## Glauberman’s and Thompson’s theorems for fusion systems

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- by Antonio Díaz, Adam Glesser, Nadia Mazza and Sejong Park
- Proc. Amer. Math. Soc.
**137**(2009), 495-503 - DOI: https://doi.org/10.1090/S0002-9939-08-09690-1
- Published electronically: September 17, 2008
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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.## References

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## Bibliographic 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**- 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