#### Vol. 312, No. 2, 2021

 Recent Issues Vol. 317: 1 Vol. 316: 1  2 Vol. 315: 1  2 Vol. 314: 1  2 Vol. 313: 1  2 Vol. 312: 1  2 Vol. 311: 1  2 Vol. 310: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Contacts Submission Guidelines Submission Form Policies for Authors ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Special Issues Author Index To Appear Other MSP Journals
Abelianization of the unit group of an integral group ring

### Andreas Bächle, Sugandha Maheshwary and Leo Margolis

Vol. 312 (2021), No. 2, 309–334
##### Abstract

For a finite group $G$ and $U:=U\left(ℤG\right)$, 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 $ℤG$ 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
Revised: 8 March 2021
Accepted: 16 March 2021
Published: 31 August 2021
##### Authors
 Andreas Bächle Vrije Universiteit Brussel Brussels Belgium Sugandha Maheshwary Indian Institute of Science Education and Research Mohali India Leo Margolis Vrije Universiteit Brussel Brussels Belgium