Primitive elements of the free groups of the varieties $$\mathfrak{A}\mathfrak{N}_n $$

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.