Vol. 312, No. 2, 2021

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Abelianization of the unit group of an integral group ring

Andreas Bächle, Sugandha Maheshwary and Leo Margolis

Vol. 312 (2021), No. 2, 309–334

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).

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integral group rings, unit group, abelianization, torsion-free rank
Mathematical Subject Classification
Primary: 16U60, 20C05
Secondary: 20F14
Received: 7 April 2020
Revised: 8 March 2021
Accepted: 16 March 2021
Published: 31 August 2021
Andreas Bächle
Vrije Universiteit Brussel
Sugandha Maheshwary
Indian Institute of Science Education and Research
Leo Margolis
Vrije Universiteit Brussel