the index of the n-center subgroup of an n-abelian group

Let n be a positive integer. a group g is said to be n-abelian, if (xy)n = xnyn, for any x; y 2 g. in 1979, fay and waals introduced the n-center subgroup of a group g, as zn(g) = fx 2 gjxyn = ynx; 8y 2 gg. in this paper, we give some conditions under which the index of the n-center subgroup of an n-abelian group is finite. moreover, we give an upper bound for the index of the n-center subgroup of an n-abelian group g, when the n-center factor subgroup is finitely generated.