Equations with one unknown in nilpotent groups
type: Item
instance of: scholarly article
title: Equations with one unknown in nilpotent groups
zbMATH Open document ID: 0536.20020
DOI: 10.1007/BF01141772
author: N. N. Repin
published in: Mathematical Notes
publication date: 1984-01-01T00:00:00Z
review text: In [Algebra Logika 16, 457-471 (1977; Zbl 0411.20021)] the reviewer proved that the solution problem for the coefficient-free equations \(f(x_ 1,x_ 2,...,x_ n)=g\) in the free nilpotent group F of class \(\ell \geq 9\) is unsolvable. The question studied by the author is that of solving equations with one unknown in a finitely generated nilpotent group. He proves: Theorem. There exists a finitely presented nilpotent group G for which the problem of recognizing the solvability of equations with one unknown is undecidable. The proof of this theorem, like that of the above-mentioned result of the reviewer is based on a simulation of Diophantine equations in a nilpotent group. At the IX. All-Union Symp. group theory, Moscow (1984), the author announced the following results: the group G in Theorem may be chosen of class 3 or may be chosen as free nilpotent group of class \(\ell \geq 10^{20}\).
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20F18
zbMATH DE Number: 3851366
zbMATH Keywords: coefficient-free equations
zbMATH Keywords: free nilpotent group
zbMATH Keywords: finitely generated nilpotent group
zbMATH Keywords: finitely presented nilpotent group
zbMATH Keywords: solvability of equations
reviewed by: Vitaliĭ Roman'kov
MaRDI profile type: Publication
cites work: Undecidable diophantine equations
description: scientific article; zbMATH DE number 3851366