For a fixed integer \(d\ge 4\), the list of groups that appear as automorphism groups of smooth plane curves whose degree is d is unknown, except for \(d=4\) 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.

中文翻译：

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自同构群是循环的光滑平面曲线自同构的阶。

对于固定整数\（d \ ge 4 \），除\（d = 4 \）或5外，作为度为d的光滑平面曲线的自同构组出现的组的列表是未知的。关于光滑平面曲线的自同构群的结构。Badr和Bars开始研究自同构的某些顺序，并尝试获得光滑平面曲线的自同构群的精确结构。在本文中，基于T. Harui的结果，我们将Badr–Bars研究扩展到不同情况和新情况，主要是针对以自同构组出现的循环组的情况。