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On the Sylow Theorem for Skew Braces

A. Caranti, I. Del Corso, M. Di Matteo, M. Ferrara, M. Trombetti

Abstract

We discuss the (first) Sylow theorem for certain classes of finite skew braces, proving it to hold true when the skew brace is two-sided, bi-skew, right nilpotent, $λ$-homomorphic or supersoluble. We also show it to hold true for soluble skew braces that are left-nilpotent, and address a number of more specialized settings, proving general Hall-type theorems.

On the Sylow Theorem for Skew Braces

Abstract

We discuss the (first) Sylow theorem for certain classes of finite skew braces, proving it to hold true when the skew brace is two-sided, bi-skew, right nilpotent, -homomorphic or supersoluble. We also show it to hold true for soluble skew braces that are left-nilpotent, and address a number of more specialized settings, proving general Hall-type theorems.

Paper Structure

This paper contains 4 sections, 15 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sylow and Hall Theorems
  4. An insoluble example

Key Result

Lemma 4

Let $(B,+,\circ)$ be a finite skew brace, and let $S$ be a Hall $\pi$-sub-skew brace for some set $\pi$ of primes. If $I$ is any ideal of $B$, then $S+I/I$ and $S\cap I$ are Hall $\pi$-sub-skew braces of $B/I$ and $I$, respectively.

Theorems & Definitions (26)

  • Remark 3
  • Lemma 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Corollary 7
  • proof
  • Lemma 8
  • proof
  • ...and 16 more