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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

Citations:

Zbl 0258.20034
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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).
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