Skip to main content
Springer Nature Link
Log in

Combinatorial geometries representable over GF(3) and GF(q). II. Dowling geometries

  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

Letq be an odd prime power not divisible by 3. In Part I of this series, it was shown that the number of points in a rank-n combinatorial geometry (or simple matroid) representable over GF(3) and GF(q) is at mostn 2. In this paper, we show that, with the exception ofn = 3, a rank-n geometry that is representable over GF(3) and GF(q) and contains exactlyn 2 points is isomorphic to the rank-n Dowling geometry based on the multiplicative group of GF(3).

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. Dowling, T.A.: Aq-analog of the partition lattice. In: A Survey of Combinatorial Theory (J.N. Srivastava, ed.), pp. 101–115. Amsterdam: North-Holland 1973

    Google Scholar 

  2. Dowling, T.A.: A class of geometric lattices based on finite groups. J. Comb. Theory (B)14, 61–86 (1973)

    Google Scholar 

  3. Heller, I.: On linear systems with integral valued solutions. Pacific J. Math.7, 1351–1354 (1957)

    Google Scholar 

  4. Kantor, W.M.: Dimension and embedding theorems for geometric lattices. J. Comb. Theory (A)17, 173–95 (1974)

    Google Scholar 

  5. Kung, J.P.S.: Combinatorial geometries representable over GF(3) and GF(q). I. The number of points. Discrete and Computational Geometry (to appear)

  6. Seymour, P.D.: Matroid representation over GF(3). J. Comb. Theory (B)26, 159–173 (1979)

    Google Scholar 

  7. Seymour, P.D.: Decomposition of regular matroids. J. Comb. Theory (B)28, 305–359 (1980)

    Google Scholar 

  8. Tutte, W.T.: A homotopy theorem for matroids, II. Trans. Amer. Math. Soc.88, 161–174 (1958)

    Google Scholar 

  9. Welsh, D.J.A.: Matroid theory. London: Academic Press 1976

    Google Scholar 

  10. White, N.L. (ed.): Theory of matroids. Cambridge: Cambridge University Press 1986

    Google Scholar 

  11. Zaslavsky, T.: Signed graphs. Discrete Appl. Math.4, 47–74 (1982): erratum, ibid.5, 248 (1983)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, North Texas State University, 76203, Denton, TX, USA

    Joseph P. S. Kung

  2. Department of Mathematics, Louisiana State University, 70803, Baton Rouge, LA, USA

    James G. Oxley

Authors
  1. Joseph P. S. Kung
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. James G. Oxley
    View author publications

    You can also search for this author inPubMed Google Scholar

Additional information

This research was partially supported by the National Science Foundation under Grants DMS-8521826 and DMS-8500494.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kung, J.P.S., Oxley, J.G. Combinatorial geometries representable over GF(3) and GF(q). II. Dowling geometries. Graphs and Combinatorics 4, 323–332 (1988). https://doi.org/10.1007/BF01864171

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01864171

Keywords