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Equivalences between fusion systems of finite groups of Lie type


Authors: Carles Broto, Jesper M. Møller and Bob Oliver
Journal: J. Amer. Math. Soc. 25 (2012), 1-20
MSC (2010): Primary 20D06; Secondary 55R37, 20D20
DOI: https://doi.org/10.1090/S0894-0347-2011-00713-3
Published electronically: July 8, 2011
MathSciNet review: 2833477
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Abstract: We prove, for certain pairs $ G,G'$ of finite groups of Lie type, that the $ p$-fusion systems $ \mathcal{F}_p(G)$ and $ \mathcal{F}_p(G')$ are equivalent. In other words, there is an isomorphism between a Sylow $ p$-subgroup of $ G$ and one of $ G'$ which preserves $ p$-fusion. This occurs, for example, when $ G=\mathbb{G}(q)$ and $ G'=\mathbb{G}(q')$ for a simple Lie ``type'' $ \mathbb{G}$, and $ q$ and $ q'$ are prime powers, both prime to $ p$, which generate the same closed subgroup of $ p$-adic units. Our proof uses homotopy-theoretic properties of the $ p$-completed classifying spaces of $ G$ and $ G'$, and we know of no purely algebraic proof of this result.


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Additional Information

Carles Broto
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain
Email: broto@mat.uab.es

Jesper M. Møller
Affiliation: Matematisk Institut, Universitetsparken 5, DK–2100 København, Denmark
Email: moller@math.ku.dk

Bob Oliver
Affiliation: LAGA, Institut Galilée, Av. J-B Clément, F–93430 Villetaneuse, France
Email: bobol@math.univ-paris13.fr

DOI: https://doi.org/10.1090/S0894-0347-2011-00713-3
Keywords: groups of Lie type, fusion systems, classifying spaces, $p$-completion
Received by editor(s): March 17, 2010
Received by editor(s) in revised form: June 14, 2011
Published electronically: July 8, 2011
Additional Notes: The first author is partially supported by FEDER-MICINN grant MTM 2010-20692
The second author was partially supported by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation
The third author was partially supported by UMR 7539 of the CNRS, and by project ANR BLAN08-2_338236, HGRT
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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