Nilpotent fusion systems. (English) Zbl 1441.20014
Summary: Let \(p\) be a prime and let \(\mathcal{F}\) be a saturated fusion system over a finite \(p\)-group \(P\). A fusion system \(\mathcal{F}\) is said to be nilpotent if \(\mathcal{F} = \mathcal{F}_P(P)\). We give various criteria for a saturated fusion system \(\mathcal{F}\) to be nilpotent, which generalize the analogous criteria for a finite group to be \(p\)-nilpotent.
MSC:
| 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 |
References:
| [1] | Asaad, M., On \(p\)-nilpotence of finite groups, J. Algebra, 277, 157-164 (2004) · Zbl 1061.20015 |
| [2] | Asaad, M., On \(p\)-nilpotence and supersolvability of finite groups, Comm. Algebra, 34, 189-195 (2006) · Zbl 1092.20020 |
| [3] | Aschbacher, M., Normal subsystems of fusion systems, Proc. Lond. Math. Soc., 97, 3, 239-271 (2008) · Zbl 1241.20026 |
| [4] | Aschbacher, M., The Generalized Fitting Subsystem of a Fusion System, Mem. Amer. Math. Soc., vol. 209(986) (2011) · Zbl 1278.20020 |
| [5] | 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 |
| [6] | Broto, C.; Castellana, N.; Grodal, J.; Levi, R.; Oliver, B., Subgroup families controlling \(p\)-local finite groups, Proc. Lond. Math. Soc., 91, 325-354 (2005) · Zbl 1090.20026 |
| [7] | 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) · Zbl 1145.55013 |
| [8] | Ballester-Bolinches, A.; Guo, X., Some results on \(p\)-nilpotence and solubility of finite groups, J. Algebra, 228, 491-496 (2000) · Zbl 0961.20016 |
| [9] | Benson, D. J.; Grodal, J.; Henke, E., Group cohomology and control of \(p\)-fusion, Invent. Math., 197, 3, 491-507 (2014) · Zbl 1311.20051 |
| [10] | Broto, C.; Levi, R.; Oliver, B., The homotopy theory of fusion systems, J. Amer. Math. Soc., 16, 4, 779-856 (2003) · Zbl 1033.55010 |
| [11] | Broué, M.; Puig, L., A Frobenius theorem for blocks, Invent. Math., 56, 2, 117-128 (1980) · Zbl 0425.20008 |
| [12] | Cantarero, J.; Scherer, J.; Viruel, A., Nilpotent \(p\)-local finite group, Ark. Mat., 52, 2, 203-225 (2014) · Zbl 1320.20024 |
| [13] | Diaz, A.; Glesser, A.; Mazza, N.; Park, S., Glauberman’s and Thompson’s theorems for fusion systems, Proc. Amer. Math. Soc., 137, 2, 495-503 (2009) · Zbl 1163.20009 |
| [14] | Diaz, A.; Glesser, A.; Park, S.; Stancu, R., Tate’s and Yoshida’s theorems on control of transfer for fusion systems, J. Lond. Math. Soc. (2), 84, 2, 475-494 (2011) · Zbl 1235.20022 |
| [15] | Dornhoff, L., \(M\)-groups and 2-groups, Math. Z., 100, 226-256 (1967) · Zbl 0157.35503 |
| [16] | Goldschmidt, D., Strongly closed 2-subgroups of finite groups, Ann. Math., 102, 3, 475-489 (1975) · Zbl 0333.20013 |
| [17] | González-Sánchez, J., A \(p\)-nilpotency criterion, Arch. Math., 94, 201-205 (2010) · Zbl 1208.20021 |
| [18] | Glesser, A., Sparse fusion systems, Proc. Edinb. Math. Soc., 56, 135-150 (2013) · Zbl 1269.20013 |
| [19] | Gorenstein, D., Finite Groups (1968), Harper and Row: Harper and Row New York · Zbl 0185.05701 |
| [20] | Guo, X.; Wei, X., The influence of \(H\)-subgroups on the structure of finite groups, J. Group Theory, 13, 267-276 (2010) · Zbl 1202.20024 |
| [21] | Gilotti, A.; Serena, L., Strongly closed subgroups and strong fusion, J. Lond. Math. Soc., 32, 103-106 (1985) · Zbl 0545.20010 |
| [22] | Guo, X.; Shum, K. P., \(p\)-nilpotence of finite groups and minimal subgroups, J. Algebra, 270, 459-470 (2003) · Zbl 1072.20020 |
| [23] | Hai, J., On \(p\)-hypercenter of finite groups, Adv. Math. (China), 29, 4, 383-384 (2000) |
| [24] | Huppert, B., Endliche Gruppen I (1967), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0217.07201 |
| [25] | Huppert, B.; Blackburn, N., Finite Groups. II, Grundlehren Math. Wiss., vol. 242 (1982), Springer-Verlag: Springer-Verlag Berlin, AMD, 44 · Zbl 0477.20001 |
| [26] | Kessar, R.; Linckelmann, M., ZJ-theorems for fusion systems, Trans. Amer. Math. Soc., 360, 6, 3093-3106 (2008) · Zbl 1144.20004 |
| [27] | Külshammer, B.; Puig, L., Extensions of nilpotent blocks, Invent. Math., 102, 1, 17-71 (1990) · Zbl 0739.20003 |
| [28] | Comm. Algebra, 26, 2453-2461 (1998) · Zbl 0911.20016 |
| [29] | Liao, J.; Liu, Y.; Zhang, J., Minimal non-nilpotent and locally nilpotent fusion systems, Algebra Colloq. (2015), in press |
| [30] | Li, R.; Zhang, Q., A note on \(p\)-supersolvable groups, Acta Math. Hungar., 116, 1-2, 27-34 (2007) · Zbl 1135.20010 |
| [31] | Onofrei, S.; Stancu, R., A characteristic subgroup for fusion systems, J. Algebra, 322, 1705-1718 (2009) · Zbl 1225.20020 |
| [32] | Puig, L., Nilpotent blocks and their source algebras, Invent. Math., 93, 1, 77-116 (1988) · Zbl 0646.20010 |
| [33] | Puig, L., The hyperfocal subalgebra of a block, Invent. Math., 141, 365-397 (2000) · Zbl 0957.20007 |
| [34] | Puig, L., Frobenius categories, J. Algebra, 303, 309-357 (2006) · Zbl 1110.20011 |
| [35] | Shi, J.; Shi, W.; Zhang, C., A note on \(p\)-nilpotence and solvability of finite groups, J. Algebra, 321, 1555-1560 (2009) · Zbl 1169.20012 |
| [36] | Tong-Viet, H. P., Influence of strongly closed 2-subgroups on the structure of finite groups, Glasg. Math. J., 53, 3, 577-581 (2011) · Zbl 1239.20022 |
| [37] | Wei, H.; Wang, Y.; Liu, X., Some necessary and sufficient conditions for \(p\)-nilpotence of finite groups, Bull. Aust. Math. Soc., 68, 371-378 (2003) · Zbl 1060.20016 |
| [38] | Wei, H.; Wang, Y.; Qian, G., Quasi-central elements and \(p\)-nilpotence of finite groups, Publ. Math. Debrecen, 77, 1-2, 233-244 (2010) · Zbl 1221.20012 |
| [39] | Zhang, G., On two theorems of Thompson, Proc. Amer. Math. Soc., 98, 4, 579-582 (1986) · Zbl 0607.20011 |
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