On the subgroup structure

of exceptional groups of Lie type

Authors:
Martin W. Liebeck and Gary M. Seitz

Journal:
Trans. Amer. Math. Soc. **350** (1998), 3409-3482

MSC (1991):
Primary 20G40, 20E28

DOI:
https://doi.org/10.1090/S0002-9947-98-02121-7

MathSciNet review:
1458329

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study finite subgroups of exceptional groups of Lie type, in particular maximal subgroups. Reduction theorems allow us to concentrate on almost simple subgroups, the main case being those with socle of Lie type in the natural characteristic. Our approach is to show that for sufficiently large (usually suffices), is contained in a subgroup of positive dimension in the corresponding exceptional algebraic group, stabilizing the same subspaces of the Lie algebra. Applications are given to the study of maximal subgroups of finite exceptional groups. For example, we show that all maximal subgroups of sufficiently large order arise as fixed point groups of maximal closed subgroups of positive dimension.

**[An]**H.H. Andersen, ``Filtrations of cohomology modules for Chevalley groups'',*Ann. Sci. Ec. Norm. Sup.***16**(1983), 495-528. MR**85k:20128****[AJL]**H.H. Andersen, J. Jorgensen and P. Landrock, ``The projective indecomposable modules of '',*Proc. London Math. Soc.***46**(1983), 38-52. MR**84f:20044****[As]**M. Aschbacher, ``On finite groups of Lie type and odd characteristic'',*J. Algebra***66**(1980), 400-424. MR**81k:20023****[Bor1]**A. Borel,*Linear Algebraic Groups*, Second Edition, Springer-Verlag, New York, 1991. MR**92d:20001****[Bor2]**A. Borel, ``Properties and linear representations of Chevalley groups'', in:*Seminar on algebraic groups and related topics*(ed. A. Borel et al.), Lecture Notes in Math. 131, Springer, Berlin, 1970, pp. 1-55. MR**41:3484****[BT]**A. Borel and J. Tits, ``Éléments unipotents et sous-groupes paraboliques de groupes réductifs'',*Invent. Math.***12**(1971), 95-104. MR**45:3419****[Bo]**N. Bourbaki,*Groupes et algèbres de Lie*(Chapters 4,5,6), Hermann, Paris, 1968. MR**39:1590****[BW]**N. Burgoyne and C. Williamson, ``Some computations involving simple Lie algebras'',*Proc. 2nd. Symp. Symbolic and Algebraic Manipulation*, (ed. S. Petrick), N.Y. Assoc. Computing Machinery, 1971, 162-171.**[CPSK]**E. Cline, B. Parshall, L. Scott and W. van der Kallen, ``Rational and generic cohomology'',*Invent. Math.***39**(1977), 143-163. MR**55:12737****[CG]**A.M. Cohen and R. Griess, ``On finite simple subgroups of the complex Lie group of type '',*Proc. Symp. Pure Math.***47**(1987), 367-405. MR**90a:20089****[DL]**D. Deriziotis and M.W. Liebeck, ``Centralizers of semisimple elements in finite twisted groups of Lie type'',*J. London Math. Soc.***31**(1985), 48-54. MR**87e:20087****[GS]**P. Gilkey and G.M. Seitz, ``Some representations of exceptional Lie algebras'',*Geom. Ded.***25**(1988), 407-416. MR**89h:20056****[GL]**D. Gorenstein and R. Lyons, ``The local structure of finite groups of characterstic type'',*Memoirs Amer. Math. Soc.***276**(1983). MR**84g:20025****[Ja]**J.C. Jantzen, ``Low dimensional representations of reductive groups are semisimple'', in*Algebraic groups and related subjects; a volume in honour of R.W. Richardson*(eds. G.I. Lehrer et al.), Austral. Math. Soc. Lecture Note Series, 1996.**[JP]**W. Jones and B. Parshall, ``On the -cohomology of finite groups of Lie type'',*Proc. Conf. on Finite Groups*(eds. W. Scott and F. Gross), Academic Press, 1976, pp. 313-328. MR**53:8272****[LT]**R. Lawther and D.M. Testerman, `` subgroups of exceptional algebraic groups'',*Trans. Amer. Math. Soc.