Constructing skew left braces whose additive group has trivial centre

TL;DR

The paper provides a complete description of all possible multiplicative groups of finite skew left braces whose additive group has trivial centre, linking the brace structure to two subgroups $X,Y\le\mathrm{Aut}(K)$ with $XY= X\mathrm{Inn}(K)=Y\mathrm{Inn}(K)$ and a compatible isomorphism $\gamma:Y/M\to X/N$. It shows that the multiplicative group of a brace is a subdirect product of $X$ and $Y$ under these data, and conversely that any such pair $X,Y$ yields a brace with additive group $K$, thereby refining previous Tsang results and answering the Ischia 2024 question in the affirmative. A key corollary (Corollary drop-split) simplifies the construction by removing the need for $X$ to split over $X\cap\mathrm{Inn}(K)$ in many cases. The included worked PSL$_2(25)$ example illustrates nontrivial intersection scenarios and confirms the viability of the approach, with computational checks supporting the theoretical framework.

Abstract

A complete description of all possible multiplicative groups of finite skew left braces whose additive group has trivial centre is shown. As a consequence, some earlier results of Tsang can be improved and an answer to an open question set by Tsang at Ischia Group Theory 2024 Conference is provided.

Figures (1)

• Figure 1: Structure of the multiplicative group in Theorem \ref{['th-trivz']}

Theorems & Definitions (6)

• Theorem 1: see Tsang23-blms
• Theorem 2: see Tsang23-blms
• Theorem A
• Corollary 4
• proof
• proof : Proof of Theorem \ref{['th-trivz']}