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Finite skew braces whose additive group is a Z-group

Marco Damele

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.

Finite skew braces whose additive group is a Z-group

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 -group. More precisely, we prove that if is a finite skew brace whose additive group is a -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 -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 -group if and only if the multiplicative group is a -group.
Paper Structure (9 sections, 10 theorems, 40 equations)

This paper contains 9 sections, 10 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Skew braces
  4. Holomorph and skew braces
  5. Z-groups and almost Sylow cyclic groups
  6. Proofs of the main results
  7. Proof of Theorem \ref{['th: generalizedRump']}
  8. Proof of Theorem \ref{['Th: B ssolv']}
  9. Proof of Theorem \ref{['Th: Multiplicative-Z-group']}

Key Result

Theorem A

Let $B$ be a finite skew brace such that $(B,+)$ is a $Z$-group. Then $(B,\cdot)$ is solvable and almost Sylow cyclic.

Theorems & Definitions (21)

  • Theorem A
  • Theorem B
  • Theorem C
  • Remark 1
  • Theorem 2.1: Guarnieri_2016
  • Theorem 2.2: Hall1959
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['th: generalizedRump']}
  • Example 1
  • ...and 11 more