{"title":"交替群与点数为平方的点基元线性空间","authors":"Haiyan Guan, Shenglin Zhou","doi":"10.1002/jcd.21879","DOIUrl":null,"url":null,"abstract":"<p>This paper is a further contribution to the classification of point-primitive finite regular linear spaces. Let <math>\n <semantics>\n <mrow>\n <mi>S</mi>\n </mrow>\n <annotation> ${\\mathscr{S}}$</annotation>\n </semantics></math> be a nontrivial finite regular linear space whose number of points <math>\n <semantics>\n <mrow>\n <mi>v</mi>\n </mrow>\n <annotation> $v$</annotation>\n </semantics></math> is squarefree. We prove that if <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n <mo>≤</mo>\n <mtext>Aut</mtext>\n <mrow>\n <mo>(</mo>\n <mi>S</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $G\\le \\text{Aut}({\\mathscr{S}})$</annotation>\n </semantics></math> is point-primitive with an alternating socle, then <math>\n <semantics>\n <mrow>\n <mi>S</mi>\n </mrow>\n <annotation> ${\\mathscr{S}}$</annotation>\n </semantics></math> is the projective space <math>\n <semantics>\n <mrow>\n <mtext>PG</mtext>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mn>3</mn>\n <mo>,</mo>\n <mn>2</mn>\n </mrow>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\text{PG}(3,2)$</annotation>\n </semantics></math>.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 5","pages":"277-286"},"PeriodicalIF":0.5000,"publicationDate":"2023-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Alternating groups and point-primitive linear spaces with number of points being squarefree\",\"authors\":\"Haiyan Guan, Shenglin Zhou\",\"doi\":\"10.1002/jcd.21879\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>This paper is a further contribution to the classification of point-primitive finite regular linear spaces. Let <math>\\n <semantics>\\n <mrow>\\n <mi>S</mi>\\n </mrow>\\n <annotation> ${\\\\mathscr{S}}$</annotation>\\n </semantics></math> be a nontrivial finite regular linear space whose number of points <math>\\n <semantics>\\n <mrow>\\n <mi>v</mi>\\n </mrow>\\n <annotation> $v$</annotation>\\n </semantics></math> is squarefree. We prove that if <math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n <mo>≤</mo>\\n <mtext>Aut</mtext>\\n <mrow>\\n <mo>(</mo>\\n <mi>S</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $G\\\\le \\\\text{Aut}({\\\\mathscr{S}})$</annotation>\\n </semantics></math> is point-primitive with an alternating socle, then <math>\\n <semantics>\\n <mrow>\\n <mi>S</mi>\\n </mrow>\\n <annotation> ${\\\\mathscr{S}}$</annotation>\\n </semantics></math> is the projective space <math>\\n <semantics>\\n <mrow>\\n <mtext>PG</mtext>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mn>3</mn>\\n <mo>,</mo>\\n <mn>2</mn>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\text{PG}(3,2)$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":15389,\"journal\":{\"name\":\"Journal of Combinatorial Designs\",\"volume\":\"31 5\",\"pages\":\"277-286\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2023-02-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Designs\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jcd.21879\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Designs","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jcd.21879","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Alternating groups and point-primitive linear spaces with number of points being squarefree
This paper is a further contribution to the classification of point-primitive finite regular linear spaces. Let be a nontrivial finite regular linear space whose number of points is squarefree. We prove that if is point-primitive with an alternating socle, then is the projective space .
期刊介绍：
The Journal of Combinatorial Designs is an international journal devoted to the timely publication of the most influential papers in the area of combinatorial design theory. All topics in design theory, and in which design theory has important applications, are covered, including:
block designs, t-designs, pairwise balanced designs and group divisible designs
Latin squares, quasigroups, and related algebras
computational methods in design theory
construction methods
applications in computer science, experimental design theory, and coding theory
graph decompositions, factorizations, and design-theoretic techniques in graph theory and extremal combinatorics
finite geometry and its relation with design theory.
algebraic aspects of design theory.
Researchers and scientists can depend on the Journal of Combinatorial Designs for the most recent developments in this rapidly growing field, and to provide a forum for both theoretical research and applications. All papers appearing in the Journal of Combinatorial Designs are carefully peer refereed.
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