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For a fixed integer    $$d\ge 4$$  d ≥ 4 , the list of groups that appear as automorphism groups of smooth plane curves whose degree is  d  is unknown, except for    $$d=4$$  d = 4 or 5. Harui showed a certain characteristic about structures of automorphism groups of smooth plane curves. Badr and Bars began to study for certain orders of automorphisms and try to obtain exact structures of automorphism groups of smooth plane curves. In this paper, based on the result of T. Harui, we extend Badr–Bars study for different and new cases, mainly for the cases of cyclic groups that appear as automorphism groups.

Orders of automorphisms of smooth plane curves for the automorphism groups to be cyclic