1.6.59 Fields, Groups, Rings

condition: Identify all points of the plane \( \mathbb{R}^{2} \) that have a non-trivial stabilizer under the action of the symmetry group of a regular \( n \)-gon and find the stabilizer of these points. Find all orbits of this action. Note. Find out what the symmetry group of a regular \( n- \) triangle is.