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Common divisor graphs for skew braces

Silvia Properzi, Arne Van Antwerpen

TL;DR

The paper studies two common divisor graphs, $\Lambda(A)$ and $\Theta(A)$, built from $\lambda$- and $\theta$-orbits of a finite skew brace to illuminate the interaction between brace structure and orbit coprimality. It proves general bounds on the graphs (at most two connected components, diameter at most four) and shows isoclinism-invariance of these graphs for equal-sized braces, enabling classification results for extreme cases: two-vertex and one-vertex graphs. It provides comprehensive classifications for one-vertex $\lambda$-graphs in both abelian-type and general cases, describing four infinite families and a complete final structure with explicit parametric families and isomorphism criteria. It also identifies the precise, restricted possibilities for two-vertex graphs, linking the combinatorial graphs to the algebraic architecture of skew braces and their associated Yang–Baxter set-theoretic solutions.

Abstract

We introduce two common divisor graphs associated with a finite skew brace, based on its $λ$- and $θ$-orbits. We prove that the number of connected components is at most two and the diameter of a connected component is at most four. Furthermore, we investigate their relationship with isoclinism. Similarly to its group theoretic inspiration, the skew braces with a graph with two disconnected vertices are very restricted and are determined. Finally, we classify all finite skew braces with a graph with one vertex, where four infinite families arise.

Common divisor graphs for skew braces

TL;DR

The paper studies two common divisor graphs, and , built from - and -orbits of a finite skew brace to illuminate the interaction between brace structure and orbit coprimality. It proves general bounds on the graphs (at most two connected components, diameter at most four) and shows isoclinism-invariance of these graphs for equal-sized braces, enabling classification results for extreme cases: two-vertex and one-vertex graphs. It provides comprehensive classifications for one-vertex -graphs in both abelian-type and general cases, describing four infinite families and a complete final structure with explicit parametric families and isomorphism criteria. It also identifies the precise, restricted possibilities for two-vertex graphs, linking the combinatorial graphs to the algebraic architecture of skew braces and their associated Yang–Baxter set-theoretic solutions.

Abstract

We introduce two common divisor graphs associated with a finite skew brace, based on its - and -orbits. We prove that the number of connected components is at most two and the diameter of a connected component is at most four. Furthermore, we investigate their relationship with isoclinism. Similarly to its group theoretic inspiration, the skew braces with a graph with two disconnected vertices are very restricted and are determined. Finally, we classify all finite skew braces with a graph with one vertex, where four infinite families arise.
Paper Structure (11 sections, 56 theorems, 126 equations, 3 tables)

This paper contains 11 sections, 56 theorems, 126 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Common divisor graph
  4. Common divisor graphs for skew braces and their subsolutions
  5. Basic properties and isoclinism of skew braces
  6. Graphs of skew braces of small order
  7. Skew braces with graphs with two disconnected vertices
  8. Skew Braces with one vertex lambda-graph: Generalities
  9. Skew braces with one-vertex lambda-graphs: Abelian type
  10. Skew braces with one-vertex lambda-graph: Finale
  11. Skew braces with one-vertex theta-graph

Key Result

Proposition 3.3

Suppose that $|G|$ and $|H|$ have the same prime factors. If $a,b\in H$ such that $|\mathcal{O}(a)|$ and $|\mathcal{O}(b)|$ are coprime, then $\operatorname{Stab}(a)\operatorname{Stab}(b)=G$ and $\mathcal{O}(a)\mathcal{O}(b)=\mathcal{O}(ab)$.

Theorems & Definitions (116)

  • Remark 2.1
  • Definition 3.1
  • Example 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • Theorem 3.6
  • ...and 106 more