On the Neumann-Shmel'kin theorem
type: Item
instance of: scholarly article
title: On the Neumann-Shmel'kin theorem
zbMATH Open document ID: 0618.20020
published in: Moscow University Mathematics Bulletin
publication date: 1986-01-01T00:00:00Z
review text: The theorem of the title asserts that any nontrivial proper variety of groups has a unique decomposition as a finite product of indecomposable varieties. The present paper considers the relationship between the product decompositions of a variety of groups (not necessarily into indecomposables) and those of a subvariety. Theorem: Let \(U\subseteq V\) be varieties of groups with decompositions \(U=U_ 1...U_ k\), \(V=V_ 1...V_{\ell}\). Then there exists an intermediate variety M, \(U\subseteq M\subseteq V\), with two decompositions \(M=M_ 1...M_ k=N_ 1...N_{\ell}\) such that \(U_ i\subseteq M_ i\) for each i and \(N_ j\subseteq V_ j\) for each j.
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 08B25
zbMATH DE Number: 4002368
zbMATH Keywords: variety of groups
zbMATH Keywords: product of indecomposable varieties
zbMATH Keywords: product decompositions
author: Alexander Y. Olshanskii
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4002368