Carmen Amarra , Alice Devillers , Cheryl E. Praeger
{"title":"Block-transitive 2-designs with a chain of imprimitive point-partitions","authors":"Carmen Amarra , Alice Devillers , Cheryl E. Praeger","doi":"10.1016/j.jcta.2024.105866","DOIUrl":null,"url":null,"abstract":"<div><p>More than 30 years ago, Delandtsheer and Doyen showed that the automorphism group of a block-transitive 2-design, with blocks of size <em>k</em>, could leave invariant a nontrivial point-partition, but only if the number of points was bounded in terms of <em>k</em>. Since then examples have been found where there are two nontrivial point partitions, either forming a chain of partitions, or forming a grid structure on the point set. We show, by construction of infinite families of designs, that there is no limit on the length of a chain of invariant point partitions for a block-transitive 2-design. We introduce the notion of an ‘array’ of a set of points which describes how the set interacts with parts of the various partitions, and we obtain necessary and sufficient conditions in terms of the ‘array’ of a point set, relative to a partition chain, for it to be a block of such a design.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000050/pdfft?md5=4350b32c706696ea2c3f719fb15cd8bc&pid=1-s2.0-S0097316524000050-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0097316524000050","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
More than 30 years ago, Delandtsheer and Doyen showed that the automorphism group of a block-transitive 2-design, with blocks of size k, could leave invariant a nontrivial point-partition, but only if the number of points was bounded in terms of k. Since then examples have been found where there are two nontrivial point partitions, either forming a chain of partitions, or forming a grid structure on the point set. We show, by construction of infinite families of designs, that there is no limit on the length of a chain of invariant point partitions for a block-transitive 2-design. We introduce the notion of an ‘array’ of a set of points which describes how the set interacts with parts of the various partitions, and we obtain necessary and sufficient conditions in terms of the ‘array’ of a point set, relative to a partition chain, for it to be a block of such a design.
期刊介绍:
The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.