Torsion-free groups with factor-groups on their hypercenter which are periodic

Assume that G is a torsion-free group, Zk(G) is the k-th term of the upper central series of G, and ¯G_k=G/Z_k(G) is a nontrivial periodic group. Then every finite subgroup of ¯G_k is nilpotent of class not higher than k; the group k 2 contains an infinite subgroup with k generators if k2 and two generators if k=1. Moreover any nontrivial invariant subgroup of ¯G_k is infinite. All elements of ¯Gk are of odd order. This assertion is generalized.Translated from Matematicheskie Zametki, Vol. 8, No. 3, pp. 373–383, September, 1970.