Primitive systems of elements in the variety$\mathfrak{A}\mathfrak{N}_2$ and some locally finite varieties of groupsand some locally finite varieties of groups

Abstract

Part of any basis of a relatively free group$F_r (\mathfrak{B})$ in the variety$\mathfrak{B}$ is called a primitive system of elements. We provide a criterion of being primitive for$F_r \left( {\mathfrak{A}_m \mathfrak{B}} \right)$, where$\mathfrak{A}_m$ is a variety of Abelian groups satisfying x^m=1, and$\mathfrak{B}$ a variety generated by a finite group. Let$\mathfrak{N}_c$ be a variety of nilpotent groups of class ≤c. It is proved that, for the group$F_2 \left( {\mathfrak{A}\mathfrak{N}_2 } \right)$, the property of being primitive for an element g is stronger than the condition of being unimodular on a vector composed of values of Fox derivatives in the ring$\mathbb{Z}F_2 \left( {\mathfrak{N}_2 } \right)$. The group$F_2 \left( {\mathfrak{A}\mathfrak{N}_2 } \right)$ is not residually finite whenever a system of elements is primitive.