{"title":"有限群的𝑝-rationality的连续性和𝑝′度特征的下界","authors":"Nguyen Hung","doi":"10.1090/tran/8926","DOIUrl":null,"url":null,"abstract":"Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a prime and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\"application/x-tex\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a finite group. We propose a strong bound for the number of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p prime\"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding=\"application/x-tex\">p’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degree irreducible characters of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\"application/x-tex\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of the commutator factor group of a Sylow <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-subgroup of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\"application/x-tex\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The bound arises from a recent conjecture of Navarro and Tiep [Forum Math. Pi 9 (2021), pp. 1–28] on fields of character values and a phenomenon called the continuity of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rationality level of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p prime\"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding=\"application/x-tex\">p’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degree characters. This continuity property in turn is predicted by the celebrated McKay-Navarro conjecture (see G. Navarro [Ann. of Math. (2) 160 (2004), pp. 1129–1140]). We achieve both the bound and the continuity property for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p equals 2\"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">p=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.","PeriodicalId":23209,"journal":{"name":"Transactions of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The continuity of 𝑝-rationality and a lower bound for 𝑝’-degree characters of finite groups\",\"authors\":\"Nguyen Hung\",\"doi\":\"10.1090/tran/8926\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p\\\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a prime and <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G\\\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a finite group. We propose a strong bound for the number of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p prime\\\"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding=\\\"application/x-tex\\\">p’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degree irreducible characters of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G\\\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of the commutator factor group of a Sylow <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p\\\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-subgroup of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G\\\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The bound arises from a recent conjecture of Navarro and Tiep [Forum Math. Pi 9 (2021), pp. 1–28] on fields of character values and a phenomenon called the continuity of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p\\\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rationality level of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p prime\\\"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding=\\\"application/x-tex\\\">p’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degree characters. This continuity property in turn is predicted by the celebrated McKay-Navarro conjecture (see G. Navarro [Ann. of Math. (2) 160 (2004), pp. 1129–1140]). We achieve both the bound and the continuity property for <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p equals 2\\\"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">p=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.\",\"PeriodicalId\":23209,\"journal\":{\"name\":\"Transactions of the American Mathematical Society\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2023-10-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions of the American Mathematical Society\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/tran/8926\",\"RegionNum\":2,\"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":"Transactions of the American Mathematical Society","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/tran/8926","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
设p p是素数，G G是有限群。利用G G的Sylow p p -子群的换易子因子群，给出了G G的p ' p '次不可约字符个数的一个强界。这个界是由Navarro和Tiep[论坛数学]最近的一个猜想产生的。Pi 9 (2021)， pp. 1-28]在字符值领域和p ' p '度字符的p ' p '理性水平的连续性现象上的研究。这种连续性特性反过来又由著名的McKay-Navarro猜想(见G. Navarro [Ann。的数学。(2) 160 (2004)， pp. 1129-1140)。我们得到了p=2的界和连续性。
The continuity of 𝑝-rationality and a lower bound for 𝑝’-degree characters of finite groups
Let pp be a prime and GG a finite group. We propose a strong bound for the number of p′p’-degree irreducible characters of GG in terms of the commutator factor group of a Sylow pp-subgroup of GG. The bound arises from a recent conjecture of Navarro and Tiep [Forum Math. Pi 9 (2021), pp. 1–28] on fields of character values and a phenomenon called the continuity of pp-rationality level of p′p’-degree characters. This continuity property in turn is predicted by the celebrated McKay-Navarro conjecture (see G. Navarro [Ann. of Math. (2) 160 (2004), pp. 1129–1140]). We achieve both the bound and the continuity property for p=2p=2.
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