Abstract
Let F be a free group with basis {xj|j ∈ J}; N a normal subgroup of F. For a given element n of N we describe an elements Dl(n), where Dl: Z(F) → Z(F) (l ∈ J) are the Fox derivations of the group ring Z(F). If r1, r2 are an elements of F/[N,N] and, for some positive integer d, r1d is in the normal closure of r2d in F/[N,N], then r1 is in the normal closure of r2 in F/[N,N]. Let F/N be a soluble group; r an element of F, R the normal closure of r in F. If, for some positive integer k, r ∉ N(k) and F/RN(k) is torsion free then F/RN(k+1) is torsion free.
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(Submitted by A. F. Krasnikov)
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Krasnikov, A.F. Some Properties of Elements of the Group F/[N,N]. Lobachevskii J Math 39, 93–96 (2018). https://doi.org/10.1134/S1995080218010171
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DOI: https://doi.org/10.1134/S1995080218010171