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The slice Burnside ring and the section Burnside ring of a finite group

Part of: Grothendieck groups and $K_0$ Categories and geometry

Published online by Cambridge University Press:  22 March 2012

Serge Bouc
Serge Bouc*
Affiliation:
CNRS-LAMFA, Université de Picardie, 33 rue St Leu, 80039, Amiens cedex 1, France (email: serge.bouc@u-picardie.fr)
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Abstract

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This paper introduces two new Burnside rings for a finite group G, called the slice Burnside ring and the section Burnside ring. They are built as Grothendieck rings of the category of morphisms of G-sets and of Galois morphisms of G-sets, respectively. The well-known results on the usual Burnside ring, concerning ghost maps, primitive idempotents, and description of the prime spectrum, are extended to these rings. It is also shown that these two rings have a natural Green biset functor structure. The functorial structure of unit groups of these rings is also discussed.

Keywords

Burnside ringGreen biset functorslicesectionunit group

MSC classification

Secondary: 19A22: Frobenius induction, Burnside and representation rings 18F30: Grothendieck groups

Information

Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 3 , May 2012 , pp. 868 - 906
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[BP07]Boltje, R. and Pfeiffer, G., An algorithm for the unit group of the Burnside ring of a finite group, in Groups St Andrews 2005, London Mathematical Society Lecture Notes Series, vol. 339 (Cambridge University Press, Cambridge, 2007), 230–236.CrossRefGoogle Scholar
[Bou96]Bouc, S., Construction de foncteurs entre catégories de G-ensembles, J. Algebra 183 (1996), 737–825.CrossRefGoogle Scholar
[Bou07]Bouc, S., The functor of units of Burnside rings for p-groups, Comment. Math. Helv. 82 (2007), 583–615.CrossRefGoogle Scholar
[Bou10]Bouc, S., Biset functors for finite groups, Lecture Notes in Mathematics, vol. 1990 (Springer, New York, 2010).CrossRefGoogle Scholar
[Cos09]Coşkun, O., Ring of subquotient of a finite group I: Linearization, J. Algebra 322 (2009), 2773–2792.CrossRefGoogle Scholar
[Dre69]Dress, A., A characterization of solvable groups, Math. Z. 110 (1969), 213–217.CrossRefGoogle Scholar
[GAP08] The GAP Group. GAP – groups, algorithms, and programming, version 4.4.12 (2008) http://www.gap-system.org.Google Scholar
[Mat82]Matsuda, T., On the unit group of Burnside rings, Japan. J. Math. 8 (1982), 71–93.CrossRefGoogle Scholar
[tomD79]tom Dieck, T., Transformation groups and representation theory, Lecture Notes in Mathematics, vol. 766 (Springer, New York, 1979).CrossRefGoogle Scholar
[Yos90]Yoshida, T., On the unit groups of Burnside rings, J. Math. Soc. Japan 42 (1990), 31–64.CrossRefGoogle Scholar