Let be a group acting on a set . Prove that the relation on defined by if and only if for some is an equivalence relation. (For each , the equivalence class of under is called the orbit of under .)
We need to show that the relation is reflexive, symmetric, and transitive.
- For all , we have so that .
- Suppose ; then we have for some . Then , so that .
- Suppose and . Then there exist such that and . Then , so that .
Thus is an equivalence relation.