a generalization of a converse of schur’s theorem in the variety of n-abelian groups

For a positive integer n, a group g is said to be n-abelian, if (xy)n = xnyn, for any x; y 2 g. fay and waals introduced the n-potent and the n-center subgroups of a group g, as gn = h[x; yn]jx; y 2 gi and zn(g) = fx 2 gjxyn = ynx; 8y 2 gg, respectively. pourmirzaei et al. extended a converse of schur’s theorem in the variety of n-abelian groups and proved that if g is n-abelian such that zng(g) is finitely generated then the finiteness of gn implies the finiteness of zng(g) and jzng(g)j ≤ jgnjd( zng(g) ). in this paper, we generalize this result and improve the mentioned upper bound.