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

Abstract

The investigation of the graph Gp 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 Gp, described their importance for the structural investigation of Gp 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≡5mod6, we show that they have a significant impact on the periodic patterns of Gp 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.