On p-groups of maximal class

Let R be the ring Z[x]/(x(p) - 1/x - 1) = Z[(x) over bar] and let a be the ideal of R generated by ((x) over bar - 1). In this paper, we discuss the structure of the Z[C-p]-module (R/a(n-1))Lambda(R/a(n-1)), which plays an important role in the theory of p-groups of maximal class (see [2]-[5]). The generators of this module allow us to obtain the defining relations of some important examples of p-groups of maximal class with Y-1 of class two. In particular we obtain the best possible estimates for the degree of commutativity of p-groups of maximal class with Y-1 of class two. (C) 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim