On Burnside's problem on periodic groups
type: Item
instance of: scholarly article
title: On Burnside's problem on periodic groups
zbMATH Open document ID: 0595.20029
DOI: 10.1007/BF01078416
publication date: 1980-01-01T00:00:00Z
review text: Burnside's problem, as is well known, consists in the following: Is every periodic group locally finite? Up to the present this question has been answered in the negative. The relevant examples were constructed first by E. S. Golod (1964) and P. S. Novikov and S. I. Adyan (1968). Later, other examples were constructed, notably Ol'shanskij's (1978), by means of which other important problems of the theory of periodic groups, namely, O. J. Schmidt's problem and that of S. N. Chernikov on groups with minimal condition, were also solved. The author gives a new, simply constructed example of a periodic group that is not locally finite. Specifically, the author constructs an infinite 2-group \(G\), generated by three involutions (i.e., elements of order 2) \(a\), \(b\) and \(c\), where \(b\) and \(c\) commute. Soon after the appearance of this paper it was established that the author's group \(G\) contains elements of arbitrarily high order (i.e., its exponential curve is not bounded) [see \textit{A. G. Ivanov} and \textit{A. V. Feklichev}, ``Unboundedness of the exponential curve of Grigorchuk's group'', Seventh All-Union Symposium on Group Theory (Shushenskoe, 1980) (Russian), Krasnoyarsk Gos. Univ., Krasnoyarsk, 1980].
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F50
zbMATH DE Number: 3957421
zbMATH Keywords: Burnside problem
zbMATH Keywords: periodic groups
zbMATH Keywords: minimal condition
zbMATH Keywords: infinite 2-group
zbMATH Keywords: involutions
author: R. I. Grigorchuk
MaRDI profile type: Publication
cites work: Q5665644
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 3957421