On soluble min-by-max groups

Abstract

We say a group G has rain (max) if it satisfies the minimal (maximal) condition on subgroups. A group which is an extension of a group with min by a group with max is said to be a rain-by-max group. A well known result of Mal'cev states that a soluble group has max if and only if its abclian subgroups have max. Cernikov proved the analogous result that a soluble group has min if and only if its abelian subgroups have rain. There is no corresponding result for soluble rain-by-max groups. We give an example of a nilpotent group of class 2 which is not rain-by-max, although all its abelian subgroups are. In many ways our example is typical of the worst that can happen. A soluble G group which is not rain-by-max contains a metabelian, hypercentral subgroup X which is not rain-by-max. Furthermore, there is an epimorphic image of X which is structurally identical with the example mentioned above. Thus an investigation of the structure of certain metabelian subgroups will determine completely whether or not a soluble group is min-by-max. We begin therefore, in § 2, by considering the structure of metabelian groups which are not rain-bymax but are such that none of their abelian subgroups have that property. From our investigations we shall then obtain conditions which are necessary and sufficient for a group to be rain-by-max.