The exact bounds for the degree of commutativity of a p-group of maximal class, II

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. Fernández-Alcober, J. Algebra 174 (1995) 523–530; A. Vera-López, J.M. Arregi, F.J. Vera-López, Comm. Algebra 23 (1995) 2765–2795, Math. Proc. Cambridge Philos. Soc. 122 (1997) 251–260]). In [A. Vera-López, J.M. Arregi, M.A. García-Sánchez, F.J. Vera-López, 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 43 have made evident that the commutator structure of G can be obtained much better, if we consider two invariants; c0 {0,1,…,p−2}, the residual class of c module p−1, and l {1,2,…,(p−3)/2}, defined by l=(1/2)min{k [2,m−c−2] [Y1,Yk]=Y1+k+c}. Besides, in [A. Vera-López, J.M. Arregi, M.A. García-Sánchez, F.J. Vera-López, 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 (c0,l), are conjectured. In [A. Vera-López, J.M. Arregi, M.A. García-Sánchez, F.J. Vera-López, 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.