论文标题
原始产品类型的基本尺寸
On base sizes for primitive groups of product type
论文作者
论文摘要
令$ g \ leqslant {\ rm sym}（ω）$为有限的排列组，并回想起$ g $的基本大小是$ g $的最小尺寸，该子集的$ω$的最小尺寸具有琐碎的点稳定器。关于原始群体的基本尺寸，有很多文献，但是原始产品类型的结果很少。在本文中，我们在这种情况下启动了对基础的系统研究。我们的第一个主要结果确定了$ l \ wr p \ wr p \ leqslant {\ rm sym}（ω）$的每个产品类型原始组的基本大小，其中$ω=γ^k $，$ω=γ^k $，$ l \ leqslant {\ rm sym}（γ）（γ）$和$ p \ p \ p \ p \ peslants s_这扩展了Burness在几乎简单的原始群体上的最新工作。我们还获得了表格$ l \ wr p $的任何产品类型组的常规亚货车数量的表达式，并且在假设$ p $是原始的，涉及由于Seress和Dolfi引起的较早结果的假设，我们将组与唯一的常规次级轨道进行了分类。我们介绍了基础两种产品类型组的SAXL图上的应用程序，并通过在一般产品类型原始组的基础尺寸上建立几个新结果来结束。
Let $G \leqslant {\rm Sym}(Ω)$ be a finite permutation group and recall that the base size of $G$ is the minimal size of a subset of $Ω$ with trivial pointwise stabiliser. There is an extensive literature on base sizes for primitive groups, but there are very few results for primitive groups of product type. In this paper, we initiate a systematic study of bases in this setting. Our first main result determines the base size of every product type primitive group of the form $L \wr P \leqslant {\rm Sym}(Ω)$ with soluble point stabilisers, where $Ω= Γ^k$, $L \leqslant {\rm Sym}(Γ)$ and $P \leqslant S_k$ is transitive. This extends recent work of Burness on almost simple primitive groups. We also obtain an expression for the number of regular suborbits of any product type group of the form $L \wr P$ and we classify the groups with a unique regular suborbit under the assumption that $P$ is primitive, which involves extending earlier results due to Seress and Dolfi. We present applications on the Saxl graphs of base-two product type groups and we conclude by establishing several new results on base sizes for general product type primitive groups.
