Verbal products of groups
type: Item
instance of: scholarly article
title: Verbal products of groups
zbMATH Open document ID: 0810.20019
DOI: 10.1007/BF02260879
author: S. V. Rychkov
published in: Algebra and Logic
publication date: 1994-05-18T00:00:00Z
review text: The author is concerned with a variety \(V\) of groups of exponent zero (namely he focuses on infinite groups that are close to free groups in this variety). The main result is the following theorem: If \(\kappa\) is a regular uncountable cardinal, then there exist \(2^ \kappa\) mutually nonisomorphic groups \(\{F_ i : i < 2^ \kappa\}\) that are strongly \(\kappa\)-free in the variety \(\mathcal F\) of all groups, such that they all have rank \(\kappa\) and, for each \(i < 2^ \kappa\), the following is valid: \(\forall \overline{\varphi} \in \text{End} (F_ i / F_ i') (\exists) \ell \in \mathbb{Z} \forall x \in F_ i/ F_ i'\) (\(\overline {\varphi} (x) = x^ \ell\)). As a corollary it is obtained that if \(\kappa\) is a regular uncountable cardinal and \(V\) an arbitrary variety of groups of exponent zero whose free groups are approximated by solvable groups, then there exist \(2^ \kappa\) mutually nonisomorphic strongly \(\kappa\)-free groups \(\{G_ i : i < 2^ \kappa\}\) in \(V\) of cardinality \(\kappa\) such that each of them is indecomposable into the \(V\)-product of two nontrivial \(V\)-factors.
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E22
zbMATH DE Number: 561012
zbMATH Keywords: verbal product
zbMATH Keywords: groups of exponent zero
zbMATH Keywords: free groups
zbMATH Keywords: regular uncountable cardinal
zbMATH Keywords: variety of groups
zbMATH Keywords: solvable groups
zbMATH Keywords: strongly \(\kappa\)-free groups
zbMATH Keywords: indecomposable
reviewed by: Dimitru Buşneag
MaRDI profile type: Publication
cites work: Varieties of groups
cites work: Almost Free Groups
cites work: A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals
cites work: Q3949041
cites work: New Criteria for Freeness in Abelian Groups. II
cites work: On constructing indecomposable groups in L
cites work: On endo-rigid, strongly \(\aleph_1\)-free Abelian groups in \(\aleph_1\).
cites work: Every Cotorsion-Free Ring is an Endomorphism Ring
cites work: Some groups all of whose proper verbal quotients are relatively free
cites work: In singular cardinality, locally free algebras are free
cites work: Almost free groups in varieties
cites work: Realization of rings by endomorphism rings
full work available at URL: https://doi.org/10.1007/bf02260879
OpenAlex ID: W1977555617
description: scientific article; zbMATH DE number 561012