Abstract
For groups of the formF/N', we find necessary and sufficient conditions for an elementg∈N/N' to belong to the normal closure of an elementh∈F/N'. It is proved that, in contrast to the case of a free metabelian group, for a free group of the variety\(\mathfrak{A}\mathfrak{N}_2 \), there exists an elementh whose normal closure contains a primitive elementg, but the elementsh andg ±1 are not conjugate. In the groupF(\(\mathfrak{A}\mathfrak{N}_2 \)), two nonconjugate elements are chosen that have equal normal closures.
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Translated fromMaternaticheskie Zametki, Vol. 61, No. 6, pp. 884–889, June, 1997.
Translated by A. I. Shtern
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Timoshenko, E.I. Primitive elements of the free groups of the varieties\(\mathfrak{A}\mathfrak{N}_n \) . Math Notes 61, 739–743 (1997). https://doi.org/10.1007/BF02361216
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DOI: https://doi.org/10.1007/BF02361216