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 of finite groups of Lie type, that the -fusion systems and are equivalent. In other words, there is an isomorphism between a Sylow -subgroup of and one of which preserves -fusion. This occurs, for example, when and for a simple Lie ``type'' , and and are prime powers, both prime to , which generate the same closed subgroup of -adic units. Our proof uses homotopy-theoretic properties of the -completed classifying spaces of and , 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.