Natural endomorphisms of Burnside rings
Abstract: The Burnside ring of a finite group G consists of formal differences of finite G-sets. is a contravariant functor from finite groups to commutative rings. We study the natural endomorphisms of this functor, of its extension to rational scalars, and of its restriction to abelian groups. Such endomorphisms are canonically associated to certain operators that assign to each group one of its conjugacy classes of subgroups. Using these operators along with a carefully constructed system of linear congruences defining the image of under its canonical embedding in a power of Z, we exhibit a multitude of natural endomorphisms of , we show that only two of them map G-sets to G-sets, and we completely describe all natural endomorphisms of .
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Keywords: Permutation group, Burnside ring, marks, natural transformation
Article copyright: © Copyright 1979 American Mathematical Society