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Mathematics of Computation

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Computing generators of free modules over orders in group algebras II


Authors: Werner Bley and Henri Johnston
Journal: Math. Comp. 80 (2011), 2411-2434
MSC (2010): Primary 11R33, 11Y40, 16Z05
DOI: https://doi.org/10.1090/S0025-5718-2011-02488-9
Published electronically: April 11, 2011
MathSciNet review: 2813368
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Abstract: Let $ E$ be a number field and $ G$ a finite group. Let $ \mathcal{A}$ be any $ \mathcal{O}_{E}$-order of full rank in the group algebra $ E[G]$ and $ X$ a (left) $ \mathcal{A}$-lattice. In a previous article, we gave a necessary and sufficient condition for $ X$ to be free of given rank $ d$ over $ \mathcal{A}$. In the case that (i) the Wedderburn decomposition $ E[G] \cong \bigoplus_{\chi} M_{\chi}$ is explicitly computable and (ii) each $ M_{\chi}$ is in fact a matrix ring over a field, this led to an algorithm that either gives elements $ \alpha_{1}, \ldots, \alpha_{d} \in X$ such that $ X=\mathcal{A}\alpha_{1} \oplus \cdots \oplus \mathcal{A}\alpha_{d}$ or determines that no such elements exist. In the present article, we generalise the algorithm by weakening condition (ii) considerably.


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Werner Bley
Affiliation: Fachbereich für Mathematik und Naturwissenschaften der Universität Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germany
Email: bley@mathematik.uni-kassel.de

Henri Johnston
Affiliation: St. John’s College, Cambridge CB2 1TP, United Kingdom
Email: H.Johnston@dpmms.cam.ac.uk

DOI: https://doi.org/10.1090/S0025-5718-2011-02488-9
Received by editor(s): June 23, 2010
Received by editor(s) in revised form: September 15, 2010
Published electronically: April 11, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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