Vol. 312, No. 2, 2021

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 314: 1  2
Vol. 313: 1  2
Vol. 312: 1  2
Vol. 311: 1  2
Vol. 310: 1  2
Vol. 309: 1  2
Vol. 308: 1  2
Vol. 307: 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(G), the group of units of the integral group ring of G, we study the implications of the structure of G on the abelianization UU of U. We pose questions on the connections between the exponent of GG and the exponent of UU as well as between the ranks of the torsion-free parts of Z(U), the center of U, and UU. We show that the units originating from known generic constructions of units in G are well-behaved under the projection from U to UU 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
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