Group_Action_Induces_Equivalence_Relation

Let $G$ be a group whose identity is $e$. Let $X$ be a set. Let $*: G \times S \to S$ be a group action. Let $\mathcal R_G$ be the relation induced by $G$, that is $x \mathrel {\mathcal R_G} y \iff y \in \Orb x$ where $\Orb x$ denotes the orbit of $x \in X$. Then $\mathcal R_G$ is an equivalence relation. The equivalence class of an element is its orbit.