Graphs, group actions and skew braces (Joint work with Arne Van Antwerpen)

A skew brace is an interlinking of two group structures on the same set, where one group (called the multiplicative group) acts via the so-called $\lambda$-action by automorphisms on the other (called the additive group). These structures play a fundamental role in the combinatorial study of solutions to the Yang–Baxter equation. Motivated by similar ideas in representation theory of finite groups and by the work of of Bertram, Herzog, and Mann, we study two common divisor graphs associated with skew braces. We first define this graph in the general setting of any action of a group by automorphisms on another group. We provide some examples and prove bound for the number of connected components and for the diameter. We then pass to the case of skew braces and we classify all skew braces with one of the graphs consisting of two disconnected vertices and with only one vertex.

Feb 22, 2024