Some Characterizations of a Normal Subgroup of a Group

TL;DR

This paper provides new characterizations of when a subgroup is normal in a group, offering a simpler proof of a key theorem without relying on the classification of finite simple groups.

Contribution

It introduces novel criteria for subgroup normality and presents an elementary proof of the main theorem in the context of finite or finite index subgroups.

Findings

New characterizations of subgroup normality

Elementary proof of the main theorem

Avoids classification of finite simple groups

Abstract

Let G be a group and H be a subgroup of G which is either finite or of finite index in G. In this note, we give some characterizations for normality of H in G. As a consequence we get a very short and elementary proof of the Main Theorem of [5], which avoids the use of the classification of finite simple groups