Finite simple groups which projectively embed in an exceptional Lie group are classified!
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- by Robert L. Griess Jr. and A. J. E. Ryba PDF
- Bull. Amer. Math. Soc. 36 (1999), 75-93 Request permission
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.References
- 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
- 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, DOI 10.1007/BF01978721
- 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
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- 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
- Arjeh M. Cohen and Robert L. Griess Jr., On finite simple subgroups of the complex Lie group of type $E_8$, 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
- 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, DOI 10.4153/CJM-1993-005-7
- Arjeh M. Cohen, Robert L. Griess Jr., and Bert Lisser, The group $L(2,61)$ embeds in the Lie group of type $E_8$, Comm. Algebra 21 (1993), no. 6, 1889–1907. MR 1215552, DOI 10.1080/00927879308824659
- 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, DOI 10.1016/1385-7258(87)90014-X
- Arjeh M. Cohen and David B. Wales, Finite subgroups of $G_{2}(\textbf {C})$, Comm. Algebra 11 (1983), no. 4, 441–459. MR 689418, DOI 10.1080/00927878308822857
- Arjeh M. Cohen and David B. Wales, Embeddings of the group $L(2,13)$ in groups of Lie type $E_6$, Israel J. Math. 82 (1993), no. 1-3, 45–86. MR 1239045, DOI 10.1007/BF02808108
- 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, DOI 10.1017/CBO9780511565823.008
- Arjeh M. Cohen and David B. Wales, Finite subgroups of $F_4(\textbf {C})$ and $E_6(\textbf {C})$, Proc. London Math. Soc. (3) 74 (1997), no. 1, 105–150. MR 1416728, DOI 10.1112/S0024611597000051
- A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8, DOI 10.1017/S0370164600012281
- Walter Feit, Characters of finite groups, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0219636
- 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, DOI 10.1155/S1073792895000195
- Darrin Frey, Conjugacy of alternating groups of degree 5 and $SL(2,5)$ subgroups of the complex Lie group of type $E_{8}$, Thesis, University of Michigan, 1995.
- Darrin Frey, Conjugacy of alternating groups of degree 5 and $SL(2,5)$ subgroups of the complex Lie group of type $E_{8}$, Memoirs of the American Mathematical Society, to appear.
- Darrin Frey, Conjugacy of alternating groups of degree 5 and $SL(2,5)$ subgroups of the complex Lie group of types $F_{4}$ and $E_{6}$, to appear in Journal of Algebra.
- D. Frey and R. Griess, The conjugacy classes of elements in the Borovik group, Journal of Algebra 203 (1998), 226-243.
- G. Frobenius, Über die cogredienten Transformationen der bilinearen Formen, S.-B. Preuss. Akad. Wiss. (Berlin) 7-16 (1896); Gesammelte Abhandlungen, II , 695-704.
- Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
- Robert L. Griess Jr., Elementary abelian $p$-subgroups of algebraic groups, Geom. Dedicata 39 (1991), no. 3, 253–305. MR 1123145, DOI 10.1007/BF00150757
- Robert L. Griess Jr., Basic conjugacy theorems for $G_2$, Invent. Math. 121 (1995), no. 2, 257–277. MR 1346206, DOI 10.1007/BF01884298
- Robert L. Griess, Jr., Twelve Sporadic Groups, Springer Mathematical Monograph, Springer Verlag, 1998.
- Robert L. Griess Jr. and A. J. E. Ryba, Embeddings of $\textrm {U}_3(8),\ \textrm {Sz}(8)$ and the Rudvalis group in algebraic groups of type $E_7$, Invent. Math. 116 (1994), no. 1-3, 215–241. MR 1253193, DOI 10.1007/BF01231561
- Robert L. Griess, Jr. and A. J. E. Ryba, Embeddings of $PGL(2,31)$ and $SL(2,32)$ in $E_8(\mathbb {C})$, Duke Math. Journal 94 (1998), 181-211.
