Algorithmic Aspects of Units in Group Rings

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.

A counterexample to the first Zassenhaus conjecture

“…This variation asks if it is true that any p-subgroup of U (ZG) consisting of elements of augmentation one is conjugate in U (QG) to a subgroup of G. This is sometimes called "(p-ZC3)" or the "Strong Sylow Theorem" for ZG. An overview of results relating to this problem can be found in [BKM16]. For the counterexample to the Zassenhaus conjecture presented in the present article it is of fundamental importance that the order of the unit is divisible by at least two different primes.…”

“…In a different vein, A. Weiss' proof of the conjecture, or even a stronger version of it, for nilpotent groups [Wei88,Wei91], was certainly a highlight of the study. The conjecture is also known to hold for a few other classes of solvable groups [Fer87,DJ96,BKM16,MR17b,MR17c,MR17a], as well as for some small groups. In particular, the conjecture holds for groups of order smaller than 144 [HK06, HS15, BHK + 17].…”