{"title":"关于字符编码度的两个结果","authors":"Yang Liu, Yong Yang","doi":"10.1142/s0219498825501580","DOIUrl":null,"url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span> be a finite group and <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">Irr</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span> be the set of irreducible characters of <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span>. The codegree of an irreducible character <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>χ</mi></math></span><span></span> of the group <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span> is defined as <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">cod</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>χ</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo>|</mo><mi>G</mi><mo>:</mo><mstyle><mtext mathvariant=\"normal\">ker</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>χ</mi><mo stretchy=\"false\">)</mo><mo>|</mo><mo stretchy=\"false\">/</mo><mi>χ</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo stretchy=\"false\">)</mo></math></span><span></span>. In this paper, we study two topics related to the character codegrees. The first result is related to the prime graph of character codegrees and we show that the codegree prime graphs of several classes of groups can be characterized only by graph theoretical terms. The second result is about the <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-parts of the codegrees and character degrees.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"135 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Two results on character codegrees\",\"authors\":\"Yang Liu, Yong Yang\",\"doi\":\"10.1142/s0219498825501580\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span><math altimg=\\\"eq-00001.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>G</mi></math></span><span></span> be a finite group and <span><math altimg=\\\"eq-00002.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mstyle><mtext mathvariant=\\\"normal\\\">Irr</mtext></mstyle><mo stretchy=\\\"false\\\">(</mo><mi>G</mi><mo stretchy=\\\"false\\\">)</mo></math></span><span></span> be the set of irreducible characters of <span><math altimg=\\\"eq-00003.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>G</mi></math></span><span></span>. The codegree of an irreducible character <span><math altimg=\\\"eq-00004.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>χ</mi></math></span><span></span> of the group <span><math altimg=\\\"eq-00005.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>G</mi></math></span><span></span> is defined as <span><math altimg=\\\"eq-00006.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mstyle><mtext mathvariant=\\\"normal\\\">cod</mtext></mstyle><mo stretchy=\\\"false\\\">(</mo><mi>χ</mi><mo stretchy=\\\"false\\\">)</mo><mo>=</mo><mo>|</mo><mi>G</mi><mo>:</mo><mstyle><mtext mathvariant=\\\"normal\\\">ker</mtext></mstyle><mo stretchy=\\\"false\\\">(</mo><mi>χ</mi><mo stretchy=\\\"false\\\">)</mo><mo>|</mo><mo stretchy=\\\"false\\\">/</mo><mi>χ</mi><mo stretchy=\\\"false\\\">(</mo><mn>1</mn><mo stretchy=\\\"false\\\">)</mo></math></span><span></span>. In this paper, we study two topics related to the character codegrees. The first result is related to the prime graph of character codegrees and we show that the codegree prime graphs of several classes of groups can be characterized only by graph theoretical terms. The second result is about the <span><math altimg=\\\"eq-00007.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>p</mi></math></span><span></span>-parts of the codegrees and character degrees.</p>\",\"PeriodicalId\":54888,\"journal\":{\"name\":\"Journal of Algebra and Its Applications\",\"volume\":\"135 1\",\"pages\":\"\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-01-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Algebra and Its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1142/s0219498825501580\",\"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":"Journal of Algebra and Its Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0219498825501580","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 是有限群,Irr(G) 是 G 的不可还原字符集。群 G 的不可还原字符 χ 的度数定义为 cod(χ)=|G:ker(χ)|/χ(1)。在本文中,我们研究了与字符编码度相关的两个课题。第一个结果与字符 codegrees 的素数图有关,我们证明了几类群的 codegree 素数图只能用图论术语来表征。第二个结果是关于密码度和字符度的 p 部分。
Let be a finite group and be the set of irreducible characters of . The codegree of an irreducible character of the group is defined as . In this paper, we study two topics related to the character codegrees. The first result is related to the prime graph of character codegrees and we show that the codegree prime graphs of several classes of groups can be characterized only by graph theoretical terms. The second result is about the -parts of the codegrees and character degrees.
期刊介绍:
The Journal of Algebra and Its Applications will publish papers both on theoretical and on applied aspects of Algebra. There is special interest in papers that point out innovative links between areas of Algebra and fields of application. As the field of Algebra continues to experience tremendous growth and diversification, we intend to provide the mathematical community with a central source for information on both the theoretical and the applied aspects of the discipline. While the journal will be primarily devoted to the publication of original research, extraordinary expository articles that encourage communication between algebraists and experts on areas of application as well as those presenting the state of the art on a given algebraic sub-discipline will be considered.