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On the Phan system of the Schur cover of SU(4, 32)

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Abstract

This article is part of the program described in Bennett et al. (in: Ivanov et al. (ed.) Groups, combinatorics, and geometry, 2003) [3]. We study the Phan amalgams and their universal completions that occur for q = 3 in rank n = 3 for the diagram A 3 = D 3, corresponding to SU(4, 32) ≅ Spin (6, 3). We show that the associated geometries admit universal 9-fold coverings, by showing that the universal completion of the Phan amalgam is the central extension of SU(4, 32) by its Schur multiplier. This information provides the last missing piece of information in the full classification of Phan amalgams and their universal completions for A n and D n .

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References

  1. 1.

    Aschbacher M. (1977) A characterization of Chevalley groups over fields of odd order, Parts I, II. Ann. Math. 106: 353–468

  2. 2.

    Buekenhout F., Cohen A.: Diagram geometry (version of 2 January 2006). http://www.win.tue.nl/~amc/buek, Accessed 31 October 2007.

  3. 3.

    Bennett C., Gramlich R., Hoffman C., Shpectorov S. (2003) Curtis-Phan-Tits theory. In: Ivanov A.A., Liebeck M.W., Saxl J. (eds) Groups, Combinatorics, and Geometry. World Scientific, River Edge, pp 13–29

  4. 4.

    Bennett C., Gramlich R., Hoffman C., Shpectorov S. (2007) Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 1: general simple connectedness. J. Algebra 312: 426–444

  5. 5.

    Bennett C., Shpectorov S. (2004) A new proof of a theorem of Phan. J. Group Theory 7: 287–310

  6. 6.

    Cohen A., Postma E. (2005) Covers of point-hyperplane graphs. J. Algebraic Combin. 22: 317–329

  7. 7.

    The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.4, http://www.gap-system.org (2005), Accessed 31 October 2007.

  8. 8.

    Gamble G., Havas G., Hulpke A., Ramsay C.: ACE: a GAP 4 interface to the Advanced Coset Enumerator of G. Havas and C. Ramsay. http://www.math.rwth-aachen.de/~Greg.Gamble/ACE/, (2000), Accessed 31 October 2007.

  9. 9.

    Gramlich R.: Phan Theory. TU Darmstadt Habilitationsschrift, (2004). http://www.mathematik.tu-darmstadt.de/~gramlich/docs/habil.pdf, Accessed 31 October 2007.

  10. 10.

    Gramlich R., Horn M., Nickel W. (2006) The complete Phan-type theorem for Sp(2n, q). J. Group Theory 9: 603–626

  11. 11.

    Gramlich R., Horn M., Nickel W. (2007) Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 2: machine computations. J. Algebra 316: 591–607

  12. 12.

    Gramlich R., Hoffman C., Nickel W., Shpectorov S. (2005) Even-dimensional orthogonal groups as amalgams of unitary groups. J. Algebra 284: 141–173

  13. 13.

    Gramlich R.: Developments in Phan Theory. http://www.mathematik.tu-darmstadt.de/~gramlich/docs/survey2.pdf, Accessed 31 October 2007.

  14. 14.

    Horn M.: Amalgams of Unitary Groups in Sp(2n, q). TU Darmstadt Diplomarbeit, (2005).

  15. 15.

    Phan K.-W. (1977) On groups generated by three-dimensional special unitary groups, I. J. Austral. Math. Soc. Ser. A 23: 67–77

  16. 16.

    Phan K.-W. (1977) On groups generated by three-dimensional special unitary groups, II. J. Austral. Math. Soc. Ser. A 23: 129–146

  17. 17.

    Serre J.-P.: Arbres, Amalgames, SL2. Soc. Math. France, Paris (1977).

  18. 18.

    Tits J. (1986) Ensembles Ordonnés, immeubles et sommes amalgamées. Bull. Soc. Math. Belg. Sér. A 38: 367–387

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Correspondence to Max Horn.

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Horn, M. On the Phan system of the Schur cover of SU(4, 32). Des. Codes Cryptogr. 47, 243–247 (2008). https://doi.org/10.1007/s10623-007-9147-5

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Keywords

  • Phan theory
  • Phan amalgam
  • Group amalgams
  • Incidence geometry
  • Computer algebra
  • GAP
  • Coset enumeration
  • ACE

AMS Classifications

  • 20G40
  • 20E42
  • 51E24
  • 57M07