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

Let $\Omega$ be a finite set and $G\leqslant \operatorname{Sym}(\Omega)$. Denote by $\operatorname{Orb}_k(G)$ the set of all orbits of the induced action of $G$ on $\Omega\times\dots\times \Omega=\Omega^k$. The $k$-closure of the permutation group $G$ is defined to be the largest subgroup $G^{(k)}$ in $\operatorname{Sym}(\Omega)$ such that $\operatorname{Orb}_k(G)=\operatorname{Orb}_k(G^{(k)})$. The group $G$ is said to be $k$-closed if $G^{(k)}=G$. The concept of $k$-closure was introduced by H. Wielandt in the framework of the method of invariant relations developed by him to study group actions [1].

In this talk we focus on $k$-closures of nilpotent groups. It is well known that every finite nilpotent group is the direct product of its nontrivial Sylow subgroups. The main result shows that $k$-closure respects this decomposition generalizing results of [2,3].

Theorem. {\it If $G$ is a finite nilpotent permutation group, and $k\geqslant 2$, then $G^{(k)}$ is the direct product of $k$-closures of Sylow subgroups of $G$.}

Corollary. For $k\geqslant 2$, a finite nilpotent permutation group $G$ is $k$-closed if and only if every Sylow subgroup of $G$ is $k$-closed.

Acknowledgement. The work is supported by the Mathematical Center in Akademgorodok under the agreement No. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation.


Список литературы

1. H. W. Wielandt, “Permutation groups through invariant relations and invariant functions”, Lecture Notes, Ohio State University, Ohio, 1969
2. D. V. Churikov, Ch. E. Praeger, “Finite totally $k$-closed groups”, Тр. ИММ УрО РАН, 27, № 1, 2021, 240–245        
3. D. Churikov, I. Ponomarenko, “On $2$-closed abelian permutation groups”, arXiv: 2011.12011