some upper bounds 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 n-abelian, if (xy)n = xnyn, for any x; y 2 g. in 1979, fay and waals introduced the n-potent and the n-center subgroups of a group g, as γ2n(g) = h[x; yn]jx; y 2 gi and zn(g) = fx 2 gjxyn = ynx; 8y 2 gg, respectively. also, the second n-center subgroup, z2n(g), is defined by z2n(g)=zn(g) = zn(g=zn(g)). in this paper, we give some upper bounds for the index of the second n-center subgroup of an n-abelian group g, in terms of the order of the n-potent subgroup, under some conditions.