Algorithmic aspects of units in group rings

TL;DR

This paper explores the structure of torsion units in integral group rings, providing algorithmic approaches and proving significant conjectures for specific classes of groups, advancing understanding in algebraic group theory.

Contribution

It proves the Zassenhaus Conjecture for Amitsur groups and characterizes torsion subgroups in units of integral group rings for Frobenius complements.

Findings

Proved the Zassenhaus Conjecture for Amitsur groups.

Characterized torsion subgroups in units of integral group rings of Frobenius complements.

Analyzed orders of torsion units in almost quasisimple groups.

Abstract

We describe the main questions connected to torsion subgroups in the unit group of integral group rings of finite groups and algorithmic methods to attack these questions. We then prove the Zassenhaus Conjecture for Amitsur groups and prove that any normalized torsion subgroup in the unit group of an integral group of a Frobenius complement is isomorphic to a subgroup of the group base. Moreover we study the orders of torsion units in integral group rings of finite almost quasisimple groups and the existence of torsion-free normal subgroups of finite index in the unit group.