Non-Local Lie Primitive Subgroups of Lie Groups

Cohen, Arjeh M. ; Jr., Robert L. Griess

Canadian journal of mathematics, Tome 45 (1993) no. 1, pp. 88-103

Résumé

Borovik found a Lie primitive subgroup of E8 (C) isomorphic to (Alt5 × Sym6) : 2. In this note, we provide a short proof of existence and his result that the conjugacy class of this subgroup is the only one among those of non-local Lie primitive subgroups of finite dimensional simple complex Lie groups having a socle with more than one simple factor.

Détail

DOI : 10.4153/CJM-1993-005-7

Mots-clés : 22E20, 22E46

Bibliographie

[Adams 1986] Adams, J.F., 2-tori in exceptional Lie groups, preprint. Google Scholar

[Aleks 1974] Alekseevskii, A.V., Finite commutative Jordan subgroups of complex simple Lie groups, Funct. Anal, and its Appl. 8(1974), 277–279. Google Scholar

[Aschb 1984] Aschbacher, M., On the maximal subgroups of the finite classical groups, Inventiones Math. 76(1984), 469–514. Google Scholar

[Atlas 1985] Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.P. and Wilson, R.A., Atlas of finite groups, Clarendon Press, Oxford, 1985. Google Scholar

[Blich 1917] Blichfeldt, H.F., Finite Collineation Groups, University of Chicago Press, 1917. Google Scholar

[Borov 1989] Borovik, A., The structure of finite subgroups of simple algebraic groups, Algebra and Logic 28(1989), 249–279. Google Scholar

[Bourb 1968] Bourbaki, N., Groupes et algèbres de Lie, Chap. 4,5 et 6, Hermann, Paris, 1968. Google Scholar

[Brauer 1968] Brauer, R.D., On simple subgroups of order 5.3a.2/?, Bull. Amer. Math. Soc. 74(1968) 900-903. Google Scholar

[CLSS 1992] Cohen, A.M., Liebeck, M.W., Saxl, J. and Seitz, G.M., The local maximal subgroups of the exceptional groups of Lie type, Proceedings London Math. Soc. (1992), to appear. Google Scholar

[CoGr 1987] Cohen, A.M. and Griess, R.L., Jr., On finite simple subgroups of the complex Lie group of type Es, Proc. of Symposia in Pure Math. 47(1987), 367–405. Google Scholar

[CoGrLi 1992] Cohen, A.M., Griess, R.L., Jr. and Bert Lisser, The group L(2,61) embeds in the Lie group of type E%, Comm. in Algebra (1992), to appear. Google Scholar

[CoSe 1987] Cohen, A.M. and Seitz, G.M., The r-rank of the groups of exceptional Lie type, Indagationes Math 90(1987), 251–259. Google Scholar

[CoWa 1983] Cohen, A.M. and Wales, D.B., Finite subgroups of G2(C), Comm. Algebra 11(1983), 441–459. Google Scholar

[CoWa 1989] Finite subgroups of £6(C) and F4(C), preprint. Google Scholar

[Dynk 1957] Dynkin, E.B., Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transi. 6(1957), 111–244. Google Scholar

[Glau 1966] Glauberman, G., Central elements in corefree groups, Jour, of Algebra 4(1966), 403–420. Google Scholar

[Gor 1968] Gorenstein, D., Finite groups, Harper and Row, New York, 1968. Google Scholar

[Gr 1991] Griess, R.L., Jr., Elementary abelian p-subgroups of algebraic Groups, Geometria Dedicata 39(1991), 253–305. Google Scholar

[GrRy 1992] Griess, R.L., Jr. and E, A.J.. Ryba, Embeddings of £/3(8), Sz(8) and the Rudvalls group in algebraic groups of type E1, (1992), submitted. Google Scholar

[Kac 1985] Kac, V., Infinite Dimensional Lie algebras, Cambridge University Press, Cambridge, 1985. Google Scholar

[KMR 1989] Kleidman, P., Meierfrankenfeld, U. & Ryba, A. Constructiono/HiS andRu in E1(5),preprint. Google Scholar

[LiSe 1989] Liebeck, M.W. and Seitz, G.M., Maximal subgroups of exceptional groups of Lie type, finite and algebraic, Geom. Dedicata 35(1989), 353–387. Google Scholar

[Tits 1959] Tits, J., Sur la trialité et certains groupes qui s'en déduisent, Publ. Math. I.H.E.S. 2(1959), 14–60. Google Scholar