# Orbit of a finite group action

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

The images of one point under all group elements form its orbit; all orbits partition the point carrier.

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



## Mathematical statement

Start with a fully qualified finite group action and enumerate G*x={g*x:g∈G} using every group element. The relation x~y exactly when some g sends x to y is an equivalence relation. Orbits are equal or disjoint and cover the carrier. One action image, finitely sampled generator steps and a stabilizer are not the whole orbit.





## Representative formula (exact TeX)

O(x)





## Assumptions


- The complete finite Cayley and action tables pass every group/action axiom.
- Orbit and equivalence witnesses enumerate every group element.





## Symbol scope


- O — group-action orbit
- x — acted-on point





## Intuition, definition, and misconceptions

An orbit uses all group elements.

Being in the same orbit is reflexive, symmetric and transitive.

All orbits partition the entire carrier.

One g*x image, a time trajectory or a stabilizer is not an orbit computation.





## Adopted derivation guide

[Complete orbits and their partition](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-group-orbit-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 action](https://ocme.evemisslab.com/en/mko/mko-finite-group-action/) — A complete valid action is required.
- [Equivalence class](https://ocme.evemisslab.com/en/mko/mko-equivalence-class/) — Orbits form equivalence classes.





## Verification boundary

Only complete finite action tables are handled; no continuous or topological group action 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 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-orbit.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-finite-group-orbit/index.md)