Realizability of fusion systems by discrete groups: II

Carles Broto; Ran Levi; Bob Oliver

doi:10.48550/arXiv.2505.16375

Realizability of fusion systems by discrete groups: II

Carles Broto
, Ran Levi
, Bob Oliver

Mathematical Science

Université Paris 13
Autonomous University of Barcelona
Mathematics Research Center

Research output: Working paper › Preprint

Abstract

We compare four different types of realizability for saturated fusion systems over discrete p-toral groups. For example, when G is a locally finite group all of whose p-subgroups are artinian (hence discrete p-toral), we show that it has ``weakly Sylow'' p-subgroups and give explicit constructions of saturated fusion systems and associated linking systems associated to G. We also show that a fusion system over a discrete p-toral group S is saturated if its set of morphisms is closed under a certain topology and the finite subgroups of S satisfy the saturation axioms, and prove a version of the Cartan-Eilenberg stable elements theorem for locally finite groups.

Original language	English
Publisher	ArXiv
Number of pages	38
DOIs	https://doi.org/10.48550/arXiv.2505.16375
Publication status	Published - 22 May 2025

Version History

[v1] Thu, 22 May 2025 08:29:41 UTC (54 KB)
[v2] Fri, 23 May 2025 17:49:18 UTC (54 KB)

Funding

C. Broto is partially supported by the Spanish State Research Agency through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M), by AEI grant PID2020-116481GB-100, and by AGAUR grant 2021-SGR-01015. B. Oliver is partially supported by UMR 7539 of the CNRS. All three authors would like to thank the Isaac Newton Institute for Mathematical Sciences and the Gaelic College on Isle of Skye for their support and hospitality during the programme “Topology, representation theory, and higher structures”, supported by EPSRC grant no. EP/R014604/1.

Funders	Funder number
Agencia Estatal de Investigación	CEX2020-001084-M, PID2020-116481GB-100
Agencia de Gestión de Ayudas Universitarias y de Investigación	2021-SGR-01015
Centre national de la recherche scientifique	UMR 7539
Engineering and Physical Sciences Research Council	EP/R014604/1

Keywords

locally finite groups
linear torsion groups
fusion
group cohomology
classifying spaces
p-compact groups

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10.48550/arXiv.2505.16375Licence: Other

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AB - We compare four different types of realizability for saturated fusion systems over discrete p-toral groups. For example, when G is a locally finite group all of whose p-subgroups are artinian (hence discrete p-toral), we show that it has ``weakly Sylow'' p-subgroups and give explicit constructions of saturated fusion systems and associated linking systems associated to G. We also show that a fusion system over a discrete p-toral group S is saturated if its set of morphisms is closed under a certain topology and the finite subgroups of S satisfy the saturation axioms, and prove a version of the Cartan-Eilenberg stable elements theorem for locally finite groups.

KW - locally finite groups

KW - linear torsion groups

KW - fusion

KW - group cohomology

KW - classifying spaces

KW - p-compact groups

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ER -

Realizability of fusion systems by discrete groups: II

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