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Finitely generated soluble groups with a condition on infinite subsets. (English) Zbl 0882.20020
Let $$\mathcal X$$ be a class of groups. A group $$G$$ is said to satisfy the condition $$(\mathcal X,\infty)$$ (resp. $$({\mathcal X},n)$$) if each infinite subset (resp. $$(n+1)$$-element subset) contains a pair of elements which generate an $$\mathcal X$$-subgroup. For the class $$\mathcal N$$ of nilpotent groups a finitely generated soluble group satisfies $$(\mathcal N,\infty)$$ if and only if it is finite-by-nilpotent [J. Lennox and J. Wiegold, J. Aust. Math. Soc., Ser. A 31, 459-463 (1981; Zbl 0492.20019)] and if it satisfies $$({\mathcal N},n)$$ then the index of the hypercentre is bounded by a function of $$n$$ [the reviewer, Publ. Math. 40, No. 3-4, 313-321 (1992; Zbl 0777.20012)]. Here the class $$\mathcal N_2$$ of nilpotent groups of class two is considered. A finitely generated soluble group satisfies $$(\mathcal N_2,\infty)$$ if and only if $$G/Z_2(G)$$ is finite. If $$G$$ satisfies $$(\mathcal N_2,n)$$ then the exponent of $$G/R_2(G)$$ is bounded by a function of $$n$$, where $$R_2(G)$$ is the subgroup of right 2-Engel elements. It is left open whether such a bound exists for the index $$|G/R_2(G)|$$.

##### MSC:
 20F16 Solvable groups, supersolvable groups 20F05 Generators, relations, and presentations of groups 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks