Definition:Orbit (Group Theory)/Set of Orbits

Definition

Let $G$ be a group.

Let $X$ be a set

Let $*: G \times S \to S$ be a group action.

Let $\Orb x$ denote the orbit of $x \in X$.

From Group Action Induces Equivalence Relation, the relation $\mathcal R_G$ defined as:

$x \mathrel {\mathcal R_G} y \iff y \in \Orb x$

is an equivalence relation.

The quotient set $X / \mathcal R_G$ is called the set of orbits of $X$ under the action of $G$.