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  3. Linear Algebraic Groups and Finite Groups of Lie Type
  • Cited by 159
Publisher:
Cambridge University Press
Online publication date:
June 2012
Print publication year:
2011
Online ISBN:
9780511994777
DOI:
https://doi.org/10.1017/CBO9780511994777
Subjects:
Mathematics (general), Mathematics, Algebra
Series:
Cambridge Studies in Advanced Mathematics (133)
Information

Book description

Originating from a summer school taught by the authors, this concise treatment includes many of the main results in the area. An introductory chapter describes the fundamental results on linear algebraic groups, culminating in the classification of semisimple groups. The second chapter introduces more specialized topics in the subgroup structure of semisimple groups and describes the classification of the maximal subgroups of the simple algebraic groups. The authors then systematically develop the subgroup structure of finite groups of Lie type as a consequence of the structural results on algebraic groups. This approach will help students to understand the relationship between these two classes of groups. The book covers many topics that are central to the subject, but missing from existing textbooks. The authors provide numerous instructive exercises and examples for those who are learning the subject as well as more advanced topics for research students working in related areas.

Reviews

"This book provides a concise introduction to the theory of linear algebraic groups over an algebraically closed field (of arbitrary charachteristic) and the closely related finite groups of Lie type. Although there are several good books covering a similar range of topics, some important recent developments are treated here for the first time.
This book is well written and the style of exposition is clear and reader-friendly, making it suitable for graduate students. The content is well organized, and the authors have sensibly avoided overloading the text with technical details."
Timothy C. Burness for Mathematical Reviews

