О разрешимости регулярных уравнений в многообразии метабелевых групп | Прикладная дискретная математика. 2017. № 36. DOI: 10.17223/20710410/36/4

We study the solvability of equations over groups within a given variety or another class of groups. The classes of nilpotent and solvable groups were considered as main classes to investigate from such point of view. The natural analogues of the famous Kervaire - Laudenbach and Levin conjectures were raised to the challenge. It was also noted that the "solvable" version of the known theorem by Brodskii is not true. In this paper, for each n Е N, n ^ 2, we prove that every regular equation over the free metabelian group Mn is solvable in the class M of all metabelian groups. Moreover, there is a metabelian group Mn that contains a solution of every unimodular equation over Mn. These results are extended to the class of rigid metabelian groups. Also, we give an example showing that there exists an equation over a locally indicable torsion-free metabelian group G that has no solution in any solvable overgroup of G. It follows that solvable versions of the Levin conjecture are not true. Another example presents an unimodular equation over a locally indicable torsion-free metabelian group G that has no solution in any metabelian overgroup of G. Hence, the Kervaire - Laudenbach conjecture is not valid for the variety of all metabelian groups. We prove that there is an unimodular equation over a finite metabelian group G that has no solutions in any finite metabelian overgroup of G. This means that analog of the famous theorem by Gerstenhaber and Rothaus (about solvability of each unimodular equation over a finite group G in some finite overgroup of G) is not valid for the class of finite metabelian groups.
  • Title О разрешимости регулярных уравнений в многообразии метабелевых групп
  • Headline О разрешимости регулярных уравнений в многообразии метабелевых групп
  • Publesher Tomask State UniversityTomsk State University
  • Issue Прикладная дискретная математика 36
  • Date:
  • DOI 10.17223/20710410/36/4
Ключевые слова
Kervaire-Laudenbach conjecture, Levin conjecture, solvable group, metabelian group, rigid group, nilpotent group, locally indicable group, regular equation, solvability over group, гипотеза Кервер-Лауденбаха, гипотеза Левина, разрешимая группа, метабелевая группа, нильпотентная группа, жесткая группа, локально отображаемая группа, регулярное уравнение, разрешимость по группе
Авторы
Ссылки
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 О разрешимости регулярных уравнений в многообразии метабелевых групп | Прикладная дискретная математика. 2017. № 36. DOI: 10.17223/20710410/36/4
О разрешимости регулярных уравнений в многообразии метабелевых групп | Прикладная дискретная математика. 2017. № 36. DOI: 10.17223/20710410/36/4