On the upper exponent of a finite \(p\)-group.
type: Item
instance of: scholarly article
title: On the upper exponent of a finite \(p\)-group.
zbMATH Open document ID: 1072.20023
author: Bettina Wilkens
published in: Journal of Algebra
publication date: 2004-09-23T00:00:00Z
review text: For a finite \(p\)-group \(G\) the upper exponent \(\overline\exp(G)\) is \(p^l\) where \(l\) is the length of the series \(G>G^p>(G^p)^p>((G^p)^p)^p>\cdots\). \textit{A. Shalev} [Q. J. Math., Oxf. II. Ser. 42, No. 167, 375-378 (1991; Zbl 0732.20004)] obtained an upper bound for \(\overline\exp(G)\) in terms of the nilpotency class \(c\) and the exponent of \(G\). Answering a question of Y. G. Berkovich the author constructs examples showing that \(\overline\exp(G)\) cannot be bounded in terms of the exponent of \(G\) alone. Actually new upper bounds are obtained that are shown to be attained on certain families of groups of arbitrarily large class. The proof uses the multilinear identities of \textit{M. R. Vaughan-Lee} in associated Lie rings of free Burnside groups of prime-power exponent [The restricted Burnside problem. (Lond. Math. Soc. Monographs. New Series. 8), Oxford: Clarendon Press (1993; Zbl 0817.20001)]. (The author mentions certain gaps in the proof for \(p=2\).)
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20F40
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 2102482
zbMATH Keywords: exponents
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: nilpotency classes
reviewed by: Evgenii I. Khukhro
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.02.033
OpenAlex ID: W1984801603
cites work: On subgroups of finite \(p\)-groups
cites work: Q4264394
cites work: On Lie algebras satisfying the Engel condition
cites work: Dimension subgroups, nilpotency indices, and the number of generators of ideals in p-group algebras
cites work: UNISERIAL PERMUTATION MODULES
cites work: Q3997555
DOI: 10.1016/J.JALGEBRA.2004.02.033
description: scientific article; zbMATH DE number 2102482