旗传递点本原的2-设计与$Sz\left(q\right)$群

Flag-Transitive Point-Primitive 2-Designs and $Sz\left(q\right)$ Groups

摘要: 群论与组合设计之间存在深刻的内在关联，这种关联通常体现为通过分析设计结构的自同构群所具有的传递性或本原性等特征来进行研究。2019年，王雨洁考虑了对于$\lambda \le 10$ 的情况下，如果自同构群$G=Sz\left(q\right)$ 是旗传递且点本原的，则一个非平凡$2-\left(v,k,\lambda \right)$ 设计是唯一确定的，其中参数为$2-\left(65,8,7\right)$ ，且相应的自同构群为$Sz\left(8\right)$ 。在本文我们将考虑在$\mathcal{D}$ 是非平凡设计$2-\left(v,k,\lambda \right)$ ，其中$11\le \lambda \le 100$ 的情况下，如果$G=Sz\left(q\right)\le Aut\left(\mathcal{D}\right)$ 是旗传递点本原群，则$G_{\alpha }\cancel{\cong }{N}_G\left(H\right)$ 。

Abstract: There exists a deep intrinsic connection between group theory and combinatorial designs, which is typically reflected by studying properties such as transitivity and primitivity of the automorphism group of the design structure. In 2019, Yujie Wang obtained that for a nontrivial $2-\left(v,k,\lambda \right)$ design, if $G=Sz\left(q\right)$ is a flag-transitive point-primitive automorphism group and $\lambda \le 10$ , then a nontrivial design is uniquely determined, with parameters 2-(65,8,7), and its corresponding automorphism group is $Sz\left(q\right)$ . In this paper, we consider the case where the design is a nontrivial $2-\left(v,k,\lambda \right)$ design with $11\le \lambda \le 100$ , if $G=Sz\left(q\right)\le Aut\left(\mathcal{D}\right)$ is a flag-transitive point-primitive group, then $G_{\alpha }\cancel{\cong }{N}_G\left(H\right)$ .