Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T14:44:09.989Z Has data issue: false hasContentIssue false
  • English
  • Français

The maximal subgroups of $E_{7}(2)$

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  01 April 2015

John Ballantyne ,
Chris Bates  and
Peter Rowley
John Ballantyne
Affiliation:
School of Mathematics, University of Manchester, Alan Turing Building, Oxford Road, Manchester M13 9PL, United Kingdom email [email protected]
Chris Bates
Affiliation:
School of Mathematics, University of Manchester, Alan Turing Building, Oxford Road, Manchester M13 9PL, United Kingdom email [email protected]
Peter Rowley
Affiliation:
School of Mathematics, University of Manchester, Alan Turing Building, Oxford Road, Manchester M13 9PL, United Kingdom email [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Here we determine up to conjugacy all the maximal subgroups of the finite exceptional group of Lie-type $E_{7}(2)$.

Supplementary materials are available with this article.

MSC classification

Primary: 20E28: Maximal subgroups 20E32: Simple groups
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 323 - 371
Copyright
© The Author(s) 2015 

References

Aschbacher, M., Finite group theory , 2nd edn, Cambridge Studies in Advanced Mathematics 10 (Cambridge University Press, Cambridge, 2000).Google Scholar
Aschbacher, M., ‘Some multilinear forms with large isometry groups’, Geom. Dedicata 25 (1988) no. 1–3, 417–465; Geometries and groups (Noordwijkerhout, 1986).CrossRefGoogle Scholar
Aschbacher, M., ‘The 27-dimensional module for E 6, I.’, Invent. Math. 89 (1987) no. 1, 159–195.Google Scholar
Aschbacher, M., ‘On the maximal subgroups of the finite classical groups’, Invent. Math. 76 (1984) no. 3, 469–514.Google Scholar
Aschbacher, M. and Seitz, G. M., ‘Involutions in Chevalley groups over fields of even order’, Nagoya Math. J. 63 (1976) 1–91.CrossRefGoogle Scholar
Bates, C., ‘The maximal subgroups of $E_{7}(2)$ and other topics’, PhD Thesis, The University of Manchester, 2006.Google Scholar
Bates, C. and Rowley, P., ‘Centralizers of real elements in finite groups’, Arch. Math. (Basel) 85 (2005) no. 6, 485–489.Google Scholar
Benson, D. J., Modular representation theory. New trends and methods , Lecture Notes in Mathematics 1081 (Springer, Berlin, 2006) Second printing of the 1984 original.Google Scholar
Benson, D. J., ‘The Loewy structure of the projective indecomposable modules for A 8 in characteristic 2’, Comm. Algebra 11 (1983) no. 13, 1395–1432.Google Scholar
Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24, (1997) no. 3–4, 235–265.Google Scholar
Bray, J. N., ‘An improved method for generating the centralizer of an involution’, Arch. Math. (Basel) 74 (2000) no. 4, 241–245.Google Scholar
Carter, R. W., Simple groups of Lie type , Wiley Classics Library (John Wiley and Sons, Inc., New York, 1989) Reprint of the 1972 original.Google Scholar
Cline, E., Parshall, B. and Scott, L., ‘Cohomology of finite groups of Lie type. I’, Publ. Math. Inst. Hautes Études Sci. 45 (1975) 169–191.Google Scholar
Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of finite groups (Clarendon Press, Oxford, 1985).Google Scholar
Cooperstein, B. N., ‘The fifty-six-dimensional module for E 7. I. A four form for E 7 ’, J. Algebra 173 (1995) no. 2, 361–389.CrossRefGoogle Scholar
Deriziotis, D. I. and Liebeck, M. W., ‘Centralizers of semisimple elements in finite twisted groups of Lie type’, J. Lond. Math. Soc. (2) 31 (1985) no. 1, 48–54.Google Scholar
Dickson, L. E., Linear groups, with an exposition of the Galois field theory (Dover, 1958) Reprint of the 1901 original.Google Scholar
Fleischmann, K. and Janiszczak, I., ‘The semisimple conjugacy classes of finite groups of Lie-type E 6 and E 7 ’, Comm. Algebra 21 (1993) no. 1, 93–161.Google Scholar
