{"title":"关于有限群法典的说明","authors":"Mark L. Lewis, Quanfu Yan","doi":"10.1007/s10468-024-10282-w","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span>\\(\\chi \\)</span> be an irreducible character of a group <i>G</i>, and <span>\\(S_c(G)=\\sum _{\\chi \\in \\textrm{Irr}(G)}\\textrm{cod}(\\chi )\\)</span> be the sum of the codegrees of the irreducible characters of <i>G</i>. Write <span>\\(\\textrm{fcod} (G)=\\frac{S_c(G)}{|G|}.\\)</span> We aim to explore the structure of finite groups in terms of <span>\\(\\textrm{fcod} (G).\\)</span> On the other hand, we determine the lower bound of <span>\\(S_c(G)\\)</span> for nonsolvable groups and prove that if <i>G</i> is nonsolvable, then <span>\\(S_c(G)\\geqslant S_c(A_5)=68,\\)</span> with equality if and only if <span>\\(G\\cong A_5.\\)</span> Additionally, we show that there is a solvable group so that it has the codegree sum as <span>\\(A_5.\\)</span></p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10282-w.pdf","citationCount":"0","resultStr":"{\"title\":\"A Note on the Codegree of Finite Groups\",\"authors\":\"Mark L. Lewis, Quanfu Yan\",\"doi\":\"10.1007/s10468-024-10282-w\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <span>\\\\(\\\\chi \\\\)</span> be an irreducible character of a group <i>G</i>, and <span>\\\\(S_c(G)=\\\\sum _{\\\\chi \\\\in \\\\textrm{Irr}(G)}\\\\textrm{cod}(\\\\chi )\\\\)</span> be the sum of the codegrees of the irreducible characters of <i>G</i>. Write <span>\\\\(\\\\textrm{fcod} (G)=\\\\frac{S_c(G)}{|G|}.\\\\)</span> We aim to explore the structure of finite groups in terms of <span>\\\\(\\\\textrm{fcod} (G).\\\\)</span> On the other hand, we determine the lower bound of <span>\\\\(S_c(G)\\\\)</span> for nonsolvable groups and prove that if <i>G</i> is nonsolvable, then <span>\\\\(S_c(G)\\\\geqslant S_c(A_5)=68,\\\\)</span> with equality if and only if <span>\\\\(G\\\\cong A_5.\\\\)</span> Additionally, we show that there is a solvable group so that it has the codegree sum as <span>\\\\(A_5.\\\\)</span></p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-08-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s10468-024-10282-w.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10468-024-10282-w\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10468-024-10282-w","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
让 \(\chi \) 是一个群 G 的不可还原字符，并且 \(S_c(G)=sum _{\chi \in \textrm{Irr}(G)}\textrm{cod}(\chi )\) 是 G 的不可还原字符的编码度之和。我们的目的是用\(\textrm{fcod} (G).\) 来探索有限群的结构。另一方面，我们确定了不可解群的\(S_c(G)\)下界，并证明了如果 G 是不可解的，那么当且仅当\(G\cong A_5.\) 时，\(S_c(G)\geqslant S_c(A_5)=68,\) 是相等的 此外，我们还证明了存在一个可解群，使得它具有\(A_5.\)的codegree sum。
Let \(\chi \) be an irreducible character of a group G, and \(S_c(G)=\sum _{\chi \in \textrm{Irr}(G)}\textrm{cod}(\chi )\) be the sum of the codegrees of the irreducible characters of G. Write \(\textrm{fcod} (G)=\frac{S_c(G)}{|G|}.\) We aim to explore the structure of finite groups in terms of \(\textrm{fcod} (G).\) On the other hand, we determine the lower bound of \(S_c(G)\) for nonsolvable groups and prove that if G is nonsolvable, then \(S_c(G)\geqslant S_c(A_5)=68,\) with equality if and only if \(G\cong A_5.\) Additionally, we show that there is a solvable group so that it has the codegree sum as \(A_5.\)
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