{"title":"非弗罗贝纽斯群中的异变","authors":"Daniele Garzoni","doi":"arxiv-2409.03305","DOIUrl":null,"url":null,"abstract":"We prove that if $G$ is a transitive permutation group of sufficiently large\ndegree $n$, then either $G$ is primitive and Frobenius, or the proportion of\nderangements in $G$ is larger than $1/(2n^{1/2})$. This is sharp, generalizes\nsubstantially bounds of Cameron--Cohen and Guralnick--Wan, and settles a\nconjecture of Guralnick--Tiep in large degree. We also give an application to\ncoverings of varieties over finite fields.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"70 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Derangements in non-Frobenius groups\",\"authors\":\"Daniele Garzoni\",\"doi\":\"arxiv-2409.03305\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove that if $G$ is a transitive permutation group of sufficiently large\\ndegree $n$, then either $G$ is primitive and Frobenius, or the proportion of\\nderangements in $G$ is larger than $1/(2n^{1/2})$. This is sharp, generalizes\\nsubstantially bounds of Cameron--Cohen and Guralnick--Wan, and settles a\\nconjecture of Guralnick--Tiep in large degree. We also give an application to\\ncoverings of varieties over finite fields.\",\"PeriodicalId\":501037,\"journal\":{\"name\":\"arXiv - MATH - Group Theory\",\"volume\":\"70 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Group Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.03305\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Group Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.03305","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We prove that if $G$ is a transitive permutation group of sufficiently large
degree $n$, then either $G$ is primitive and Frobenius, or the proportion of
derangements in $G$ is larger than $1/(2n^{1/2})$. This is sharp, generalizes
substantially bounds of Cameron--Cohen and Guralnick--Wan, and settles a
conjecture of Guralnick--Tiep in large degree. We also give an application to
coverings of varieties over finite fields.