{"title":"迈向π-激进的巴尔-铃木锐定理：小等级的特殊群","authors":"Zh. Wang, W. Guo, D. O. Revin","doi":"10.1007/s10469-023-09720-3","DOIUrl":null,"url":null,"abstract":"<p>Let π be a proper subset of the set of all prime numbers. Denote by <i>r</i> the least prime number not in π, and put <i>m</i> = <i>r</i>, if <i>r</i> = 2, 3, and m = <i>r</i> − 1 if <i>r</i> ≥ 5. We look at the conjecture that a conjugacy class <i>D</i> in a finite group <i>G</i> generates a π-subgroup in <i>G</i> (or, equivalently, is contained in the π-radical) iff any m elements from <i>D</i> generate a π-group. Previously, this conjecture was confirmed for finite groups whose every non-Abelian composition factor is isomorphic to a sporadic, alternating, linear or unitary simple group. Now it is confirmed for groups the list of composition factors of which is added up by exceptional groups of Lie type <sup>2</sup><i>B</i><sub>2</sub>(<i>q</i>), <sup>2</sup><i>G</i><sub>2</sub>(<i>q</i>), <i>G</i><sub>2</sub>(<i>q</i>), and <sup>3</sup><i>D</i><sub>4</sub>(<i>q</i>).</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Toward a Sharp Baer–Suzuki Theorem for the π-Radical: Exceptional Groups of Small Rank\",\"authors\":\"Zh. Wang, W. Guo, D. O. Revin\",\"doi\":\"10.1007/s10469-023-09720-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let π be a proper subset of the set of all prime numbers. Denote by <i>r</i> the least prime number not in π, and put <i>m</i> = <i>r</i>, if <i>r</i> = 2, 3, and m = <i>r</i> − 1 if <i>r</i> ≥ 5. We look at the conjecture that a conjugacy class <i>D</i> in a finite group <i>G</i> generates a π-subgroup in <i>G</i> (or, equivalently, is contained in the π-radical) iff any m elements from <i>D</i> generate a π-group. Previously, this conjecture was confirmed for finite groups whose every non-Abelian composition factor is isomorphic to a sporadic, alternating, linear or unitary simple group. Now it is confirmed for groups the list of composition factors of which is added up by exceptional groups of Lie type <sup>2</sup><i>B</i><sub>2</sub>(<i>q</i>), <sup>2</sup><i>G</i><sub>2</sub>(<i>q</i>), <i>G</i><sub>2</sub>(<i>q</i>), and <sup>3</sup><i>D</i><sub>4</sub>(<i>q</i>).</p>\",\"PeriodicalId\":7422,\"journal\":{\"name\":\"Algebra and Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2024-01-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Algebra and Logic\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10469-023-09720-3\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"LOGIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra and Logic","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10469-023-09720-3","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 0
摘要
设 π 是所有素数集合的一个适当子集。用 r 表示不在π中的最小素数，如果 r = 2，3，则设 m = r，如果 r ≥ 5，则设 m = r - 1。我们研究这样一个猜想：如果 D 中的任意 m 个元素生成一个π群，那么有限群 G 中的共轭类 D 就会在 G 中生成一个π子群（或者，等价地，包含在π激元中）。在此之前，这一猜想是在有限群中得到证实的，这些群中的每个非阿贝尔组成因子都与零星群、交替群、线性群或单元简单群同构。现在，这个猜想对于由列类型为 2B2(q)、2G2(q)、G2(q) 和 3D4(q) 的特殊群相加而成的群也得到了证实。
Toward a Sharp Baer–Suzuki Theorem for the π-Radical: Exceptional Groups of Small Rank
Let π be a proper subset of the set of all prime numbers. Denote by r the least prime number not in π, and put m = r, if r = 2, 3, and m = r − 1 if r ≥ 5. We look at the conjecture that a conjugacy class D in a finite group G generates a π-subgroup in G (or, equivalently, is contained in the π-radical) iff any m elements from D generate a π-group. Previously, this conjecture was confirmed for finite groups whose every non-Abelian composition factor is isomorphic to a sporadic, alternating, linear or unitary simple group. Now it is confirmed for groups the list of composition factors of which is added up by exceptional groups of Lie type 2B2(q), 2G2(q), G2(q), and 3D4(q).
期刊介绍：
This bimonthly journal publishes results of the latest research in the areas of modern general algebra and of logic considered primarily from an algebraic viewpoint. The algebraic papers, constituting the major part of the contents, are concerned with studies in such fields as ordered, almost torsion-free, nilpotent, and metabelian groups; isomorphism rings; Lie algebras; Frattini subgroups; and clusters of algebras. In the area of logic, the periodical covers such topics as hierarchical sets, logical automata, and recursive functions.
Algebra and Logic is a translation of ALGEBRA I LOGIKA, a publication of the Siberian Fund for Algebra and Logic and the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences.
All articles are peer-reviewed.
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