Bounding the exponent of a verbal subgroup.
type: Item
instance of: scholarly article
title: Bounding the exponent of a verbal subgroup.
zbMATH Open document ID: 1306.20034
author: Eloisa Detomi
author: Marta Morigi
author: Pavel Shumyatsky
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2014-10-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1301.4093
review text: Let \(w=w(x_1,x_2,\ldots,x_m)\) be a group word and \(G\) be a group; the verbal subgroup of \(G\) defined by \(w\) is \(w(G)=\langle w(g_1,g_2,\ldots,g_m)\mid g_1,g_2,\ldots,g_m\in G\rangle\). In the paper under review the authors deal with the following conjecture. If \(G\) is a finite group in which any nilpotent subgroup generated by \(w\)-values has exponent dividing \(e\), then the exponent of \(w(G)\) is bounded in terms of \(w\) and \(e\) only. In Theorem A the previous conjecture is proved to be true in the case where \(w\) is either the \(n\)-th Engel word or the word \([x^r,y_1,y_2,\ldots,y_n]\). In Theorem B the authors are able to prove that for any positive integer \(e\) there exists a number \(k=k(e)\) such that if \(w\) is a word and \(G\) is a finite group in which any nilpotent subgroup generated by products of \(k\) values of the word \(w\) has exponent dividing \(e\), then the exponent of \(w(G)\) is bounded in terms of \(w\) and \(e\) only.
reviewed by: Enrico Jabara
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F45
Mathematics Subject Classification ID: 20F12
zbMATH DE Number: 6363178
zbMATH Keywords: group words
zbMATH Keywords: verbal subgroup
zbMATH Keywords: commutator words
zbMATH Keywords: Engel words
zbMATH Keywords: exponent
zbMATH Keywords: finite groups
zbMATH Keywords: nilpotent subgroups
MaRDI profile type: Publication
OpenAlex ID: W1982714209
arXiv ID: 1301.4093
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DOI: 10.1007/S10231-013-0336-8
description: scientific article; zbMATH DE number 6363178