\(\mathbb Q\)-completions of free solvable groups.
type: Item
instance of: scholarly article
title: \(\mathbb Q\)-completions of free solvable groups.
zbMATH Open document ID: 1337.20037
author: Nikolai S. Romanovskiy
author: Chander K. Gupta
published in: Algebra and Logic
publication date: 2015-10-27T00:00:00Z
review text: A group \(G\) is said to be \(m\)-rigid if \(G\) contains a normal series of the form \(G=G_1>G_2>\cdots>G_m>G_{m+1}=1\), whose quotients \(G_i/G_{i+1}\) are abelian and are torsion-free as right \(\mathbb Z[G/G_i]\)-modules. Such a series (if it exists) is defined by the group \(G\) uniquely, and the derived length of \(G\) is exactly \(m\). It is proved that every \(m\)-rigid group is embedded in a complete \(m\)-rigid group. Under the specified condition of the embedding, the \(\mathbb Q\)-completion of the group \(G\) in the class of rigid groups is defined uniquely up to \(G\)-isomorphism. It is stated that the centralizer of any element of such \(\mathbb Q\)-completion of a free solvable group, which does not lie in the last nontrivial member of a rigid series of this completion, is isomorphic to the additive group of \(\mathbb Q\).
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20F70
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 6498556
zbMATH Keywords: complete groups
zbMATH Keywords: \(\mathbb Q\)-completions
zbMATH Keywords: rigid groups
zbMATH Keywords: centralizers
zbMATH Keywords: solvable groups
reviewed by: Vitaliĭ Roman'kov
MaRDI profile type: Publication
full work available at URL: s10469-015-9332-y
OpenAlex ID: W1192718685
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DOI: 10.1007/S10469-015-9332-Y
description: scientific article; zbMATH DE number 6498556