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

Abstract

Our main result states that if $G$ is a finitely generated soluble group having a normal Abelian subgroup $A$, such that $G/A$ and $\left\langle x,a\right\rangle $ are nilpotent (respectively, finite-by-nilpotent, periodic-by-nilpotent, nilpotent-by-finite, finite-by-supersoluble, supersoluble-by-finite) for all $(x,a)\in G\times A$, then so is $G$. We deduce that if $\mathfrak{X}$ is a subgroup and quotient closed class of groups and if all $2$-generated Abelian-by-cyclic groups of $\mathfrak{X}$ 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 $\mathfrak{X}$. 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.