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  • 2024HallPhD

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

Research output: Thesis › Doctoral Thesis

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Structures in the Burnside ring of profinite groups: Ideals, idempotents and F-stable subrings of Burnside rings of profinite groups. / Hall, Zac.
Lancaster University, 2024. 159 p.

Research output: Thesis › Doctoral Thesis

Harvard

Hall, Z 2024, 'Structures in the Burnside ring of profinite groups: Ideals, idempotents and F-stable subrings of Burnside rings of profinite groups', PhD, Lancaster University. https://doi.org/10.17635/lancaster/thesis/2403

APA

Hall, Z. (2024). Structures in the Burnside ring of profinite groups: Ideals, idempotents and F-stable subrings of Burnside rings of profinite groups. [Doctoral Thesis, Lancaster University]. Lancaster University. https://doi.org/10.17635/lancaster/thesis/2403

Vancouver

Hall Z. Structures in the Burnside ring of profinite groups: Ideals, idempotents and F-stable subrings of Burnside rings of profinite groups. Lancaster University, 2024. 159 p. doi: 10.17635/lancaster/thesis/2403

Author

Hall, Zac. / Structures in the Burnside ring of profinite groups : Ideals, idempotents and F-stable subrings of Burnside rings of profinite groups. Lancaster University, 2024. 159 p.

Bibtex

@phdthesis{f17f2b8b4570425d8437643b0726d1eb,
title = "Structures in the Burnside ring of profinite groups: Ideals, idempotents and F-stable subrings of Burnside rings of profinite groups",
abstract = "The purpose of this thesis is to take established results and structuresfor the Burnside ring of finite groups and to create an analogue in the casewhere we take the Burnside ring of profinite groups. Since every finite groupis a profinite group, we create these structures in mind of ensuring that theycoincide on the Burnside ring finite profinite groups. The main differencebeing that in the Burnside ring of profinite groups, we consider almost finiteG-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 apro-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 inthe 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.",
keywords = "Algebra, Topology, Group theory, Fusion systems, Ring theory",
author = "Zac Hall",
year = "2024",
doi = "10.17635/lancaster/thesis/2403",
language = "English",
publisher = "Lancaster University",
school = "Lancaster University",

}

RIS

TY - BOOK

T1 - Structures in the Burnside ring of profinite groups

T2 - Ideals, idempotents and F-stable subrings of Burnside rings of profinite groups

AU - Hall, Zac

PY - 2024

Y1 - 2024

N2 - The purpose of this thesis is to take established results and structuresfor the Burnside ring of finite groups and to create an analogue in the casewhere we take the Burnside ring of profinite groups. Since every finite groupis a profinite group, we create these structures in mind of ensuring that theycoincide on the Burnside ring finite profinite groups. The main differencebeing that in the Burnside ring of profinite groups, we consider almost finiteG-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 apro-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 inthe 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.

AB - The purpose of this thesis is to take established results and structuresfor the Burnside ring of finite groups and to create an analogue in the casewhere we take the Burnside ring of profinite groups. Since every finite groupis a profinite group, we create these structures in mind of ensuring that theycoincide on the Burnside ring finite profinite groups. The main differencebeing that in the Burnside ring of profinite groups, we consider almost finiteG-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 apro-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 inthe 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.

KW - Algebra

KW - Topology

KW - Group theory

KW - Fusion systems

KW - Ring theory

U2 - 10.17635/lancaster/thesis/2403

DO - 10.17635/lancaster/thesis/2403

M3 - Doctoral Thesis

PB - Lancaster University

ER -