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Designs, Codes and Cryptography

, Volume 34, Issue 2–3, pp 265–281 | Cite as

Square-Free Non-Cayley Numbers. On Vertex-Transitive Non-Cayley Graphs of Square-Free Order

  • Ákos SeressEmail author
Article

Abstract

A complete classification is given of finite primitive permutation groups which contain a regular subgroup of square-free order. Then a collection \({\cal P}{\cal N}{\cal C}\) of square-free numbers n is obtained such that there exists a vertex-primitive non-Cayley graph on n vertices if and only if n is a member of \({\cal P}{\cal N}{\cal C}\).

Keywords

Data Structure Information Theory Discrete Geometry Permutation Group Complete Classification 
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.

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References

  1. Alspach, B., Parsons, T. 1982A construction for vertex-transitive graphsCanad. J. Math.34307318zbMATHMathSciNetGoogle Scholar
  2. Alspach, B., Sutcliffe, R. 1979Vertex-transitive graphs of order 2p Ann. New York Acad. Sci.3191827MathSciNetGoogle Scholar
  3. Biggs, N. L. 1993Algebraic Graph Theory, Third ed.Cambridge University PressCambridgeGoogle Scholar
  4. Borel, A., Tits, J. 1971Elements unipotent et sous-groupes paraboliques de groupes reductifs IInvent. Math.1295104zbMATHMathSciNetGoogle Scholar
  5. Conway, J., Curtis, R., Norton, S., Parker, R., Wilson, R. 1985Atlas of Finite GroupsOxford University PressOxfordzbMATHGoogle Scholar
  6. Dixon, J. D., Mortimer, B. 1996Permutation GroupsSpringer-VerlagNew York, BerlinzbMATHGoogle Scholar
  7. Gamble, G., Praeger, C. E. 2000Vertex-primitive groups and graphs of order twice the product of two distinct odd primesJ. Group Theory3247269zbMATHMathSciNetGoogle Scholar
  8. The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.3; 2002, (http://www.gap-system.org).Google Scholar
  9. Hassani, A., Iranmanesh, M., Praeger, C. E. 1998On vertex-imprimitive graphs of order a product of three distinct odd primesJ. Combin. Math. Combin. Comput.28187213zbMATHMathSciNetGoogle Scholar
  10. Huppert, B., Blackborn, N. 1982Finite Groups IIISpringer-VerlagNew York, BerlinzbMATHGoogle Scholar
  11. Iranmanesh, M., Praeger, C. E. 2001On non-Cayley vertex-transitive graphs of order a product of three primesJ. Combin. Theory Ser. B81119zbMATHMathSciNetGoogle Scholar
  12. Kantor, W. 1972 k-homogeneous groupsMath. Z.124261265zbMATHMathSciNetGoogle Scholar
  13. Li, C. H. 2003The finite primitive permutation groups containing and abelian regular subgroup, Proc.London Math. Soc.87725747zbMATHGoogle Scholar
  14. Li, C. H., Seress, A. 2003The finite primitive permutation groups of square-free degree, BullLondon Math. Soc.35635644zbMATHMathSciNetGoogle Scholar
  15. H. L. Li, J. Wang, L. Y. Wang and M. Y. Xu, Vertex primitive graphs of order containing a large prime factor, Comm. Algebra, Vol. 22, No. 9 (1994) pp. 3449–3477.Google Scholar
  16. Liebeck, M., Praeger, C. E., Saxl, J. 1990The maximal factorizations of the finite simple groups and their automorphism groupsMem. Amer. Math. Soc.86432MathSciNetGoogle Scholar
  17. Livingstone, D., Wagner, A. 1965Transitivity of finite permutation groups on unordered setsMath. Z.90393403zbMATHMathSciNetGoogle Scholar
  18. McKay, B., Praeger, C. E. 1994Vertex-transitive graphs which are not Cayley graphs IJ. Austral. Math. Soc.565363CrossRefzbMATHMathSciNetGoogle Scholar
  19. McKay, B., Praeger, C. E. 1994Vertex-transitive graphs which are not Cayley graphs IIJ. Graph Theory22321334MathSciNetGoogle Scholar
  20. Marušič, D. 1983Cayley graphs of vertex symmetric graphsArs Combin.16297302Google Scholar
  21. Marušič, D., Scapellato, R. 1992Characterizing vertex-transitive pq-graphs with an imprimitive automorphism subgroupJ. Graph Theory16375387zbMATHMathSciNetGoogle Scholar
  22. Marušič, D., Scapellato, R., Zgrablić, B. 1995On quasiprimitive pqr-graphsAlgebra Coll.2295314zbMATHGoogle Scholar
  23. Miller, A. A., Praeger, C. E. 1994Non-Cayley vertex-transitive graphs of order twice the product of two odd primesJ. Algebraic Combin.377111zbMATHMathSciNetGoogle Scholar
  24. Neumann, P. 1994Helmut Wielandt on permutation groupsHuppert, B.Schneider, H. eds. Helmut WielandtMathematical Works, Walter De GruyterBerlin, New York320Google Scholar
  25. Praeger, C.E., Xu, M. Y. 1993Vertex-primitive graphs of order a product of two distinct primesJ. Combin. Theory Ser. B59245266zbMATHMathSciNetGoogle Scholar
  26. Seress, Á. 1998On vertex-transitive, non-Cayley graphs of order pqr Discrete Math.182279292zbMATHMathSciNetGoogle Scholar
  27. Zassenhaus, H. 1936Über endliche FastkörperAbh. Math. Sem. Univ. Hamburg11187220Google Scholar
  28. Zsigmondy, K. 1892Zur Theorie der PotenzresteMonatsh. Math. Phys.3265284MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA
  2. 2.Department of Mathematics and StatisticsThe University of Western AustraliaPerthAustralia

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