Stabilizer of a finite group action

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

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 — The complete action model is required.
• Subgroups and group homomorphisms — 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.