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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Central units of integral group rings


Authors: Eric Jespers, Gabriela Olteanu, Ángel del Río and Inneke Van Gelder
Journal: Proc. Amer. Math. Soc. 142 (2014), 2193-2209
MSC (2010): Primary 16S34, 16U60, 16U70
Published electronically: March 27, 2014
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Abstract: We give an explicit description for a basis of a subgroup of finite index in the group of central units of the integral group ring $ \mathbb{Z} G$ of a finite abelian-by-supersolvable group such that every cyclic subgroup of order not a divisor of 4 or 6 is subnormal in $ G$. The basis elements turn out to be a natural product of conjugates of Bass units. This extends and generalizes a result of Jespers, Parmenter and Sehgal showing that the Bass units generate a subgroup of finite index in the center $ \mathcal {Z} (\mathcal {U} (\mathbb{Z} G))$ of the unit group $ \mathcal {U} (\mathbb{Z} G)$ in case $ G$ is a finite nilpotent group. Next, we give a new construction of units that generate a subgroup of finite index in $ \mathcal {Z}(\mathcal {U}(\mathbb{Z} G))$ for all finite strongly monomial groups $ G$. We call these units generalized Bass units. Finally, we show that the commutator group $ \mathcal {U}(\mathbb{Z} G)/\mathcal {U}(\mathbb{Z} G)'$ and $ \mathcal {Z}(\mathcal {U}(\mathbb{Z} G))$ have the same rank if $ G$ is a finite group such that $ \mathbb{Q} G$ has no epimorphic image which is either a non-commutative division algebra other than a totally definite quaternion algebra or a two-by-two matrix algebra over a division algebra with center either the rationals or a quadratic imaginary extension of $ \mathbb{Q}$. This allows us to prove that in this case the natural images of the Bass units of $ \mathbb{Z} G$ generate a subgroup of finite index in $ \mathcal {U}(\mathbb{Z} G)/\mathcal {U}(\mathbb{Z} G)'$.


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Additional Information

Eric Jespers
Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
Email: efjesper@vub.ac.be

Gabriela Olteanu
Affiliation: Department of Statistics-Forecasts-Mathematics, Babeş-Bolyai University, Strada T. Mihali 58-60, 400591 Cluj-Napoca, Romania
Email: gabriela.olteanu@econ.ubbcluj.ro

Ángel del Río
Affiliation: Departamento de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain
Email: adelrio@um.es

Inneke Van Gelder
Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
Email: ivgelder@vub.ac.be

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-11958-7
PII: S 0002-9939(2014)11958-7
Keywords: Group ring, central unit, generators
Received by editor(s): March 1, 2012
Received by editor(s) in revised form: July 6, 2012
Published electronically: March 27, 2014
Additional Notes: This research was partially supported by Ministerio de Ciencia y Tecnología of Spain and Fundación Séneca of Murcia, the Research Foundation Flanders (FWO - Vlaanderen), Onderzoeksraad Vrije Universiteit Brussel and by the grant PN-II-RU-TE-2009-1 project ID_303 financed by the Romanian Ministry of National Education, CNCS-VEFISCDI
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.