Galois, E., ‘Letter to A. Chevallier’, translated by A. R. Singh and C. S. Yogonanda,http://link.springer.com/content/pdf/10.1007%2FBF02834265.pdf.Google Scholar
The GAP group, ‘GAP—groups, algorithms, and programming, Version 4.4.12’, 2008,http://www.gap-system.org.Google Scholar
Gorenstein, D., Finite groups , 2d edn (Chelsea Publishing Co., New York, 1980).Google Scholar
Hartley, R. W., ‘Determination of the ternary collineation groups whose coefficients lie in the GF (2 n )’, Ann. of Math. (2) 27 (1925) 140–158.CrossRefGoogle Scholar
Jansen, C., Lux, K., Parker, R. and Wilson, R., An atlas of Brauer characters , London Mathematical Society Monographs 11 (Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995) Appendix 2 by T. Breuer and S. Norton.Google Scholar
Kleidman, P. B. and Wilson, R. A., ‘The maximal subgroups of E 6(2) and Aut(E 6(2))’, Proc. Lond. Math. Soc. (3) 60 (1990) no. 2, 266–294.Google Scholar
Lawther, R., ‘Sublattices generated by root differences’, J. Algebra 412 (2014) 255–263.Google Scholar
Liebeck, M. W. and Saxl, J., ‘On the orders of maximal subgroups of the finite exceptional groups of Lie type’, Proc. Lond. Math. Soc. (3) 55 (1987) no. 2, 299–330.Google Scholar
Liebeck, M. W., Saxl, J. and Seitz, G. M., ‘Subgroups of maximal rank in finite exceptional groups of Lie type’, Proc. Lond. Math. Soc. (3) 65 (1992) no. 2, 297–325.Google Scholar
Liebeck, M. W., Saxl, J. and Testerman, D. M., ‘Simple subgroups of large rank in groups of Lie type’, Proc. Lond. Math. Soc. (3) 72 (1996) no. 2, 425–457.CrossRefGoogle Scholar
Liebeck, M. W. and Seitz, G. M., ‘Maximal subgroups of large rank in exceptional groups of Lie type’, J. Lond. Math. Soc. (2) 71 (2005) no. 2, 345–361.Google Scholar
Liebeck, M. W. and Seitz, G. M., ‘A survey of maximal subgroups of exceptional groups of Lie type’, Groups, combinatorics and geometry (Durham, 2001) (World Scientific, Singapore, 2003).Google Scholar
Liebeck, M. W. and Seitz, G. M., ‘On finite subgroups of exceptional algebraic groups’, J. Reine Angew. Math. 515 (1999) 25–72.CrossRefGoogle Scholar
Liebeck, M. W. and Seitz, G. M., ‘On the subgroup structure of exceptional groups of Lie type’, Trans. Amer. Math. Soc. 350 (1998) no. 9, 3409–3482.CrossRefGoogle Scholar
Liebeck, M. W. and Seitz, G. M., ‘Maximal subgroups of exceptional groups of Lie type, finite and algebraic’, Geom. Dedicata 36 (1990) 353–387.Google Scholar
Lübeck, F., ‘Conjugacy classes and character degrees of $E_{7}(2)$ ’,http://www.math.rwth-aachen.de/∼Frank.Luebeck/chev/E72.html.Google Scholar
Lübeck, F., ‘Small degree representations of finite Chevalley groups in defining characteristic’, LMS J. Comput. Math. 4 (2001) 135–169.CrossRefGoogle Scholar
Mitchell, H. H., ‘Determination of the ordinary and modular ternary linear groups’, Trans. Amer. Math. Soc. 12 (1911) 207–242.Google Scholar
Parker, C. and Saxl, J., ‘Two maximal subgroups of E 8(2)’, Israel J. Math. 153 (2006) 307–318.Google Scholar
Seitz, G. M., ‘Maximal subgroups of finite exceptional groups’, Groups and geometries (Siena, 1996) , Trends in Mathematics (Birkhäuser, Basel, 1998) 155–161.CrossRefGoogle Scholar
Wilson, R. A., The finite simple groups , Graduate Texts in Mathematics 251 (Springer, London, 2009).Google Scholar
Wilson, R., Walsh, P., Tripp, J., Suleiman, I., Parker, R., Norton, S., Nickerson, S., Linton, S., Bray, J. and Abbot, R., ‘Atlas of finite group representations, version 3’,http://brauer.maths.qmul.ac.uk/Atlas/v3/.Google Scholar
Supplementary material: File

Ballantyne supplementary material S1

Ballantyne supplementary material

Download Ballantyne supplementary material S1(File)
File 3.4 MB