Homology of group extensions with divisible abelian kernel

Abstract

We shall consider group extensions D >-, G + Q such that D is a divisible abelian group. Our object is to investigate relations between the integral homology groups H,G and the structure of the extension. An important source of extensions with kernel a divisible abelian torsion group is the class of cernikov groups. Recall that a Cernikov group is a finite extension of an abelian group with the minimal condition (min); the intersection of all the subgroups of finite index in a Cernikov group is a characteristic, divisible abelian group with min whose quotient group is finite. For this reason we call an extension D * G --u Q a eernikov extension if D is a divisible abelian group with min and Q is finite. Central Cernikov extensions also arise naturally: for example, Baer has shown that a nil- potent group with min is a central cernikov group, i.e. one whose maximal divisible abelian subgroup lies in the centre