Let G be a p-group of maximal class. Since the pioneer work of Blackburn in 1958 (cf. [N. Blackburn, Acta Math. 100 (1958) 45-92]), several authors have obtained information about the degree of commutativity c of G, in order to precise which the defining relations of G are (cf. [N. Blackburn, Acta Math. 100 (1958) 45-92; R. Shepherd, PhD Thesis, University of Chicago, 1970; C.R. Leedham-Green, S. McKay, Quart. J. Math. Oxford Ser. (2) 27 (1976) 297-311, Quart. J. Math. Oxford Ser. (2) 29 (1978) 175-186, 281-299; G.A. Fernandez-Alcober, J. Algebra 174 (1995) 523-530; A. Vera-Lopez, J.M. Arregi, F.J. Vera-Lopez, Comm. Algebra 23 (1995) 2765-2795, Math. Proc. Cambridge Philos. Soc. 122 (1997) 251-260]). In [A. Vera-Lopez, J.M. Arregi, M.A. Garcia-Sanchez, F.J. Vera-Lopez, R. Esteban-Romero, An algorithm for the computation of bounds for the degree of commutativity of a p-group of maximal class, submitted for publication], computational calculations for G, when p less than or equal to, 43 have made evident that the commutator structure of G can be obtained much better, if we consider two invariants; c(0) is an element of {0, 1,..., p - 2}, the residual class of c module p-1, and l is an element of {1, 2,..., (p-3)/2}, defined by l = (1/2) min{k is an element of [2, m-c-2] \ [Y-1, Y-k] = Yl+k+c}. Besides, in [A. Vera-Lopez, J.M. Arregi, M.A. Garcia-Sanchez, F.J. Vera-Lopez, R. Esteban-Romero, An algorithm for the computation of bounds for the degree of commutativity of a p-group of maximal class, submitted for publication] six functions, which covers almost all possible values of (c(0), 1), are conjectured. In [A. Vera-Lopez, J.M. Arregi, M.A. Garcia-Sanchez, FT Vera-Lopez, R. Esteban-Romero, J. Algebra 256 (2002) 375-401], it is proved the validity of two of them. In this paper, we prove that the rest of conjectured holds. (C) 2004 Elsevier Inc. All rights reserved.
