Skip to main content
SpringerLink
Log in

Exceptional Groups

  • Chapter
  • First Online:
Cherlin’s Conjecture for Finite Primitive Binary Permutation Groups

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2302))

  • 260 Accesses

Abstract

In this chapter we prove Cherlin’s conjecture for exceptional groups of Lie type. Our main result shows that, if G is an almost simple primitive group having socle an exceptional group of Lie type, then G is not binary. The proof uses some of the results in the previous chapters together with a detailed analysis of the maximal subgroups of G.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 19.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 29.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. J. Ballantyne, C. Bates, P. Rowley, The maximal subgroups of E 7(2). LMS J. Comput. Math. 18, 323–371 (2015)

    Article  MathSciNet  Google Scholar 

  2. N. Bourbaki, Elements of Mathematics. Lie Groups and Lie Algebras. Chapters 4–6. Transl. from the French by A. Pressley (Springer, Berlin, 2002)

    Google Scholar 

  3. T.C. Burness, A.R. Thomas, On the involution fixity of exceptional groups of Lie type. Int. J. Algebra Comput. 28(3), 411–466 (2018)

    Article  MathSciNet  Google Scholar 

  4. J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, Atlas of Finite Groups (Oxford University Press, Oxford, 1985)

    MATH  Google Scholar 

  5. D.A. Craven, Maximal PSL 2 subgroups of exceptional groups of lie type. To appear in Mem. Amer. Math. Soc.

    Google Scholar 

  6. D.A. Craven, On medium-rank lie primitive and maximal subgroups of exceptional groups of lie type. To appear in Mem. Amer. Math. Soc.

    Google Scholar 

  7. D.A. Craven, Alternating subgroups of exceptional groups of Lie type. Proc. Lond. Math. Soc. 115(3), 449–501 (2017)

    Article  MathSciNet  Google Scholar 

  8. D. Craven, D. Stewart, A. Thomas, A new maximal subgroup of E 8 in characteristic 3. To appear in Proc. Amer. Math. Soc.

    Google Scholar 

  9. F.Dalla Volta, N. Gill, P. Spiga, Cherlin’s conjecture for sporadic simple groups. Pac. J. Math. 297(1), 47–66 (2018)

    Article  MathSciNet  Google Scholar 

  10. D.I. Deriziotis, The centralizers of semisimple elements of the Chevalley groups E 7 and E 8. Tokyo J. Math. 6, 191–216 (1983)

    Article  MathSciNet  Google Scholar 

  11. D.I. Deriziotis, M.W. Liebeck, Centralizers of semisimple elements in finite twisted groups of Lie type. J. Lond. Math. Soc. 31, 48–54 (1985)

    Article  MathSciNet  Google Scholar 

  12. H. Enomoto, The characters of the finite Chevalley group G 2(q), q = 3f. Japan. J. Math. 2, 191–248 (1976)

    Article  MathSciNet  Google Scholar 

  13. M. Geck, Generalized Gelfand-Graev characters for Steinberg’s triality groups and their applications. Commun. Algebra 19(12), 3249–3269 (1991)

    Article  MathSciNet  Google Scholar 

  14. N. Gill, F. Hunt, P. Spiga, Cherlin’s conjecture for almost simple groups of Lie rank 1. Math. Proc. Camb. Philos. Soc. 167(3), 417–435 (2019)

    Article  MathSciNet  Google Scholar 

  15. D. Gorenstein, R. Lyons, R. Solomon, The Classification of the Finite Simple Groups. Number 3. Part I. Chapter A. Mathematical Surveys and Monographs, vol. 40 (American Mathematical Society, Providence, 1998)

    Google Scholar 

  16. F. Himstedt, Character tables of parabolic subgroups of Steinberg’s triality groups 3 D 4(2n). J. Algebra 316(1), 254–283 (2007)

    Article  MathSciNet  Google Scholar 

  17. W.M. Kantor. Á. Seress, Prime power graphs for groups of Lie type. J. Algebra 247(2), 370–434 (2002)

    Google Scholar 

  18. P.B. Kleidman, The maximal subgroups of the Steinberg triality groups \({{ }^3\!D_4}(q)\) and their automorphism groups. J. Algebra 115(1), 182–199 (1988)

    Google Scholar 

  19. P.B. Kleidman, R.A. Wilson, The maximal subgroups of E 6(2) and Aut(E 6(2)). Proc. London Math. Soc. 60(2), 266–294 (1990)

