Skip to main content
SpringerLink
Log in

The Hurwitz subgroups of \(E_6(2)\)

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

We prove that the exceptional group \(E_6(2)\) is not a Hurwitz group. In the course of proving this, we complete the classification up to conjugacy of all Hurwitz subgroups of \(E_6(2)\), in particular, those isomorphic to \(L_2(8)\) and \(L_3(2)\).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Conder, M.: An update on Hurwitz groups. Groups Complex. Cryptol. 2, 35–49 (2010)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  3. Frobenius, F. G.: Über Gruppencharaktere, Sitzungsber. Kön. Preuss. Akad. Wiss. Berlin, 985–1021 (1896)

  4. Howlett, R.B., Rylands, L.J., Taylor, D.E.: Matrix generators for exceptional groups of Lie type. J. Symb. Comput. 31, 429–445 (2001)

    Article  MathSciNet  Google Scholar 

  5. Hurwitz, A.: Ueber algebraische Gebilde mit eindeutigen Transformationen in sich. Math. Ann. 41, 403–442 (1892)

    Article  MathSciNet  Google Scholar 

  6. Jones, G.A.: Ree groups and Riemann surfaces. J. Algebra 165(1), 41–62 (1994)

    Article  MathSciNet  Google Scholar 

  7. Kleidman, P.B., Wilson, R.A.: The maximal subgroups of \(E_6(2)\) and Aut\((E_6(2))\). Proc. Lond. Math. Soc. (3) 60, 266–294 (1990)

  8. Malle, G.: Hurwitz groups and \(G_2(q)\). Can. Math. Bull. 33, 349–357 (1990)

    Article  Google Scholar 

  9. Malle, G.: Small rank exceptional Hurwitz groups, groups of Lie type and their geometries (Como: 1993). In: London Mathematical Society Lecture Note Series, vol. 207, pp. 173–183. Cambridge University Press, Cambridge (1993)

  10. The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.7.8, The GAP Group. http://www.gap-system.org (2015)

Download references

Acknowledgements

The author is grateful to Rob Wilson and Alastair Litterick for many helpful conversations in the preparation of this paper and to Marston Conder for assistance in the verification of these results. The author also thanks the referee for their thorough reading of the manuscript.

This work was supported by the SFB 701 and by ARC Grant DP140100416.

Author information

Author notes

  1. Emilio Pierro

    Present address: Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA,  6009, Australia

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, 33602, Bielefeld, Germany

    Emilio Pierro

Authors
  1. Emilio Pierro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emilio Pierro.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pierro, E. The Hurwitz subgroups of \(E_6(2)\). Arch. Math. 111, 457–468 (2018). https://doi.org/10.1007/s00013-018-1211-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-018-1211-z

Keywords

Mathematics Subject Classification

Search

Navigation