Saturated fusion systems with parabolic families. (English) Zbl 1261.20017
Summary: Let \(G\) be a group; a finite \(p\)-subgroup \(S\) of \(G\) is a Sylow \(p\)-subgroup if every finite \(p\)-subgroup of \(G\) is conjugate to a subgroup of \(S\). In this paper, we examine the relations between the fusion system over \(S\) which is given by conjugation in \(G\) and a certain chamber system \(\mathcal C\), on which \(G\) acts chamber transitively with chamber stabilizer \(N_G(S)\).
Next, we introduce the notion of a fusion system with a parabolic family and we show that a chamber system can be associated to such a fusion system. We determine some conditions the chamber system has to fulfill in order to assure the saturation of the underlying fusion system. We give an application to fusion systems with parabolic families of classical type.
Next, we introduce the notion of a fusion system with a parabolic family and we show that a chamber system can be associated to such a fusion system. We determine some conditions the chamber system has to fulfill in order to assure the saturation of the underlying fusion system. We give an application to fusion systems with parabolic families of classical type.
MSC:
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 20J15 | Category of groups |
| 55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |
| 20E42 | Groups with a \(BN\)-pair; buildings |
Keywords:
saturated fusion systems; parabolic systems; chamber systems; diagrams of finite groups; Sylow subgroupsReferences:
| [1] | Aschbacher, M.; Kessar, R.; Oliver, B., Fusion Systems in Algebra and Topology, London Math. Soc. Lecture Note Ser., vol. 391 (2011), Cambridge University Press · Zbl 1255.20001 |
| [2] | M. Aschbacher, \(S_3\); M. Aschbacher, \(S_3\) · Zbl 1278.20021 |
| [3] | Aschbacher, M., 3-Transposition Groups, Cambridge Tracts in Math., vol. 124 (1997), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0883.20010 |
| [4] | Aschbacher, M., Normal subsystems of fusion systems, Proc. Lond. Math. Soc. (3), 97, 1, 239-271 (2008) · Zbl 1241.20026 |
| [5] | Broto, C.; Castellana, N.; Grodal, J.; Levi, R.; Oliver, B., Subgroup families controlling \(p\)-local finite groups, Proc. Lond. Math. Soc. (3), 91, 2, 325-354 (2005) · Zbl 1090.20026 |
| [6] | Broto, C.; Castellana, N.; Grodal, J.; Levi, R.; Oliver, B., Extensions of \(p\)-local finite groups, Trans. Amer. Math. Soc., 359, 8, 3791-3858 (2007), (electronic) · Zbl 1145.55013 |
| [7] | Broto, C.; Levi, R.; Oliver, B., The homotopy theory of fusion systems, J. Amer. Math. Soc., 16, 4, 779-856 (2003), (electronic) · Zbl 1033.55010 |
| [8] | Broto, C.; Levi, R.; Oliver, B., The theory of \(p\)-local groups: A survey, (Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic \(K\)-Theory. Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic \(K\)-Theory, Contemp. Math., vol. 346 (2004), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 51-84 · Zbl 1063.55013 |
| [9] | Broto, C.; Levi, R.; Oliver, B., A geometric construction of saturated fusion systems, (An Alpine Anthology of Homotopy Theory. An Alpine Anthology of Homotopy Theory, Contemp. Math., vol. 399 (2006), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 11-40 · Zbl 1100.55005 |
| [10] | Fukshansky, A.; Stroth, G.; Wiedorn, C., The semiclassical parabolic systems, Geom. Dedicata, 81, 1-3, 87-139 (2000) · Zbl 0959.20018 |
| [11] | Libman, A., Webbʼs conjecture for fusion systems, Israel J. Math., 167, 141-154 (2008) · Zbl 1177.20017 |
| [12] | Linckelmann, M., Simple fusion systems and the Solomon 2-local groups, J. Algebra, 296, 2, 385-401 (2006) · Zbl 1097.20014 |
| [13] | Linckelmann, M., Introduction to fusion systems, (Group Representation Theory (2007), EPFL Press: EPFL Press Lausanne), 79-113, revised version 2010 · Zbl 1161.20007 |
| [14] | Leary, I. J.; Stancu, R., Realising fusion systems, Algebra Number Theory, 1, 1, 17-34 (2007) · Zbl 1131.20012 |
| [15] | Meixner, T., Groups acting transitively on locally finite classical Tits chamber systems, (Finite Geometries, Buildings, and Related Topics. Finite Geometries, Buildings, and Related Topics, Pingree Park, CO, 1988. Finite Geometries, Buildings, and Related Topics. Finite Geometries, Buildings, and Related Topics, Pingree Park, CO, 1988, Oxford Sci. Publ. (1990), Oxford Univ. Press: Oxford Univ. Press New York), 45-65 · Zbl 0754.20008 |
| [16] | Parker, C.; Rowley, P., Symplectic Amalgams, Springer Monogr. Math. (2002), Springer: Springer London · Zbl 1003.20013 |
| [17] | Puig, L., Frobenius categories, J. Algebra, 303, 1, 309-357 (2006) · Zbl 1110.20011 |
| [18] | Puig, L., Frobenius categories versus Brauer blocks, (The Grothendieck Group of the Frobenius Category of a Brauer Block. The Grothendieck Group of the Frobenius Category of a Brauer Block, Progr. Math., vol. 274 (2009), Birkhäuser: Birkhäuser Basel) · Zbl 1192.20008 |
| [19] | Quillen, D., Homotopy properties of the poset of nontrivial \(p\)-subgroups of a group, Adv. Math., 28, 2, 101-128 (1978) · Zbl 0388.55007 |
| [20] | Robinson, G. R., Amalgams, blocks, weights, fusion systems and finite simple groups, J. Algebra, 314, 2, 912-923 (2007) · Zbl 1184.20024 |
| [21] | Ronan, M., Coverings and automorphisms of chamber systems, European J. Combin., 1, 3, 259-269 (1980) · Zbl 0553.51004 |
| [22] | Ronan, M., Lectures on Buildings, Perspect. Math., vol. 7 (1989), Academic Press Inc.: Academic Press Inc. Boston, MA · Zbl 0694.51001 |
| [23] | Scharlau, R., Buildings, (Handbook of Incidence Geometry (1995), North-Holland: North-Holland Amsterdam), 477-645 · Zbl 0841.51005 |
| [24] | R. Stancu, Equivalent definitions of fusion systems, preprint, 2004.; R. Stancu, Equivalent definitions of fusion systems, preprint, 2004. |
| [25] | Stancu, R., Control of fusion in fusion systems, J. Algebra Appl., 5, 6, 817-837 (2006) · Zbl 1118.20020 |
| [26] | Timmesfeld, F. G., Tits geometries and parabolic systems in finitely generated groups, I, Math. Z., 184, 3, 377-396 (1983) · Zbl 0521.20007 |
| [27] | Timmesfeld, F. G., Tits geometries and revisionism of the classification of finite simple groups of characteristic 2-type, (Proceedings of the Rutgers Group Theory Year. Proceedings of the Rutgers Group Theory Year, New Brunswick, NJ, 1983-1984 (1985), Cambridge University Press: Cambridge University Press Cambridge), 229-242 · Zbl 0653.20017 |
| [28] | Timmesfeld, F. G., Locally finite classical Tits chamber systems of large order, Invent. Math., 87, 3, 603-641 (1987) · Zbl 0607.20019 |
| [29] | Tits, J., A local approach to buildings, (The Geometric Vein (1981), Springer: Springer New York), 519-547 · Zbl 0496.51001 |
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