Connections between properties of the additive and the multiplicative groups of a two-sided skew brace

TL;DR

This paper explores the relationship between the additive and multiplicative groups of two-sided skew braces, establishing conditions under which properties like solvability and nilpotency are transferred between these groups, and solving open problems from the Kourovka notebook.

Contribution

It generalizes previous results by linking properties of additive and multiplicative groups in two-sided skew braces and solves two open problems from the Kourovka notebook.

Findings

If the additive group is finite solvable, the multiplicative group is solvable.

If the multiplicative group is nilpotent of class k, the additive group is solvable of class at most 2k.

The paper solves two open problems related to skew braces from the Kourovka notebook.

Abstract

We study relations between the additive and the multiplicative groups of a two-sided skew brace. In particular, we prove that if the additive group of a two-sided skew brace is finite solvable (respectively, finitely generated nilpotent, finitely generated residually nilpotent, finitely generated residually finite), then the multiplicative group of this skew brace is solvable (respectively, solvable, residually solvable, residually finite). Also, we prove that if the multiplicative group of a two-sided skew brace is nilpotent of nilpotency class $k$, then the additive group of this skew brace is solvable of class at most $2k$. The letter result generalizes the result of Byott which says that if the multiplicative group of a finite skew brace is abelian, then the additive group of this skew brace is solvable. In addition, we solve two problems (Problem 19.49 and Problem 19.90(a)) concerning skew braces which are formulated in the Kourovka notebook.