In this paper, we determine the number of isotopic classes of transversals of a subgroup of order 2 in D_2n (n is a positive odd integer greater than 1), where isotopism classes are formed with respect to the induced right loop structures. We also determine the cyclic index of the affine group Aff\((1,p^{2})\), where p is an odd prime.

中文翻译：

---

二面体群D 2n，n奇数中的横向同位素类

在本文中，我们确定D _{2 n}（n是大于1的正奇整数）中2阶次子集的横向同位素类的数量，其中相对于诱导的右环结构形成了同位素类。我们还确定仿射组Aff \（（（1，p ^ {2}）\）的循环索引，其中p是奇质数。