On totally k-closed nilpotent groups

Abstract

A group $G$ is said to be totally $k$-closed for a positive integer $k$ if, in each of its faithful permutation representations on a set $\Omega^k$, $G$ is the largest subgroup of the symmetric group $\operatorname{Sym}(\Omega)$ that preserves every $k$-orbit in the induced action on the set $\Omega\times\dots\times \Omega=\Omega^k$. We prove that for $k\geq1$, every finite nilpotent group with Sylow subgroups of orders at most $p^k$ for all primes $p$ dividing $|G|$ is totally $k$-closed if and only if it does not contain an elementary abelian subgroup $\mathbb{Z}_p^k$ for every prime $p$.