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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