Skip to main content
SpringerLink
Log in

A translation plane of order 192 admitting SL(2,5), obtained by 12-nest replacement

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

The problem of determining two-dimensional translation planes admitting SL(2,5) in the translation complement has been intensively studied. Most examples arise from multiple-derivation of a Desarguesian plane. In this paper, we construct two translation planes of order 192 admitting SL(2,5), one of which is obtained by 12-nest replacement.

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. Abatangelo V (1984) A translation plane of order 81 and its full collineation group. Bull Aust Math Soc 29:19–34

    Article  MATH  MathSciNet  Google Scholar 

  2. Abatangelo V, Larato B (1986) Translation planes with an automorphism group isomorphic to SL(2,5). Ann Discrete Math 30:1–8

    MathSciNet  Google Scholar 

  3. Baker RD, Ebert GL (1996) Filling the nest gaps. Finite Fields Appl 2:42–61

    Article  MATH  MathSciNet  Google Scholar 

  4. Billiotti M, Korchmaros G (1990) Some finite translation planes arising from A 6-invariant ovoids of the Klein Quadric. J Geom 37:29–47

    Article  MathSciNet  Google Scholar 

  5. Bonisoli A, Korchmaros G, Szonyi T (1997) Some multiply derived translation planes with SL(2,5) as an inherited collineation group in the translation complement. Des Codes Cryptogr 10:109–114

    Article  MATH  MathSciNet  Google Scholar 

  6. Bruck RH (1969) Construction problems of finite projective planes. In: Proceedings of the conference combinatorial mathematics. University of North Carolina Press, Chapel Hill, pp 426–514

  7. Bruen AA (1978) Inversive geometry and some translation planes, I. Geom Dedicata 7:81–98

    MATH  MathSciNet  Google Scholar 

  8. Charnes C, Dempwolff U (1995) Spreads, ovoids and S 5. Geom Dedicata 56:129–143

    Article  MATH  MathSciNet  Google Scholar 

  9. Huppert B (1967) Endliche Gruppen I. Springer, Berlin Heidelberg

    MATH  Google Scholar 

  10. Johnson NL, Prince AR (2005) The translation planes of order 81 that admit SL(2,5), generated by affine elations. J Geom 84:73–82

    Article  MATH  MathSciNet  Google Scholar 

  11. Johnson NL, Prince AR (2005) Orbit constructions for translation planes of order 81 admitting SL(2,5). Note Mat 24:41–57

    MATH  MathSciNet  Google Scholar 

  12. Mason G, Shult EE (1986) The Klein correspondence and the ubiquity of certain translation planes. Geom Dedicata 21:29–50

    Article  MATH  MathSciNet  Google Scholar 

  13. Prohaska O (1977) Konfigurationen einander meidender Kreise in miquelschen Möbiusebenen ungerader Ordnung. Arch Math 28:550–556

    Article  MATH  MathSciNet  Google Scholar 

  14. Weida RA (1989) An extension of Bruen chains. Geom Dedicata 30:11–21

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    Alan R. Prince

Authors
  1. Alan R. Prince
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alan R. Prince.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Prince, A.R. A translation plane of order 192 admitting SL(2,5), obtained by 12-nest replacement. Des. Codes Cryptogr. 44, 25–30 (2007). https://doi.org/10.1007/s10623-007-9045-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-007-9045-x

Keywords

AMS Classification

Search

Navigation