Finite simple groups which projectively embed in an exceptional Lie group are classified!

Griess, Robert, Jr.; Ryba, A.

doi:10.1090/S0273-0979-99-00771-5

Finite simple groups which projectively embed in an exceptional Lie group are classified!

Authors: Robert L. Griess Jr. and A. J. E. Ryba Jr.
Journal: Bull. Amer. Math. Soc. 36 (1999), 75-93
MSC (1991): Primary 17Bxx, 20Bxx, 20Cxx, 20Dxx, 20Exx, 22Exx
DOI: https://doi.org/10.1090/S0273-0979-99-00771-5
MathSciNet review: 1653177
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Abstract: Since finite simple groups are the building blocks of finite groups, it is natural to ask about their occurrence ``in nature''. In this article, we consider their occurrence in algebraic groups and moreover discuss the general theory of finite subgroups of algebraic groups.

[Alek] A. V. Alekseevskiĭ, Jordan finite commutative subgroups of simple complex Lie groups, Funkcional. Anal. i Priložen. 8 (1974), no. 4, 1–4 (Russian). MR 0379748
[Bor89] A. V. Borovik, The structure of finite subgroups of simple algebraic groups, Algebra i Logika 28 (1989), no. 3, 249–279, 366 (Russian); English transl., Algebra and Logic 28 (1989), no. 3, 163–182 (1990). MR 1066315, https://doi.org/10.1007/BF01978721
[Bor90] A. V. Borovik, Finite subgroups of simple algebraic groups, Dokl. Akad. Nauk SSSR 309 (1989), no. 4, 784–786 (Russian); English transl., Soviet Math. Dokl. 40 (1990), no. 3, 570–573. MR 1037652
[BS] A. Borel and J.-P. Serre, Sur certains sous-groupes de Lie compacts, Comment. Hath. Helv., 27(1953), 128-139. MR 14:948d
[Car] Roger W. Carter, Simple groups of Lie type, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1989. Reprint of the 1972 original; A Wiley-Interscience Publication. MR 1013112
[CG] Arjeh M. Cohen and Robert L. Griess Jr., On finite simple subgroups of the complex Lie group of type 𝐸₈, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 367–405. MR 933426
[CG93] Arjeh M. Cohen and Robert L. Griess Jr., Nonlocal Lie primitive subgroups of Lie groups, Canad. J. Math. 45 (1993), no. 1, 88–103. MR 1200322, https://doi.org/10.4153/CJM-1993-005-7
[CGL] Arjeh M. Cohen, Robert L. Griess Jr., and Bert Lisser, The group 𝐿(2,61) embeds in the Lie group of type 𝐸₈, Comm. Algebra 21 (1993), no. 6, 1889–1907. MR 1215552, https://doi.org/10.1080/00927879308824659
[CS] Arjeh M. Cohen and Gary M. Seitz, The 𝑟-rank of the groups of exceptional Lie type, Nederl. Akad. Wetensch. Indag. Math. 49 (1987), no. 3, 251–259. MR 914084
[CW83] Arjeh M. Cohen and David B. Wales, Finite subgroups of 𝐺₂(𝐶), Comm. Algebra 11 (1983), no. 4, 441–459. MR 689418, https://doi.org/10.1080/00927878308822857
[CW93] Arjeh M. Cohen and David B. Wales, Embeddings of the group 𝐿(2,13) in groups of Lie type 𝐸₆, Israel J. Math. 82 (1993), no. 1-3, 45–86. MR 1239045, https://doi.org/10.1007/BF02808108
[CW95] Arjeh M. Cohen and David B. Wales, Finite simple subgroups of semisimple complex Lie groups—a survey, Groups of Lie type and their geometries (Como, 1993) London Math. Soc. Lecture Note Ser., vol. 207, Cambridge Univ. Press, Cambridge, 1995, pp. 77–96. MR 1320515, https://doi.org/10.1017/CBO9780511565823.008
[CW97] Arjeh M. Cohen and David B. Wales, Finite subgroups of 𝐹₄(𝐶) and 𝐸₆(𝐶), Proc. London Math. Soc. (3) 74 (1997), no. 1, 105–150. MR 1416728, https://doi.org/10.1112/S0024611597000051
[Cox] Coxeter, H. S. M., Integral Cayley numbers, Duke Math. J. 13 (1946), 561-578. MR 8:370b
[Fe] Walter Feit, Characters of finite groups, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0219636
[FoG] Paul Fong and Robert L. Griess Jr., An infinite family of elementwise-conjugate nonconjugate homomorphisms, Internat. Math. Res. Notices 5 (1995), 249–252. MR 1333751, https://doi.org/10.1155/S1073792895000195
[F1] Darrin Frey, Conjugacy of alternating groups of degree 5 and subgroups of the complex Lie group of type $E_{8}$ , Thesis, University of Michigan, 1995.
[F2] Darrin Frey, Conjugacy of alternating groups of degree 5 and subgroups of the complex Lie group of type $E_{8}$ , Memoirs of the American Mathematical Society, to appear.
[F3] Darrin Frey, Conjugacy of alternating groups of degree 5 and subgroups of the complex Lie group of types $F_{4}$ and $E_{6}$ , to appear in Journal of Algebra.
[FrG] D. Frey and R. Griess, The conjugacy classes of elements in the Borovik group, Journal of Algebra 203 (1998), 226-243. CMP 98:12
