On the commutator lengths of certain classes of finitely presented groups.
type: Item
instance of: scholarly article
title: On the commutator lengths of certain classes of finitely presented groups.
zbMATH Open document ID: 1123.20028
DOI: 10.1155/IJMMS/2006/74981
author: P. P. Campbell
author: Hossein Doostie
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2007-06-13T00:00:00Z
full work available at URL: https://eudml.org/doc/53527
review text: Summary: For a finite group \(G=\langle X\rangle\) (\(X\neq G\)), the least positive integer \(\text{ML}_X(G)\) is called the maximum length of \(G\) with respect to the generating set \(X\) if every element of \(G\) may be represented as a product of at most \(\text{ML}_X(G)\) elements of \(X\). The maximum length of \(G\), denoted by \(\text{ML}(G)\), is defined to be the minimum of \(\{\text{ML}_X(G)\mid G=\langle X\rangle\), \(X\neq G\), \(X\neq G-\{1_G\}\}\). The well-known commutator length of a group \(G\), denoted by \(c(G)\), satisfies the inequality \(c(G)\leq\text{ML}(G')\), where \(G'\) is the derived subgroup of \(G\). In this paper we study the properties of \(\text{ML}(G)\) and by using this inequality we give upper bounds for the commutator lengths of certain classes of finite groups. In some cases these upper bounds involve the interesting sequences of Fibonacci and Lucas numbers.
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 5163997
zbMATH Keywords: finite groups
zbMATH Keywords: generating sets
zbMATH Keywords: commutator lengths
zbMATH Keywords: Fibonacci numbers
zbMATH Keywords: Lucas numbers
MaRDI profile type: Publication
cites work: A note on the Todd-Coxeter coset enumeration algorithm
cites work: Q3684288
cites work: On \((2,n)\)-groups related to Fibonacci groups
cites work: A class of deficiency zero soluble groups of derived length 4
cites work: Q3050627
cites work: Q3726349
cites work: Some Remarks on Commutators
cites work: Commutators in the Special and General Linear Groups
cites work: Q3977768
description: scientific article; zbMATH DE number 5163997