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Contents

Page 1 of 2

Contents

Page 1 of 2

References
References
[1] M., Aschbacher, On the maximal subgroups of the finite classical groups. Invent. Math. 76 (1984 Google Scholar), 469–514.
[2] M., Aschbacher, Finite Group Theory. Second edition. Cambridge Studies in Advanced Mathematics, 10. Cambridge University Press, Cambridge, 2000 Google Scholar.
[3] H. B., Azad, M., Barry, G. M., Seitz, On the structure of parabolic subgroups. Comm. Algebra 18 (1990 Google Scholar), 551–562.
[4] A., Borel, Linear Algebraic Groups. Second edition. Graduate Texts in Mathematics, 126. Springer-Verlag, New York, 1991 Google Scholar.
[5] A., Borel, J., de Siebenthal, Les sous-groupes fermés de rang maximum des groupes de Lie clos. Comment. Math. Helv. 23 (1949 Google Scholar), 200–221.
[6] A., Borel, J., Tits, Groupes, réductifs. Inst. Hautes Études Sci. Publ. Math. 27 (1965 Google Scholar), 55–150.
[7] A., Borel, J., Tits, Eléments unipotents et sous-groupes paraboliques de groupes réductifs. I. Invent. Math. 12 (1971 Google Scholar), 95–104.
[8] A. V., Borovik, The structure of finite subgroups of simple algebraic groups. (Russian)Algebra i Logika 28 (1989 Google Scholar), 249–279, 366; translation in Algebra and Logic28 (1989), 163–182 (1990).
[9] N., Bourbaki, Groupes et Algèbres de Lie. IV, V, VI, Hermann, Paris, 1968 Google Scholar.
[10] M., Broué, G., Malle, Théorèmes de Sylow génériques pour les groupes réductifs sur les corps finis. Math. Ann. 292 (1992 Google Scholar), 241–262.
[11] J., Brundan, Double coset density in classical algebraic groups. Trans. Amer. Math. Soc. 352 (2000 Google Scholar), 1405–1436.
[12] M., Cabanes, Unicité du sous-groupe abélien distingué maximal dans certains sous-groupes de Sylow. C. R. Acad. Sci. Paris Sér. I Math. 318 (1994 Google Scholar), 889–894.
[13] R. W., Carter, Simple Groups of Lie Type. John Wiley & Sons, London, 1972 Google Scholar.
[14] R. W., Carter, Finite Groups of Lie Type—Conjugacy Classes and Complex Characters. John Wiley & Sons, New York, 1985 Google Scholar.
[15] C., Chevalley, Classification des Groupes Algébriques Semi-simples. Collected Works. Vol. 3. Springer-Verlag, Berlin, 2005 Google Scholar.
[16] E., Cline, B., Parshall, L., Scott, On the tensor product theorem for algebraic groups. J. Algebra 63 (1980 Google Scholar), 264–267.
[17] A. M., Cohen, M. W., Liebeck, J., Saxl, G. M., Seitz, The local maximal subgroups of exceptional groups of Lie type, finite and algebraic. Proc. London Math. Soc.(3) 64 (1992 Google Scholar), 21–48.
[18] B. N., Cooperstein, Maximal subgroups of G2(2n). J. Algebra 70 (1981 Google Scholar), 23–36.
[19] C. W., Curtis, I., Reiner, Representation Theory of Finite Groups and Associative Algebras. John Wiley & Sons, New York, 1962 Google Scholar.
[20] D. I., Deriziotis, Centralizers of semisimple elements in a Chevalley group. Comm. Algebra 9 (1981 Google Scholar), 1997–2014.
[21] J. A., Dieudonné, La Géométrie des Groupes Classiques. Third edition. Springer-Verlag, Berlin, 1971 Google Scholar.
[22] F., Digne, J., Michel, Fonctions L des Variétés de Deligne–Lusztig et Descente de Shintani. Mém. Soc. Math. France (N.S.) 20 (1985 Google Scholar).
[23] E. B., Dynkin, Semisimple subalgebras of semisimple Lie algebras. Amer. Math. Soc., Transl., II. Ser. 6 (1957 Google Scholar), 111–243.
[24] E. B., Dynkin, Maximal subgroups of the classical groups. Amer. Math. Soc., Transl., II. Ser. 6 (1957 Google Scholar), 245–378.
[25] A., Fröhlich, M. J., Taylor, Algebraic Number Theory. Cambridge Studies in Advanced Mathematics, 27. Cambridge University Press, Cambridge, 1993 Google Scholar.
[26] M., Geck, An Introduction to Algebraic Geometry and Algebraic Groups. Oxford Graduate Texts in Mathematics, 10. Oxford University Press, Oxford, 2003 Google Scholar.
[27] R., Goodman, N., Wallach, Representations and Invariants of the Classical Groups. Encyclopedia of Mathematics and its Applications, 68. Cambridge University Press, Cambridge, 1998 Google Scholar.
