Skip to main content
SpringerLink
Log in

Regular Sets and Geometric Groups

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

If G is a permutation group acting on a set Ω, a subset Λ of Ω is called a regular set for G if the set-stabilizer of Λ in G is the identity subgroup. We show here that the projective and affine semi-linear groups acting in the natural way as permutation groups on their respective finite geometries, have, in general, for all finite dimensions and all finite fields, regular sets of points. The exceptions to this are found, and an extension of the results to infinite fields is discussed.

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. D. Betten, Geometrische Permutationsgruppen, Mitt. Math. Gesellschaft Hamburg 10 (1977) 317–324.

    MathSciNet  MATH  Google Scholar 

  2. P. J. Cameron, P. M. Neumann and J. Saxl, On groups with no regular orbits on the set of subsets, Arch. Math. 43 (1984) 295–296.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. J. Cannon, “A Language for Group Theory”, Cayley Manual, Sydney University, July 1982.

  4. F. Dalla Volta, Private communication.

  5. P. Dembowski, “Finite Geometries”, Springer, Berlin, 1968.

  6. D. Gluck, Trivial set-stabilizers in finite permutation groups, Can. J. Math. Vol. XXXV, No. 1 (1983) 59–67.

    Article  MathSciNet  Google Scholar 

  7. N. F. J. Inglis, On orbit equivalent permutation groups, Arch. Math. 43 (1984) 297–300.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. D. Key and J. Siemons, On the k-closure of finite linear groups, Bollettino UMI (to appear).

  9. J. D. Key and J. Siemons, Closure properties of the special linear groups, Ars Combinatoria (to appear).

  10. J. D. Key, J. Siemons and A. Wagner, Regular sets on the projective line, J. Geometry (to appear).

  11. R. S. Pierce, Permutation representations with trivial set-stabilizers, J. Algebra 95 (1985) 88–95.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Siemons, On partitions and permutation groups on unordered sets, Arch. Math. 38 (1982) 391–403.

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Siemons and A. Wagner, On finite permutation groups with the same orbits on unordered sets, Arch. Math. 45 (1985) 492–500.

    Article  MathSciNet  MATH  Google Scholar 

  14. C. C. Sims, Primitive groups of degree less than 50, (unpublished).

  15. H. Wielandt, “Finite Permuation Groups”, Academic Press, New York, 1964.

    Google Scholar 

  16. H. Wielandt, “Permutation Groups through Invariant Relations and Invariant Functions”, Lecture Notes, Ohio State University, 1969.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Birmingham, PO Box 363, Birmingham, B15 2TT, UK

    Jennifer D. Key

  2. School of Mathematics and Physics, University of East Anglia, University Plain, Norwich, NR4 7TJ, UK

    Johannes Siemons

Authors
  1. Jennifer D. Key
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Johannes Siemons
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Financial support from SERC is acknowledged. e]19850531

Rights and permissions

Reprints and permissions

About this article

Cite this article

Key, J.D., Siemons, J. Regular Sets and Geometric Groups. Results. Math. 11, 97–116 (1987). https://doi.org/10.1007/BF03323262

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03323262

Keywords

Search

Navigation