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On the structure of some one-generator nilpotent braces

Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya. Subbotin

TL;DR

The paper investigates the structure of one-generator Smoktunowicz-nilpotent left braces by constructing explicit models D(1,2) and D(1,3) realizing zl-values 2 and 3, respectively. It proves that any non-abelian one-generator brace with zl(A)=2 or zl(A)=3 is an epimorphic image of the corresponding model, and shows abelianness of A^2 in the zl=3 case with an epimorphic-description via D(1,3). These results advance the classification of one-generator, star-nilpotent braces and clarify how the upper star-central series governs their epimorphic images and internal abelian substructures.

Abstract

This article provides a detailed description of some nilpotent left braces generated by one element.

On the structure of some one-generator nilpotent braces

TL;DR

The paper investigates the structure of one-generator Smoktunowicz-nilpotent left braces by constructing explicit models D(1,2) and D(1,3) realizing zl-values 2 and 3, respectively. It proves that any non-abelian one-generator brace with zl(A)=2 or zl(A)=3 is an epimorphic image of the corresponding model, and shows abelianness of A^2 in the zl=3 case with an epimorphic-description via D(1,3). These results advance the classification of one-generator, star-nilpotent braces and clarify how the upper star-central series governs their epimorphic images and internal abelian substructures.

Abstract

This article provides a detailed description of some nilpotent left braces generated by one element.
Paper Structure (4 sections, 12 theorems, 50 equations)

This paper contains 4 sections, 12 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. The Proof of Theorem \ref{['t:maina']}
  4. The Proofs of Theorems \ref{['t:mainb']} and \ref{['t:mainc']}

Key Result

Theorem A

Theorems & Definitions (22)

  • Theorem A
  • Theorem B
  • Theorem C
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 12 more