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Journal of Mathematical Sciences

, Volume 237, Issue 3, pp 485–495 | Cite as

On k-Transitivity Conditions of a Product of Regular Permutation Groups

  • A. V. ToktarevEmail author
Article

Abstract

The paper analyzes the product of m regular permutation groups G1· . . . · Gm, where m ≥ 2 is a natural number. Each of the regular permutation groups is a subgroup of the symmetric permutation group S(Ω) of degree |Ω| for the set Ω. M. M. Glukhov proved that for k = 2 and m = 2, 2-transitivity of the product G1· G2 is equivalent to the absence of zeros in the corresponding square matrix with the number of rows and columns equal to |Ω| − 1. Also M. M. Glukhov has given necessary conditions of 2-transitivity of such a product of regular permutation groups.

In this paper, we consider the general case for any natural m and k such that m ≥ 2 and k ≥ 2. It is proved that k-transitivity of the product of regular permutation groups G1· . . . · Gm is equivalent to the absence of zeros in the square matrix with the number of rows and columns equal to (|Ω| − 1)!/(|Ω| − k)!. We obtain correlation between the number of arcs corresponding to this matrix and a natural number l such that the product (PsQt)l is 2-transitive, where P,Q ⊆ S(Ω) are some regular permutation groups and the permutation st is an (|Ω| − 1)-cycle. We provide an example of the building of AES ciphers such that their round transformations are k-transitive on a number of rounds.

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References

  1. 1.
    M. M. Glukhov, “On 2-transitive products of regular permutation groups,” Tr. Diskr. Mat., 3, 37–52 (2000).Google Scholar
  2. 2.
    R. Levingston and D. E. Taylor, “The theorem of Marggraff on primitive permutation groups which contain a cycle,” Bull. Austral. Math. Soc., 15, No. 1, 125–128 (1976).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

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