Abstract
In this article we present some recent results concerning the subgroups of the simple algebraic groups of exceptional type, and of the corresponding finite groups of Lie type. There are six sections. The first contains some general observations, while in the second we focus on connected subgroups. The third section contains results on infinite closed subgroups, and in the last three sections we discuss finite subgroups.
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References
Azad, H., Barry, M. and Seitz, G.M. (1990) On the structure of parabolic subgroups, Comm. in Alg. 18, 551–562.
Borel, A. and Tits, J. (1971) Éléments unipotents et sousgroupes paraboliques de groupes réductifs, Invent. Math. 12, 95–104.
Borel, A. and de Siebenthal, J. (1971) Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Heiv. 23, 200–221.
Borovik, A. (1989) The structure of finite subgroups of simple algebraic groups, Algebra and Logic 28, 249–279 (in Russian).
Bourbaki, N. (1968) Groupes et algæbres de Lie (Chapters 4,5 and 6), Hermann, Paris.
Carter, R.W. (1972) Conjugacy classes in the Weyl group, Compositio Math. 25, 1–59.
Cohen, A.M. and Griess, R.L. (1987) On finite simple subgroups of the complex Lie group of type Ea, Proc. Symp. Pure Math. 47, 367–405.
Cohen, A.M., Liebeck, M.W., Saxl, J. and Seitz, G.M. (1992) The local maximal subgroups of exceptional groups of Lie type, finite and algebraic, Proc. London Math. Soc. 64, 21–48.
Cohen, A.M. and Wales, D.B. (1983) Finite subgroups of G2(C), Comm. in Alg. 11, 441–459.
Cohen, A.M. and Wales, D.B. (1997) Finite subgroups of E6((C) and F4(C), Proc. London Math. Soc. 74, 105–150.
Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A. and Wilson, R.A. (1985) Atlas of Finite Groups, Oxford University Press.
Dynkin, E.B. (1957) Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Translations 6, 111–244.
Gorenstein, D. and Lyons, R. (1983) The local structure of finite groups of characteristic 2 type, Mem. Amer. Math. Soc. 276, 1–731.
Griess, R.L. and Ryba, A.J.E. (1994) Embeddings of U3 (8), Sz(8) and the Rudvalis group in algebraic groups of type E 7, Invent. Math. 116, 215–241.
. Griess, R.L. and Ryba, A.J.E. (in press) Embeddings of PSL(2,32) and PGL(2,31) in E8 (C), Duke Math. J..
. Griess, R.L. and Ryba, A.J.E. (in press) Embeddings of PSL(2,41) and PGL(2,49) in E8 (C), Duke Math. J..
Griess, R.L. and Ryba, A.J.E. (to appear) Embedding of Sz(8) in E8 (ℂ).
Janko, Z. (1966) A new finite simple group with abelian Sylow 2-subgroups and its characterization, J. Algebra 3, 147–186.
Jansen, C., Lux, K., Parker, R.A. and Wilson, R. (1995) An Atlas of Brauer Characters,Clarendon Press, Oxford.
Jantzen, J.C. (1997) Low dimensional representations of reductive groups are semisimple, in G. Lehrer et al. (eds.), Algebraic groups and Lie groups, Austral. Math. Soc. Lecture Series 9, pp. 255–266.
Kleidman, P.B. and Wilson, R.A. (1993) Sporadic simple subgroups of finite exceptional groups of Lie type, J. Algebra 137, 316–330.
. Kleidman, P.B., Meierfrankenfeld, U. and Ryba, A.J.E. (in press) HS < E) 7 (5),J. London Math. Soc..
Landazuri, V. and Seitz, G.M. (1974) On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra 32, 418–443.
. Lawther, R. and Testerman, D.M. (in press) Al subgroups of exceptional algebraic groups, Trans. Amer. Math. Soc..
Liebeck, M.W. and Seitz, G.M. (1990) Maximal subgroups of exceptional groups of Lie type, finite and algebraic, Geom. Dedicata 36, 353–387.
Liebeck, M.W. and Seitz, G.M. (1994) Subgroups generated by root elements in groups of Lie type, Annals of Math. 139, 293–361.
. Liebeck, M.W. and Seitz, G.M. (1996) Reductive subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc. 580, 1-111.
. Liebeck, M.W. and Seitz, G.M. (in press) On the subgroup structure of exceptional groups of Lie type, Trans. Amer. Math. Soc..
Liebeck, M.W. and Seitz, G.M. (to appear) On finite subgroups of exceptional algebraic groups.
Saxl, J., Wales, D.B. and Wilson, R.A. (to appear) Embeddings of Sz(32) in E8 (F).
Seitz, G.M. (1991) Maximal subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc. 441,1-197.
Seitz, G.M. and Testerman, D.M. (1990) Extending morphisms from finite to algebraic groups, J. Algebra 131, 559–574.
Serre, J.-P. (1996) Exemples de plongements des groupes PSL2(F r ) dans des groupes de Lie simples, Invent. Math. 124, 525–562.
Slodowy, P. (1997) Two notes on a finiteness problem in the representation theory of finite groups, in G. Lehrer et al. (eds.), Algebraic groups and Lie groups, Austral. Math. Soc. Lecture Series 9, pp. 331–348.
Springer, T.A. and Steinberg, R. (1970) Conjugacy classes, in A. Borel et al. (eds.), Seminar on algebraic groups and related topics, Lecture Notes in Math. 131, Springer, Berlin, pp. 168–266.
Thompson, J.G. (1976) A simple subgroup of E8(3), in N. Iwahori (ed.) Finite Groups, Japan Soc. for the Promotion of Science, Tokyo, pp. 113–116.
Weil, A. (1964) Remarks on the cohomology of groups, Annals of Math. 80, 149–157.
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Liebeck, M.W. (1998). Subgroups of Exceptional Groups. In: Carter, R.W., Saxl, J. (eds) Algebraic Groups and their Representations. NATO ASI Series, vol 517. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5308-9_15
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DOI: https://doi.org/10.1007/978-94-011-5308-9_15
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