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

2021/09/20 by A. Cant, Cant, Alexander, Heiko Dietrich +5
Mathematics · #20D15 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2109.09355

openalex publication_date 2021/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 and 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 ≡ 5 \bmod 6, we show that they have a significant impact on the periodic patterns of 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.

Related

• Modeling rooted in-trees by finite p-groups
• Tree representations of Galois groups
• The Elementary Type Conjecture for Maximal Pro-p Galois groups
• Combinatorial Techniques in the Galois Theory of p-Extensions
• Explicit computation of Galois p-groups unramified at p