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

Abstract

The paper analyzes the product of m regular permutation groups G₁· . . . · G_m, 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· . . . · G_m 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.