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Computational aspects of Burnside rings, part I: the ring structure

Abstract

The Burnside ring \(\mathcal {B}(G)\) 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.

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Correspondence to Martin Kreuzer.

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Kreuzer, M., Patil, D.P. Computational aspects of Burnside rings, part I: the ring structure. Beitr Algebra Geom 58, 427–452 (2017). https://doi.org/10.1007/s13366-016-0324-4

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  • DOI: https://doi.org/10.1007/s13366-016-0324-4

Keywords

  • Burnside ring
  • Table of marks
  • Prime ideal graph
  • Spectrum
  • Connected component
  • Quasi-idempotent
  • Idempotent

Mathematics Subject Classification

  • Primary 19A22
  • Secondary 13F99
  • 13P99
  • 20C40