Definition:Orbit (Group Theory)/Definition 2

Definition

Let $G$ be a group acting on a set $X$.

Let $\mathcal R$ be the relation on $X$ defined as:

$\forall x, y \in X: x \mathrel {\mathcal R} y \iff \exists g \in G: y = g * x$

where $*$ denotes the group action.

From Group Action Induces Equivalence Relation, $\mathcal R$ is an equivalence relation.

The orbit of $x$, denoted $\Orb x$, is the equivalence class of $x$ under $\mathcal R$.