The Separation Theorem for Group Actions

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


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.


Permutation Group Finite Subset Separation Theorem Transitive Group Combinatorial Design 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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