{"title":"Lie型例外群的极大值< 0.05𝑆𝐿2 >子群","authors":"David A. Craven","doi":"10.1090/memo/1355","DOIUrl":null,"url":null,"abstract":"<p>We study embeddings of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper P normal upper S normal upper L Subscript 2 Baseline left-parenthesis p Superscript a Baseline right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">P</mml:mi>\n <mml:mi mathvariant=\"normal\">S</mml:mi>\n <mml:mi mathvariant=\"normal\">L</mml:mi>\n </mml:mrow>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>p</mml:mi>\n <mml:mi>a</mml:mi>\n </mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {PSL}_2(p^a)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> into exceptional groups <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G left-parenthesis p Superscript b Baseline right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>p</mml:mi>\n <mml:mi>b</mml:mi>\n </mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">G(p^b)</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 G equals upper F 4 comma upper E 6 comma squared upper E 6 comma upper E 7\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:msub>\n <mml:mi>F</mml:mi>\n <mml:mn>4</mml:mn>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>E</mml:mi>\n <mml:mn>6</mml:mn>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n\n </mml:mrow>\n <mml:mn>2</mml:mn>\n </mml:msup>\n <mml:mspace width=\"negativethinmathspace\" />\n <mml:msub>\n <mml:mi>E</mml:mi>\n <mml:mn>6</mml:mn>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>E</mml:mi>\n <mml:mn>7</mml:mn>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">G=F_4,E_6,{}^2\\!E_6,E_7</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=\"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> a prime with <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"a comma b\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>a</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>b</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">a,b</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> positive integers. With a few possible exceptions, we prove that any almost simple group with socle <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper P normal upper S normal upper L Subscript 2 Baseline left-parenthesis p Superscript a Baseline right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">P</mml:mi>\n <mml:mi mathvariant=\"normal\">S</mml:mi>\n <mml:mi mathvariant=\"normal\">L</mml:mi>\n </mml:mrow>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>p</mml:mi>\n <mml:mi>a</mml:mi>\n </mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {PSL}_2(p^a)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, that is maximal inside an almost simple exceptional group of Lie type <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F 4\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>F</mml:mi>\n <mml:mn>4</mml:mn>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">F_4</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E 6\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>E</mml:mi>\n <mml:mn>6</mml:mn>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">E_6</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"squared upper E 6\">\n <mml:semantics>\n <mml:mrow>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n\n </mml:mrow>\n <mml:mn>2</mml:mn>\n </mml:msup>\n <mml:mspace width=\"negativethinmathspace\" />\n <mml:msub>\n <mml:mi>E</mml:mi>\n <mml:mn>6</mml:mn>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">{}^2\\!E_6</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 E 7\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>E</mml:mi>\n <mml:mn>7</mml:mn>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">E_7</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A 1\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>A</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">A_1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> inside the algebraic group.</p>\n\n<p>Together with a recent result of Burness and Testerman for <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> the Coxeter number plus one, this proves that all maximal subgroups with socle <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper P normal upper S normal upper L Subscript 2 Baseline left-parenthesis p Superscript a Baseline right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">P</mml:mi>\n <mml:mi mathvariant=\"normal\">S</mml:mi>\n <mml:mi mathvariant=\"normal\">L</mml:mi>\n </mml:mrow>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>p</mml:mi>\n <mml:mi>a</mml:mi>\n </mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {PSL}_2(p^a)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> inside these finite almost simple groups are known, with three possible exceptions (<inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p Superscript a Baseline equals 7 comma 8 comma 25\">\n <mml:semantics>\n <mml:mrow>\n <mml:msup>\n <mml:mi>p</mml:mi>\n <mml:mi>a</mml:mi>\n </mml:msup>\n <mml:mo>=</mml:mo>\n <mml:mn>7</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mn>8</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mn>25</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">p^a=7,8,25</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 E 7\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>E</mml:mi>\n <mml:mn>7</mml:mn>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">E_7</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>).