An Upper Bound for the Index of the Second n-Center Subgroup of An n-Abelian Group

Let n be a positive integer. A group G is said to be nabelian, if (xy)n = xnyn, for any x, y ∈ G. In 1979, Fay and Waals introduced the n potent and the ncenter subgroups of a group G, as Gn = ⟨[x, yn]|x, y ∈ G⟩, Zn(G) = {x ∈ G|xyn = ynx, ∀y ∈ G}, respectively. Also, the second ncenter subgroup, Zn ∈ (G), is defined by Zn 2 (G)/Zn(G) = Zn(G/Zn(G)). In this paper, we give an upper bound for the index of the second ncenter subgroup of any nabelian group G in terms of the order of npotent subgroup Gn.