Group Action Induces Equivalence Relation

Theorem
Let $G$ be a group whose identity is $e$.

Let $X$ be a set

Let $G * X$ be a group action.

Let $\operatorname{Orb} \left({x}\right)$ be the orbit of $x \in X$.

Then the relation $\mathcal R_G$ defined as $x \mathop {\mathcal R_G} y \iff y \in \operatorname{Orb} \left({x}\right)$ is an equivalence relation.

That is, the orbit of an element is an equivalence class.

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

Proof
Let $x \mathop {\mathcal R_G} y \iff y \in \operatorname{Orb} \left({x}\right)$.

Checking in turn each of the critera for equivalence:

Reflexivity
$x = e * x \implies x \in \operatorname{Orb} \left({x}\right)$ from the definition of group action.

Thus $\mathcal R_G$ is reflexive.

Symmetry
Thus $\mathcal R_G$ is symmetric.

Transitivity
Thus $\mathcal R_G$ is transitive.

So $\mathcal R_G$ has been shown to be an equivalence relation.

Hence the result, by definition of an equivalence class.