A graph associated with finite skew braces

Inspired by a work of Bertram, Herzog, and Mann, we introduce a graph associated with the $\lambda$-classes of a given finite skew brace. Given a finite skew brace A, its graph of non-trivial $\lambda$-classes has as vertices the non-trivial $\lampda$-orbits of A and two vertices are adjacent if their sizes are not coprime. This graph has similar properties to the one defined for groups by Bertram, Herzog, and Mann: it has at most two connected components, some bounds on the diameter and there is only one skew brace whose graph consists of two disconnected vertices. On the other hand, in contrast to the group version, there exist graphs with precisely one vertex, and skew braces with this property can be classified completely.

Apr 18, 2023