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
M. M. Glukhov, “On 2-transitive products of regular permutation groups,” Tr. Diskr. Mat., 3, 37–52 (2000).
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).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 21, No. 3, pp. 217–231, 2016.
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Toktarev, A.V. On k-Transitivity Conditions of a Product of Regular Permutation Groups. J Math Sci 237, 485–495 (2019). https://doi.org/10.1007/s10958-019-04173-5
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DOI: https://doi.org/10.1007/s10958-019-04173-5