Isotopic classes of transversals in Dihedral group $ D_{2n} $, $n$ odd

TL;DR

This paper classifies isotopic classes of transversals in dihedral groups of order 2n for odd n and computes the cyclic index of a related affine group, advancing understanding of algebraic structures and symmetries.

Contribution

It determines the number of isotopic classes of transversals in dihedral groups of order 2n for odd n and calculates the cyclic index of the affine group Aff(1,p^2).

Findings

Number of isotopic classes of transversals in D_{2n} for odd n is established.

Cyclic index of Aff(1,p^2) is computed.

Provides new insights into the structure of right loops and affine groups.

Abstract

In this article 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.