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

TL;DR

This paper investigates the structure of $k$-closures of finite nilpotent permutation groups, revealing that such closures decompose into direct products of the $k$-closures of their Sylow subgroups.

Contribution

It establishes that the $k$-closure of a finite nilpotent permutation group is the direct product of the $k$-closures of its Sylow subgroups, providing a structural insight.

Findings

$k$-closure of a finite nilpotent permutation group decomposes into Sylow subgroup closures.

The structure of $k$-closures aligns with the direct product of Sylow subgroup closures.

Provides a foundational result for understanding symmetries in 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.