On finite totally k-closed groups

Let G be a finite group acting faithfully on a finite set. For a positive integer k, G acts naturally on the Catesian product ᵏ: =. . . . 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.