Block-transitive 3-designs Associated with Alternating Groups  

交错群上的区传递3-设计

作　　者：Gan Yunsong Liu Weijun 甘芸松;刘伟俊(中南大学数学与统计学院,长沙410075;广东科技学院通识教育学院,东莞523083)

机构地区：[1]School of Mathematics and Statistics,Central South University,Changsha 410075,China [2]College of General Education,Guangdong University of Science and Technology,Dongguan 523083,China

出　　处：《数学理论与应用》2024年第4期19-30,共12页Mathematical Theory and Applications

基　　金：supported by the National Natural Science Foundation of China(No.12471022)。

摘　　要：A t-(v,k,λ)design is said to be G-point-primitive or G-block-transitive,if its automorphism group G acts primitively on the point set or transitively on the block set,respectively.In this paper we begin by extending some results on block-transitive Steiner 2-designs to block-transitive 3-designs,and then based on these results,investigate the G-point-primitive block-transitive 3-(v,k,λ)designs for alternating or symmetric groups G.We prove that when n≥min{λ^(2),30}the point stabilizer in G must be of intransitive type,and specifically,when n≥30 there exists no nontrivial G-point-primitive block-transitive 3-(v,k,2)design.一个t-(v,k,λ)~设计分别称为G-点本原或G-区传递的,如果它的自同构群G在点集上的作用本原或在区集上的作用传递.本文首先将区传递斯坦纳2-设计的一些结论推广到一般的区传递3-设计.在此基础上,我们研究当G为交错群或对称群时的G-点本原区传递3-(v,k,λ)~设计.我们证明:当n≥min{λ^(2),30}~时,G在点集上作用的点稳定子只可能为非传递型子群;特别地,当n≥30时,不存在G-点本原区传递非平凡3-(v,k,2)设计.

关 键 词：Block-transitive design Primitive group Alternating group 

分 类 号：O15[理学—数学]