An upper bound for the order of the $n$-potent subgroup of an $n$-abelian nilpotent group

Let $n$ be a positive integer. The $n$-potent and the $n$-center subgroups of a group $G$ are defined as $G_n = langle [x,y^n] | x,y in G angle$, $Z^n(G)= { x in G | xy^n =y^nx, forall y in G }$, respectively. In this paper, we give an upper bound for the order of $n$-potent subgroup in terms of the index of $n$-center subgroup, for an $n$-abelian group which is finite nilpotent.