Comparison of addition and multiplication in a skew brace

TL;DR

This paper investigates the relationship between addition and multiplication in finite skew braces, providing conditions under which the multiplicative group is solvable, and confirms the conjecture in cases where the additive derived subgroup is cyclic.

Contribution

It proves that if the additive derived subgroup of a finite skew brace is cyclic, then its multiplicative group is solvable, supporting a conjecture about skew brace structures.

Findings

If the additive derived subgroup is cyclic, the multiplicative group is solvable.

Under certain conditions, the images of addition and multiplication coincide in quotient groups.

The conjecture holds for skew braces with cyclic additive derived subgroup.

Abstract

A. Smoktunowicz and L. Vendramin conjectured that if $A=(A,\oplus,\odot)$ is a finite skew brace with solvable additive group $A_{\oplus}$, then the multiplicative group $A_{\odot}$ of $A$ is also solvable. Proving or disproving this conjecture is currently an open problem. The interest to the conjecture of A. Smoktunowicz and L. Vendramin is due to the fact that, despite the fact that the addition and multiplication in a skew brace are related to each other, they can be very different. The present work focuses on comparing addition and multiplication in a skew brace. The results presented in the paper say that if $B$ is a characteristic subgroup of $A_{\oplus}$, then under certain conditions on elements $a,b\in A$ the images of $a\odot b$ and $a\oplus b$ coincide in $A_{\oplus}/B$. As a corollary we conclude that if $A$ is a finite skew brace such that the derived subgroup $A_{\oplus}^{\prime}$ is cyclic, then $A_{\odot}$ is solvable. This statement gives a positive answer to the conjecture of A. Smoktunowicz and L. Vendramin in the case when $A_{\oplus}^{\prime}$ is a cyclic group.