The exact lower bound for the degree of commutativity of a \(p\)-group of maximal class
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 \(c(G)\).
scientific article; zbMATH DE number 783874