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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 \(G\).
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