The closures of wreath products in product action

Abstract

Let m be a positive integer and let Ω be a finite set. The mclosure of G ≤ Sym(Ω) is the largest permutation group on Ω having the same orbits as G in its induced action on the Cartesian product Ω 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.