Definition:Equivalence Relation Induced by Group Action

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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:

$x \mathrel {\mathcal R_G} y \iff y \in \Orb x$


$\Orb x$ denotes the orbit of $x \in X$.

That is:

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

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