The converse of Baer s theorem for two-nilpotent variety

In this paper the generalization of the converse of Baer’s theorem for two-nilpotent variety of class row (n, m) is carried out. Baer proved that finiteness of G/Zn(G) implies that γn+1(G) is finite. Hekster proved the converse of the Baer’s theorem with the assumption that G can be finitely generated. The Baer’s theorem can be considered as a result of a classical theorem by Schur denoting that finiteness of G/Z(G) leads to the finiteness of G′. The converse of the Baer’s theorem has been proved conditionally by Taghavi et al. (2019), as well. In the Main Theorem, we prove that, if γm,n(G) ∩ Zn,m(G) = 1 and γm,n+i(G) is finite for some n, i, m ≥ 0. Then G/Zn,m(G) is finite in which γm,n(G) and Zn,m(G) denote verbal and marginal subgroups with respect to two-nilpotent variety of class row (n, m). Thus the generalization of the converse of Baer’s theorem for two-nilpotent variety of groups valids by considering i = 0. In this article some other results are attained by the converse of the Baer’s theorem. It is also concluded that when n = m = 1. Similar results are obtained for variety of the soluble groups. In addition, the converse of the Schur’s theorem which proved by Halasi and Podoski is concluded in this paper, for two-nilpotent variety. We have also obtained some similar results of Chakaneh et al. (2019) for (n, m)-isoclinic family of groups and (1, m)-stem groups.