On the torsion-free ranks of finitely generated nilpotent groups and of their abelian subgroups
Copyright policy )On the torsion-free ranks of finitely generated nilpotent groups and of their abelian subgroups
Denote by \(f(n)\) the greatest integer \(h\) such that there exists a finitely generated nilpotent group of torsion-free rank \(h\) such that the torsion-free ranks of all Abelian subgroups of this group are not greater than \(n\). The author proves that the function \(f(n)\) satisfies the inequality \(f(n)\geq\tfrac18(n^2-4)+n\). Proving this theorem, for each positive integer \(s\) the author constructs a torsion-free nilpotent group of class 2 and of torsion-free rank at least \(\tfrac18(s^2-4)+s\) such that any Abelian subgroup can be generated by at most \(s\) elements.
Generators, relations, and presentations of groups, torsion-free ranks, quadratic lower bounds, Nilpotent groups, Subgroup theorems; subgroup growth, Abelian subgroups, nilpotent groups
Generators, relations, and presentations of groups, torsion-free ranks, quadratic lower bounds, Nilpotent groups, Subgroup theorems; subgroup growth, Abelian subgroups, nilpotent groups
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