The exact bounds for the degree of commutativity of a \(p\)-group of maximal class. I
The goal of the series of papers of which the one under review is the first, which continue previous work on the subject by (some of) the authors, is to provide best possible estimates for the degree of commutativity of \(p\)-groups of maximal class. Recall that a group of order \(p^m\), where \(p\) is a prime and \(m\geq 4\), is said to be of maximal class if it has nilpotency class \(m-1\). The degree of commutativity of such a group is the greatest integer \(k\) such that \([G_i,G_j]\leq G_{i+j+k}\) for all \(i,j\geq 1\). Here \(G_i\) is the \(i\)-th term of the lower central series of \(G\), for \(i\geq 2\), whereas \(G_1=C_G(G_2/G_4)\). To obtain sharp estimates, the parameter space has to be divided into six regions; we refer to the paper for the details. Here, estimates for two of the regions are proved. In a forthcoming paper by Antonio Vera-López and A.~Jaikin-Zapirain, examples are constructed to show that these estimates are indeed sharp.
scientific article; zbMATH DE number 1868036