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

TL;DR

This paper investigates the automorphism groups of skew braces, establishing conditions for triviality and constructing examples with trivial automorphism groups for specific orders.

Contribution

It extends the understanding of automorphism groups in skew braces by proving triviality conditions and providing explicit constructions for certain orders.

Findings

Groups of order 1 or 2 have trivial automorphism groups

Certain families of skew braces also have trivial automorphism groups

Constructed skew braces of order 2p^3 with trivial automorphism groups for odd primes p

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.