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Computational aspects of Burnside rings. I: The ring structure. (English) Zbl 1390.19001

Authors’ abstract: The Burnside ring \(B(G)\) of a finite group \(G\), a classical tool in group theory and representation theory, is studied from the point of view of computational commutative algebra. Starting from a table of marks, we describe efficient algorithms for computing a presentation, the image of the mark homomorphism, the prime ideals and the prime ideal graph, the singular locus, the conductor in its integral closure, the connected components of its spectrum, and its idempotents. On the way, we provide methods for identifying p-residual subgroups, direct products of subgroups of coprime order, commutator subgroups, and perfect subgroups.

MSC:

19A22 Frobenius induction, Burnside and representation rings
13F99 Arithmetic rings and other special commutative rings
13P99 Computational aspects and applications of commutative rings
20C40 Computational methods (representations of groups) (MSC2010)

Software:

GAP; ApCoCoA
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Full Text: DOI

References:

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