Isotopic classes of Transversals

TL;DR

This paper characterizes normal subgroups in finite nilpotent groups via isotopy classes of transversals and counts isotopy classes of transversals in specific dihedral groups, advancing understanding of subgroup structures.

Contribution

It establishes a new criterion for normality in finite nilpotent groups based on isotopy classes of transversals and determines the number of such classes in certain dihedral groups.

Findings

Normality characterized by isotopy classes of transversals

Number of isotopy classes in dihedral groups of order 2p

Isotopism classes formed with respect to right loop structures

Abstract

Let $G$ be a finite group and $H$ be a subgroup of $G$. In this paper, we prove that if $G$ is a finite nilpotent group and $H$ a subgroup of $G$, then $H$ is normal in $G$ if and only if all normalized right transversals of $H$ in $G$ are isotopic, where the isotopism classes are formed with respect to induced right loop structures. We have also determined the number isotopy classes of transversals of a subgroup of order 2 in $D_{2p}$, the dihedral group of order $2p$, where $p$ is an odd prime.