On finite and locally finite subgroups of free Burnside groups of large even exponents
type: Item
instance of: scholarly article
title: On finite and locally finite subgroups of free Burnside groups of large even exponents
zbMATH Open document ID: 0918.20031
published in: Journal of Algebra
publication date: 1999-05-25T00:00:00Z
review text: The Burnside Problem asks whether the free \(m\)-generator (Burnside) group \(B(m,n)\) of exponent \(n\) is finite. It was proved by P. S. Novikov and S. I. Adyan, 1968, that \(B(m,n)\) is infinite for any \(m\geq 2\) and any odd \(n\geq 665\). For sufficiently large even exponents \(n\) infinite groups \(B(m,n)\) were constructed only recently in the works of S. V. Ivanov, 1994, and I. G. Lysenok, 1996. The structure of \(B(m,n)\) for even \(n\) seems to be more complicated than for \(n\) odd. For example, for odd \(n\geq 665\) all finite or abelian subgroups of \(B(m,n)\) are cyclic. But this cannot be true for even \(n\), since every infinite \(2\)-group has infinite abelian subgroups by a theorem of D. Held and O. H. Kegel. Analyzing finite subgroups of \(B(m,n)\) for even \(n\) turned out to be an important part of proving the infiniteness of those groups. In an earlier article, the first author proved that if \(n=n_1n_2\geq 2^{48}\) with \(n_1\) odd and \(n_2=2^k\geq 2^9\), then any finite subgroup of \(B(m,n)\) is isomorphic to a subgroup of \(D(2n_1)\times D(2n_2)^l\), where \(D(2a)\) denotes the dihedral group of order \(2a\), and \(D(2n_2)^l\) is the \(l\)th direct power of \(D(2n_2)\). In the present article the authors prove, in particular, that all such groups do indeed appear as subgroups of \(B(m,n)\), where \(m\geq 2\) and \(n\) satisfies the above conditions. For \(m\) and \(n\) as above, the authors give a concise description of all infinite locally finite subgroups of \(B(m,n)\) (up to isomorphism). It is proved that any infinite locally finite subgroup \(S\) of \(B(m,n)\) is contained in a unique maximal infinite locally finite subgroup, while any finite \(2\)-subgroup of \(B(m,n)\) is contained in continuously many pairwise non-isomorphic maximal locally finite subgroups. In addition, \(S\) is locally conjugate to a maximal locally finite subgroup. Proofs are based on the analysis of the centralizers of subgroups of \(B(m,n)\). For example, it i…
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E28
zbMATH DE Number: 1076274
zbMATH Keywords: Burnside problem
zbMATH Keywords: groups of even exponent
zbMATH Keywords: locally finite groups
zbMATH Keywords: Abelian subgroups
zbMATH Keywords: finitely generated periodic groups
zbMATH Keywords: centralizers
zbMATH Keywords: finite subgroups
author: S. V. Ivanov
author: Alexander Y. Olshanskii
reviewed by: Evgenii I. Khukhro
MaRDI profile type: Publication
full work available at URL: jabr.1996.6941
OpenAlex ID: W2164736002
cites work: New estimates of odd exponents of infinite Burnside groups.
cites work: Q3139887
cites work: Sur les groupes hyperboliques d'après Mikhael Gromov. (On the hyperbolic groups à la M. Gromov)
cites work: Q3961016
cites work: Q3772406
cites work: On Groups in Which Every Element has Finite Order
cites work: Q5515413
cites work: On Cancellation in Groups
cites work: On the Burnside problem on periodic groups
cites work: THE FREE BURNSIDE GROUPS OF SUFFICIENTLY LARGE EXPONENTS
cites work: Hyperbolic groups and their quotients of bounded exponents
cites work: Q3270631
cites work: Q5513066
cites work: Q5628181
cites work: Varieties of groups
cites work: Q3994685
cites work: Conditional identities in finite groups
cites work: Q4317890
cites work: Q3338438
DOI: 10.1006/JABR.1996.6941
description: scientific article; zbMATH DE number 1076274