Saturated fusion systems over 2-groups

Oliver, Bob; Ventura, Joana

doi:10.1090/S0002-9947-09-04881-8

Saturated fusion systems over $2$-groups
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by Bob Oliver and Joana Ventura PDF

Trans. Amer. Math. Soc. 361 (2009), 6661-6728 Request permission

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

Emil Artin, The orders of the linear groups, Comm. Pure Appl. Math. 8 (1955), 355–365. MR 70642, DOI 10.1002/cpa.3160080302
Michael Aschbacher, Finite group theory, Cambridge Studies in Advanced Mathematics, vol. 10, Cambridge University Press, Cambridge, 1986. MR 895134
Michael Aschbacher, Sporadic groups, Cambridge Tracts in Mathematics, vol. 104, Cambridge University Press, Cambridge, 1994. MR 1269103, DOI 10.1017/CBO9780511665585
Helmut Bender, Transitive Gruppen gerader Ordnung, in denen jede Involution genau einen Punkt festläßt, J. Algebra 17 (1971), 527–554 (German). MR 288172, DOI 10.1016/0021-8693(71)90008-1
David J. Benson, Cohomology of sporadic groups, finite loop spaces, and the Dickson invariants, Geometry and cohomology in group theory (Durham, 1994) London Math. Soc. Lecture Note Ser., vol. 252, Cambridge Univ. Press, Cambridge, 1998, pp. 10–23. MR 1709949, DOI 10.1017/CBO9780511666131.003
Richard Brauer and Michio Suzuki, On finite groups of even order whose $2$-Sylow group is a quaternion group, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 1757–1759. MR 109846, DOI 10.1073/pnas.45.12.1757
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
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
C. Broto, N. Castellana, J. Grodal, R. Levi, and B. Oliver, Extensions of $p$-local finite groups, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3791–3858. MR 2302515, DOI 10.1090/S0002-9947-07-04225-0
J. H. Conway, Three lectures on exceptional groups, Finite simple groups (Proc. Instructional Conf., Oxford, 1969) Academic Press, London, 1971, pp. 215–247. MR 0338152
Larry Finkelstein, The maximal subgroups of Conway’s group $C_{3}$ and McLaughlin’s group, J. Algebra 25 (1973), 58–89. MR 346046, DOI 10.1016/0021-8693(73)90075-6
Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
Daniel Gorenstein and Koichiro Harada, A characterization of Janko’s two new simple groups, J. Fac. Sci. Univ. Tokyo Sect. I 16 (1969), 331–406 (1970). MR 283075
Dieter Held, The simple groups related to $M_{24}$, J. Algebra 13 (1969), 253–296. MR 249500, DOI 10.1016/0021-8693(69)90074-X
Bertram Huppert and Norman Blackburn, Finite groups. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 242, Springer-Verlag, Berlin-New York, 1982. AMD, 44. MR 650245
Zvonimir Janko, Some new simple groups of finite order. I, Symposia Mathematica (INDAM, Rome, 1967/68) Academic Press, London, 1969, pp. 25–64. MR 0244371
Ran Levi and Bob Oliver, Construction of 2-local finite groups of a type studied by Solomon and Benson, Geom. Topol. 6 (2002), 917–990. MR 1943386, DOI 10.2140/gt.2002.6.917
Ran Levi and Bob Oliver, Correction to: “Construction of 2-local finite groups of a type studied by Solomon and Benson” [Geom. Topol. 6 (2002), 917–990 (electronic); MR1943386], Geom. Topol. 9 (2005), 2395–2415. MR 2209376, DOI 10.2140/gt.2005.9.2395
Saunders MacLane, Homology, 1st ed., Die Grundlehren der mathematischen Wissenschaften, Band 114, Springer-Verlag, Berlin-New York, 1967. MR 0349792
Lluis Puig, Frobenius categories, J. Algebra 303 (2006), no. 1, 309–357. MR 2253665, DOI 10.1016/j.jalgebra.2006.01.023
A. Ruiz, Exotic subsystems of finite index in the fusion systems of general linear groups, J. London Math. Soc. (to appear)
Albert Ruiz and Antonio Viruel, The classification of $p$-local finite groups over the extraspecial group of order $p^3$ and exponent $p$, Math. Z. 248 (2004), no. 1, 45–65. MR 2092721, DOI 10.1007/s00209-004-0652-1
Ronald Solomon, Finite groups with Sylow $2$-subgroups of type $3$, J. Algebra 28 (1974), 182–198. MR 344338, DOI 10.1016/0021-8693(74)90031-3
Michio Suzuki, On a class of doubly transitive groups, Ann. of Math. (2) 75 (1962), 105–145. MR 136646, DOI 10.2307/1970423
Michio Suzuki, Group theory. II, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 248, Springer-Verlag, New York, 1986. Translated from the Japanese. MR 815926, DOI 10.1007/978-3-642-86885-6
K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. Phys. 3 (1892), no. 1, 265–284 (German). MR 1546236, DOI 10.1007/BF01692444