On some infinite groups
type: Item
instance of: scholarly article
title: On some infinite groups
zbMATH Open document ID: 0727.20025
DOI: 10.1007/BF00970163
published in: Siberian Mathematical Journal
publication date: 1991-01-01T00:00:00Z
review text: A group G with a finite set of generators X and a set of defining relations \(R_ i=1\), \(i\in I\), can be represented by means of graphs using van Kampen diagrams. Following these ideas \textit{A. Yu Ol'shanskij} found examples of groups with many extraordinary properties [Geometry of defining relations in groups (Moscow, Nauka, 1989; Zbl 0676.20014)]. The author gives for sufficiently large odd integers \(n>2\cdot 10^{77}\) a series of examples of infinite simple groups G of exponent n with almost arbitrary lattices of subgroups. In particular, there exists an infinite simple group G of exponent n with descending chain condition for subgroups which does not satisfy the weak ascending chain condition for subgroups. There exists an infinite simple group G of exponent n which satisfies the ascending chain condition for subgroups and does not satisfy the descending chain condition. Similar examples exist in the class of torsion-free groups.
Mathematics Subject Classification ID: 20F06
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F50
zbMATH DE Number: 4200478
zbMATH Keywords: generators
zbMATH Keywords: relations
zbMATH Keywords: van Kampen diagrams
zbMATH Keywords: infinite simple groups
zbMATH Keywords: exponent n
zbMATH Keywords: lattices of subgroups
zbMATH Keywords: descending chain condition
zbMATH Keywords: ascending chain condition
author: Viatcheslav N. Obraztsov
reviewed by: Vyacheslav A. Artamonov
MaRDI profile type: Publication
cites work: Q5343408
description: scientific article; zbMATH DE number 4200478