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 Algebra i Logika. Sem., 1967, Volume 6, Number 3, Pages 13–24 (Mi al1103)  On the central series of the free groups of varieties

Yu. M. Gorčakov

Abstract: 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 $\mathfrak{N}_{K_1}, \mathfrak{N}_{K_2}, …, \mathfrak{N}_{K_s}$ 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 $s=1,2,…,t$ of the varieties have been constructed. Then we define $K_{t+1}$ as the class consisted of the varieties $(\mathfrak{N}_{K_1}\cap \mathfrak{M}_1)(\mathfrak{N}_{K_2}\cap \mathfrak{M}_2)…(\mathfrak{N}_{K_r}\cap \mathfrak{M}_r$) 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. Full text: PDF file (378 kB)

Bibliographic databases: Citation: Yu. M. Gorčakov, “On the central series of the free groups of varieties”, Algebra i Logika. Sem., 6:3 (1967), 13–24 Citation in format AMSBIB
\Bibitem{Gor67} \by Yu.~M.~Gor{\v{c}}akov \paper On the central series of the free groups of varieties \jour Algebra i Logika. Sem. \yr 1967 \vol 6 \issue 3 \pages 13--24 \mathnet{http://mi.mathnet.ru/al1103} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=0220808} 

• http://mi.mathnet.ru/eng/al1103
• http://mi.mathnet.ru/eng/al/v6/i3/p13

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This publication is cited in the following articles:
1. L. E. Krop, B. I. Plotkin, “Magnus varieties in group representations”, Math. USSR-Sb., 24:4 (1974), 487–510    •   Contact us: math-net2021_11 [at] mi-ras ru Terms of Use Registration to the website Logotypes © Steklov Mathematical Institute RAS, 2021