On the multiplicative group of a two-sided skew brace of solvable type

TL;DR

This paper proves that for two-sided skew braces with a solvable additive group, all finite quotients of their multiplicative group are solvable, extending known results to a broader class of algebraic structures.

Contribution

It extends Nasybullov's theorem from finite to arbitrary two-sided skew braces of solvable type, showing all finite quotients of the multiplicative group are solvable.

Findings

Finite quotients of the multiplicative group are solvable

Extends previous finite case results to infinite cases

Provides new insights into the structure of skew braces

Abstract

We prove that if $(B,+,\cdot)$ is a two-sided skew brace whose additive group is solvable, then every finite quotient of the multiplicative group $(B,\cdot)$ is solvable. In particular, our result recovers Nasybullov's theorem in the finite case ~\cite[Theorem~4.3(1)]{Nas} and extends it to arbitrary two-sided skew braces of solvable type.