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Extensions of  -local finite groups
Author(s):
C.
Broto;
N.
Castellana;
J.
Grodal;
R.
Levi;
B.
Oliver
Journal:
Trans. Amer. Math. Soc.
359
(2007),
3791-3858.
MSC (2000):
Primary 55R35;
Secondary 55R40, 20D20
Posted:
March 20, 2007
MathSciNet review:
2302515
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Additional information
Abstract:
A  -local finite group consists of a finite  -group  , together with a pair of categories which encode ``conjugacy'' relations among subgroups of  , and which are modelled on the fusion in a Sylow  -subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as  -completed classifying spaces of finite groups. In this paper, we study and classify extensions of  -local finite groups, and also compute the fundamental group of the classifying space of a  -local finite group.
References:
-
- [AM]
- A. Adem & J. Milgram, Cohomology of finite groups, Springer-Verlag (1994) MR 1317096 (96f:20082)
- [Al]
- J. Alperin, Sylow intersections and fusion, J. Algebra 6 (1967), 222-241 MR 0215913 (35:6748)
- [BK]
- P. Bousfield & D. Kan, Homotopy limits, completions, and localizations, Lecture notes in math. 304, Springer-Verlag (1972) MR 0365573 (51:1825)
- [5A1]
- 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
- [BLO1]
- C. Broto, R. Levi, & B. Oliver, Homotopy equivalences of
 -completed classifying spaces of finite groups, Invent. math. 151 (2003), 611-664 MR 1961340 (2004c:55031) - [BLO2]
- C. Broto, R. Levi, & B. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), 779-856 MR 1992826 (2004k:55016)
- [DKS]
- W. Dwyer, D. Kan, & J. Smith, Towers of fibrations and homotopical wreath products, J. Pure Appl. Algebra 56 (1989), 9-28 MR 0974710 (89k:55009)
- [GZ]
- P. Gabriel & M. Zisman, Calculus of fractions and homotopy theory, Springer-Verlag (1967) MR 0210125 (35:1019)
- [Gl]
- G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403-420 MR 0202822 (34:2681)
- [GJ]
- P. Goerss & R. Jardine, Simplicial homotopy theory, Birkhäuser (1999) MR 1711612 (2001d:55012)
- [Go]
- D. Gorenstein, Finite groups, Harper & Row (1968) MR 0231903 (38:229)
- [Hf]
- G. Hoff, Cohomologies et extensions de categories, Math. Scand. 74 (1994), 191-207. MR 1298361 (95k:18012)
- [JMO]
- S. Jackowski, J. McClure, & B. Oliver, Homotopy classification of self-maps of
 via  -actions, Annals of Math. 135 (1992), 184-270 - [May]
- P. May, Simplicial objects in algebraic topology, Van Nostrand (1967) MR 0222892 (36:5942)
- [O1]
- B. Oliver, Equivalences of classifying spaces completed at odd primes, Math. Proc. Camb. Phil. Soc. 137 (2004), 321-347 MR 2092063 (2006g:55010)
- [O2]
- B. Oliver, Equivalences of classifying spaces completed at the prime two, Amer. Math. Soc. Memoirs 848 (2006) MR 2203209 (2007c:55014)
- [Pu1]
- Ll. Puig, Unpublished notes (ca. 1990)
- [Pu2]
- Ll. Puig, The hyperfocal subalgebra of a block, Invent. math. 141 (2000), 365-397 MR 1775217 (2001h:20012)
- [Pu3]
- L. Puig, Full Frobenius systems and their localizing categories, preprint (2001)
- [St]
- R. Stancu, Equivalent definitions of fusion systems, preprint
- [Suz2]
- M. Suzuki, Group theory II, Springer-Verlag (1986) MR 0815926 (87e:20001)
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Additional Information:
C.
Broto
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, E--08193 Bellaterra, Spain
Email:
broto@mat.uab.es
N.
Castellana
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, E--08193 Bellaterra, Spain
Email:
natalia@mat.uab.es
J.
Grodal
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Address at time of publication:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 København Ø, Denmark
Email:
jg@math.uchicago.edu, jg@math.ku.dk
R.
Levi
Affiliation:
Department of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen AB24 3UE, United Kingdom
Email:
ran@maths.abdn.ac.uk
B.
Oliver
Affiliation:
LAGA, Institut Galilée, Av. J-B Clément, 93430 Villetaneuse, France
Email:
bob@math.univ-paris13.fr
DOI:
10.1090/S0002-9947-07-04225-0
PII:
S 0002-9947(07)04225-0
Keywords:
Classifying space,
$p$-completion,
finite groups,
fusion.
Received by editor(s):
July 11, 2005
Posted:
March 20, 2007
Additional Notes:
The first author was partially supported by MCYT grant BFM2001--2035
The second author was partially supported by MCYT grant BFM2001--2035
The third author was partially supported by NSF grants DMS-0104318 and DMS-0354633
The fourth author was partially supported by EPSRC grant GR/M7831.
The fifth author was partially supported by UMR 7539 of the CNRS
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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