A theorem on freedom for groups with one defining relation in the variety of polynilpotent groups (Q1820241)

For group G denote by \(G_ k\) the k-th member of a lower central series in G. Put \(G_{k,n,...,\ell}=(...(G_ k)_ n...)_{\ell}\). A group G is polynilpotent of class \((c_ 1,...,c_ n)\) if \(G_{c_ 1+1,...,c_ n+1}=1\). Let G be a free polynilpotent group of class \((c_ 1,...,c_ n)\) with basis \(x,x_ 1,...,x_ m\), and \(r\in H_ j\setminus H_{j+1}\), where \(H_ j=G_{c_ 1+1,...,c_ i+1}\) for some \(0\leq i<n\), \(0\leq j\leq c_{i+1}\). Then the group U generated by \(x_ 1,...,x_ m\) modulo r is a free polynilpotent group of the same class \((c_ 1,...,c_ n)\) if and only if r is not conjugate modulo \(H_{j+1}\) with any element of U. - This result generalizes a theorem of Magnus for free groups (case \(n=0)\) and a theorem of \textit{N. S. Romanovskij} [Mat. Sb., Nov. Ser. 89(131), 93-99 (1972; Zbl 0258.20034)] for free nilpotent and solvable groups.