Galois trees in the graph of p-groups of maximal class

The investigation of the graph G(p) associated with the finite p- groups of maximal class was initiated by Blackburn (1958) and became a deep and interesting research topic since then. Leedham-Green & McKay (1976-1984) introduced skeletons of G(p), described their importance for the structural investigation of G(p) and exhibited their relation to algebraic number theory. Here we go one step further: we partition the skeletons into so-called Galois trees and study their general shape. In the special case p >= 7 and p equivalent to 5 mod 6, we show that they have a significant impact on the periodic patterns of G(p) conjectured by Eick, Leedham-Green, Newman & O'Brien (2013). In particular, we use Galois trees to prove a conjecture by Dietrich (2010) on these periodic patterns. (c) 2022 Elsevier Inc. All rights reserved.