On two nilpotent variety

In this paper, first we study relation of k-st center and marginal subgroup in two-nilpotent variety. Moreover, we prove that if G is a finitely generated group such that n;m ≥ 1 and γm+1(G / Zn(G)) is finite then |G / Zn,m(G)| is finite, where Zn(G), γn+1(G) and Zn,m(G) are nth term of the upper central series of G, (n+1)st term of the lower central series of G and Zn,m(G) = {a ∈ G | [a,mG] ⊆ Zn (γm+1(G))}; marginal subgroup of G with respect to twonilpotent variety, respectively. Also, we show that finiteness of γn+1(γm+1(G)), verbal subgroup of G with respect to two-nilpotent variety, doesn’t imply that γm+1 (G / Zn(G)) is of finite order, if G is finitely generated.