://ocme.evemisslab.com/en/mko/mko-set/) — The point carrier is explicit.





## Verification boundary

Only complete finite tables with nonempty carriers in this prototype are handled; no infinite-group, continuous-action or representation-theoretic property 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 orbits, stabilizers and the orbit-stabilizer theorem.





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