Finite Groups Which are Almost Groups of Lie Type in Characteristic 𝐩

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS

ACS Applied Bio Materials Pub Date : 2023-12-01 DOI:10.1090/memo/1452

Chris Parker, Gerald Pientka, Andreas Seidel, G. Stroth

{"title":"Finite Groups Which are Almost Groups of Lie Type in Characteristic 𝐩","authors":"Chris Parker, Gerald Pientka, Andreas Seidel, G. Stroth","doi":"10.1090/memo/1452","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=\"p\">\n <mml:semantics>\n <mml:mi>p</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be a prime. In this paper we investigate finite <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper K Subscript StartSet 2 comma p EndSet\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">K</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mi>p</mml:mi>\n <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal K_{\\{2,p\\}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-groups <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> which have a subgroup <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H less-than-or-equal-to upper G\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>H</mml:mi>\n <mml:mo>≤</mml:mo>\n <mml:mi>G</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">H \\le G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> such that <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K less-than-or-equal-to upper H equals upper N Subscript upper G Baseline left-parenthesis upper K right-parenthesis less-than-or-equal-to upper A u t left-parenthesis upper K right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>K</mml:mi>\n <mml:mo>≤</mml:mo>\n <mml:mi>H</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:msub>\n <mml:mi>N</mml:mi>\n <mml:mi>G</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>K</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>≤</mml:mo>\n <mml:mi>Aut</mml:mi>\n <mml:mo>⁡</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>K</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">K \\le H = N_G(K) \\le \\operatorname {Aut}(K)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> a simple group of Lie type in characteristic <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\n <mml:semantics>\n <mml:mi>p</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue upper G colon upper H EndAbsoluteValue\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">|</mml:mo>\n </mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:mo>:</mml:mo>\n <mml:mi>H</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">|</mml:mo>\n </mml:mrow>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">|G:H|</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is coprime to <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\n <mml:semantics>\n <mml:mi>p</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. If <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> is of local characteristic <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\n <mml:semantics>\n <mml:mi>p</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, then <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> is called almost of Lie type in characteristic <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\n <mml:semantics>\n <mml:mi>p</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Here <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> is of local characteristic <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\n <mml:semantics>\n <mml:mi>p</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> means that for all nontrivial <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\n <mml:semantics>\n <mml:mi>p</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-subgroups <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P\">\n <mml:semantics>\n <mml:mi>P</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">P</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> 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>, and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Q\">\n <mml:semantics>\n <mml:mi>Q</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">Q</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> the largest normal <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\n <mml:semantics>\n <mml:mi>p</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-subgroup in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N Subscript upper G Baseline left-parenthesis upper P right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>N</mml:mi>\n <mml:mi>G</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>P</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">N_G(P)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> we have the containment <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Subscript upper G Baseline left-parenthesis upper Q right-parenthesis less-than-or-equal-to upper Q\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>C</mml:mi>\n <mml:mi>G</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>Q</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>≤</mml:mo>\n <mml:mi>Q</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">C_G(Q)\\le Q</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We determine details of the structure of groups which are almost of Lie type in characteristic <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\n <mml:semantics>\n <mml:mi>p</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. In particular, in the case that the rank of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is at least <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\">\n <mml:semantics>\n <mml:mn>3</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> we prove that <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G equals upper H\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:mo>=</mml:m","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/memo/1452","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}

引用次数: 0

Abstract

Let p p be a prime. In this paper we investigate finite K { 2 , p } \mathcal K_{\{2,p\}} -groups G G which have a subgroup H ≤ G H \le G such that K ≤ H = N G ( K ) ≤ Aut ⁡ ( K ) K \le H = N_G(K) \le \operatorname {Aut}(K) for K K a simple group of Lie type in characteristic p p , and | G : H | |G:H| is coprime to p p . If G G is of local characteristic p p , then G G is called almost of Lie type in characteristic p p . Here G G is of local characteristic p p means that for all nontrivial p p -subgroups P P of G G , and Q Q the largest normal p p -subgroup in N G ( P ) N_G(P) we have the containment C G ( Q ) ≤ Q C_G(Q)\le Q . We determine details of the structure of groups which are almost of Lie type in characteristic p p . In particular, in the case that the rank of K K is at least 3 3 we prove that G =

查看原文本刊更多论文

特征𝐩中几乎是列类型群的有限群

设p是素数。本文研究有限K {2，p} \数学K_{\{2,p\}} -群G G，其子群H≤G H \le G使得K≤H = N G(K)≤Aut (K) K \le H = N_G(K) \le \算子名{Aut}(K)对于K K是特征为p p的李型的简单群，且| G:H| |G:H|是p p的素数。如果G G具有局部特征p p，则G G在特征p p上几乎是李氏型。这里G G具有局部特征p p意味着对于所有非平凡p -子群p p (G G)和Q Q (N G(p) N_G(p)中最大的正规p -子群)我们有包含C G(Q)≤Q C_G(Q)\le Q。我们确定了特征p p几乎为李氏型的群的结构细节。特别地，在K K的秩至少为33的情况下，我们证明了G =

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