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On a Cauchy theorem for finite skew braces

Marco Damele, Vicent Pérez Calabuig

Abstract

One of the major problems in the structural theory of skew braces consists in the classification of skew braces of finite order up to isomorphism. In this light, the open question of the existence of a Cauchy theorem for finite skew braces is of great interest. We prove a positive answer for the classes of finite two-sided skew braces and bi-skew braces. Consequences and related structural results are also outlined.

On a Cauchy theorem for finite skew braces

Abstract

One of the major problems in the structural theory of skew braces consists in the classification of skew braces of finite order up to isomorphism. In this light, the open question of the existence of a Cauchy theorem for finite skew braces is of great interest. We prove a positive answer for the classes of finite two-sided skew braces and bi-skew braces. Consequences and related structural results are also outlined.
Paper Structure (4 sections, 9 theorems, 21 equations)

This paper contains 4 sections, 9 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['teo:two-sided']}
  4. Proof of Theorem \ref{['teo:bi-skew']}

Key Result

Theorem A

Let $B$ be a two-sided skew brace. For every prime divisor $p$ of $|B|$, there exists a subbrace $S$ of $B$ with $|S| = p$.

Theorems & Definitions (20)

  • Theorem A
  • Theorem B
  • Lemma 2.1
  • proof
  • Remark 1
  • Lemma 2.2
  • proof
  • Remark 2
  • Lemma 2.3
  • proof
  • ...and 10 more