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

TL;DR

This paper investigates the solvability of unimodular equations in solvable groups of higher derived lengths, revealing that such equations may lack solutions even in larger groups of the same class.

Contribution

It extends known results about unimodular equations from abelian and metabelian groups to more complex solvable groups, demonstrating new limitations.

Findings

Unimodular equations do not always have solutions in solvable groups of higher derived lengths.

Some unimodular equations lack solutions in any larger solvable group of the same derived length.

The paper provides proofs of these non-solvability results.

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.