[论文解读] On the triviality and non-triviality of the automorphism group of a skew brace

Cindy Tsang|arXiv (Cornell University)|Mar 18, 2026

Finite Group Theory Research被引用 0

一句话总结

本文分析 Aut(A) 在斜撑结构中何时为平凡，并证明对于任意奇素数 p，存在阶为 2p^3 的斜撑结构其自同构群平凡；同时也证明若干族群的 Aut(A) 不平凡。

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.

研究动机与目标

• Motivate the study of automorphism groups of skew braces and understand when Aut(A) must be non-trivial.
• Provide conditions under which Aut(A) is non-trivial for two-sided skew braces, bi-skew braces, or finite nilpotent structures.
• Demonstrate a construction method to produce skew braces of order 2p^3 with trivial automorphism groups for any odd prime p.

提出的方法

• Review known relations between the two group operations in a skew brace and their automorphisms.
• Prove Theorem 1.1 by analyzing two-sided, bi-skew, and nilpotent cases to show non-trivial Aut(A).
• Introduce and apply a constructive framework (Theorem 4.1) to build new skew braces from given pieces B and C with compatibility conditions.
• Provide explicit constructions (Theorem 4.1) and verify automorphism-triviality in explicit cases (Theorem 1.2).
• Compute and leverage explicit automorphism descriptions for examples (e.g., skew brace of order 24) to illustrate the method.

实验结果

研究问题

• RQ1 Under what conditions is Aut(A) trivial or non-trivial for a skew brace A?
• RQ2 Can one construct skew braces of a given order with a trivial automorphism group, particularly for order 2p^3 with p odd?
• RQ3 How do two-sided or bi-skew properties influence the automorphism group of skew braces?
• RQ4 What mechanisms extend existing constructions to produce new skew braces with prescribed Aut(A) behavior?

主要发现

• Aut(A) is non-trivial for skew braces of size at least 3 that are two-sided with a non-abelian (A,∘), or are bi-skew braces, or are finite with both operations nilpotent.
• For any odd prime p, there exists a skew brace of order 2p^3 whose automorphism group is trivial.
• A constructive method (Theorem 4.1) extends prior results to build skew braces from a normal ideal B and a sub-skew brace C with compatible actions, yielding controllable Aut(A).
• An explicit 24-element skew brace (SmallSkewbrace(24,855)) with trivial Aut(A) is described and analyzed using the construction, including an explicit automorphism group computation.
• The paper provides a second explicit construction for order 2p^3 skew braces with trivial Aut(A), using a parametrized family with conditions ensuring Aut(A) is trivial.