{"title":"关于经典群的卡莱图直径，其生成集包含一个横切面","authors":"Martino Garonzi, Zoltán Halasi, Gábor Somlai","doi":"10.1007/s11856-024-2605-1","DOIUrl":null,"url":null,"abstract":"<p>A well-known conjecture of Babai states that if <i>G</i> is any finite simple group and <i>X</i> is a generating set for <i>G</i>, then the diameter of the Cayley graph Cay(<i>G</i>, <i>X</i>) is bounded by log ∣<i>G</i>∣<sup><i>c</i></sup> for some universal constant <i>c</i>. In this paper, we prove such a bound for Cay(<i>G</i>, <i>X</i>) for <i>G</i> = PSL(<i>n</i>, <i>q</i>), PSp(<i>n</i>, <i>q</i>) or PSU(<i>n</i>, <i>q</i>) where <i>q</i> is odd, under the assumptions that <i>X</i> contains a transvection and <i>q</i> ≠ 9 or 81.</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the diameter of Cayley graphs of classical groups with generating sets containing a transvection\",\"authors\":\"Martino Garonzi, Zoltán Halasi, Gábor Somlai\",\"doi\":\"10.1007/s11856-024-2605-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A well-known conjecture of Babai states that if <i>G</i> is any finite simple group and <i>X</i> is a generating set for <i>G</i>, then the diameter of the Cayley graph Cay(<i>G</i>, <i>X</i>) is bounded by log ∣<i>G</i>∣<sup><i>c</i></sup> for some universal constant <i>c</i>. In this paper, we prove such a bound for Cay(<i>G</i>, <i>X</i>) for <i>G</i> = PSL(<i>n</i>, <i>q</i>), PSp(<i>n</i>, <i>q</i>) or PSU(<i>n</i>, <i>q</i>) where <i>q</i> is odd, under the assumptions that <i>X</i> contains a transvection and <i>q</i> ≠ 9 or 81.</p>\",\"PeriodicalId\":14661,\"journal\":{\"name\":\"Israel Journal of Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-04-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Israel Journal of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s11856-024-2605-1\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Israel Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11856-024-2605-1","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
摘要
Babai 的一个著名猜想指出，如果 G 是任意有限单纯群，X 是 G 的一个生成集，那么对于某个普遍常数 c，Cayley 图 Cay(G, X) 的直径以 log ∣G∣c 为界。在本文中，我们将证明对于 G = PSL(n，q)、PSp(n，q) 或 PSU(n，q)（其中 q 为奇数）的 Cay(G，X)，在 X 包含一个横切面且 q ≠ 9 或 81 的假设条件下，Cay(G，X) 的这样一个约束。
On the diameter of Cayley graphs of classical groups with generating sets containing a transvection
A well-known conjecture of Babai states that if G is any finite simple group and X is a generating set for G, then the diameter of the Cayley graph Cay(G, X) is bounded by log ∣G∣c for some universal constant c. In this paper, we prove such a bound for Cay(G, X) for G = PSL(n, q), PSp(n, q) or PSU(n, q) where q is odd, under the assumptions that X contains a transvection and q ≠ 9 or 81.
期刊介绍：
The Israel Journal of Mathematics is an international journal publishing high-quality original research papers in a wide spectrum of pure and applied mathematics. The prestigious interdisciplinary editorial board reflects the diversity of subjects covered in this journal, including set theory, model theory, algebra, group theory, number theory, analysis, functional analysis, ergodic theory, algebraic topology, geometry, combinatorics, theoretical computer science, mathematical physics, and applied mathematics.
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