On the basis of the Burnside ring of a fusion system

Gelvin, Matthew; Reeh, Sune Precht; Yalcin, Ergun

doi:10.1016/j.jalgebra.2014.10.026

Cited by 4 publications

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References 13 publications

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“…So far, Theorem 1 provides the only known way to circumvent the absence of a biset in the compact case. As shown in work by several authors (see for example Díaz and Libman [13; 14], Ragnarsson and Stancu [32], Gelvin and Reeh [21], and Gelvin, Reeh and Yalçın [22]) this biset is closely related to the Burnside ring of the corresponding p-local finite group. In this sense, it would be interesting to analyze the Burnside ring of a p-local compact group G D .S; F; L/, and compare it with the Burnside rings of a given approximation of G by p-local finite groups.…”

Section: D20 55r35 55r40mentioning

confidence: 71%

Finite approximations of p–local compact groups

Gonzalez

2016

Geom. Topol.

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Section: D20 55r35 55r40mentioning

confidence: 71%

Finite approximations of p–local compact groups

Gonzalez

2016

Geom. Topol.

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“…So far, Theorem 1 provides the only known way to circumvent the absence of a biset in the compact case. As shown in work by several authors, see for example [13], [14], [32], [21] and [22], this biset is closely related to the Burnside ring of the corresponding p-local finite group. In this sense, it would be interesting to analyze the Burnside ring of a p-local compact group G = (S, F, L), and compare it with the Burnside rings of a given approximation of G by p-local finite groups.…”

mentioning

confidence: 71%

“…22 Let G = (S, F, L) be a p-local compact group, and let A ≤ S.(a) If A is fully F -centralized, the centralizer p-local compact group of A in G is the triple…”

mentioning

confidence: 99%

Finite approximations of $p$-local compact groups

Gonzalez

2015

Preprint

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In this paper we show how every p-local compact group can be described as a telescope of p-local finite groups. As a consequence, we deduce several corollaries, such as a Stable Elements Theorem for the mod p cohomology of their classifying spaces, and a generalized Dwyer-Zabrodsky description of certain related mapping spaces. 55R35; 55R40, 20D20For a fixed prime p, the concept of p-local compact group was introduced in the last decade by C. Broto, R. Levi and B. Oliver in [7]. It provides a unifying, categorical language to study several classes of groups from a p-local point of view, including finite groups, compact Lie groups, p-compact groups and algebraic groups over algebraic closures of fields of characteristic other than p.In view of the many classes of groups and other mathematical objects that p-local compact groups model, it is no surprise that this theory is not very well understood yet. Nevertheless several structural results have already been conjectured since its introduction in [7], of which two are of special relevance in this work. The first conjecture asks about the existence of a version of the Stable Elements Theorem by H. Cartan and S. Eilenberg [10, Theorem 10.1] for p-local compact groups, while the second conjecture suggests the extension of the classic result of W. Dwyer and A. Zabrodsky [17, Theorem 1.1] on mapping spaces to p-local compact groups.A reader familiar with the theory of p-local finite groups will know that the proofs of these results in the finite case, as well as many others, depend ultimately on a certain biset, which is at the core this theory. And it is there where the main obstacle to study p-local compact groups resides, since there is no such biset in the context of p-local compact groups.An alternative to the biset is necessary, and evidence that such alternative may exist is found in a result by E. M. Friedlander and G. Mislin,[19,20], that relates compact Lie groups to towers of finite groups. In this paper we prove a version of the result of Friedlander and Mislin for p-local compact groups.Many statements for p-local finite groups extend to p-local compact groups as a consequence of this approximation result that replaces the use of the biset by arguments on Mapping spaces 54Bibliography 69Acknowledgements. The author is specially grateful to C. Broto, R. Levi and B. Oliver for numerous inspirational conversations and their support. The author would like to thank also N. Castellana, A. Libman and R. Stancu for their help in various stages of this work, and A. Chermak and R. Molinier for their support during the final stages of this work. Finally, the author is truly indebted to the referee, without whose assistance and thorough job the reader's experience with this paper would be greatly diminished. Background on p-local compact groupsLet p a prime, to remain fixed for the rest of the paper unless otherwise stated. In this section we review all the definitions and results about p-local compact groups that we Finally, we say that P is fully F -centralized, resp...

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“…Let p be a prime number and fix an S ∈ Syl p (G). Let F S (G) denote the Frobenius fusion system on S induced by G. Following [6], we define the Burnside ring of F S (G) to be the subring B(F S (G)) of B(S) consisting of the elements a ∈ B(S) such that |a P | = |a Q | whenever P, Q are subgroups of S that are conjugate in G. More generally, for any saturated fusion system F on S, we define the Burnside ring of F to be the subring B(F) of B(S) consisting of the elements a ∈ B(S) such that |a P | = |a Q | whenever P and Q are isomorphic in F. The first main result about B(F) is due to Sune Reeh. This theorem also gives a canonical basis of B(F) indexed by the F-conjugacy classes of S, but we use this result only in the next example, so we omit a general explanation.…”

Section: Burnside Ring Of a Fusion Systemmentioning

confidence: 99%

Burnside rings of fusion systems and their unit groups

Barsotti

Carman

2020

Journal of Group Theory

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AbstractFor a saturated fusion system {\mathcal{F}} on a p-group S, we study the Burnside ring of the fusion system {B(\mathcal{F})}, as defined by Matthew Gelvin and Sune Reeh, which is a subring of the Burnside ring {B(S)}. We give criteria for an element of {B(S)} to be in {B(\mathcal{F})} determined by the {\mathcal{F}}-automorphism groups of essential subgroups of S. When {\mathcal{F}} is the fusion system induced by a finite group G with S as a Sylow p-group, we show that the restriction of {B(G)} to {B(S)} has image equal to {B(\mathcal{F})}. We also show that, for {p=2}, we can gain information about the fusion system by studying the unit group {B(\mathcal{F})^{\times}}. When S is abelian, we completely determine this unit group.

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On the basis of the Burnside ring of a fusion system

Cited by 4 publications

References 13 publications

Finite approximations of p–local compact groups

Finite approximations of p–local compact groups

Finite approximations of $p$-local compact groups

Burnside rings of fusion systems and their unit groups

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