Fernández-Alcober, Gustavo A. The exact lower bound for the degree of commutativity of a \(p\)-group of maximal class. (English) Zbl 0835.20030 J. Algebra 174, No. 2, 523-530 (1995). Let \(G\) be a \(p\)-group of maximal class of order \(p^m\), \(m\geq 4\). 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 \(p=2,3,5\). G. Shepherd [“\(p\)-groups of maximal class”, Ph. D. thesis, Univ. Chicago (1970)] and C. R. Leedham-Green and 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 \(c(G)\). Reviewer: N.Yu.Makarenko (Novosibirsk) Cited in 4 ReviewsCited in 8 Documents MSC: 20D15 Finite nilpotent groups, \(p\)-groups 20F14 Derived series, central series, and generalizations for groups Keywords:\(p\)-groups of maximal class; lower central series; degree of commutativity Citations:Zbl 0083.24802; Zbl 0353.20020 PDFBibTeX XMLCite \textit{G. A. Fernández-Alcober}, J. Algebra 174, No. 2, 523--530 (1995; Zbl 0835.20030) Full Text: DOI