{"title":"三阶共三次方有限毛方环中的同余可解性","authors":"Aleš Drápal, Petr Vojtěchovský","doi":"10.1007/s00025-024-02231-2","DOIUrl":null,"url":null,"abstract":"<p>We prove that a normal subloop <i>X</i> of a Moufang loop <i>Q</i> induces an abelian congruence of <i>Q</i> if and only if <span>\\(u(xy) = (uy)x\\)</span> for all <span>\\(x,y\\in X\\)</span> and <span>\\(u\\in Q\\)</span>. This characterization is then used to show that classically solvable finite 3-divisible Moufang loops are congruence solvable.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Congruence Solvability in Finite Moufang Loops of Order Coprime to Three\",\"authors\":\"Aleš Drápal, Petr Vojtěchovský\",\"doi\":\"10.1007/s00025-024-02231-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We prove that a normal subloop <i>X</i> of a Moufang loop <i>Q</i> induces an abelian congruence of <i>Q</i> if and only if <span>\\\\(u(xy) = (uy)x\\\\)</span> for all <span>\\\\(x,y\\\\in X\\\\)</span> and <span>\\\\(u\\\\in Q\\\\)</span>. This characterization is then used to show that classically solvable finite 3-divisible Moufang loops are congruence solvable.</p>\",\"PeriodicalId\":54490,\"journal\":{\"name\":\"Results in Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2024-07-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Results in Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00025-024-02231-2\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Results in Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00025-024-02231-2","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Congruence Solvability in Finite Moufang Loops of Order Coprime to Three
We prove that a normal subloop X of a Moufang loop Q induces an abelian congruence of Q if and only if \(u(xy) = (uy)x\) for all \(x,y\in X\) and \(u\in Q\). This characterization is then used to show that classically solvable finite 3-divisible Moufang loops are congruence solvable.
期刊介绍：
Results in Mathematics (RM) publishes mainly research papers in all fields of pure and applied mathematics. In addition, it publishes summaries of any mathematical field and surveys of any mathematical subject provided they are designed to advance some recent mathematical development.
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