Stabiliser and Orbit Properties
For an action of on , and the orbits form a partition of .
Proof
For the stabiliser, closure follows from , the identity satisfies , and implies . Thus is a subgroup.
Every lies in its orbit because . If , write . Then , and every lies in . Hence , and symmetry gives equality.
Related
Stated in
- Proposition 4.7§4.2 Orbit-Stabiliser Theorem