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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Equivalences between fusion systems of finite groups of Lie type
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by Carles Broto, Jesper M. Møller and Bob Oliver PDF
J. Amer. Math. Soc. 25 (2012), 1-20 Request permission

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
  • MR Author ID: 42005
  • Email: broto@mat.uab.es
  • Jesper M. Møller
  • Affiliation: Matematisk Institut, Universitetsparken 5, DK–2100 København, Denmark
  • ORCID: 0000-0003-4053-2418
  • Email: moller@math.ku.dk
  • Bob Oliver
  • Affiliation: LAGA, Institut Galilée, Av. J-B Clément, F–93430 Villetaneuse, France
  • MR Author ID: 191965
  • Email: bobol@math.univ-paris13.fr
  • 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
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2833477