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Glauberman's and Thompson's theorems for fusion systems


Authors: Antonio Díaz, Adam Glesser, Nadia Mazza and Sejong Park
Journal: Proc. Amer. Math. Soc. 137 (2009), 495-503
MSC (2000): Primary 20C20
DOI: https://doi.org/10.1090/S0002-9939-08-09690-1
Published electronically: September 17, 2008
MathSciNet review: 2448569
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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

DOI: https://doi.org/10.1090/S0002-9939-08-09690-1
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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