This article is cited in 3 papers

The closures of wreath products in product action

A. V. Vasilev^ab, I. N. Ponomarenko<sup>cb

a</sup> Novosibirsk State University
^b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
^c St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: Let $m$ be a positive integer and let $\Omega$ be a finite set. The $m$-closure of $G\le{\rm Sym} (\Omega)$ is the largest permutation group $G^{(m)}$ on $\Omega$ having the same orbits as $G$ in its induced action on the Cartesian product $\Omega^m$. An exact formula for the $m$-closure of the wreath product in product action is given. As a corollary, a sufficient condition is obtained for this $m$-closure to be included in the wreath product of the $m$-closures of the factors.

Keywords: right-symmetric ring, left-symmetric algebra, pre-Lie algebra, prime ring, Pierce decomposition, $(1,1)$-superalgebra.

UDC: 512.542.7

Received: 20.07.2021
Revised: 18.10.2021

DOI: 10.33048/alglog.2021.60.302

 English version: Algebra and Logic, 2021, 60:3, 188–195

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