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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Natural endomorphisms of Burnside rings

Author: Andreas Blass
Journal: Trans. Amer. Math. Soc. 253 (1979), 121-137
MSC: Primary 20B05
MathSciNet review: 536938
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Abstract: The Burnside ring $ \mathcal{B}(G)$ of a finite group G consists of formal differences of finite G-sets. $ \mathcal{B}$ is a contravariant functor from finite groups to commutative rings. We study the natural endomorphisms of this functor, of its extension $ \textbf{Q} \otimes \mathcal{B}$ to rational scalars, and of its restriction $ \mathcal{B} \upharpoonright {\text{Ab}}$ 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 $ \mathcal{B}(G)$ under its canonical embedding in a power of Z, we exhibit a multitude of natural endomorphisms of $ \mathcal{B}$, we show that only two of them map G-sets to G-sets, and we completely describe all natural endomorphisms of $ \mathcal{B} \upharpoonright {\text{Ab}}$.

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Keywords: Permutation group, Burnside ring, marks, natural transformation
Article copyright: © Copyright 1979 American Mathematical Society