On the Use of the Lazard Correspondence in the Classification of p-Groups of Maximal Class,

Let G be a p-group of maximal class of order pm, p an odd prime, and m ≥ 4. In this work we reduce the construction of this group to the consideration of certain elements of HomS(R/ m − 2 R/ m − 2, R/ m − 4), where R = [x]/(1 + ••• + xp − 1) = [ ], = ( − 1), and S = [x]/(xp − 1). As an application of this result we prove that if we consider in G the generators s, s1,…,sm − 1 introduced by Blackburn, then the structure of G is determined by the (p − 3)/2 commutators [si, si + 1] for 1 ≤ i ≤ (p − 3)/2 and the three integers 0 ≤ u, v, w ≤ p − 1, where u and v are defined by the condition sp = sum − 1, (ss1)p = svm − 1, and w = cm − 1 is the exponent of sm − 1 in the commutator [s(p − 1)/2, s(p + 1)/2] = scpp•••scm − 1m − 1, 0 ≤ ci ≤ p − 1.