Finite skew braces with solvable additive group

TL;DR

This paper advances the understanding of finite skew braces by showing that minimal examples with solvable additive groups cannot have simple, non-solvable multiplicative groups, supporting a conjecture in the field.

Contribution

It proves the conjecture for cases where the order of the skew brace is not divisible by 3 and provides insights into the structure of minimal skew braces.

Findings

If A is a minimal finite skew brace with solvable additive group and non-solvable multiplicative group, then the multiplicative group is not simple.

The conjecture holds when the order of A is not divisible by 3.

Abstract

A. Smoktunowicz and L. Vendramin conjectured that if $A$ is a finite skew brace with solvable additive group, then the multiplicative group of $A$ is solvable. In this short note we make a step towards positive solution of this conjecture proving that if $A$ is a minimal finite skew brace with solvable additive group and non-solvable multiplicative group, then the multiplicative group of $A$ is not simple. On the way to obtaining this result, we prove that the conjecture of A. Smoktunowicz and L. Vendramin is correct in the case when the order of $A$ is not divisible by $3$.