Writing units of integral group rings of finite abelian groups as a product of Bass units
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- by Eric Jespers, Ángel del Río and Inneke Van Gelder PDF
- Math. Comp. 83 (2014), 461-473 Request permission
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 $b^{\varphi (|G|)}$ as a product of a trivial unit and powers of at most two units in this basis $B$.References
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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
- 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): 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 - © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 83 (2014), 461-473
- MSC (2010): Primary 16U60, 16S34, 13P99; Secondary 20C05
- DOI: https://doi.org/10.1090/S0025-5718-2013-02718-4
- MathSciNet review: 3120600