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

Abstract

The investigation of the graph ${\mathcal{G}}_p$ 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 and McKay (1976-1984) introduced skeletons of ${\mathcal{G}}_p$, described their importance for the structural investigation of ${\mathcal{G}}_p$ 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 $p\ge 7$ and $p\equiv 5\mathrm{mod}6$, we show that they have a significant impact on the periodic patterns of ${\mathcal{G}}_p$ conjectured by Eick, Leedham-Green, Newman and O'Brien (2013). In particular, we use Galois trees to prove a conjecture by Dietrich (2010) on these periodic patterns.