[Frob] G. Frobenius, Über die cogredienten Transformationen der bilinearen Formen, S.-B. Preuss. Akad. Wiss. (Berlin) 7-16 (1896); Gesammelte Abhandlungen, II , 695-704.
[Gor] Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
[GrElAb] Robert L. Griess Jr., Elementary abelian 𝑝-subgroups of algebraic groups, Geom. Dedicata 39 (1991), no. 3, 253–305. MR 1123145, https://doi.org/10.1007/BF00150757
[GrG2] Robert L. Griess Jr., Basic conjugacy theorems for 𝐺₂, Invent. Math. 121 (1995), no. 2, 257–277. MR 1346206, https://doi.org/10.1007/BF01884298
[GrTwelve] Robert L. Griess, Jr., Twelve Sporadic Groups, Springer Mathematical Monograph, Springer Verlag, 1998.
[GRU] Robert L. Griess Jr. and A. J. E. Ryba, Embeddings of 𝑈₃(8),𝑆𝑧(8) and the Rudvalis group in algebraic groups of type 𝐸₇, Invent. Math. 116 (1994), no. 1-3, 215–241. MR 1253193, https://doi.org/10.1007/BF01231561
[GR31] Robert L. Griess, Jr. and A. J. E. Ryba, Embeddings of and in $E_8(\mathbb{C})$ , Duke Math. Journal 94 (1998), 181-211. CMP 98:16
[GR41] Robert L. Griess, Jr. and A. J. E. Ryba, Embeddings of and in $E_8(\mathbb{C})$ , to appear in Journal of Symbolic Computation.
[GRQ] Robert L. Griess, Jr. and A. J. E. Ryba, The finite quasisimple groups which embed in exceptional Lie groups. Preprint.
[GR8] Robert L. Griess, Jr. and A. J. E. Ryba, Embeddings of into exceptional Lie groups. Preprint.
[Hup] B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
[I] I. Martin Isaacs, Character theory of finite groups, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Pure and Applied Mathematics, No. 69. MR 0460423
[J] Zvonimir Janko, A new finite simple group with abelian Sylow 2-subgroups and its characterization, J. Algebra 3 (1966), 147–186. MR 193138, https://doi.org/10.1016/0021-8693(66)90010-X
[K] Michael J. Kantor, and Subgroups of $E_8(\mathbb{C})~$ and their Actions on a Maximal Torus, Thesis, University of Michigan, 1996.
[KR] Peter B. Kleidman and A. J. E. Ryba, Kostant’s conjecture holds for 𝐸₇:𝐿₂(37)<𝐸₇(𝐶), J. Algebra 161 (1993), no. 2, 535–540. MR 1247371, https://doi.org/10.1006/jabr.1993.1234
[Kul] B. Kulshammer, Algebraic representations of finite groups, Universität Augsburg, 1992.
[Lar1] Michael Larsen, On the conjugacy of element-conjugate homomorphisms, Israel J. Math. 88 (1994), no. 1-3, 253–277. MR 1303498, https://doi.org/10.1007/BF02937514
[Lar2] Michael Larsen, On the conjugacy of element-conjugate homomorphisms. II, Quart. J. Math. Oxford Ser. (2) 47 (1996), no. 185, 73–85. MR 1380951, https://doi.org/10.1093/qmath/47.1.73
[Mal] A. I. Mal'cev, Semisimple subgroups of Lie groups, Amer. Math. Soc. Translations 1, 172-273 (1962).
[McK] John McKay (ed.), Finite groups—coming of age, Contemporary Mathematics, vol. 45, American Mathematical Society, Providence, RI, 1985. MR 822230
[Sep] Mark R. Sepanski, Kostant’s conjecture and 𝐿₂(𝑞) invariant theory in the rank two Lie groups, Comm. Algebra 24 (1996), no. 6, 1915–1938. MR 1386020, https://doi.org/10.1080/00927879608825680
[S96] Jean-Pierre Serre, Exemples de plongements des groupes 𝑃𝑆𝐿₂(𝐹_{𝑝}) dans des groupes de Lie simples, Invent. Math. 124 (1996), no. 1-3, 525–562 (French). MR 1369427, https://doi.org/10.1007/s002220050062
[SP] J-P. Serre, Personal Communication, 1998.
[Slo] Peter Slodowy, Two notes on a finiteness problem in the representation theory of finite groups, Hamburger Beiträge zur Mathematik, aus dem Mathematischen Seminar, Heft 21; 1993. Published in ``Algebraic Groups and Lie Groups" (A volume of papers in honour of the late R.W. Richardson), Ed. G.I. Lehrer, Australian Math. Soc. Lecture Series No. 9, Cambridge University Press, Cambridge, 1997; pages 331-346. CMP 98:16
[Spr] T. A. Springer, Regular elements of finite reflection groups, Invent. Math. 25 (1974), 159–198. MR 354894, https://doi.org/10.1007/BF01390173
[Tits55] Jacques Tits, Sous-algèbres des algèbres de Lie semi-simples (d'apres V. Morosov, A. Malcev, E.Dynkin et F. Karpelevitch), Séminaire Bourbaki, No. 119, May 1955. CMP 98:09
[W] David B. Wales, Finite linear groups of degree seven. I, Canadian J. Math. 21 (1969), 1042–1056. MR 248237, https://doi.org/10.4153/CJM-1969-115-9
[Weil] André Weil, Remarks on the cohomology of groups, Ann. of Math. (2) 80 (1964), 149–157. MR 169956, https://doi.org/10.2307/1970495