Properties of Abelian-by-cyclic shared by soluble finitely generated groups.

Our main result states that if G is a finitely generated soluble group having a normal Abelian subgroup A, such that G/A and <x, a> are nilpotent (respectively, finite-by-nilpotent, periodic-by-nilpotent, nilpotent-by-finite, finite-by-supersoluble, supersoluble-by-finite) for all (x, a) ∈ G × A, then so is G. We deduce that if 픛 is a subgroup and quotient closed class of groups and if all 2-generated Abelian-by-cyclic groups of 픛 are nilpotent (respectively, finite-by-nilpotent, periodic-by-nilpotent, nilpotent-by-finite, finite-by-supersoluble, supersoluble-by-finite), then so are all finitely generated soluble groups of 픛. We give examples that show that our main result is not true for other classes of groups, like the classes of Abelian, supersoluble, and FC-groups. [ABSTRACT FROM AUTHOR]