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 sub- groups 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 S1F (the class of soluble by finite minimax groups). More precisely, this paper is dedicated of the study of S1F groups whose non polycyclic by finite subgroups are nearly normal.