A Study of Groups through Transversals

TL;DR

This paper investigates conditions under which the normalizer of a core-free subgroup in a finite group is normal, using transversals to derive new insights and results in group theory.

Contribution

It provides a necessary and sufficient condition for the normalizer of a core-free subgroup to be normal, utilizing transversal methods to extend known finite group results.

Findings

Established a criterion for normalizers of core-free subgroups to be normal.

Derived a known finite group result using transversal techniques.

Enhanced understanding of subgroup normality conditions in finite groups.

Abstract

In this note, a necessary and sufficient condition for the normalizer of a core-free subgroup $H$ of a finite group $G$ to be normal in $G$ is obtained. Also, a known result of finite groups is obtained through transversal.