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Characterisation of finitely generated soluble finite-by-nilpotent groups. (English) Zbl 1119.20039
For a class of groups $$\mathfrak X$$, let $$(\mathfrak X,\infty)$$ denote the class of groups in which every infinite subset contains two distinct elements generating a subgroup in $$\mathfrak X$$. B. H. Neumann [J. Aust. Math. Soc., Ser. A 21, 467-472 (1976; Zbl 0333.05110)] proved, for the class $$\mathfrak A$$ of Abelian groups, that the groups in $$(\mathfrak A,\infty)$$ are exactly the center-by-finite groups. J. C. Lennox and J. Wiegold [ibid. 31, 459-463 (1981; Zbl 0492.20019)] showed, for the class $$\mathfrak N$$ of nilpotent groups, that a finitely generated soluble group is in $$(\mathfrak N,\infty)$$ iff it is finite-by-nilpotent.
Here are the main results of the paper under review. For $$k>0$$, let $$\Omega_k$$ be the class of groups $$H$$ such that $$\gamma_{k-1}(H)>\gamma_k(H)=\gamma_{k+1}(H)$$ (where $$\gamma_i(H)$$ is the $$i$$-th member of the lower central series of $$H$$), and $$\Omega=\bigcup_k\Omega_k$$. Let $$G$$ be a finitely generated soluble group. Then (1) $$G\in(\Omega,\infty)$$ iff $$G$$ is finite-by-nilpotent, and (2) $$G\in(\Omega_k,\infty)$$ iff $$G$$ is finite-by-$$\mathfrak N_{k,2}$$, where $$\mathfrak N_{k,2}$$ is the class of groups in which every two-generated subgroup is $$k$$-step nilpotent.

##### MSC:
 20F16 Solvable groups, supersolvable groups 20F05 Generators, relations, and presentations of groups 20F14 Derived series, central series, and generalizations for groups
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