Carter subgroups, characters and composition series

Transactions of the American Mathematical Society, Series B Pub Date : 2022-05-27 DOI:10.1090/btran/93

I. Isaacs

{"title":"Carter subgroups, characters and composition series","authors":"I. Isaacs","doi":"10.1090/btran/93","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be a finite solvable group. We construct a set <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper H\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">H</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {H}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of irreducible characters of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> such that if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C\">\n <mml:semantics>\n <mml:mi>C</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">C</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is a Carter subgroup of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, then the members of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper H\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">H</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {H}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> behave well with respect to <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C\">\n <mml:semantics>\n <mml:mi>C</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">C</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-composition series for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, and we show that <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper H\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">H</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {H}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is in bijective correspondence with the set of linear characters of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C\">\n <mml:semantics>\n <mml:mi>C</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">C</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Also, if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\n <mml:semantics>\n <mml:mi>A</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is a group that acts coprimely on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, then analogously, we characterize in terms of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\n <mml:semantics>\n <mml:mi>A</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-composition series for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, the set of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\n <mml:semantics>\n <mml:mi>A</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-invariant characters of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> that have a linear Glauberman-Isaacs correspondent.</p>","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"31 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2022-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/btran/93","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

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Abstract

Let G G be a finite solvable group. We construct a set H \mathcal {H} of irreducible characters of G G such that if C C is a Carter subgroup of G G , then the members of H \mathcal {H} behave well with respect to C C -composition series for G G , and we show that H \mathcal {H} is in bijective correspondence with the set of linear characters of C C . Also, if A A is a group that acts coprimely on G G , then analogously, we characterize in terms of A A -composition series for G G , the set of A A -invariant characters of G G that have a linear Glauberman-Isaacs correspondent.

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卡特子组、人物和构图系列

设G G是有限可解群。构造了G的不可约字符的集合H \mathcal {H}，使得如果C C是G G的Carter子群，则H \mathcal {H}的成员对于G G的C C复合级数表现良好，并证明了H \mathcal {H}与C C的线性字符集双射对应。同样，如果A A是作用于G G的群，那么类似地，我们用G G的A A -复合级数来表示，G G的A A -不变特征的集合，它们有一个线性的Glauberman-Isaacs对应。

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Transactions of the American Mathematical Society, Series B

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1.70

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