AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Overgroups of root groups in classical groups
About this Title
Michael Aschbacher, Department of Mathematics, California Institute of Technology, Pasadena, California 91125
Publication: Memoirs of the American Mathematical Society
Publication Year:
2016; Volume 241, Number 1140
ISBNs: 978-1-4704-1845-8 (print); 978-1-4704-2873-0 (online)
DOI: https://doi.org/10.1090/memo/1140
Published electronically: December 10, 2015
Keywords: Finite groups,
linear groups
MSC: Primary 20D06, 20E28, 20G40
Table of Contents
Chapters
- Introduction
- 1. 3-transpositions
- 2. The $(V,f)$-setup
- 3. Direct sum decompositions
- 4. Subfield structures
- 5. Modules for alternating groups
- 6. Modules with $p=2$
- 7. The orthogonal space $\mathbf {F}_2^n$
- 8. Overgroups of long root subgroups
- 9. Maximal overgroups of long root subgroups
- 10. Subgroups containing long root elements
- 11. Overgroups of short root subgroups
- 12. Short root subgroups in symplectic groups of characteristic 2
- 13. Overgroups of subgroups in $\mathbf {R}_c$ in III
- 14. Overgroups of subgroups in $\mathbf {R}_c$ in III when $q>3$
- 15. A special case for $q=3$ in III
- 16. Overgroups of subgroups in $\mathbf {R}_c$ in III when $q=3$
- 17. A result of Stellmacher
- 18. More case III with $q=3$
- 19. The proof of Theorem 1
- 20. A characterization of alternating groups
- 21. Orthogonal groups with $q=2$
- 22. The proof of Theorem 2
- 23. Symplectic and unitary groups
- 24. Symplectic and unitary groups with $q$ odd
- 25. The proof of Theorem 3
- 26. Unitary groups with $q$ even
- 27. The proofs of Theorems A and B
- References
Abstract
We extend results of McLaughlin and Kantor on overgroups of long root subgroups and long root elements in finite classical groups. In particular we determine the maximal subgroups of this form. We also determine the maximal overgroups of short root subgroups in finite classical groups, and the maximal overgroups in finite orthogonal groups of c-root subgroups.- Michael Aschbacher, A characterization of Chevalley groups over fields of odd order, Ann. of Math. (2) 106 (1977), no.ย 2, 353โ398. MR 498828, DOI 10.2307/1971100
- Michael Aschbacher, A characterization of Chevalley groups over fields of odd order. II, Ann. of Math. (2) 106 (1977), no.ย 3, 399โ468. MR 498829, DOI 10.2307/1971063
- Michael Aschbacher, Thin finite simple groups, J. Algebra 54 (1978), no.ย 1, 50โ152. MR 511458, DOI 10.1016/0021-8693(78)90022-4
- M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), no.ย 3, 469โ514. MR 746539, DOI 10.1007/BF01388470
- Michael Aschbacher, Overgroups of Sylow subgroups in sporadic groups, Mem. Amer. Math. Soc. 60 (1986), no.ย 343, iv+235. MR 831891, DOI 10.1090/memo/0343
- Michael Aschbacher, Overgroup lattices in finite groups of Lie type containing a parabolic, J. Algebra 382 (2013), 71โ99. MR 3034474, DOI 10.1016/j.jalgebra.2013.01.034
- Michael Aschbacher, On finite groups generated by odd transpositions. I, Math. Z. 127 (1972), 45โ56. MR 310058, DOI 10.1007/BF01110103
- Michael Aschbacher, Finite groups generated by odd transpositions, Finite groups โ72 (Proc. Gainesville Conf., Univ. Florida, Gainesville, Fla., 1972) North-Holland, Amsterdam, 1973, pp.ย 8โ11. North-Holland Math. Studies, Vol. 7. MR 0360793
- Michael Aschbacher and Gary M. Seitz, Involutions in Chevalley groups over fields of even order, Nagoya Math. J. 63 (1976), 1โ91. MR 422401
- Michael Aschbacher and Stephen D. Smith, The classification of quasithin groups. I, Mathematical Surveys and Monographs, vol. 111, American Mathematical Society, Providence, RI, 2004. Structure of strongly quasithin $K$-groups. MR 2097623
- Leonard Eugene Dickson, Linear groups: With an exposition of the Galois field theory, Dover Publications, Inc., New York, 1958. With an introduction by W. Magnus. MR 0104735
- Michael Aschbacher, Finite group theory, Cambridge Studies in Advanced Mathematics, vol. 10, Cambridge University Press, Cambridge, 1986. MR 895134
- Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1994. MR 1303592
- Robert M. Guralnick and Gunter Malle, Classification of $2F$-modules. II, Finite groups 2003, Walter de Gruyter, Berlin, 2004, pp.ย 117โ183. MR 2125071
- Robert M. Guralnick and Jan Saxl, Generation of finite almost simple groups by conjugates, J. Algebra 268 (2003), no.ย 2, 519โ571. MR 2009321, DOI 10.1016/S0021-8693(03)00182-0
- Wayne Jones and Brian Parshall, On the $1$-cohomology of finite groups of Lie type, Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975) Academic Press, New York, 1976, pp.ย 313โ328. MR 0404470
- William M. Kantor, Subgroups of classical groups generated by long root elements, Trans. Amer. Math. Soc. 248 (1979), no.ย 2, 347โ379. MR 522265, DOI 10.1090/S0002-9947-1979-0522265-1
- Peter Kleidman and Martin Liebeck, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990. MR 1057341
- Martin W. Liebeck and Gary M. Seitz, Subgroups generated by root elements in groups of Lie type, Ann. of Math. (2) 139 (1994), no.ย 2, 293โ361. MR 1274094, DOI 10.2307/2946583
- Martin W. Liebeck and Gary M. Seitz, Subgroups of simple algebraic groups containing elements of fundamental subgroups, Math. Proc. Cambridge Philos. Soc. 126 (1999), no.ย 3, 461โ479. MR 1684243, DOI 10.1017/S0305004199003497
- Jack McLaughlin, Some groups generated by transvections, Arch. Math. (Basel) 18 (1967), 364โ368. MR 222184, DOI 10.1007/BF01898827
- Jack McLaughlin, Some subgroups of $\textrm {SL}_{n}\,(\textbf {F}_{2})$, Illinois J. Math. 13 (1969), 108โ115. MR 237660
- V. V. Nesterov, Generation of pairs of short root subgroups in Chevalley groups, Algebra i Analiz 16 (2004), no.ย 6, 172โ208 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 16 (2005), no.ย 6, 1051โ1077. MR 2117453, DOI 10.1090/S1061-0022-05-00890-3
- A. A. Premet and I. D. Suprunenko, Quadratic modules for Chevalley groups over fields of odd characteristics, Math. Nachr. 110 (1983), 65โ96. MR 721267, DOI 10.1002/mana.19831100107
- Michael Aschbacher, 3-transposition groups, Cambridge Tracts in Mathematics, vol. 124, Cambridge University Press, Cambridge, 1997. MR 1423599
- F. G. Timmesfeld, Groups generated by root-involutions. I, II, J. Algebra 33 (1975), 75โ134; ibid. 35 (1975), 367โ441. MR 372019, DOI 10.1016/0021-8693(75)90133-7
- F. Timmesfeld, Groups generated by Root-Involutions, II, J. Alg. 35 (1975), 367โ441.
- F. G. Timmesfeld, Subgroups of Lie type groups containing a unipotent radical, J. Algebra 323 (2010), no.ย 5, 1408โ1431. MR 2584962, DOI 10.1016/j.jalgebra.2009.12.006