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

HTML articles powered by AMS MathViewer

- 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.## References

- J. L. Alperin,
*Sylow intersections and fusion*, J. Algebra**6**(1967), 222–241. MR**215913**, DOI 10.1016/0021-8693(67)90005-1 - J. Alperin and Michel Broué,
*Local methods in block theory*, Ann. of Math. (2)**110**(1979), no. 1, 143–157. MR**541333**, DOI 10.2307/1971248 - Carles Broto, Natàlia Castellana, Jesper Grodal, Ran Levi, and Bob Oliver,
*Subgroup families controlling $p$-local finite groups*, Proc. London Math. Soc. (3)**91**(2005), no. 2, 325–354. MR**2167090**, DOI 10.1112/S0024611505015327 - Carles Broto, Ran Levi, and Bob Oliver,
*The homotopy theory of fusion systems*, J. Amer. Math. Soc.**16**(2003), no. 4, 779–856. MR**1992826**, DOI 10.1090/S0894-0347-03-00434-X - George Glauberman,
*Weakly closed elements of Sylow subgroups*, Math. Z.**107**(1968), 1–20. MR**251141**, DOI 10.1007/BF01111043 - G. Glauberman,
*Global and local properties of finite groups*, Finite simple groups (Proc. Instructional Conf., Oxford, 1969) Academic Press, London, 1971, pp. 1–64. MR**0352241** - Radha Kessar and Markus Linckelmann,
*$ZJ$-theorems for fusion systems*, Trans. Amer. Math. Soc.**360**(2008), no. 6, 3093–3106. MR**2379788**, DOI 10.1090/S0002-9947-08-04275-X - Markus Linckelmann,
*Introduction to fusion systems*, Group representation theory, EPFL Press, Lausanne, 2007, pp. 79–113. MR**2336638** - Markus Linckelmann,
*Simple fusion systems and the Solomon 2-local groups*, J. Algebra**296**(2006), no. 2, 385–401. MR**2201048**, DOI 10.1016/j.jalgebra.2005.09.024 - Lluis Puig,
*Frobenius categories*, J. Algebra**303**(2006), no. 1, 309–357. MR**2253665**, DOI 10.1016/j.jalgebra.2006.01.023 - Radu Stancu,
*Control of fusion in fusion systems*, J. Algebra Appl.**5**(2006), no. 6, 817–837. MR**2286725**, DOI 10.1142/S0219498806002034 - John G. Thompson,
*Normal $p$-complements for finite groups*, J. Algebra**1**(1964), 43–46. MR**167521**, DOI 10.1016/0021-8693(64)90006-7

## 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**- 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