On the theories of free solvable groups
type: Item
instance of: scholarly article
title: On the theories of free solvable groups
zbMATH Open document ID: 0927.20015
DOI: 10.1016/S0022-4049(98)00024-3
author: Olivier Chapuis
published in: Journal of Pure and Applied Algebra
publication date: 1999-10-11T00:00:00Z
review text: In 1960, Malcev proved that a noncyclic free solvable group \(S\) of class \(\geq 2\) has an undecidable theory. The author proves that if Hilbert's 10th problem has a negative answer for the field of rationals (still a famous open question), then the universal theory of \(S\) in the case of class \(\geq 3\) is undecidable. Remind that the universal theory of a free metabelian group is decidable by the author's known theorem. Note that by the reviewer's result Hilbert's 10th problem for the rationals is equivalent to the problem of the decidability of the universal theory of a noncyclic free nilpotent group of class \(\geq 2\). To derive his main theorems the author proves that the terms of the derived series of a free solvable group are definable by existential formulae. Malcev defined them by universal ones.
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20A15
zbMATH DE Number: 1308222
zbMATH Keywords: free solvable groups
zbMATH Keywords: existential formulae
zbMATH Keywords: universal theories
zbMATH Keywords: definability
zbMATH Keywords: decidability
zbMATH Keywords: universal formulae
reviewed by: Vitaliĭ Roman'kov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1308222