Comparison of addition and multiplication in a skew brace
Baojun Li, Timur Nasybullov, Vyacheslav Zadvornov
TL;DR
This work examines the relationship between the additive and multiplicative structures in finite skew braces, motivated by a standing conjecture on solvability. It develops a framework based on quotients by characteristic subgroups and the action of automorphisms to compare $a\odot b$ and $a\oplus b$ modulo a subgroup. A key contribution is the demonstration that, when $A_{\oplus}'$ is cyclic, the multiplicative group $A_{\odot}$ is solvable, providing a positive case of the conjecture. The approach combines quotient-based arguments, centralizers of cyclic subgroups, and known results about nilpotent/additive groups to propagate solvability from the additive to the multiplicative structure. Overall, the paper advances understanding of how closely related the two operations are in skew braces and identifies conditions under which their associated groups share solvability properties.
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.