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On the triviality and non-triviality of the automorphism group of a skew brace

Cindy Tsang

Abstract

It is a simple fact that a group has a trivial automorphism group if and only if it is of order $1$ or $2$. We prove that the same holds for certain families of skew braces, and given any odd prime $p$, we construct a skew brace of order $2p^3$ that has a trivial automorphism group.

On the triviality and non-triviality of the automorphism group of a skew brace

Abstract

It is a simple fact that a group has a trivial automorphism group if and only if it is of order or . We prove that the same holds for certain families of skew braces, and given any odd prime , we construct a skew brace of order that has a trivial automorphism group.
Paper Structure (9 sections, 11 theorems, 116 equations)

This paper contains 9 sections, 11 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm']}
  4. Part (1)
  5. Part (2)
  6. Part (3)
  7. A method of constructing skew braces
  8. An explicit description of the skew brace of order $24$ with a trivial automorphism group
  9. Proof of Theorem \ref{["thm'"]}

Key Result

Theorem 1.1

Let $A= (A,\cdot,\circ)$ be a skew brace with $|A|\geq 3$ such that Then $\mathop{\mathrm{Aut}}\nolimits(A)$ is non-trivial.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 3.1
  • Theorem 4.1
  • proof
  • ...and 12 more