The Separation Theorem for Group Actions

  • Cheryl E. Praeger
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 354)

Abstract

Let G be a group of permutations of a set Ω, and let Γ and Δ be (not necessarily distinct) subsets of Ω. Under what conditions on G, or on Γ and Δ is it possible to separate Γ from Δ by an element of G, that is to have Γ g ⋂Δ = \(\not 0\) for some gG? In 1976 Peter M. Neumann [25, Lemma 2.3] proved that every finite subset can be separated from itself in this way provided that all the G-orbits in Ω are infinite.

Keywords

Oates 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Ja. G. Berkovich, G. A. Freiman and C. E. Praeger, Small squaring and cubing properties for finite groups, Bull. Austral. Math. Soc. 44 (1991), 429–450.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    B. J. Birch, R. G. Burns, S. O. Macdonald and P. M. Neumann, On the orbit sizes of permutation groups containing elements separating finite subsets, Bull. Austral. Math. Soc. 14 (1976), 7–10.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    L. Brailovsky, Structure of quasi-invariant sets, Arch. Math. (Basel) 59 (1992), 322–326.MathSciNetMATHGoogle Scholar
  4. [4]
    L. Brailovsky, A characterisation of abelian groups, Proc. Amer. Math. Soc. 117 (1993), 627–629.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    L. Brailovsky, On the small squaring and commutativity, Bull. London Math. Soc. 25 (1993), 330–336.MathSciNetCrossRefGoogle Scholar
  6. [6]
    L. Brailovsky, Combinatorial conditions forcing commutativity of an infinite group, J. Algebra, 165 (1994), 394–400.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    L. Brailovsky, D. V. Pasechnik and C. E. Praeger, Classification of 2-quasi- mvariant sets, Ars Combin., to appear.Google Scholar
  8. [8]
    L. Brailovsky, D. V. Pasechnik and C. E. Praeger, Subsets close to invariant subsets under group actions, Proc. Amer. Math. Soc., to appear.Google Scholar
  9. [9]
    A. R. Calderbank, Symmetric designs as the solution of an extremal problem in combinatorial set theory, European J. Combin. 8 (1987), 171–173.MathSciNetGoogle Scholar
  10. [10]
    J. R. Cho, P. S. Kim and C. E. Praeger, The maximum number of orbits of a permutation group with bounded movement, (in preparation).Google Scholar
  11. [11]
    P. Dembowski, Finite Geometries, Springer-Verlag, New York, (1967).Google Scholar
  12. [12]
    P. Frankl and Z. Füredi, Finite projective spaces and intersecting hypergraphs, Combinatorica 6 (1986), 335–354.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    G. A. Freiman, On two and three-element subsets of groups, Aeq. Math. 22 (1981), 140–152.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Z. Füredi, Maximum degree and fractional matchings in uniform hypergraphs, Combinatorica 1 (1981), 155–162.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Z. Füredi, Intersecting designs from linear programming and graphs of diameter two, Discrete Maths 127 (1994), 187–207.MATHCrossRefGoogle Scholar
  16. [16]
    A. Gardiner and C. E. Praeger, Transitive permutation groups with bounded movement, J. Algebra 168 (1994), 798–803.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    A. Gardiner and C. E. Praeger, Bounds on orbit sizes for permutation groups admitting subsets with restricted movement, (preprint, 1995 ).Google Scholar
  18. [18]
    B. Ganter, J. Pelikan and L. Tierlinck, Some sprawling systems of equicardmal sets, Ars Combin. 4 (1977), 133–142.MATHGoogle Scholar
  19. [19]
    A. S. Hedayatt, G. B. Khosrovshahi and D. Majumdar, A prospect for a general method of constructing t-designs, Discrete Appd. Math. 42 (1993), 31–50.CrossRefGoogle Scholar
  20. [20]
    M. Herzog, P. Longobardi and M. Maj, On a combinatorial problem in group theory, Israel J. Math. 82 (1993), 329–340.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    G. B. Khosrovshahi and C. E. Praeger, On the intersection problem for combinatorial designs, (in preparation).Google Scholar
  22. [22]
    P. Longobardi and M. Maj, The classification of groups with the small squaring property on 3-sets, Bull. Austral. Math. Soc. 46 (1992), 263–270.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    A. Mann and C. E. Praeger, Transitive permutation groups of minimal movement, J. Algebra, to appear.Google Scholar
  24. [24]
    B. H. Neumann, Groups covered by permutable subsets, J. London Math. Soc. 29 (1954), 236–248.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    P. M. Neumann, The structure of finitary permutation groups, Arch. Math. (Basel) 27 (1976), 3–17.MathSciNetMATHGoogle Scholar
  26. [26]
    P. M. Neumann, A combinatorial problem in group theory, (unpublished manuscript, 1989 ).Google Scholar
  27. [27]
    P. M. Neumann and C. E. Praeger, An inequality for tactical configurations, Bull. London Math. Soc. (to appear).Google Scholar
  28. [28]
    C. E. Praeger, On permutation groups with bounded movement, J. Algebra 144 (1991), 436–442.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    C. E. Praeger, Restricted movement for intransitive group actions, Proceedings of Groups Korea 1994, (to appear).Google Scholar
  30. [30]
    C. E. Praeger, Movement and separation of subsets of points under group actions, (preprint, 1995 ).Google Scholar
  31. [31]
    A. P. Street and D. J. Street, Combinatorics of Experimental Design, Clarendon Press, Oxford, 1987.MATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Cheryl E. Praeger
    • 1
  1. 1.University of Western AustraliaNedlandsAustralia

Personalised recommendations