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

Abstract

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