A Note on Subsoluble Groups

Abstract

A group is said to be hyperabelian if each of its non-trivial quotient groups has a non-trivial abelian normal subgroup, and subsoluble if each of its non-trivial quotient groups has a non-trivial abelian subnormal subgroup. In this note we settle a point raised by Robinson ([2], p. 87) by showing that subsoluble groups satisfying Min-n, the minimal condition for normal subgroups, need not be hyperabelian. More exactly, we construct a group G whose normal subgroups are well-ordered by inclusion, of order-type ω + 1, having a perfect minimal normal subgroup N which is generated by its abelian normal subgroups, such that G/N is locally soluble and hyperabelian; G is obviously a group satisfying Min-n which is subsoluble but not hyperabelian. Our construction uses the notion of the treble product rower of a family of groups introduced in [1].