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