On finite totally k-closed groups

Let $G$ be a finite group acting faithfully on a finite set $\Omega$. For a positive integer $k$, $G$ acts naturally on the Catesian product $\Omega^k := \Omega \times ...\times \Omega$. In this paper, we prove that finite nilpotent group $G$ with $2\nmid |G|$ is a totally $k$-closed group if and only if $G$ is abelian with $n(G)\leq k-1$ or cyclic, where $n(G)$ is the number of invariant factors in the invariant factor decomposition of $G$.