Definition:Equivalence Relation Induced by Group Action

Definition
Let $G$ be a group acting on a set $X$.

The equivalence relation on $X$ induced by the action is the relation $\mathcal R_G$ defined as:
 * $x \mathrel {\mathcal R_G} y \iff y \in \operatorname{Orb} \left({x}\right)$

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

That is:
 * $x \mathrel {\mathcal R_G} y \iff \exists g \in G : y = g*x$

Also see

 * Definition:Induced Equivalence, of which this is a special case
 * Orbit is Equivalence Class
 * Definition:Saturation (Group Action)