A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver. (English) Zbl 1213.20017

C. Broto, R. Levi and B. Oliver [J. Am. Math. Soc. 16, No. 4, 779-856 (2003; Zbl 1033.55010)] introduced the notion of a \(p\)-local finite group \(\mathcal S\), consisting of a finite \(p\)-group \(S\) and a pair of categories \(\mathcal F\) and \(\mathcal L\) (the fusion system and centric linking system) whose objects are subgroups of \(S\), and which satisfy axioms which reflect the structure of a finite group having \(S\) as Sylow \(p\)-subgroup. If \(G\) is a finite group with Sylow \(p\)-subgroup \(S\), then there is a canonical construction which associates to \(G\) a \(p\)-local finite group \(\mathcal S=\mathcal S_S(G)\), such that the \(p\)-completed nerve of \(\mathcal L\) is homotopically equivalent to the \(p\)-completed classifying space of \(G\). A \(p\)-local finite group \(\mathcal S\) is said to be exotic if \(\mathcal S\) is not equal to \(\mathcal S_S(G)\) for any finite group \(G\) with Sylow subgroup \(S\).
In the present paper the notion of a \(p\)-local finite group is extended to the notion of a \(p\)-local group. The authors define morphisms of \(p\)-local groups, obtaining thereby a category, and introduce the notion of a representation of a \(p\)-local group via signalizer functors associated with groups. They construct a chain \(\mathcal G=(\mathcal S_0\to\mathcal S_1\to\cdots)\) of 2-local finite groups, via a representation of a chain \(\mathcal G^*=(G_0\to G_1\to\cdots)\) of groups, such that \(\mathcal S_0\) is the 2-local finite group of the third Conway sporadic group \(Co_3\), and for \(n>0\), \(\mathcal S_n\) is one of the 2-local finite groups constructed by R. Levi and B. Oliver [in Geom. Topol. 6, 917-990 (2002; Zbl 1021.55010)]. The authors show that the direct limit \(\mathcal S\) of \(\mathcal G\) exists in the category of 2-local groups, and that it is the 2-local group of the union of the chain \(\mathcal G^*\). The 2-completed classifying space of \(\mathcal S\) is shown to be the classifying space \(BDI(4)\) of the exotic 2-compact group of W. G. Dwyer and C. W. Wilkerson [J. Am. Math. Soc. 6, No. 1, 37-64 (1993; Zbl 0769.55007)].


20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D15 Finite nilpotent groups, \(p\)-groups
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
55R37 Maps between classifying spaces in algebraic topology
20J15 Category of groups
55P99 Homotopy theory
Full Text: DOI Link


