An application of Burnside rings in elementary finite group theory

Abstract

A canonical map from the Burnside ring Ω(C) of a finite cyclic group C into the Burnside ring Ω(G) of any finite group G of the same order is exhibited and it is shown that many results from elementary finite group theory, in particular those claiming certain congruence relations, are simple consequences of the existence of this map. In addition, it is shown that this map defines an isomorphism from Ω(C) onto the subring Ω 0(G) of Ω(G), consisting of those virtual G-sets x which have the same number of invariants for every two subgroups U, V of G of the same order, if and only if G is nilpotent. Finally, a rather natural extension to profinite groups is indicated.