Structures in the Burnside ring of profinite groups : Ideals, idempotents and F-stable subrings of Burnside rings of profinite groups

Hall, Zac and Mazza, Nadia (2024) Structures in the Burnside ring of profinite groups : Ideals, idempotents and F-stable subrings of Burnside rings of profinite groups. PhD thesis, Lancaster University.

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Abstract

The purpose of this thesis is to take established results and structures for the Burnside ring of finite groups and to create an analogue in the case where we take the Burnside ring of profinite groups. Since every finite group is a profinite group, we create these structures in mind of ensuring that they coincide on the Burnside ring finite profinite groups. The main difference being that in the Burnside ring of profinite groups, we consider almost finite G-spaces, and so we can have infinite series within the Burnside ring representing infinite G-spaces. We begin with taking a pro-fusion system over a pro-p group S and considering the F-stable S-spaces as a subring of Bb(S). We show Bb(F) ∼= lim←−i(B(Fi)) ∼= Bb(lim←−iFi ) and use this to construct a basis for the subring. For prime ideals, we show that there exists an equivalent to the prime ideals in the finite case and that we have prime ideals arising in the infinite case that differ in construction from those in the finite. Finally, we derive expressions for idempotents, showing that they are either finite, and therefore an inflation of an idempotent in B(G/N), or they are infinite.