Original Research Article

Article volume = 2024 and issue = 1

Pages: 52–59

Article publication Date: December 09, 2023

You can download PDF file of the article here: Download

Visited 155 times and downloaded 78 times

Transitive permutation groups with elements of movement three consecutive integers

Bahman Askari

Department of Mathematics, Qorveh Branch, Islamic Azad University, Qorveh, Iran.


Abstract:

Let G be a permutation group on a set Ω with no fixed point in Ω and let m be a positive integer. If for each subset Γ of Ω the size |Γg \ Γ | is bounded, for g ∈ G, we define the movement of g as the max |Γg \ Γ | over all subsets Γ of Ω, and the movement of G is defined as the maximum of move(g) over all non-identity elements of g ∈ G. In this paper we classify all transitive permutation groups with bounded movement equal to m that are not a 2−group, but in which every non-identity element has the movement m, m−1 or m−2.

Keywords:

Permutation group, transitive, bounded movement, fixed point free element.


References:
  • [1] Alaeiyan, M., Improvement on the bounds of permutation groups with bounded movement, Bull. Austral. Math. Soc., 67 (2003) 246-257.
  • [2] Alaeiyan, M., and Askari, B., Transitive permutation groups with elements of movement m or m−1, Mathematical Reports, To appear.
  • [3] Alaeiyan, M., and Askari, B., Transitive permutation groups with elements of movement m or m−2, Submitted. 3
  • [4] Alaeiyan, M., and Tavallaee, H.A., Permutation groups with the same movement, Carpathian J. Math., 25 (2009) 147-156.
  • [5] Alaeiyan, M., and Yoshiara, S., Permutation groups of minimal movement, Arch. Math., 85 (2005) 211-226.
  • [6] Brandl, R., Finite groups all of whose elements are of prime power order, Bull. U. M. I., 5 18-A (1981) 491-493.
  • [7] Cho, J. R., Kim, P. S. and Praeger, C. E., The maximal number of orbits of a permutation groups with bounded movement, J. Algebra, 214 (1999), 625-630.
  • [8] Fein, B., Kantor, W. M. and Schacher, M., Relative Brauer groups, II, J. Reine Angew., 328 (1981), 39-57.
  • [9] Hassani, A., Alaeiyan (Khayaty), M., Khukhro, E. I. and Praeger, C. E., Transitive permutation groups with bounded movement having maximal degree, J. Algebra, 214 (1999), 317-337. 1, 2.1, 3
  • [10] Higman, G., Finite groups in which every element has prime power order, J. London. Math. Soc., 32 (1957), 335-342.
  • [11] Huppert, B., Endliche Gruppen I, Springer-Verlag, Berlin and New York, 1967.
  • [12] Mann, A. and Praeger, C. E., Transitive permutation groups of minimal movement, J. Algebra, 181 (1996), 903-911.
  • [13] Praeger, C. E., On permutation groups with bounded movement, J. Algebra, 144 (1991), 436-442. 1, 1.1
  • [14] Praeger, C. E., Movement and separation of subsets of points under group action, J. London Math. Soc., 56 (2) (1997) 519-528.
  • [15] Rotman, J. J., An Introduction to the Theory of Groups, 3rd ed., Allyn and Bacon, Boston 1984. 2
  • [16] Suzuki, M., On a class of doubly transitive groups, Ann. Math. 75 (1962) 105-145.
  • [17] Tsuzuku, T., Finite Groups and Finite Geometries, Cambridge University Press, 1982.
Cite this article as:
  • Bahman Askari, Transitive permutation groups with elements of movement three consecutive integers, Communications in Combinatorics, Cryptography & Computer Science, 2024(1), PP.52–59, 2023
  • Export citation to BibTeX