[28] D., Gorenstein, Finite Groups. Chelsea Publishing Company, New York, 1980 Google Scholar.
[29] D., Gorenstein, R., Lyons, R., Solomon, The Classification of Finite Simple Groups, Number 3. Mathematical Surveys and Monographs, 40. American Mathematical Society, Providence, RI, 1998 Google Scholar.
[30] L., Grove, Classical Groups and Geometric Algebra. Graduate Studies in Mathematics, 39. American Mathematical Society, Providence, RI, 2002 Google Scholar.
[31] G., Hiss, Die adjungierten Darstellungen der Chevalley-Gruppen. Arch. Math. (Basel) 42 (1984 Google Scholar), 408–416.
[32] J., Humphreys, Linear Algebraic Groups. Graduate Texts in Mathematics, 21. Springer-Verlag, New York, 1975 Google Scholar.
[33] J., Humphreys, Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, Second printing, 1980 Google Scholar.
[34] J., Humphreys, Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge, 1992 Google Scholar.
[35] J., Humphreys, Conjugacy Classes in Semisimple Algebraic Groups. Mathematical Surveys and Monographs, 43. American Mathematical Society, Providence, RI, 1995 Google Scholar.
[36] J., Humphreys, Modular Representations of Finite Groups of Lie Type. LMS Lecture Notes Series, 326. Cambridge University Press, Cambridge, 2006 Google Scholar.
[37] B., Huppert, Endliche Gruppen. I. Grundlehren der Mathematischen Wissenschaften, 134. Springer-Verlag, Berlin, 1967 Google Scholar.
[38] B., Huppert, N., Blackburn, Finite Groups. II. Grundlehren der Mathematis-chen Wissenschaften, 242. Springer-Verlag, Berlin, 1982 Google Scholar.
[39] I. M., Isaacs, Character Theory of Finite Groups. Dover, New York, 1994 Google Scholar.
[40] J.C., Jantzen, Darstellungen halbeinfacher algebraischer Gruppen und zugeordnete kontravariante Formen. Bonner math. Schr. 67 (1973 Google Scholar).
[41] J.C., Jantzen, Representations of Algebraic Groups. Second edition. Mathematical Surveys and Monographs, 107. American Mathematical Society, Providence, RI, 2003 Google Scholar.
[42] P., Kleidman, The maximal subgroups of the Steinberg triality groups 3D4(q) and of their automorphism groups. J. Algebra 115 (1988 Google Scholar), 182–199.
[43] P., Kleidman, The maximal subgroups of the Chevalley groups G2(q)with q odd, the Ree groups 2G2(q), and their automorphism groups. J. Algebra 117 (1988 Google Scholar), 30–71.
[44] P., Kleidman, M. W., Liebeck, The Subgroup Structure of the Finite Classical Groups. London Mathematical Society Lecture Note Series, 129. Cambridge University Press, Cambridge, 1990 Google Scholar.
[45] V., Landazuri, G. M., Seitz, On the minimal degrees of projective representations of the finite Chevalley groups. J. Algebra 32 (1974 Google Scholar), 418–443.
[46] R., Lawther Google Scholar, Sublattices generated by root differences, preprint.
[47] G. I., Lehrer, D.E., Taylor, Unitary Reflection Groups. Australian Mathematical Society Lecture Series, 20. Cambridge University Press, Cambridge, 2009 Google Scholar.
[48] M. W., Liebeck, G. M., Seitz, Maximal subgroups of exceptional groups of Lie type, finite and algebraic. Geom. Dedicata 35 (1990 Google Scholar), 353–387.
[49] M. W., Liebeck, G. M., Seitz, Reductive Subgroups of Exceptional Algebraic Groups. Memoirs Amer. Math. Soc., 121 (1996 Google Scholar).
[50] M. W., Liebeck, G. M., Seitz, On the subgroup structure of classical groups. Invent. Math. 134 (1998 Google Scholar), 427–453.
[51] M. W., Liebeck, G. M., Seitz, On the subgroup structure of exceptional groups of Lie type. Trans. Amer. Math. Soc. 350 (1998 Google Scholar), 3409–3482.
[52] M. W., Liebeck, G.M., Seitz, A survey of maximal subgroups of exceptional groups of Lie type. Groups, Combinatorics & Geometry (Durham, 2001), World Sci. Publ., River Edge, NJ, 2003 Google Scholar, pp. 139–146.
[53] M.W., Liebeck, G.M., Seitz, The Maximal Subgroups of Positive Dimension in Exceptional Algebraic Groups. Memoirs Amer. Math. Soc., 802 (2004 Google Scholar).
