Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

Saturated fusion systems over $ 2$-groups

Author(s): Bob Oliver; Joana Ventura
Journal: Trans. Amer. Math. Soc. 361 (2009), 6661-6728.
MSC (2000): Primary 20D20; Secondary 20D45, 20D08
Posted: July 21, 2009
MathSciNet review: 2538610
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We develop methods for listing, for a given 2-group $ S$, all nonconstrained centerfree saturated fusion systems over $ S$. These are the saturated fusion systems which could, potentially, include minimal examples of exotic fusion systems: fusion systems not arising from any finite group. To test our methods, we carry out this program over four concrete examples: two of order $ 2^7$ and two of order $ 2^{10}$. Our long term goal is to make a wider, more systematic search for exotic fusion systems over 2-groups of small order.

References:

[Ar]
E. Artin, The orders of the linear groups, Comm. Pure Appl. Math. 8 (1955), 355-365 MR 0070642 (17:12d)

[A1]
M. Aschbacher, Finite group theory, Cambridge Univ. Press (1986) MR 895134 (89b:20001)

[A2]
M. Aschbacher, Sporadic groups, Cambridge Univ. Press (1994) MR 1269103 (96e:20020)

[Bd]
H. Bender, Transitive Gruppen gerader Ordnung, in denen jede Involution genau einen Punkt festlässt, J. Algebra 17 (1971), 527-554 MR 0288172 (44:5370)

[Bs]
D. Benson, Cohomology of sporadic groups, finite loop spaces, and the Dickson invariants, Geometry and cohomology in group theory, London Math. Soc. Lecture Notes Ser. 252, Cambridge Univ. Press (1998), 10-23 MR 1709949 (2001i:55017)

[BS]
R. Brauer and M. Suzuki, On finite groups of even order whose 2-Sylow group is a quaternion group, Proc. Natl. Acad. Sci. 45 (1959), 1757-1759 MR 0109846 (22:731)

[BLO2]
C. Broto, R. Levi, and B. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), 779-856 MR 1992826 (2004k:55016)

[BCGLO1]
C. Broto, N. Castellana, J. Grodal, R. Levi, and B. Oliver, Subgroup families controlling $ p$-local finite groups, Proc. London Math. Soc. 91 (2005), 325-354 MR 2167090 (2007e:20111)

[BCGLO2]
C. Broto, N. Castellana, J. Grodal, R. Levi, and B. Oliver, Extensions of $ p$-local finite groups, Trans. Amer. Math. Soc. 359 (2007), 3791-3858 (electronic) MR 2302515 (2008i:55013)

[Co]
J. Conway, Three lectures on exceptional groups, Finite simple groups (M. Powell and G. Higman, eds.), Academic Press (1971), 215-247 MR 0338152 (49:2918)

[Fi]
L. Finkelstein, The maximal subgroups of Conway's group $ C_3$ and McLaughlin's group, J. Algebra 25 (1973), 58-89 MR 0346046 (49:10772)

[G]
D. Gorenstein, Finite groups, Harper & Row (1968) MR 0231903 (38:229)

[GH]
D. Gorenstein and K. Harada, A charaterization of Janko's two new simple groups, J. Fac. Sci. Univ. Tokyo 16 (1970), 331-406 MR 0283075 (44:308)

[He]
D. Held, The simple group related to $ M_{24}$, J. Algebra 13 (1969), 253-279 MR 0249500 (40:2745)

[HB3]
B. Huppert and N. Blackburn, Finite groups III, Springer-Verlag (1982) MR 662826 (84i:20001b)

[J]
Z. Janko, Some new simple groups of finite order I, Symposia Mathematica 1 (1969), 25-64 MR 0244371 (39:5686)

[LO]
R. Levi and B. Oliver, Construction of 2-local finite groups of a type studied by Solomon and Benson, Geometry & Topology 6 (2002), 917-990 MR 1943386 (2003k:55016)

[LO2]
R. Levi and B. Oliver, Correction to: Construction of 2-local finite groups of a type studied by Solomon and Benson, Geometry & Topology 9 (2005), 2395-2415 MR 2209376 (2006j:55018)

[Mc]
S. Mac Lane, Homology, Springer-Verlag (1975) MR 0349792 (50:2285)

[Pg]
L. Puig, Frobenius categories, J. Algebra 303 (2006), 309-357 MR 2253665 (2007j:20011)

[Rz]
A. Ruiz, Exotic subsystems of finite index in the fusion systems of general linear groups, J. London Math. Soc. (to appear)

[RV]
A. Ruiz and A. Viruel, The classification of $ p$-local finite groups over the extraspecial group of order $ p^3$ and exponent $ p$, Math. Z. 248 (2004), 45-65 MR 2092721 (2005f:20036)

[Sol]
R. Solomon, Finite groups with Sylow 2-subgroups of type $ .3$, J. Algebra 28 (1974), 182-198 MR 0344338 (49:9077)

[Sz]
M. Suzuki, On a class of doubly transitive groups, Annals of Math. 75 (1962), 105-145 MR 0136646 (25:112)

[Sz2]
M. Suzuki, Group theory II, Springer-Verlag (1986) MR 815926 (87e:20001)

[Z]
K. Zsigmondy, Zur Theorie Potenzreste, Monatsh. Math. Phys. 3 (1892), 265-284 MR 1546236

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20D20, 20D45, 20D08

Retrieve articles in all Journals with MSC (2000): 20D20, 20D45, 20D08

Additional Information:

Bob Oliver
Affiliation: Laboratoire d'analyse, géométrie et applications, Institut Galilée, Av. J-B Clément, 93430 Villetaneuse, France
Email: bobol@math.univ-paris13.fr

Joana Ventura
Affiliation: Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Email: jventura@math.ist.utl.pt

DOI: 10.1090/S0002-9947-09-04881-8
PII: S 0002-9947(09)04881-8
Keywords: Finite groups, 2-groups, fusion, simple groups.
Received by editor(s): February 29, 2008
Posted: July 21, 2009
Additional Notes: The first author was partially supported by UMR 7539 of the CNRS
The second author was partially supported by FCT/POCTI/FEDER and grant PDCT/MAT/58497/2004.
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia