On finite totally k-closed groups

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