Natural endomorphisms of Burnside rings

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 $\mathcal {B} \upharpoonright {\text {Ab}}$.