Group Action Induces Equivalence Relation

Theorem
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:
 * $x \mathrel {\mathcal R_G} y \iff y \in \operatorname{Orb} \left({x}\right)$

where:
 * $\operatorname{Orb} \left({x}\right)$ denotes the orbit of $x \in X$.

Then:
 * $\mathcal R_G$ is an equivalence relation.
 * The equivalence class of an element is its orbit.

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

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.

Also see

 * Definition:Set of Orbits
 * Definition:Equivalence Relation Induced by Group Action
 * Set of Orbits is Partition