On the central series of the free groups of varieties

Let be the variety of all nilpotent groups which have the nilpotent class $\leqslant k$. A. I. Malcev formulated the problem: does the free group $A$ of the variety constructed from the varieties $\mathfrak{N}_{K_1}, \mathfrak{N}_{K_2}, \dots, \mathfrak{N}_{K_s}$ by means of intersections and multiplications satisfy the following conditions:

• $\bigcap\limits_n\gamma_n(A)=\{1\}$ where $\gamma_n(A)$ is the $n$ member of the descending central series of $A$, $n$ is a natural number,
• the factors $\gamma_n(A)/\gamma_{n+1}(A)$ are free abelian groups?

In this paper the problem is solved for $K$-varieties defined below.
Let $K_1$ be the class of the varieties $\mathfrak{N}_k$ for all natural numbers $k$. We assume that the classes $K_s$ for $s=1,2,\dots,t$ of the varieties have been constructed. Then we define $K_{t+1}$ as the class consisted of the varieties $(\mathfrak{N}_{K_1}\cap \mathfrak{M}_1)(\mathfrak{N}_{K_2}\cap \mathfrak{M}_2)\dots(\mathfrak{N}_{K_r}\cap \mathfrak{M}_r$) where $\mathfrak{M}_i\in K_s$ for $s\leqslant t$. Suppose $K=\bigcup\limits_s K_s$. We shall call any variety of $K$ by $K$-variety.

Received: 10.04.1967