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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 X is a subgroup and quotient closed class of groups and if all 2 -generated Abelian-by-cyclic groups of X 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 X . 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.