{"title":"因式化有限群的形成残差","authors":"X. Li, X. Wu","doi":"10.1007/s10474-024-01470-7","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>G</i> be a finite group and <i>G</i> be a split extension of <i>A</i> by <i>B</i>, that is, <i>G</i> is a semidirect product: <span>\\(G=A\\rtimes B\\)</span>, where <i>A</i> and <i>B</i> are subgroups of <i>G</i>. Under the condition that <i>B</i> permutes with every maximal subgroup of Sylow subgroups of <i>A</i>, every maximal subgroup of <i>A</i> or every nontrivial normal subgroup of <i>A</i>, we prove that the supersolvable residual of <i>G</i> is the product of the supersolvable residuals of <i>A</i> and <i>B</i>.</p></div>","PeriodicalId":50894,"journal":{"name":"Acta Mathematica Hungarica","volume":"174 1","pages":"202 - 208"},"PeriodicalIF":0.6000,"publicationDate":"2024-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The formation residual of factorized finite groups\",\"authors\":\"X. Li, X. Wu\",\"doi\":\"10.1007/s10474-024-01470-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <i>G</i> be a finite group and <i>G</i> be a split extension of <i>A</i> by <i>B</i>, that is, <i>G</i> is a semidirect product: <span>\\\\(G=A\\\\rtimes B\\\\)</span>, where <i>A</i> and <i>B</i> are subgroups of <i>G</i>. Under the condition that <i>B</i> permutes with every maximal subgroup of Sylow subgroups of <i>A</i>, every maximal subgroup of <i>A</i> or every nontrivial normal subgroup of <i>A</i>, we prove that the supersolvable residual of <i>G</i> is the product of the supersolvable residuals of <i>A</i> and <i>B</i>.</p></div>\",\"PeriodicalId\":50894,\"journal\":{\"name\":\"Acta Mathematica Hungarica\",\"volume\":\"174 1\",\"pages\":\"202 - 208\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-10-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica Hungarica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10474-024-01470-7\",\"RegionNum\":3,\"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":"Acta Mathematica Hungarica","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10474-024-01470-7","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
设 G 是一个有限群，G 是 A 的 B 的分裂扩展，即 G 是一个半径积：\在 B 与 A 的 Sylow 子群的每个最大子群、A 的每个最大子群或 A 的每个非琐正则子群发生包络的条件下，我们证明 G 的可超分解残差是 A 和 B 的可超分解残差的乘积。
The formation residual of factorized finite groups
Let G be a finite group and G be a split extension of A by B, that is, G is a semidirect product: \(G=A\rtimes B\), where A and B are subgroups of G. Under the condition that B permutes with every maximal subgroup of Sylow subgroups of A, every maximal subgroup of A or every nontrivial normal subgroup of A, we prove that the supersolvable residual of G is the product of the supersolvable residuals of A and B.
期刊介绍：
Acta Mathematica Hungarica is devoted to publishing research articles of top quality in all areas of pure and applied mathematics as well as in theoretical computer science. The journal is published yearly in three volumes (two issues per volume, in total 6 issues) in both print and electronic formats. Acta Mathematica Hungarica (formerly Acta Mathematica Academiae Scientiarum Hungaricae) was founded in 1950 by the Hungarian Academy of Sciences.
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