On the order of the n-center factor subgroup of an n-abelian group

A group G is said to be n-abelian, if (xy)n=xnyn, for any x,y ∈in G and a positive integer n. In 1979, Fay and Waals introduced the n-potent and the n-center subgroups of a group G, denoted by Gn and Zn(G), respectively. In this paper, we show that the index of the n-center is bounded by an order power of the n-potent subgroup, for some classes of groups. In fact for all n-abelian groups G with finite n-potent subgroup, we prove that if G/Zn(G) is finitely generated, then [G : Zn(G)] ≤ |Gn|d(G/Zn(G)). Moreover, we conclude that [G : Zn(G)] ≤ |Gn|2log2|Gn|, for some n-capable group G.