On k-Transitivity Conditions of a Product of Regular Permutation Groups
- 作者: Toktarev A.1
-
隶属关系:
- Moscow State University
- 期: 卷 237, 编号 3 (2019)
- 页面: 485-495
- 栏目: Article
- URL: https://journals.rcsi.science/1072-3374/article/view/242378
- DOI: https://doi.org/10.1007/s10958-019-04173-5
- ID: 242378
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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.