Skip to main content
SpringerLink
Log in

Regulus-free Spreads of PG(3,q)

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

Abstract

An old conjecture of Bruck and Bose is that every spreadof Σ = PG(3,q) could be obtained by startingwith a regular spread and reversing reguli. Although it was quicklyrealized that this conjecture is false, at least for qeven, there still remains a gap in the spaces for which it isknown that there are spreads which are regulus-free. In severalpapers Denniston, Bruen, and Bruen and Hirschfeld constructedspreads which were regulus–free, but none of these dealtwith the case when q is a prime congruent to onemodulo three. This paper closes that gap by showing that forany odd prime power q, spreads of PG(3,q) yielding nondesarguesian flag-transitive planes are regulus–free.The arguments are interesting in that they are based on elementarylinear algebra and the arithmetic of finite fields.

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. J. André, Uber nicht-Desarguesian Ebenen mit transitiver Translationgruppe, Math. Z., Vol. 60 (1954) pp. 156–186.

    Google Scholar 

  2. R. D. Baker and G. L. Ebert, Construction of two-dimensional flag-transitive planes, Geom. Dedicata, Vol. 27 (1988) pp. 9–14.

    Google Scholar 

  3. R. D. Baker and G. L. Ebert, Enumeration of two-dimensional flag-transitive planes, Algebras, Groups and Geometries, Vol. 3 (1985) pp. 248–257.

    Google Scholar 

  4. R. H. Bruck, Construction problems of finite projective planes, Combinatorial Mathematics and Its Appli-cations, North Carolina Press (1969) pp. 426–514.

  5. R. H. Bruck and R. C. Bose, The construction of translation planes from projective spaces, J. Alg., Vol. 1 (1964) pp. 85–102.

    Google Scholar 

  6. A. A. Bruen, Spreads and a conjecture of Bruck and Bose, J. Alg., Vol. 23 (1972) pp. 1–19.

    Google Scholar 

  7. A. A. Bruen and J. W. P. Hirschfeld, Applications of line geometry over finite fields I: The twisted cubic, Geom. Dedicata, Vol. 6 (1977) pp. 495–509.

    Google Scholar 

  8. R. H. F. Denniston, Spreads which are not subregular, Glasnik Mat., Vol. 8 (1973) pp. 3–5.

    Google Scholar 

  9. G. L. Ebert, Partitioning projective geometries into caps, Canad. J. Math., Vol. 37 (1985) pp. 1163–1175.

    Google Scholar 

  10. G. L. Ebert, Spreads obtained from ovoidal fibrations, Finite Geometries, Lecture Notes in Pure and Applied Mathematics, Vol. 103, (C. A. Baker and L. M. Batten, eds.), Marcel Dekker (1985) pp. 117–126.

  11. D. A. Foulser, The flag transitive collineation groups of the finite Desarguesian affine planes, Canad. J. Math., Vol. 16 (1964) pp. 443–472.

    Google Scholar 

  12. D. A. Foulser, Solvable flag transitive affine groups, Math. Z., Vol. 86 (1964) pp. 191–204.

    Google Scholar 

  13. C. Hering, A new class of quasifields, Math. Z., Vol. 118 (1970) pp. 56–57.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. West Virginia StateCollege, Institute, WV, 25112-1000

    R. D. Baker

  2. Department of Mathematical Sciences, Universityof Delaware, Newark, DE, 19716

    G. L. Ebert

Authors
  1. R. D. Baker
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. L. Ebert
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Baker, R.D., Ebert, G.L. Regulus-free Spreads of PG(3,q). Designs, Codes and Cryptography 8, 79–89 (1996). https://doi.org/10.1023/A:1018024723184

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1018024723184

Search

Navigation