Definition:Equivalence Relation Induced by Group Action

Definition
Let $G$ be a group.

Let $X$ be a set.

Let $\phi : G\times X \to X$ be a group action.

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

where:
 * $\Orb x$ 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

 * Group Action Induces Equivalence Relation
 * Definition:Saturation (Group Action)
 * Definition:Equivalence Relation Induced by Mapping, of which this is a special case (using the quotient mapping)