On the structure of some locally nilpotent groups without contranormal subgroups

TL;DR

This paper investigates the structure of certain locally nilpotent groups, establishing conditions under which they are nilpotent and analyzing the role of contranormal subgroups in hypercentral groups.

Contribution

It proves that nilpotent-by-finite groups with no proper contranormal subgroups are nilpotent and studies hypercentral groups with finite proper contranormal subgroups.

Findings

Nilpotent-by-finite groups without proper contranormal subgroups are nilpotent.

Existence of locally nilpotent groups with proper contranormal subgroups.

Structural analysis of hypercentral groups with finite contranormal subgroups.

Abstract

Following J.S. Rose, a subgroup H of a group G is said contranormal in G if G = H^G . In a certain sense, contranormal subgroups are antipodes to subnormal subgroups. It is well known that a finite group is nilpotent if and only if it has no proper contranormal subgroups. We prove that a nilpotent-by-finite group with no proper contranormal subgroup is nilpotent. There are locally nilpotent groups with a proper contranormal subgroup. We study the structure of hypercentral groups with a finite proper contranormal subgroup.