Geodesic
Groups, in which almost all subgroups are near to normal
Algebra and discrete mathematics, no. 2 (2004), pp. 92-113 
M. M. Semko; S. M. Kuchmenko. Groups, in which almost all subgroups are near to normal. Algebra and discrete mathematics, no. 2 (2004), pp. 92-113. http://geodesic.mathdoc.fr/item/ADM_2004_2_a9/
@article{ADM_2004_2_a9,
     author = {M. M. Semko and S. M. Kuchmenko},
     title = {Groups, in which almost all subgroups are near to normal},
     journal = {Algebra and discrete mathematics},
     pages = {92--113},
     year = {2004},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2004_2_a9/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.