Orbit-stabilizer theorem

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

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 — The complete orbit is required.
• Stabilizer of a finite group action — Use the stabilizer of the same point.
• Cardinality comparison of sets — 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.