On Subnormality in Soluble Minimax Groups

Abstract

Finiteness conditions associated with subnormal subgroups are in general fairly difficult to handle. In this note we refer in particular to two restrictions of this type. The first is the so-called subnormal intersection property, which demands that the intersection of any family of subnormal subgroups should again he a subnormal subgroup. The second condition entails the existence of an upper bound for the subnormal indices of all subnormal subgroups. Since their introduction by Robinson almost ten years ago, these conditions have prompted a number of investigations aimed at elucidating the structure of groups (usually soluble) with such nations on the behaviour of their subnormal subgroups. In his recent treatise [7, 8] — to which we refer the reader for background and terminology — Robinson remarks on the apparent difficulty of such investigations. Cases in point are Robinson’s proof, in [5], that finitely generated soluble groups with the subnormal intersection property are finite-by-nilpotent, and McDougall’s relatively incomplete description of soluble minimax groups with the subnormal intersection property ([4]).