Abstract

In the present paper we start to study the soluble groups in which every non-subnormal subgroup has finite special rank.

Introduction

Let G be a group, Lnon–sn(G) the set of all non-subnormal subgroups of G. If Lnon–sn(G) = Ø then we obtain a group, every subgroup of which is subnormal. The study of these groups was very fruitful and has brought many interesting results (see the books [12], [13], [8]). The groups G in which the set Lnon–sn(G) is “very small” in some sense is the natural next consideration. For many domains of Infinite Group Theory “to be very small” means to satisfy some finiteness conditions. The first natural finiteness conditions in Group Theory were the classical minimal and maximal conditions. The groups with minimal condition for non-subnormal subgroups have been considered by S. Franciosi and F. de Giovanni [1]. The groups with dual maximal condition for non-subnormal subgroups have been studied by L.A. Kurdachenko and H. Smith [4]. The groups with maximal condition for non-subnormal subgroups is nearly allied to the groups, in which the set Lnon–sn(G) consists only of the finitely generated subgroups. Such groups have been considered. The results of papers [1], [4] have been extended on groups with weak minimal and maximal condition on non-subnormal subgroups [5], [6]. The weak minimal and maximal conditions are connected with the concept of special rank (or Mal'tsev-Prüfer rank).