# Orbit-stabilizer theorem

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

For a complete finite group action, the orbit size and stabilizer size of the same point multiply to the group order.

mko-orbit-stabilizer-theorem · theorem · v0.10.0



## Mathematical statement

Fix the same fully certified action of G on X and point x. The action map phi_x(g)=g*x is surjective onto the orbit. If g0*x=y, its fiber over y is exactly the left coset g0*Stab_G(x); hence |G|=|G*x|*|Stab_G(x)|. The action need not be free, faithful or transitive. A proper group sample, a different point binding or a coincidentally equal numerical product is insufficient.





## Representative formula (exact TeX)

c(G)=c(O(x))*c(S(x))





## Assumptions


- The complete finite group and total action tables satisfy every group and action axiom.
- Orbit, stabilizer and every fiber are completely enumerated for the same specified point.





## Symbol scope


- c — finite cardinality
- G — acting finite group
- O — orbit operator
- x — acted-on point
- S — stabilizer operator





## Intuition, definition, and misconceptions

Action-map fibers are left cosets of the point stabilizer.

Each fiber has equal size, and the orbit is the image.

The counting identity does not add free or transitive hypotheses.

A numerical product alone does not verify the same action/point binding or the fiber-coset bijection.





## Adopted derivation guide

[Fibers as left cosets of the stabilizer](https://ocme.evemisslab.com/en/derivations/derivation-mko-orbit-stabilizer-theorem-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Orbit of a finite group action](https://ocme.evemisslab.com/en/mko/mko-finite-group-orbit/) — The complete orbit is required.
- [Stabilizer of a finite group action](https://ocme.evemisslab.com/en/mko/mko-finite-group-stabilizer/) — Use the stabilizer of the same point.
- [Cardinality comparison of sets](https://ocme.evemisslab.com/en/mko/mko-cardinality-comparison/) — Count finite cosets and fibers.





## Verification boundary

Only complete finite actions are handled; sampled group elements do not prove the general theorem.

not_formalized

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





## Next obligation

Add Cauchy theorem and homomorphism kernels.





## 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/orbit-stabilizer-theorem.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-orbit-stabilizer-theorem/index.md)