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

Checking in turn each of the critera for equivalence:

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

Symmetric
$$ $$ $$ $$ $$ $$

Transitive
$$ $$ $$ $$ $$

Hence the result.