Finite skew braces whose additive group is a Z-group

TL;DR

This paper extends known results about finite skew braces with cyclic additive groups to those with Z-groups, showing they have solvable, almost Sylow cyclic, and supersolvable multiplicative groups, and establishing a Z-group equivalence for odd order braces.

Contribution

It generalizes the classification of finite skew braces from cyclic to Z-groups, demonstrating their multiplicative groups are solvable, almost Sylow cyclic, and supersolvable, and proves a Z-group equivalence for odd order braces.

Findings

Finite skew braces with Z-group additive groups have solvable, almost Sylow cyclic multiplicative groups.

Such skew braces are supersolvable and 2-nilpotent.

For odd order skew braces, the additive and multiplicative groups are Z-groups if and only if each other.

Abstract

Rump proved in \cite[Theorem~1]{Rump2018ClassificationOC} that if a finite skew brace has cyclic additive group, then its multiplicative group is solvable and almost Sylow cyclic. In this paper we show that this rigidity persists when the additive group is a \(Z\)-group. More precisely, we prove that if \(B\) is a finite skew brace whose additive group is a \(Z\)-group, then \((B,\cdot)\) is solvable and almost Sylow cyclic. In addition, we show that every such skew brace is supersolvable; in particular, \((B,\cdot)\) is \(2\)-nilpotent. This extends \cite[Theorem~3.8]{ballesterbolinches2024finiteskewbracessquarefree} and recovers, in this broader setting, another result of Rump \cite[Proposition 13]{Rump2018ClassificationOC}. Finally, we prove that for skew braces of odd order the additive group is a \(Z\)-group if and only if the multiplicative group is a \(Z\)-group.