On conciseness of some commutator words
type: Item
instance of: scholarly article
title: On conciseness of some commutator words
zbMATH Open document ID: 1429.20024
author: Costantino Delizia
author: Antonio Tortora
author: Maria Tota
author: Pavel Shumyatsky
published in: Archiv der Mathematik
publication date: 2019-01-23T00:00:00Z
review text: Let $w =w(x_{1}, \dots, x_{n})$ be a group-word in variables $x_{1}, \dots, x_{n}$ and let $G$ be a group. The verbal subgroup $w(G)$ is the subgroup of $G$ generated by the set $G_{w}=\{w(g_{1},\dots,g_{n}) \mid g_{1},\dots,g_{n} \in G \}$. A word $w$ is \textit{concise} if, for every group $G$, whenever $G_{w}$ is finite the subgroup $w(G)$ is finite. Philip Hall conjectured that every word is concise, and he proved this for every non-commutator word (a word is non-commutator if the sum of exponents of some variables involved in it is non-zero), and for lower central words. \par The purpose of the article under review is to prove Theorem 1.1: For any non-commutator words $w$ and $w'$, the word $[w,w']$ is concise.
reviewed by: Egle Bettio
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20F24
zbMATH DE Number: 7005656
zbMATH Keywords: concise word
zbMATH Keywords: verbal subgroup
zbMATH Keywords: commutator word
MaRDI profile type: Publication
full work available at URL: s00013-018-1215-8
OpenAlex ID: W2810942257
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DOI: 10.1007/S00013-018-1215-8
description: scientific article; zbMATH DE number 7005656