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Subgroup families controlling p-local finite groups

2005; Wiley; Volume: 91; Issue: 02 Linguagem: Inglês

10.1112/s0024611505015327

1460-244X

Carles Broto, Natàlia Castellana, Jesper Grodal, Ran Levi, Bob Oliver,

Algebraic structures and combinatorial models

Proceedings of the London Mathematical SocietyVolume 91, Issue 2 p. 325-354 Articles Subgroup Families Controlling p-Local Finite Groups† Carles Broto, Carles Broto [email protected] Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, SpainSearch for more papers by this authorNatàlia Castellana, Natàlia Castellana [email protected] Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, SpainSearch for more papers by this authorJesper Grodal, Jesper Grodal [email protected] Department of Mathematics, University of Chicago, Chicago, IL, 60637 USASearch for more papers by this authorRan Levi, Ran Levi [email protected] Department of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen, AB24 3UE United KingdomSearch for more papers by this authorBob Oliver, Bob Oliver [email protected] LAGA, Institut Galilée, Avenue J.-B. Clément, 93430 Villetaneuse, FranceSearch for more papers by this author Carles Broto, Carles Broto [email protected] Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, SpainSearch for more papers by this authorNatàlia Castellana, Natàlia Castellana [email protected] Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, SpainSearch for more papers by this authorJesper Grodal, Jesper Grodal [email protected] Department of Mathematics, University of Chicago, Chicago, IL, 60637 USASearch for more papers by this authorRan Levi, Ran Levi [email protected] Department of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen, AB24 3UE United KingdomSearch for more papers by this authorBob Oliver, Bob Oliver [email protected] LAGA, Institut Galilée, Avenue J.-B. Clément, 93430 Villetaneuse, FranceSearch for more papers by this author First published: 23 December 2016 https://doi.org/10.1112/S0024611505015327Citations: 58 † C. Broto and N. Castellana were partially supported by MCYT grant BFM2001–2035. J. Grodal was partially supported by NSF grant DMS-0104318. R. Levi was partially supported by EPSRC grant GR/M7831. B. Oliver was partially supported by UMR 7539 of the CNRS. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract A p-local finite group consists of a finite p-group S, together with a pair of categories which encode 'conjugacy' relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we examine which subgroups control this structure. More precisely, we prove that the question of whether an abstract fusion system F over a finite p-group S is saturated can be determined by just looking at smaller classes of subgroups of S. We also prove that the homotopy type of the classifying space of a given p-local finite group is independent of the family of subgroups used to define it, in the sense that it remains unchanged when that family ranges from the set of F-centric F-radical subgroups (at a minimum) to the set of F-quasicentric subgroups (at a maximum). Finally, we look at constrained fusion systems, analogous to p-constrained finite groups, and prove that they in fact all arise from groups. 2000 Mathematics Subject Classification 20J99 (primary), 55R35, 20D20 (secondary). Citing Literature Volume91, Issue2September 2005Pages 325-354 RelatedInformation