Triply Factorised Groups and the Structure of Skew Left Braces

Abstract

The algebraic structure of skew left brace has proved to be useful as a source of set-theoretic solutions of the Yang–Baxter equation. We study in this paper the connections between left and right $\pi $-nilpotency and the structure of finite skew left braces. We also study factorisations of skew left braces and their impact on the skew left brace structure. As a consequence of our study, we define a Fitting-like ideal of a left brace. Our approach depends strongly on a description of a skew left brace in terms of a triply factorised group obtained from the action of the multiplicative group of the skew left brace on its additive group.

Keywords

Skew left brace / Trifactorised group / Triply factorised group / Left nilpotent skew left brace / Right nilpotent skew left brace / Ideal / Left Fitting ideal / Factorised skew left brace