The divisible radical of a group.
type: Item
instance of: scholarly article
title: The divisible radical of a group.
zbMATH Open document ID: 1200.20025
published in: Central European Journal of Mathematics
publication date: 2009-11-24T00:00:00Z
review text: The author investigates the notion of ``divisible radical'' for non-Abelian groups. The divisible radical subgroup of an arbitrary group, modeled on the Abelian case, is defined to be its unique maximal divisible normal subgroup, \textit{whenever it exists}. If such a radical for a group \(G\) exists, it will be denoted by \(\delta(G)\). The main theme is to find conditions on a group under which such a radical exists and after the existence guaranteed, properties of the radical are investigated. However, the author provides groups for which such a radical does not exist. The main results are the following. Theorem 1.2. Let \(G\) be any group. a) \(G/\tau(G)\) has a unique maximal, locally nilpotent, divisible normal subgroup, where \(\tau(G)\) is the unique maximal locally finite normal subgroup of \(G\). b) The subgroup of \(G\), generated by all divisible normal subgroups of \(G\) that are hypercentral groups, is locally nilpotent and divisible. c) If \(G\) is locally nilpotent and torsion-free, then \(\delta(G)\) exists, \(\delta(G/\delta(G))=\langle 1\rangle\) and \(\delta(G)=\langle D\leq G:D\text{ is divisible}\rangle\). Theorem 1.3. There exists a non-divisible torsion-free metabelian group that is the product of two of its divisible normal subgroups. In particular \(\delta(G)\) for such a group \(G\) does not exist. Theorem 1.4. For every prime \(p\) there exists a non-divisible metabelian \(p\)-group \(G\) that is the product of two of its divisible normal subgroups. Again \(\delta(G)\) for this group does not exist. A group \(G\) is called `semi-divisible' if \(G=G^m=\langle g^m:g\in G\rangle\) for every positive integer \(m\). Theorem 1.5. There exists a non-divisible, torsion-free, locally nilpotent group \(T\) that is perfect and semi-divisible. In particular \(T/\delta(T)\) is neither hypo Abelian nor hypo (of finite exponent).
reviewed by: Alireza Abdollahi
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F19
zbMATH DE Number: 5637490
zbMATH Keywords: divisible normal subgroups
zbMATH Keywords: locally nilpotent groups
zbMATH Keywords: metabelian groups
zbMATH Keywords: divisible radical
author: B. A. F. Wehrfritz
MaRDI profile type: Publication
full work available at URL: s11533-009-0022-7
OpenAlex ID: W2025253443
cites work: Q5609457
cites work: Q5612538
cites work: Q5328221
cites work: Q5654235
cites work: On hypercentral groups.
cites work: The Abelianization of hypercyclic groups.
cites work: On Hyper (Abelian of Finite Rank) Groups
cites work: ON HYPER AND HYPO ABELIAN-OF-FINITE-RANK GROUPS
DOI: 10.2478/S11533-009-0022-7
description: scientific article; zbMATH DE number 5637490