On varieties arising from the solution of the restricted Burnside problem
type: Item
instance of: scholarly article
title: On varieties arising from the solution of the restricted Burnside problem
zbMATH Open document ID: 1007.20025
DOI: 10.1016/S0022-4049(01)00115-3
published in: Journal of Pure and Applied Algebra
publication date: 2002-08-22T00:00:00Z
review text: Let \(h\) and \(n\) be positive integers, and let \(w\) be a multilinear commutator. Then let \({\mathfrak X}(w,n,h)\) be the class of groups \(G\) such that, first, the verbal subgroup \(w(G)\) has Hirsch-Plotkin height \(h\) (that is, \(w(G)\) has a normal series of length \(h\) with locally nilpotent factors) and, second, \(G\) satisfies the identity \(w^n=1\). It is proved that \({\mathfrak X}(w,n,h)\) is a variety. This is another generalization of the positive solution of the Restricted Burnside Problem. Earlier [Math. Proc. Camb. Philos. Soc. 132, No. 2, 193-196 (2002; see the preceding review Zbl 1007.20024)] the author proved this in the special case when \(w\) is a multilinear simple commutator. He also proposes the following interesting problems: let \(w\) be any group word and \(n\) a positive integer; is the class of groups \(G\) satisfying the identity \(w^n=1\) and having \(w(G)\) locally nilpotent (or locally finite) a variety? The proof in the present paper relies on \textit{E. I. Zel'manov}'s theorem [Lond. Math. Soc. Lect. Note Ser. 212, 567-585 (1995; Zbl 0860.20031)] stating that a finitely generated PI Lie algebra in which all commutators in the generators are ad-nilpotent is nilpotent. This powerful result is applied to the Lazard-Zassenhaus Lie algebra defined via dimension subgroups, which is a PI-algebra by a result of \textit{J. S. Wilson} and \textit{E. I. Zel'manov} [J. Pure Appl. Algebra 81, No. 1, 103-109 (1992; Zbl 0851.17007)].
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20F40
Mathematics Subject Classification ID: 20F50
zbMATH DE Number: 1787440
zbMATH Keywords: locally nilpotent groups
zbMATH Keywords: restricted Burnside problem
zbMATH Keywords: Lie PI-algebras
zbMATH Keywords: multilinear commutators
zbMATH Keywords: varieties of groups
zbMATH Keywords: Lazard-Zassenhaus Lie algebras
zbMATH Keywords: Hirsch-Plotkin height
zbMATH Keywords: locally finite groups
zbMATH Keywords: verbal subgroups
zbMATH Keywords: group words
zbMATH Keywords: dimension subgroups
author: Pavel Shumyatsky
reviewed by: Evgenii I. Khukhro
MaRDI profile type: Publication
cites work: On the p -Length of p -Soluble Groups and Reduction Theorems for Burnside's Problem
cites work: Q3934684
cites work: Strong finiteness conditions in locally finite groups
cites work: Bounding the exponent of a finite group with automorphisms
cites work: Sur les groupes nilpotents et les anneaux de Lie
cites work: Q4177758
cites work: Applications of Lie ring methods to group theory
cites work: Verbal subgroups in residually finite groups
cites work: A (locally nilpotent)-by-nilpotent variety of groups
cites work: Identities for Lie algebras of pro-\(p\) groups
cites work: SOLUTION OF THE RESTRICTED BURNSIDE PROBLEM FOR GROUPS OF ODD EXPONENT
cites work: Q3362124
cites work: Q4520878
cites work: Q4891348
Recommended article: Q3362125
Recommended article: THE RESTRICTED BURNSIDE PROBLEM FOR VARIETIES OF SEMIGROUPS
Recommended article: The word problem for the Burnside varieties
Recommended article: Q4309585
Recommended article: Q4636059
Recommended article: Q4227245
Recommended article: Q4011125
Recommended article: On the restricted Burnside problem
Recommended article: Q3487599
Recommended article: PROBLEMS OF BURNSIDE TYPE AND THE FINITE BASIS PROPERTY IN VARIETIES OF SEMIGROUPS
description: scientific article; zbMATH DE number 1787440