- Robert L. Griess, Jr. and A. J. E. Ryba, Embeddings of $PSL(2,41)$ and $PSL(2,49)$ in $E_8(\mathbb {C})$, to appear in Journal of Symbolic Computation.
- Robert L. Griess, Jr. and A. J. E. Ryba, The finite quasisimple groups which embed in exceptional Lie groups. Preprint.
- Robert L. Griess, Jr. and A. J. E. Ryba, Embeddings of $Sz(8)$ into exceptional Lie groups. Preprint.
- B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703, DOI 10.1007/978-3-642-64981-3
- I. Martin Isaacs, Character theory of finite groups, Pure and Applied Mathematics, No. 69, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0460423
- Zvonimir Janko, A new finite simple group with abelian Sylow $2$-subgroups and its characterization, J. Algebra 3 (1966), 147–186. MR 193138, DOI 10.1016/0021-8693(66)90010-X
- Michael J. Kantor, $SL(2,7)$ and $PSL(2,7)$ Subgroups of $E_8(\mathbb {C})$ and their Actions on a Maximal Torus, Thesis, University of Michigan, 1996.
- Peter B. Kleidman and A. J. E. Ryba, Kostant’s conjecture holds for $E_7\colon \ L_2(37)<E_7(\textbf {C})$, J. Algebra 161 (1993), no. 2, 535–540. MR 1247371, DOI 10.1006/jabr.1993.1234
- B. Kulshammer, Algebraic representations of finite groups, Universität Augsburg, 1992.
- Michael Larsen, On the conjugacy of element-conjugate homomorphisms, Israel J. Math. 88 (1994), no. 1-3, 253–277. MR 1303498, DOI 10.1007/BF02937514
- Michael Larsen, On the conjugacy of element-conjugate homomorphisms. II, Quart. J. Math. Oxford Ser. (2) 47 (1996), no. 185, 73–85. MR 1380951, DOI 10.1093/qmath/47.1.73
- A. I. Mal’cev, Semisimple subgroups of Lie groups, Amer. Math. Soc. Translations 1, 172-273 (1962).
- John McKay (ed.), Finite groups—coming of age, Contemporary Mathematics, vol. 45, American Mathematical Society, Providence, RI, 1985. MR 822230, DOI 10.1090/conm/045
- Mark R. Sepanski, Kostant’s conjecture and $L_2(q)$ invariant theory in the rank two Lie groups, Comm. Algebra 24 (1996), no. 6, 1915–1938. MR 1386020, DOI 10.1080/00927879608825680
- Jean-Pierre Serre, Exemples de plongements des groupes $\textrm {PSL}_2(\textbf {F}_p)$ dans des groupes de Lie simples, Invent. Math. 124 (1996), no. 1-3, 525–562 (French). MR 1369427, DOI 10.1007/s002220050062
- J-P. Serre, Personal Communication, 1998.
- 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.
- T. A. Springer, Regular elements of finite reflection groups, Invent. Math. 25 (1974), 159–198. MR 354894, DOI 10.1007/BF01390173
- 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.
- David B. Wales, Finite linear groups of degree seven. I, Canadian J. Math. 21 (1969), 1042–1056. MR 248237, DOI 10.4153/CJM-1969-115-9
- André Weil, Remarks on the cohomology of groups, Ann. of Math. (2) 80 (1964), 149–157. MR 169956, DOI 10.2307/1970495
Additional Information
- Robert L. Griess Jr.
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1003
- Email: rlg@math.lsa.umich.edu
- A. J. E. Ryba
- Affiliation: Department of Mathematics, Marquette University, Milwaukee, WI 53201-1881
- Address at time of publication: Department of Mathematics, Queens College, CUNY, Flushing, NY 11367-1597
- Email: alexr@sylow.mscs.mu.edu
- Received by editor(s): April 13, 1998
- Received by editor(s) in revised form: May 19, 1998, and October 16, 1998
- © Copyright 1999 American Mathematical Society
- 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