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Skew braces with no proper left ideals

Cindy Tsang

TL;DR

The paper investigates finite left-simple skew braces, defined as nontrivial braces with no nontrivial left ideals, and shows they correspond to minimal Hopf–Galois structures on finite Galois extensions. It proves a complete characterization when the additive group is abelian or when the additive group is a single simple factor ($n=1$), showing in the abelian case the brace is merely $(\mathbb{F}_p,+)$ and in the $n=1$ case the brace must be almost trivial; for nonabelian simple factors and $n\ge 2$, it derives strong restrictions via group-factorization arguments and the action on $S_n$. The results bridge skew-brace theory with Hopf–Galois theory, giving a partial classification of minimal Hopf–Galois structures on finite Galois extensions and highlighting how transitivity of permutation actions arises from left-simplicity. Overall, the work tightens the understanding of when a Hopf–Galois structure is minimal by translating the problem into the language of skew braces and modular representations, with concrete consequences for the abelian and non-abelian cases.

Abstract

A skew brace $A = (A,\cdot,\circ)$ is said to be \textit{left-simple} if $A\neq1$ and it has no left ideal other than $1$ and $A$. The purpose of this paper is to give a partial classification of the finite left-simple skew braces. A result of Stefanello and Trappeniers implies that finite left-simple skew braces correspond precisely to minimal Hopf--Galois structures on finite Galois extensions of fields.

Skew braces with no proper left ideals

TL;DR

The paper investigates finite left-simple skew braces, defined as nontrivial braces with no nontrivial left ideals, and shows they correspond to minimal Hopf–Galois structures on finite Galois extensions. It proves a complete characterization when the additive group is abelian or when the additive group is a single simple factor (), showing in the abelian case the brace is merely and in the case the brace must be almost trivial; for nonabelian simple factors and , it derives strong restrictions via group-factorization arguments and the action on . The results bridge skew-brace theory with Hopf–Galois theory, giving a partial classification of minimal Hopf–Galois structures on finite Galois extensions and highlighting how transitivity of permutation actions arises from left-simplicity. Overall, the work tightens the understanding of when a Hopf–Galois structure is minimal by translating the problem into the language of skew braces and modular representations, with concrete consequences for the abelian and non-abelian cases.

Abstract

A skew brace is said to be \textit{left-simple} if and it has no left ideal other than and . The purpose of this paper is to give a partial classification of the finite left-simple skew braces. A result of Stefanello and Trappeniers implies that finite left-simple skew braces correspond precisely to minimal Hopf--Galois structures on finite Galois extensions of fields.
Paper Structure (6 sections, 7 theorems, 42 equations)

This paper contains 6 sections, 7 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm2']}
  4. The case when $n=1$
  5. The case when $n\geq 2$.
  6. Connection with minimal Hopf--Galois structures

Key Result

Theorem 1.2

Let $A=(A,\cdot,\circ)$ be a finite skew brace such that $(A,\cdot)\simeq \mathbb{F}_p^n$ for some prime $p$ and $n\in \mathbb{N}$. Then $A$ is left-simple if and only if $n=1$.

Theorems & Definitions (13)

  • Definition 1.1
  • Theorem 1.2
  • proof
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 3 more