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A condition of finitely generated soluble groups. (English) Zbl 0942.20014

In this note the authors prove that if \(G\) is a finitely generated soluble group, then every infinite subset of \(G\) contains two elements generating a nilpotent group of class \(\leq k\) if and only if \(G\) is an extension of a finite group by a group in which every 2-generator subgroup is nilpotent of class \(\leq k\).

MSC:

20F16 Solvable groups, supersolvable groups
20F18 Nilpotent groups
20F05 Generators, relations, and presentations of groups
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References:

[1] Delizia, C. 1994.Finitely Generated Soluble Groups with a Condition on Infinite Subsets, Vol. A128, 201–208. Instito Lombardo. Rend. Sc · Zbl 0882.20020
[2] DOI: 10.1080/00927879608825768 · Zbl 0878.20021
[3] DOI: 10.1017/S1446788700024253
[4] DOI: 10.1017/S1446788700019303
[5] DOI: 10.1017/S1446788700015469 · Zbl 0274.20033
[6] Robinson D.J.S., Finiteness Conditions and Generalized Soluble Groups (1972) · Zbl 0243.20032
[7] Robinson D.J.S., A Course in the Theory of Groups (1982) · Zbl 0483.20001
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