Groups, in which almost all subgroups are near to normal

Abstract

A subgroup \(H\) of a group \(G\) is said to be nearly normal, if \(H\) has a finite index in its normal closure. These subgroups have been introduced by B.H. Neumann. In a present paper is studied the groups whose non polycyclic by finite subgroups are nearly normal. It is not hard to show that under some natural restrictions these groups either have a finite derived subgroup or belong to the class \(S_{1}F\) (the class of soluble by finite minimax groups). More precisely, this paper is dedicated of the study of \(S_{1}F\) groups whose non polycyclic by finite subgroups are nearly normal.