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.