A note on torsion length and torsion subgroups

TL;DR

This paper constructs polycyclic groups with any torsion length and provides examples where quotients by torsion subgroups are not finitely presented, addressing specific open questions in group theory.

Contribution

It introduces new constructions of polycyclic groups with arbitrary torsion lengths and examples of finitely presented groups with non-finitely presented torsion quotient groups.

Findings

Polycyclic groups with arbitrary torsion lengths constructed.

Examples of finitely presented groups with non-finitely presented torsion quotients.

Addresses open questions from the Kourovka notebook.

Abstract

Answering Questions 19.23 and 19.24 from the Kourovka notebook we construct polycyclic groups with arbitrary torsion lengths and give examples of finitely presented groups whose quotients by their torsion subgroups are not finitely presented.