On $p$-nonsingular systems of equations over solvable groups

TL;DR

This paper investigates the structure of certain solvable groups with specific torsion properties, demonstrating how they can embed into larger groups where particular systems of equations are solvable, and determines the minimal order of a metabelian group with an unsolvable unimodular equation.

Contribution

It introduces a method to embed groups with specific subnormal series into larger groups where $p$-nonsingular systems are solvable, and finds the minimal order of a metabelian group with an unsolvable unimodular equation.

Findings

Groups with specified subnormal series can embed into larger groups with preserved properties.

Any $p$-nonsingular system over such groups is solvable within the group.

The minimal order of a metabelian group with an unsolvable unimodular equation is 42.

Abstract

Any group that has a subnormal series, in which all factors are abelian and all except the last one are $p'$-torsion-free, can be embedded into a group with a subnormal series of the same length, with the same properties and such that any $p$-nonsingular system of equations over this group is solvable in this group itself. This helps us to prove that the minimal order of a metabelian group, over which there is a unimodular equation that is unsolvable in metabelian groups, is 42.