# Stabilizer 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 stabilizer of a specified point consists of every group element fixing that point and forms a subgroup.

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



## Mathematical statement

For the same fully certified action of G on X, Stab_G(x)={g:g*x=x} requires enumeration of every g. Check identity membership, product closure and inverse closure completely in the same group. One fixing element does not give the full stabilizer. A point stabilizer, an orbit and the action kernel are different constructions, even when particular underlying sets happen to agree.





## Representative formula (exact TeX)

S(x)





## Assumptions


- The complete finite group and action tables satisfy every group and action axiom.
- The specified point belongs to that action carrier, and inverses come from the same group table.





## Symbol scope


- S — point stabilizer
- x — acted-on point





## Intuition, definition, and misconceptions

Membership means g*x=x.

Identity, every product and every inverse must be checked.

The stabilizer depends on the point; it must not be conflated with an orbit or the action kernel.

Finding one fixing group element does not compute the entire stabilizer.





## Adopted derivation guide

[All group elements fixing point zero](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-group-stabilizer-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/) — The complete action model is required.
- [Subgroups and group homomorphisms](https://ocme.evemisslab.com/en/mko/mko-subgroup-homomorphism/) — Subgroup closure has its own obligations.





## Verification boundary

Only complete finite action tables are handled.

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