Groups with small deviation for non-subnormal subgroups

Abstract

Abstract:We introduce the notion of the non-subnormal deviation of a group G. If the deviation is 0 then G satisfies the minimal condition for nonsubnormal subgroups, while if the deviation is at most 1 then G satisfies the so-called weak minimal condition for such subgroups (though the converse does not hold). Here we present some results on groups G that are either soluble or locally nilpotent and that have deviation at most 1. For example, a torsion-free locally nilpotent with deviation at most 1 is nilpotent, while a Baer group with deviation at most 1 has all of its subgroups subnormal. MSC:20E15, 20F16, 20F19

Groups with Subnormal Deviation

“…The aim of this paper is to give a further contribution to the theory of groups with many subnormal subgroups, characterizing periodic groups with a subnormal deviation, at least within the universe of soluble groups. The structure of locally soluble groups with subnormal deviation at most 1 was described in [8,9], where it was specifically proved that such groups are soluble. Our main result proves that if a periodic soluble group has a subnormal deviation, then only the extreme (and unavoidable) cases can occur.…”