Geodesic
Primitive elements of the free groups of the varieties $\mathfrak A\mathfrak N_n$
Matematičeskie zametki, Tome 61 (1997) no. 6, pp. 884-889

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For groups of the form $F/N'$, we find necessary and sufficient conditions for an element $g\in N/N'$ to belong to the normal closure of an element $h\in F/N'$. 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 element $h$ whose normal closure contains a primitive element $g$, but the elements $h$ and $g^{\pm1}$ are not conjugate. In the group $F(\mathfrak A\mathfrak N_2)$, two nonconjugate elements are chosen that have equal normal closures. 
@article{MZM_1997_61_6_a8,
     author = {E. I. Timoshenko},
     title = {Primitive elements of the free groups of the varieties $\mathfrak A\mathfrak N_n$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {884--889},
     publisher = {mathdoc},
     volume = {61},
     number = {6},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_61_6_a8/}
}
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E. I. Timoshenko. Primitive elements of the free groups of the varieties $\mathfrak A\mathfrak N_n$. Matematičeskie zametki, Tome 61 (1997) no. 6, pp. 884-889. http://geodesic.mathdoc.fr/item/MZM_1997_61_6_a8/