*, to appear.**[Li]**M.W. Liebeck, ``On the orders of maximal subgroups of the finite classical groups'',*Proc. London Math. Soc.***50**(1985), 426-446. MR**87a:20046****[LS1]**M.W. Liebeck and G.M. Seitz, ``Maximal subgroups of exceptional groups of Lie type, finite and algebraic'',*Geom. Dedicata***36**(1990), 353-387. MR**91f:20032****[LS2]**M.W. Liebeck and G.M. Seitz, ``Reductive subgroups of exceptional algebraic groups'',*Memoirs Amer. Math. Soc.*, No. 580 (1996). MR**96i:20059****[LS3]**M.W. Liebeck and G.M. Seitz, ``Subgroups generated by root elements in groups of Lie type'',*Ann. of Math.***139**(1994), 293-361. MR**95d:20078****[LSS]**M.W. Liebeck, J. Saxl and G.M. Seitz, ``Subgroups of maximal rank in finite exceptional groups of Lie type'',*Proc. London Math. Soc.***65**(1992), 297-325. MR**93e:20046****[LST]**M.W. Liebeck, J. Saxl and D.M. Testerman, ``Simple subgroups of large rank in groups of Lie type'',*Proc. London Math. Soc.***72**(1996), 425-457. MR**96k:20087****[Pr]**O. Premet, ``Weights of infinitesimally irreducible representations of Chevalley groups over a field of prime characteristic'',*Math. USSR Sbornik***61**(1988), 167-183. MR**88h:20051****[Se1]**G.M. Seitz, ``The maximal subgroups of classical algebraic groups'',*Mem. Amer. Math. Soc.*, No. 365 (1987). MR**88g:20092****[Se2]**G.M. Seitz, ``Maximal subgroups of exceptional algebraic groups'',*Memoirs Amer. Math. Soc.*, No. 441 (1991). MR**91g:20038****[Se3]**G.M. Seitz, ``Representations and maximal subgroups of finite groups of Lie type'',*Geom. Dedicata***25**(1988), 391-406. MR**89d:20035****[Se4]**G.M. Seitz, ``Flag-transitive subgroups of Chevalley groups'',*Annals of Math.***97**(1973), 27-56. MR**49:5201****[ST1]**G.M. Seitz and D.M. Testerman, ``Extending morphisms from finite to algebraic groups'',*J. Algebra***131**(1990), 559-574. MR**91f:20054****[ST2]**G.M. Seitz and D.M. Testerman, ``Subgroups of type containing semiregular unipotent elements'',*J. Algebra***196**(1997), 595-619. CMP**98:02****[Sin]**P. Sin, ``Extensions of simple modules for and '',*Proc. London Math. Soc.***65**(1992), 265-296. MR**93e:20021****[SS]**T.A. Springer and R. Steinberg, ``Conjugacy classes'', in:*Seminar on algebraic groups and related topics*(ed. A. Borel et al.), Lecture Notes in Math. 131, Springer, Berlin, 1970, pp. 168-266. MR**42:3091****[St]**R. Steinberg,*Lectures on Chevalley Groups*, notes by J. Faulkner and R. Wilson, Yale University (1968). MR**57:6215****[Te1]**D.M. Testerman, ``Irreducible subgroups of exceptional algebraic groups'',*Mem. Amer. Math. Soc.*, No. 390 (1988). MR**90a:20082****[Te2]**D.M. Testerman, ``A construction of certain maximal subgroups of the algebraic groups and '',*J. Algebra***122**(1989), 299-322. MR**90d:20082**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
20G40,
20E28

Retrieve articles in all journals with MSC (1991): 20G40, 20E28

Additional Information

**Martin W. Liebeck**

Affiliation:
Department of Mathematics, Imperial College, London SW7 2BZ, United Kingdom

Email:
m.liebeck@ic.ac.uk

**Gary M. Seitz**

Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403

Email:
seitz@math.uoregon.edu

DOI:
https://doi.org/10.1090/S0002-9947-98-02121-7

Received by editor(s):
October 11, 1996

Additional Notes:
The authors acknowledge the support of NATO Collaborative Research Grant CRG 931394. The second author also acknowledges the support of an NSF Grant

Article copyright:
© Copyright 1998
American Mathematical Society