---
tag: 00F2JFA41J
kind: definition
name: Orbit
msc: [20B05]
defines: 0YQPZB5Q1G
uses: [0YSE802YCQ]
sources:
  - url: https://en.wikipedia.org/wiki/Group_action
    locator: Orbits and stabilizers
---

## Definition

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$, and let $x \in X$.
The orbit of $x$ is the set $G \cdot x = \{g \cdot x : g \in G\}$.
The set of all orbits is written $X/G$.

## Remarks

The point $x$ lies in its own orbit, because $e \cdot x = x$.
The [[orbits partition|0ZC7FYE0NZ]] the set $X$.
For the action of the rotation group of a square on its corners, there is one orbit, which contains all four corners.

## Explanations

### L1 human

The orbit of a point collects all the places where the group can move the point.
Two points in the same orbit can be moved onto each other.
Points in different orbits are never moved onto each other.