The exact lower bound for the degree of commutativity of a \(p\)-group of maximal class
type: Item
instance of: scholarly article
title: The exact lower bound for the degree of commutativity of a \(p\)-group of maximal class
zbMATH Open document ID: 0835.20030
author: Gustavo A. Fernández-Alcober
published in: Journal of Algebra
publication date: 1995-08-29T00:00:00Z
review text: Let \(G\) be a \(p\)-group of maximal class of order \(p^m\), \(m\geq 4\). \textit{N. Blackburn} [Acta Math. 100, 45-92 (1958; Zbl 0083.24802)] introduced the important invariant \(c(G)\) which can be defined as \(c(G)=\max\{\kappa\leq m-2\mid[G_i,G_j]\leq G_{i+j+\kappa}\}\), where \(G_i=\gamma_i(G)\) for \(i\geq 2\) and \(G_1=C_G(G_2/G_4)\). This invariant is a measure of the commutativity among the members of the lower central series of \(G\). In the paper mentioned above Blackburn found the exact bound of the degree of commutativity for \(p=2,3,5\). \textit{G. Shepherd} [``\(p\)-groups of maximal class'', Ph. D. thesis, Univ. Chicago (1970)] and \textit{C. R. Leedham-Green} and \textit{S. McKay} [Q. J. Math., Oxf. II. Ser. 27, 297-311 (1976; Zbl 0353.20020)] proved that \(2c(G)\geq m-3p+6\) for \(p\geq 7\). They also constructed examples showing that there exists \(p\)-groups of maximal class such that \(m-3\geq c(G)\geq (m-5)/2\). The aim of the present paper is to prove that \(2c(G)\geq m-2p+5\) for a \(p\)-group \(G\) of maximal class of order \(p^m\), \(p\geq 7\). So the author found the exact lower bound for \(c(G)\).
reviewed by: N. Yu. Makarenko
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20F14
zbMATH DE Number: 783874
zbMATH Keywords: \(p\)-groups of maximal class
zbMATH Keywords: lower central series
zbMATH Keywords: degree of commutativity
MaRDI profile type: Publication
full work available at URL: jabr.1995.1137
OpenAlex ID: W1990309883
DOI: 10.1006/JABR.1995.1137
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description: scientific article; zbMATH DE number 783874