Structure of $k$-closures of finite nilpotent permutation groups

Abstract

Let $G$ be a permutation group on a set $\Omega$, and $k$ a positive integer. The $k$-closure $G^{(k)}$ of $G$ is the largest subgroup of $\operatorname{Sym}(\Omega)$, with the same as $G$ orbits of componentwise action on $\Omega^k$. We prove that the $k$-closure of a finite nilpotent permutation group is the direct product of $k$-closures of its Sylow subgroups.