On locally nilpotent groups (Q1363832)

In 1969 B. I. Plotkin proposed the following problem: ``Can every locally nilpotent group be represented as a homomorphic image of a locally nilpotent torsion free group?''. Positive answers to this question in the classes of locally finite groups and countable groups were given by E. M. Levich, A. I. Tokarenko and N. S. Romanovskij. To solve the general case A. Yu. Ol'shanskij formulated the following question. ``Let \(X\) be a finite set, and \(f\) be a mapping from the set of subsets of \(X\) to the set of natural numbers. Consider the following condition: in the group generated by \(X\) the subgroup generated by a subset \(Y\) of \(X\) is nilpotent of class less than or equal to \(f(Y)\). Is it true that the group \(G_f\) free with respect to this condition is torsion free?'' V. V. Bludov gave a negative answer to this question in 1989. The present paper contains new results about the groups \(G_f\) and a negative answer to the analogue of Plotkin's question for Lie rings.