[54] F., Lübeck, Small degree representations of finite Chevalley groups in defining characteristic. LMS J. Comput. Math. 4 (2001 Google Scholar), 135–169.
[55] G., Malle, The maximal subgroups of 2F4(q2). J. Algebra 139 (1991 Google Scholar), 52–69.
[56] G., Malle, Height 0 characters of finite groups of Lie type. Represent. Theory 11 (2007 Google Scholar), 192–220.
[57] R., Ree, A family of simple groups associated with the simple Lie algebra of type (F4). Amer.J.Math. 83 (1961 Google Scholar) 401–420.
[58] R., Ree, A family of simple groups associated with the simple Lie algebra of type (G2). Amer.J.Math. 83 (1961 Google Scholar), 432–462.
[59] R. W., Richardson, Finiteness theorems for orbits of algebraic groups. Nederl. Akad. Wetensch. Indag. Math. 47 (1985 Google Scholar), 337–344.
[60] G. M., Seitz, The Maximal Subgroups of Classical Algebraic Groups. Memoirs Amer. Math. Soc., 67 (1987 Google Scholar).
[61] G. M., Seitz, Maximal Subgroups of Exceptional Algebraic Groups. Memoirs Amer. Math. Soc., 90 (1991 Google Scholar).
[62] G. M., Seitz, D. M., Testerman, Extending morphisms from.nite to algebraic groups. J. Algebra 131 (1990 Google Scholar), 559–574.
[63] S. D., Smith, Irreducible modules and parabolic subgroups. J. Algebra 75 (1982 Google Scholar), 286–289.
[64] N., Spaltenstein, Classes Unipotentes et Sous-Groupes de Borel. Lecture Notes in Mathematics, 946. Springer-Verlag, Berlin, 1982 Google Scholar.
[65] T. A., Springer, Regular elements of.nite re.ection groups. Invent. Math. 25 (1974 Google Scholar), 159–198.
[66] T. A., Springer, Linear Algebraic Groups. Second edition. Progress in Mathematics, 9. Birkhäuser, Boston, 1998 Google Scholar.
[67] T. A., Springer, R., Steinberg, Conjugacy classes. In: Seminar on Algebraic Groups and Related Finite Groups. Lecture Notes in Mathematics, 131. Springer-Verlag, Berlin, 1970 Google Scholar, pp. 167–266.
[68] R., Steinberg, Variations on a theme of Chevalley. Paci.c J. Math. 9 (1959 Google Scholar), 875–891.
[69] R., Steinberg, Automorphisms of classical Lie algebras. Paci.c J. Math. 11 (1961 Google Scholar), 1119–1129.
[70] R., Steinberg, Representations of algebraic groups. Nagoya Math. J. 22 (1963 Google Scholar), 33–56.
[71] R., Steinberg, Regular elements of semisimple algebraic groups. Inst. Hautes ´ Etudes Sci. Publ. Math. 25 (1965 Google Scholar), 49–80.
[72] R., Steinberg, Endomorphisms of Linear Algebraic Groups. Memoirs Amer. Math. Soc., 80 (1968 Google Scholar).
[73] R., Steinberg, Lectures on Chevalley Groups. Notes prepared by J., Faulkner and R., Wilson. Yale University, New Haven, Conn., 1968 Google Scholar.
[74] R., Steinberg, Torsion in reductive groups. Advances in Math. 15 (1975 Google Scholar), 63–92.
[75] R., Steinberg, Conjugacy Classes in Algebraic Groups. Lecture Notes in Mathematics, Vol. 366. Springer-Verlag, Berlin, 1974 Google Scholar.
[76] R., Steinberg, On theorems of Lie–Kolchin, Borel, and Lang. In: Contributions to Algebra (Collection of Papers Dedicated to Ellis Kolchin). Academic Press, New York, 1977 Google Scholar, pp. 349–354.
[77] I., Suprunenko, Conditions on the irreducibility of restrictions of irreducible representations of the group SL(n, K) to connected algebraic subgroups. Preprint #13, (222), Ins. Mat. Akad.Nauk BSSR (1985 Google Scholar) (in Russian).
[78] M., Suzuki, On a class of doubly transitive groups. Ann. of Math. (2) 75 (1962 Google Scholar), 105–145.
[79] D. E., Taylor, The Geometry of the Classical Groups. Heldermann Verlag, Berlin, 1992 Google Scholar.
[80] D., Testerman, Irreducible Subgroups of Exceptional Algebraic Groups. Memoirs Amer. Math. Soc., 75 (1988 Google Scholar).
[81] D., Testerman, A construction of certain maximal subgroups of the algebraic groups E 6 and F 4. J. Algebra 122 (1989 Google Scholar), 299–322.
[82] J., Tits, Algebraic and abstract simple groups. Ann. of Math. (2) 80 (1964 Google Scholar), 313–329.

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