Unimodular equations which do not preserve the derived length of a group

2025/05/19 by Mikheenko, Mikhail A.
#20F16 (Secondary) #20F70 (Primary) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2505.12783

Abstract

It is a known fact that any unimodular equation over an abelian group has a solution in that group itself. It is also known that for metabelian groups this does not hold; moreover, there is a unimodular equation over some metabelian group which has no solutions in any larger metabelian group. Here we present the proof of an analagous fact for solvable groups of higher derived lengths.

Citations

• Infinite systems of equations in abelian and nilpotent groups
• On singular equations over torsion-free groups
• On solvability of certain equations of arbitrary length over\n torsion-free groups
• Universal solvability of group equations
• Finitary approximations of groups and their applications

Related

• Infinite systems of equations in abelian and nilpotent groups
• On the length of non-solutions to equations with constants in some linear groups
• Universal solvability of group equations
• On solvability of certain equations of arbitrary length over\n torsion-free groups
• Subrack lattice determines the derived length