The automorphism group of Cycle C_n

The automorphism group usually reflects the symmetry of a graph.To determine the construction of the group of a graph is one of the focus problems in algebraic graph theory.Intuitionally,the authors think that the order of Aut(Cn) is 2.But there is never a machinery proof for now.In this paper,the problem will be solved by the orbit equation based on the action of a group on the vertex set of cycle Cn.Therefore,the authors will know that the automorphism group of a cycle on n vertices is isomorphic to a dihedral group.