The closures of wreath products in product action

TL;DR

This paper derives an exact formula for the m-closure of wreath products in product action, providing insights into their structure and conditions for inclusion in wreath products of factors.

Contribution

It presents a precise formula for the m-closure of wreath products in product action and establishes a sufficient condition for their inclusion in wreath products of the factors' m-closures.

Findings

Exact formula for m-closure of wreath products in product action.

Sufficient condition for m-closure inclusion in wreath products of factors.

Enhanced understanding of the structure of wreath product closures.

Abstract

Let $m$ be a positive integer and let $\Omega$ be a finite set. The $m$-closure of $G\le$Sym$(\Omega)$ is the largest permutation group on $\Omega$ having the same orbits as $G$ in its induced action on the Cartesian product $\Omega^m$. The 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.