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Writing units of integral group rings of finite abelian groups as a product of Bass units


Authors: Eric Jespers, Ángel del Río and Inneke Van Gelder
Journal: Math. Comp. 83 (2014), 461-473
MSC (2010): Primary 16U60, 16S34, 13P99; Secondary 20C05
Published electronically: May 30, 2013
MathSciNet review: 3120600
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Abstract: We give a constructive proof of the theorem of Bass and Milnor saying that if $ G$ is a finite abelian group then the Bass units of the integral group ring $ \mathbb{Z} G$ generate a subgroup of finite index in its unit group $ \mathcal {U}(\mathbb{Z} G)$. Our proof provides algorithms to represent some units that contribute to only one simple component of $ \mathbb{Q} G$ and generate a subgroup of finite index in $ \mathcal {U}(\mathbb{Z} G)$ as product of Bass units. We also obtain a basis $ B$ formed by Bass units of a free abelian subgroup of finite index in $ \mathcal {U}(\mathbb{Z} G)$ and give, for an arbitrary Bass unit $ b$, an algorithm to express $ b^{\varphi (\vert G\vert)}$ as a product of a trivial unit and powers of at most two units in this basis $ B$.


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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

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/S0025-5718-2013-02718-4
Keywords: Integral group rings, units, finite abelian groups
Received by editor(s): November 15, 2011
Received by editor(s) in revised form: May 8, 2012
Published electronically: May 30, 2013
Additional Notes: The first and second authors have been partially supported by the Ministerio de Ciencia y Tecnología of Spain MTM2009-07373, Fundación Séneca of Murcia 04555/GERM/06 and Fonds FEDER
The first author is partially supported by Fonds voor Wetenschappelijk Onderzoek Vlaanderen-Belgium and Onderzoeksraad Vrije Universiteit Brussel.
The third author is supported by Fonds voor Wetenschappelijk Onderzoek Vlaanderen-Belgium
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.