    Article  MathSciNet  Google Scholar 

  20. R. Lawther, Unipotent classes in maximal subgroups of exceptional algebraic groups. J. Algebra 322(1), 270–293 (2009)

    Article  MathSciNet  Google Scholar 

  21. R. Lawther, Sublattices generated by root differences. J. Algebra 412, 255–263 (2014)

    Article  MathSciNet  Google Scholar 

  22. M.W. Liebeck, J. Saxl, The primitive permutation groups of odd degree. J. Lond. Math. Soc. 31, 250–264 (1985)

    Article  MathSciNet  Google Scholar 

  23. M.W. Liebeck, G.M. Seitz, Subgroups generated by root elements in groups of Lie type. Ann. Math. 139(2), 293–361 (1994)

    Article  MathSciNet  Google Scholar 

  24. M.W. Liebeck, G.M. Seitz, On finite subgroups of exceptional algebraic groups. J. Reine Angew. Math. 515, 25–72 (1999)

    Article  MathSciNet  Google Scholar 

  25. M.W. Liebeck, G.M. Seitz, A survey of maximal subgroups of exceptional groups of Lie type, in Groups, Combinatorics and Geometry. Proceedings of the L. M. S. Durham Symposium, Durham, 2001 (World Scientific, River Edge, 2003), pp. 139–146

    Google Scholar 

  26. M.W. Liebeck, G.M. Seitz, The Maximal Subgroups of Positive Dimension in Exceptional Algebraic Groups. Memoirs of the American Mathematical Society, vol. 802 (American Mathematical Society, Providence, 2004)

    Google Scholar 

  27. M.W. Liebeck, C.E. Praeger, J. Saxl, The Maximal Factorizations of the Finite Simple Groups and Their Automorphism Groups. Memoirs of the American Mathematical Society, vol. 432 (American Mathematical Society, Providence, 1990)

    Google Scholar 

  28. M.W. Liebeck, J. Saxl, G.M. Seitz, Subgroups of maximal rank in finite exceptional groups of Lie type. Proc. Lond. Math. Soc. 65(2), 297–325 (1992)

    Article  MathSciNet  Google Scholar 

  29. F. Lübeck, Conjugacy classes and character degrees of E 7(2) (2019). http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/E72.html?LANG=en. Accessed 13 March 2019

  30. S.P. Norton, R.A. Wilson, The maximal subgroups of F 4(2) and its automorphism group. Comm. Algebra 17(11), 2809–2824 (1989)

    Article  MathSciNet  Google Scholar 

  31. G.M. Seitz, Maximal Subgroups of Exceptional Algebraic Groups. Memoirs of the American Mathematical Society, vol. 441 (American Mathematical Society, Providence, 1991)

    Google Scholar 

  32. K. Shinoda, The conjugacy classes of Chevalley groups of type (F 4) over finite fields of characteristic 2. J. Fac. Sci., Univ. Tokyo, Sect. I A 21, 133–159 (1974)

    Google Scholar 

  33. K. Shinoda, The conjugacy classes of the finite Ree groups of type (F 4). J. Fac. Sci., Univ. Tokyo, Sect. I A 22, 1–15 (1975)

    Google Scholar 

  34. C.C. Sims, Graphs and finite permutation groups. Math. Z. 95, 76–86 (1967)

    Article  MathSciNet  Google Scholar 

  35. A.R. Thomas, Simple irreducible subgroups of exceptional algebraic groups. J. Algebra 423, 190–238 (2015)

    Article  MathSciNet  Google Scholar 

  36. R.A. Wilson, Maximal subgroups of 2 E 6(2) and its automorphism groups (2018). Preprint available: https://arxiv.org/abs/1801.08374

  37. J. Wiscons, A reduction theorem for primitive binary permutation groups. Bull. Lond. Math. Soc. 48(2), 291–299 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, The Open University, Milton Keynes, UK

    Nick Gill

  2. Department of Mathematics, Imperial College London, London, UK

    Martin W. Liebeck

  3. Dipartimento di Matematica pura e Applicata, University of Milano-Bicocca, Milano, Italy

    Pablo Spiga

Authors
  1. Nick Gill
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin W. Liebeck
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pablo Spiga
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Gill, N., Liebeck, M.W., Spiga, P. (2022). Exceptional Groups. In: Cherlin’s Conjecture for Finite Primitive Binary Permutation Groups. Lecture Notes in Mathematics, vol 2302. Springer, Cham. https://doi.org/10.1007/978-3-030-95956-2_3

Download citation

Publish with us

Policies and ethics

Search

Navigation