Abstract
Let G be a simple Lie type group of rank ≥ 2 different from \({^2F_4, P_J}\) a maximal parabolic of G and \({\Delta_J = A_h^{P_J}}\), where A h is the root subgroup corresponding to the highest root of the root system. (\({\Delta_J\subseteq Z(U_J)}\), since \({A_h \subseteq Z(U)}\), U the unipotent subgroup.) Then we describe the subgroups of G containing \({\Delta_J}\), which generalizes the main result of Timmesfeld (J Algebra 323:1408–1431, 2010). For finite G this can be used to obtain results on the subgroup structure of G.
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References
Ajad H., Barry M., Seitz G.: On the structure of parabolic subgroups. Commun. Algebra 18(2), 551–562 (1990)
Aschbacher, M.: 3-Transposition groups. In: Cambridge Tracts in Mathematics, vol. 124. Cambridge University Press, Cambridge (1997)
Aschbacher M., Hall M. Jr.: Groups generated by a class of elements of order 3. J. Algebra 24, 591–612 (1973)
Cooperstein B.: The geometry of root subgroups in exceptional groups, I, II. Geom. Dedicata 8, 317–381 (1979)
Cooperstein B.: The geometry of root subgroups in exceptional groups, I, II. Geom. Dedicata 15, 1–45 (1983)
Kantor W.M.: Subgroups of classical groups generated by long root elements. Trans. Am. Math. Soc. 248, 347–379 (1979)
Liebeck M.W., Seitz G.: Subgroups generated by root elements in groups of Lie type. Ann. Math., II Ser. 139, 293–361 (1994)
Steinbach, A.: Subgroups of classical groups generated by transvections or Siegel transvections I,II. Geom. Dedicata 68, 281–322, 323–557 (1997)
Steinbach A.: Groups of Lie type generated by long root elements in F 4(K). Algebra 255, 463–488 (2002)
Steinbach A.: Subgroups of the Chevalley groups of type F 4 arising from a polar space. Adv. Geom. 3, 73–100 (2003)
Timmesfeld F.G.: Abstract root subgroups and simple groups of lie type. Monographs in Math., Vol. 95. Birkhäuser Verlag, Basel (2001)
Timmesfeld F.G.: Subgroups of Lie type groups containing a unipotent radical. J. Algebra 323, 1408–1431 (2010)
Timmesfeld F.G.: A remark on presentations of certain Chevalley groups. Arch. Math. 79, 404–407 (2002)
Timmesfeld F.G.: Steinberg-type presentation for Lie type groups. J. Algebra 300(2), 806–819 (2006)
Timmesfeld F.G.: The Curtis-Tits-presentation. Adv. Math. 189, 38–67 (2004)
Tits, J.: Buildings of spherical type and finite BN-pairs. In: Lecture Notes in Math., vol. 386, Springer Verlag, Berlin (1974)
Weiss R.: The Structure of Spherical Buildings. Princeton University Press, Princeton (2003)
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Timmesfeld, F.G. Subgroups of Lie type groups containing the center of a unipotent radical. Beitr Algebra Geom 53, 311–347 (2012). https://doi.org/10.1007/s13366-012-0109-3
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DOI: https://doi.org/10.1007/s13366-012-0109-3