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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Glauberman’s and Thompson’s theorems for fusion systems
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by Antonio Díaz, Adam Glesser, Nadia Mazza and Sejong Park PDF
Proc. Amer. Math. Soc. 137 (2009), 495-503 Request permission

Abstract:

We prove analogues of results of Glauberman and Thompson for fusion systems. Namely, given a (saturated) fusion system $\mathcal {F}$ on a finite $p$-group $S$, and in the cases where $p$ is odd or $\mathcal {F}$ is $S_4$-free, we show that $\mathrm {Z}(\mathrm {N}_{\mathcal {F}}(\mathrm {J}(S))) =\mathrm {Z}(\mathcal {F})$ (Glauberman) and that if $\mathrm {C}_{\mathcal {F}} (\mathrm {Z}(S))=\mathrm {N}_{\mathcal {F}}(\mathrm {J}(S))=\mathcal {F}_S(S)$, then $\mathcal {F}=\mathcal {F}_S(S)$ (Thompson). As a corollary, we obtain a stronger form of Frobenius’ theorem for fusion systems, applicable under the above assumptions and generalizing another result of Thompson.
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Additional Information
  • Antonio Díaz
  • Affiliation: Department of Mathematical Sciences, King’s College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
  • Address at time of publication: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
  • Email: adiaz@math.ku.dk
  • Adam Glesser
  • Affiliation: Department of Mathematical Sciences, King’s College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
  • Address at time of publication: Mathematics and Computer Science Department, Suffolk University, Fenton Building, Room 621, 32 Derne Street, Boston, Massachusetts 02114
  • Email: aglesser@suffolk.edu
  • Nadia Mazza
  • Affiliation: Department of Mathematical Sciences, King’s College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
  • Address at time of publication: Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4FY, United Kingdom
  • Sejong Park
  • Affiliation: Department of Mathematical Sciences, King’s College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
  • Email: s.park@maths.abdn.ac.uk
  • Received by editor(s): February 7, 2008
  • Published electronically: September 17, 2008
  • Additional Notes: The first author was supported by EPSRC grant EP/D506484/1 and partially supported by MEC grant MTM2007-60016.
    The third author’s research was supported by Swiss National Research Fellowship PA002-113164/1.
  • Communicated by: Jonathan I. Hall
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 495-503
  • MSC (2000): Primary 20C20
  • DOI: https://doi.org/10.1090/S0002-9939-08-09690-1
  • MathSciNet review: 2448569