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 Fundam. Prikl. Mat.: Year: Volume: Issue: Page: Find

 Fundam. Prikl. Mat., 2016, Volume 21, Issue 3, Pages 217–231 (Mi fpm1743)

On $k$-transitivity conditions of a product of regular permutation groups

A. V. Toktarev

Lomonosov Moscow State University

Abstract: The paper analyses the product of $m$ regular permutation groups ${G_1}\cdot\ldots\cdot{G_{m}}$, where $m \geq 2$ is natural number. Each of regular permutation groups is the subgroup of symmetric permutation group $S(\Omega)$ of degree $|\Omega|$ for the set $\Omega$. M. M. Glukhov proved that for $k=2$ and $m=2$, $2$-transitivity of the product ${G_1}\cdot{G_{2}}$ is equivalent to the absence of zeros in the corresponding square matrix with number of rows and columns equal to $|\Omega|-1$. Also by M. M. Glukhov necessary conditions of $2$-transitivity of such product of regular permutation groups are given.
In this paper, we consider the general case for any natural $m$ and $k$ such that $m \geq 2$ and $k \geq 2$. It is proved that $k$-transitivity of product of regular permutation groups ${G_1}\cdot\ldots\cdot{G_{m}}$ is equivalent to the absence of zeros in the square matrix with number of rows and columns equal to $(|\Omega | - 1)!/(|\Omega | - k)!$. 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 $P,Q \subseteq S(\Omega )$ are some regular permutation groups and permutation $st$ is $(|\Omega | - 1)$-loop. We provide an example of the building of AES ciphers such that their round transformation are $k$-transitive on a number of rounds.

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English version:
Journal of Mathematical Sciences (New York), 2019, 237:3, 485–495

UDC: 512.542.72

Citation: A. V. Toktarev, “On $k$-transitivity conditions of a product of regular permutation groups”, Fundam. Prikl. Mat., 21:3 (2016), 217–231; J. Math. Sci., 237:3 (2019), 485–495

Citation in format AMSBIB
\Bibitem{Tok16} \by A.~V.~Toktarev \paper On $k$-transitivity conditions of a product of regular permutation groups \jour Fundam. Prikl. Mat. \yr 2016 \vol 21 \issue 3 \pages 217--231 \mathnet{http://mi.mathnet.ru/fpm1743} \transl \jour J. Math. Sci. \yr 2019 \vol 237 \issue 3 \pages 485--495 \crossref{https://doi.org/10.1007/s10958-019-04173-5}