Kolmakov, Yu. A. A theorem on freedom for groups with one defining relation in the variety of polynilpotent groups. (English. Russian original) Zbl 0614.20026 Sib. Math. J. 27, 523-537 (1986); translation from Sib. Mat. Zh. 27, No. 4(158), 67-83 (1986). 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 N. S. Romanovskij [Mat. Sb., Nov. Ser. 89(131), 93-99 (1972; Zbl 0258.20034)] for free nilpotent and solvable groups. Reviewer: V.A.Artamonov MSC: 20F19 Generalizations of solvable and nilpotent groups 20F14 Derived series, central series, and generalizations for groups 20F05 Generators, relations, and presentations of groups 20E05 Free nonabelian groups Keywords:one-relator group; lower central series; free polynilpotent group Citations:Zbl 0258.20034 PDFBibTeX XMLCite \textit{Yu. A. Kolmakov}, Sib. Math. J. 27, 523--537 (1986; Zbl 0614.20026); translation from Sib. Mat. Zh. 27, No. 4(158), 67--83 (1986) Full Text: DOI References: [1] N. S. Romanovskii, ?A theorem on freeness for groups with one defining relation in varieties of solvable and nilpotent groups of given degrees,? Mat. Sb.,89, 93-99 (1972). · Zbl 0258.20034 · doi:10.1070/SM1972v018n01ABEH001614 [2] G. G. Yabanzhi ?A theorem on freeness for groups with one defining relation in the variety ?c \(\mathfrak{A}\) ,? Sib. Mat. Zh.,21, 215-222 (1980). [3] R. A. Sarkisyan, ?Conjugacy in free polynilpotent groups,? Algebra Logika,11, 694-710 (1972). [4] Yu. A. Kolmakov, ?A generalization of Magnus’s embedding theorem,? Deposited at VINITI, No. 3445-82. [5] W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory, 2nd ed., Dover, New York (1976). · Zbl 0362.20023 [6] Yu. A. Kolmakov, ?Finite approximability with respect to conjugacy of free polynilpotent groups,? Mat. Sb.,122, 313-340 (1983). · Zbl 0536.20017 [7] P. Hall, ?Nilpotent group,? Matematika,12, No. 1, 3-36 (1968). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.