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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.

20F16 Solvable groups, supersolvable groups
20F05 Generators, relations, and presentations of groups
20F14 Derived series, central series, and generalizations for groups
Full Text: EuDML
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