[1] M. Aschbacher, Finite Group Theory, Cambridge: Cambridge Univ. Press, 1986. · Zbl 0583.20001
[2] D. Benson, ”Conway’s group \({ Co}_3\) and the Dickson invariants,” Manuscripta Math., vol. 85, iss. 2, pp. 177-193, 1994. · Zbl 0853.55018 · doi:10.1007/BF02568192
[3] C. Broto, N. Castellana, J. Grodal, R. Levi, and B. Oliver, ”Subgroup families controlling \(p\)-local finite groups,” Proc. London Math. Soc., vol. 91, iss. 2, pp. 325-354, 2005. · Zbl 1090.20026 · doi:10.1112/S0024611505015327
[4] C. Broto, R. Levi, and B. Oliver, ”The homotopy theory of fusion systems,” J. Amer. Math. Soc., vol. 16, iss. 4, pp. 779-856, 2003. · Zbl 1033.55010 · doi:10.1090/S0894-0347-03-00434-X
[5] C. Broto, R. Levi, and B. Oliver, Discrete models for the \(p\)-local homotopy theory of compact Lie groups, 2005. · Zbl 1135.55008 · doi:10.2140/gt.2007.11.315
[6] C. Chevalley, The Algebraic Theory of Spinors and Clifford Algebras, New York: Springer-Verlag, 1997. · Zbl 0899.01032
[7] A. Chermak, B. Oliver, and S. Shpectorov, The simple connectivity of \({ Sol}(q)\), preprint, 2006.
[8] W. G. Dwyer, ”Classifying spaces and homology decompositions,” in Homotopy Theoretic Methods in Group Cohomology, Dwyer, W. G. and Hans-Werner, H., Eds., Basel: Birkhäuser Verlag, 2001, pp. 3-53.
[9] W. G. Dwyer and C. W. Wilkerson, ”A new finite loop space at the prime two,” J. Amer. Math. Soc., vol. 6, iss. 1, pp. 37-64, 1993. · Zbl 0769.55007 · doi:10.2307/2152794
[10] L. Finkelstein, ”The maximal subgroups of Conway’s group \(C_3\) and McLaughlin’s group,” J. Algebra, vol. 25, pp. 58-89, 1973. · Zbl 0263.20010 · doi:10.1016/0021-8693(73)90075-6
[11] D. M. Goldschmidt, ”A conjugation family for finite groups,” J. Algebra, vol. 16, pp. 138-142, 1970. · Zbl 0198.04306 · doi:10.1016/0021-8693(70)90046-3
[12] D. M. Goldschmidt, ”Automorphisms of trivalent graphs,” Ann. of Math., vol. 111, iss. 2, pp. 377-406, 1980. · Zbl 0475.05043 · doi:10.2307/1971203
[13] D. Gorenstein, R. Lyons, and R. Solomon, The classification of the finite simple groups, Providence, RI: Amer. Math. Soc., 1994. · Zbl 1069.20011
[14] A. Hatcher, Algebraic topology, Cambridge: Cambridge Univ. Press, 2002. · Zbl 1044.55001
[15] N. Jacobson, Basic Algebra. II, San Francisco, CA: W. H. Freeman and Co., 1980. · Zbl 0441.16001
[16] W. M. Kantor, ”Some exceptional \(2\)-adic buildings,” J. Algebra, vol. 92, iss. 1, pp. 208-223, 1985. · Zbl 0562.51006 · doi:10.1016/0021-8693(85)90155-3
[17] H. Kurzweil and B. Stellmacher, The Theory of Finite Groups. An Introduction, New York: Universitext Springer-Verlag, 2004. · Zbl 1047.20011 · doi:10.1007/b97433
[18] I. Leary and R. Stancu, Realizing fusion systems, preprint, 2006.
[19] R. Levi and B. Oliver, ”Construction of 2-local finite groups of a type studied by Solomon and Benson,” Geom. Topol., vol. 6, pp. 917-990, 2002. · Zbl 1021.55010 · doi:10.2140/gt.2002.6.917
[20] L. Puig, ”Frobenius categories,” J. Algebra, vol. 303, iss. 1, pp. 309-357, 2006. · Zbl 1110.20011 · doi:10.1016/j.jalgebra.2006.01.023
[21] G. R. Robinson, ”Amalgams, blocks, weights, fusion systems and finite simple groups,” J. Algebra, vol. 314, iss. 2, pp. 912-923, 2007. · Zbl 1184.20024 · doi:10.1016/j.jalgebra.2007.05.010
[22] J. Serre, Trees, New York: Springer-Verlag, 1980. · Zbl 1013.20001
[23] R. Solomon, ”Finite groups with Sylow \(2\)-subgroups of type \(3\),” J. Algebra, vol. 28, pp. 182-198, 1974. · Zbl 0293.20022 · doi:10.1016/0021-8693(74)90031-3
[24] R. Steinberg, Lecture Notes on Chevalley Groups, Yale Univ. Lecture Notes, New Haven, CT, 1967.
[25] J. Tits, ”Sur le groupe des automorphismes d’un arbre,” in Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), New York: Springer-Verlag, 1970, pp. 188-211. · Zbl 0214.51301
[26] B. A. F. Wehrfritz, Infinite Linear Groups. An Account of the Group-Theoretic Properties of Infinite Groups of Matrices, New York: Springer-Verlag, 1973, vol. 76. · Zbl 0261.20038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.