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A nilpotency condition for finitely generated soluble groups. (English) Zbl 0928.20029
Assume that \(G\) is a finitely generated soluble group such that every infinite set of elements of \(G\) contains a pair that generates a nilpotent subgroup of class at most \(k\), a fixed integer. The author shows for this situation that \(G\) is an extension of a finite group by a torsionfree \(k\)-Engel group, and that there is an integer \(n\) depending on \(k\) and the derived length of \(G\) such that \(| G:Z_n(G)|\) is finite. For \(k\leq 3\) the integer \(n\) does no longer depend on the derived length of \(G\).
MSC:
20F16 Solvable groups, supersolvable groups
20F45 Engel conditions
20F19 Generalizations of solvable and nilpotent groups
20E07 Subgroup theorems; subgroup growth
20F14 Derived series, central series, and generalizations for groups
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