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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Central units of integral group rings
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by Eric Jespers, Gabriela Olteanu, Ángel del Río and Inneke Van Gelder PDF
Proc. Amer. Math. Soc. 142 (2014), 2193-2209 Request permission

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
  • MR Author ID: 94560
  • 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
  • MR Author ID: 288713
  • Email: adelrio@um.es
  • Inneke Van Gelder
  • Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
  • Email: ivgelder@vub.ac.be
  • 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
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 142 (2014), 2193-2209
  • MSC (2010): Primary 16S34, 16U60, 16U70
  • DOI: https://doi.org/10.1090/S0002-9939-2014-11958-7
  • MathSciNet review: 3195747