Abelianization of the unit group of an integral group ring

Andreas Bächle, Sugandha Maheshwary and Leo Margolis

Vol. 312 (2021), No. 2, 309–334

DOI: 10.2140/pjm.2021.312.309

Abstract

For a finite group $G$ and $U:=U\left(\mathbb{Z}G\right)$, the group of units of the integral group ring of $G$, we study the implications of the structure of $G$ on the abelianization $U∕U^{\prime }$ of $U$. We pose questions on the connections between the exponent of $G∕G^{\prime }$ and the exponent of $U∕U^{\prime }$ as well as between the ranks of the torsion-free parts of $Z\left(U\right)$, the center of $U$, and $U∕U^{\prime }$. We show that the units originating from known generic constructions of units in $\mathbb{Z}G$ are well-behaved under the projection from $U$ to $U∕U^{\prime }$ and that our questions have a positive answer for many examples. We then exhibit an explicit example which shows that the general statement on the torsion-free part does not hold, which also answers questions from (Bächle et al. 2018b).

Keywords

integral group rings, unit group, abelianization, torsion-free rank

Mathematical Subject Classification

Primary: 16U60, 20C05

Secondary: 20F14

Milestones

Received: 7 April 2020

Revised: 8 March 2021

Accepted: 16 March 2021

Published: 31 August 2021