\begin{definition}[Definition:Equivalence Relation Induced by Group Action]
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 $\RR_G$ defined as:
:$x \mathrel {\RR_G} y \iff y \in \Orb x$
where:
:$\Orb x$ denotes the orbit of $x \in X$.
That is:
:$x \mathrel {\RR_G} y \iff \exists g \in G: y = g*x$
\end{definition}