Group Action Induces Equivalence Relation

Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $X$ be a set.

Let $*: G \times S \to S$ be a group action.

Let $\RR_G$ be the relation induced by $G$, that is:
 * $\forall x, y \in X: x \mathrel {\RR_G} y \iff y \in \Orb x$

where:
 * $\Orb x$ denotes the orbit of $x \in X$.

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

Proof
By, it is taken for granted that $\struct {G, \circ}$ is closed under $\circ$.

Let $x \mathrel {\RR_G} y \iff y \in \Orb x$.

Checking in turn each of the criteria for equivalence:

Reflexivity
Thus $\RR_G$ is reflexive.

Symmetry
Thus $\RR_G$ is symmetric.

Transitivity
Thus $\RR_G$ is transitive.

So $\RR_G$ 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 forms Partition