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

TL;DR

This paper studies Galois trees within the skeletons of graphs associated with finite p-groups of maximal class, revealing their influence on periodic patterns and proving a related conjecture.

Contribution

It introduces the concept of Galois trees in the skeletons of $ ext{G}_p$ and demonstrates their role in understanding periodic patterns, including proving a conjecture.

Findings

Galois trees significantly influence the periodic patterns of $ ext{G}_p$

Proof of Dietrich's conjecture on these patterns

Identification of the shape and structure of Galois trees

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 $p \geq 7$ and $p \equiv 5 \bmod 6$, 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.