Primitive elements of the free groups of the varieties $\mathfrak A\mathfrak N_n$

E. I. Timoshenko

Matematičeskie zametki, Tome 61 (1997) no. 6, pp. 884-889

Résumé

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.