Orbit of a finite group action

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

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 — A complete valid action is required.
• 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.