论文标题
关于萨克斯图的价值问题
On Valency Problems of Saxl Graphs
论文作者
论文摘要
让$ g $是集合$ω$的排列组，并回想起$ g $的基础是$ω$的子集，因此其点稳定器是微不足道的。在最近的一篇论文中，伯恩斯和朱迪奇推出了$ g $的saxl图，表示为$σ（g）$，而顶点套装$ω$，如果它们形成了基础，则两个顶点。如果$ g $是传递的，则$σ（g）$是顶点传递的，并且自然考虑其价值（我们称为$ g $的价值）。在本文中，我们提出了一种计算任何有限及时式群体价值的通用方法，并使用它来计算每个具有稳定器的原始群体的确切价值，它是带有循环内核的Frobenius组。作为应用程序，我们计算了每个几乎简单的原始群体的价值，并具有交替的SOCLE和可溶性稳定剂，并使用它来扩展Burness和Giudici的结果，这些结果几乎是具有Prime Power或Odd Valency的几乎简单的原始群体。
Let $G$ be a permutation group on a set $Ω$ and recall that a base for $G$ is a subset of $Ω$ such that its pointwise stabiliser is trivial. In a recent paper, Burness and Giudici introduced the Saxl graph of $G$, denoted $Σ(G)$, with vertex set $Ω$ and two vertices adjacent if they form a base. If $G$ is transitive, then $Σ(G)$ is vertex-transitive and it is natural to consider its valency (which we refer to as the valency of $G$). In this paper we present a general method for computing the valency of any finite transitive group and we use it to calculate the exact valency of every primitive group with stabiliser a Frobenius group with cyclic kernel. As an application, we calculate the valency of every almost simple primitive group with an alternating socle and soluble stabiliser and we use this to extend results of Burness and Giudici on almost simple primitive groups with prime-power or odd valency.
