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  • Cited by 2
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    This chapter has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Bouc, Serge 2012. The slice Burnside ring and the section Burnside ring of a finite group. Compositio Mathematica, Vol. 148, Issue. 03, p. 868.

    Barker, Laurence 2010. Tornehave Morphisms I: Resurrecting the Virtual Permutation Sets Annihilated by Linearization. Communications in Algebra, Vol. 39, Issue. 1, p. 355.

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  • Print publication year: 2007
  • Online publication date: May 2010

An algorithm for the unit group of the Burnside ring of a finite group

Summary

Abstract

In this note we present an algorithm for the construction of the unit group of the Burnside ring Ω(G) of a finite group G from a list of representatives of the conjugacy classes of subgroups of G.

Introduction

Let G be a finite group. The Burnside ring Ω(G) of G is the Grothendieck ring of the isomorphism classes [X] of the finite left G-sets X with respect to disjoint union and direct product. It has a ℤ-basis consisting of the isomorphism classes of the transitive G-sets G/H, where H runs through a system of representatives of the conjugacy classes of subgroups of G.

The ghost ring of G is the set of functions f from the set of subgroups of G into ℤ which are constant on conjugacy classes of subgroups of G. For any finite G-set X, the function φX which maps a subgroup H of G to the number of its fixed points on X, i.e., φX(H) = #{x ∈ X : h.x = x for all h ∈ H}, belongs to (G). By a theorem of Burnside, the map φ : [X] → φX is an injective homomorphism of rings from Ω (G) to (G). We identify Ω (G) with its image under φ in (G), i.e., for x ∈ Ω(G), we write x(H) = φ (x)(H) = φH(x).

The ghost ring has a natural basis consisting of the characteristic functions of the conjugacy classes of subgroups of G.

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Groups St Andrews 2005
  • Online ISBN: 9780511721212
  • Book DOI: https://doi.org/10.1017/CBO9780511721212
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