Lie rings related to the $p$-groups of maximal class

TL;DR

This paper explores Lie rings associated with maximal class p-groups, proving their finiteness and nilpotency, thereby filling a key gap in understanding their algebraic structure.

Contribution

It establishes that these Lie rings are always finite and nilpotent of small class, advancing the theoretical understanding of their properties.

Findings

Lie rings are always finite

Lie rings are nilpotent of small class

Fills a gap in prior research (Eick, Komma & Saha 2025)

Abstract

The Lazard correspondence induces a close relation between the $p$-groups of maximal class and a certain type of Lie ring constructed from $p$-adic number fields. Our aim here is to investigate such Lie rings. In particular, we show that they are always finite. It then follows that they are nilpotent of small class. These results close an important gap in (Eick, Komma \& Saha 2025).