Finite skew braces with isomorphic non-abelian characteristically simple additive and circle groups

Abstract

A skew brace is a triplet $(A,\cdot ,\circ )$, where $(A,\cdot )$ and $(A,\circ )$ are groups such that the brace relation $x\circ (y\cdot z)=(x\circ y)\cdot x^{-1}\cdot (x\circ z)$ holds for all $x,y,z\in A$. In this paper, we study the number of finite skew braces $(A,\cdot ,\circ )$, up to isomorphism, such that $(A,\cdot )$ and $(A,\circ )$ are both isomorphic to $T^n$ with $T$ non-abelian simple and $n\in \mathbb{N}$. We prove that it is equal to the number of unlabeled directed graphs on $n+1$ vertices, with one distingusihed vertex, and whose underlying undirected graph is a tree. In particular, it depends only on $n$ and is independent of $T$.

slightly revised based on referee's comments

References in corpus (4)

• On skew braces (with an appendix by N. Byott and L. Vendramin)
• Computing skew left braces of small orders
• Non-existence of Hopf-Galois structures and bijective crossed homomorphisms
• Hopf-Galois structures of isomorphic type on a non-abelian characteristically simple extension

Cited by in corpus (1)

• Rota-Baxter groups, skew left braces, and the Yang-Baxter equation