The construction of infinite finitely generated periodic groups using wreath products
type: Item
instance of: scholarly article
title: The construction of infinite finitely generated periodic groups using wreath products
zbMATH Open document ID: 0641.20031
DOI: 10.1016/0021-8693(88)90287-6
author: Martyn Dixon
author: Thomas A. Fournelle
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let n be a natural number, \(n\geq 4\), and let \(\pi\) be the set of the prime divisors of n. With the help of generalized wreath products, an infinite periodic \(\pi\)-group \(E_ n\) is constructed with two generators of order n and with the following additional properties: (i) some member of the transfinite derived series of \(E_ n\) is equal to the identity, (ii) the group \(E_ n\) is residually finite, (iii) \(Z(E_ n)=1\), (iv) \(E_ n\) contains an isomorphic copy of every countable residually finite locally soluble FC-\(\pi\)-group. Furthermore, if n divides m then the groups \(E_ n\) and \(E_ m\) may be constructed such that \(E_ n\leq E_ m\). It is to be noted that an example of a 2-generated periodic group of Aleshin type containing an element of arbitrary finite order was constructed by \textit{A. V. Rozhkov} [Mat. Zametki 40, No.5, 572-589 (1986; Zbl 0614.20019)].
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20E26
zbMATH DE Number: 4044090
zbMATH Keywords: generalized wreath products
zbMATH Keywords: infinite periodic \(\pi \)-group
zbMATH Keywords: generators
zbMATH Keywords: residually finite locally soluble FC-\(\pi \)-group
zbMATH Keywords: 2-generated periodic group
reviewed by: Ju. I. Merzljakov
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90287-6
OpenAlex ID: W2061683212
cites work: Finite automata and Burnside's problem for periodic groups
cites work: Q3344117
cites work: Wreath products indexed by partially ordered sets
cites work: Q3772405
cites work: On Burnside's problem on periodic groups
cites work: On the Burnside problem for periodic groups
cites work: Some infinite \(p\)-groups
cites work: Contributions to Group Theory
cites work: Q3845383
cites work: Q3318953
cites work: Q5654235
description: scientific article; zbMATH DE number 4044090