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

TL;DR

This paper classifies finite skew braces where both additive and circle groups are isomorphic to a power of a non-abelian simple group, linking their enumeration to certain labeled tree graphs.

Contribution

It provides a classification of such skew braces and establishes a surprising connection to labeled tree graphs, independent of the specific simple group involved.

Findings

Number of skew braces equals the count of certain labeled trees

Classification depends only on the exponent n, not on the simple group T

Establishes a combinatorial enumeration for these algebraic structures

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 $T$.