</p>\n\n<p>In the three remaining cases we provide considerable information about a potential maximal subgroup.</p>","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"9","resultStr":"{\"title\":\"Maximal 𝑃𝑆𝐿₂ Subgroups of Exceptional Groups of Lie Type\",\"authors\":\"David A. Craven\",\"doi\":\"10.1090/memo/1355\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We study embeddings of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper P normal upper S normal upper L Subscript 2 Baseline left-parenthesis p Superscript a Baseline right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"normal\\\">P</mml:mi>\\n <mml:mi mathvariant=\\\"normal\\\">S</mml:mi>\\n <mml:mi mathvariant=\\\"normal\\\">L</mml:mi>\\n </mml:mrow>\\n <mml:mn>2</mml:mn>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:msup>\\n <mml:mi>p</mml:mi>\\n <mml:mi>a</mml:mi>\\n </mml:msup>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathrm {PSL}_2(p^a)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> into exceptional groups <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G left-parenthesis p Superscript b Baseline right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>G</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:msup>\\n <mml:mi>p</mml:mi>\\n <mml:mi>b</mml:mi>\\n </mml:msup>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">G(p^b)</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 G equals upper F 4 comma upper E 6 comma squared upper E 6 comma upper E 7\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>G</mml:mi>\\n <mml:mo>=</mml:mo>\\n <mml:msub>\\n <mml:mi>F</mml:mi>\\n <mml:mn>4</mml:mn>\\n </mml:msub>\\n <mml:mo>,</mml:mo>\\n <mml:msub>\\n <mml:mi>E</mml:mi>\\n <mml:mn>6</mml:mn>\\n </mml:msub>\\n <mml:mo>,</mml:mo>\\n <mml:msup>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n\\n </mml:mrow>\\n <mml:mn>2</mml:mn>\\n </mml:msup>\\n <mml:mspace width=\\\"negativethinmathspace\\\" />\\n <mml:msub>\\n <mml:mi>E</mml:mi>\\n <mml:mn>6</mml:mn>\\n </mml:msub>\\n <mml:mo>,</mml:mo>\\n <mml:msub>\\n <mml:mi>E</mml:mi>\\n <mml:mn>7</mml:mn>\\n </mml:msub>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">G=F_4,E_6,{}^2\\\\!E_6,E_7</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=\\\"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> a prime with <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"a comma b\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>a</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>b</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">a,b</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> positive integers. With a few possible exceptions, we prove that any almost simple group with socle <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper P normal upper S normal upper L Subscript 2 Baseline left-parenthesis p Superscript a Baseline right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"normal\\\">P</mml:mi>\\n <mml:mi mathvariant=\\\"normal\\\">S</mml:mi>\\n <mml:mi mathvariant=\\\"normal\\\">L</mml:mi>\\n </mml:mrow>\\n <mml:mn>2</mml:mn>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:msup>\\n <mml:mi>p</mml:mi>\\n <mml:mi>a</mml:mi>\\n </mml:msup>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathrm {PSL}_2(p^a)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, that is maximal inside an almost simple exceptional group of Lie type <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper F 4\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mi>F</mml:mi>\\n <mml:mn>4</mml:mn>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">F_4</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper E 6\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mi>E</mml:mi>\\n <mml:mn>6</mml:mn>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">E_6</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"squared upper E 6\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msup>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n\\n </mml:mrow>\\n <mml:mn>2</mml:mn>\\n </mml:msup>\\n <mml:mspace width=\\\"negativethinmathspace\\\" />\\n <mml:msub>\\n <mml:mi>E</mml:mi>\\n <mml:mn>6</mml:mn>\\n </mml:msub>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">{}^2\\\\!E_6</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 E 7\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mi>E</mml:mi>\\n <mml:mn>7</mml:mn>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">E_7</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper A 1\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mi>A</mml:mi>\\n <mml:mn>1</mml:mn>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">A_1</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> inside the algebraic group.</p>\\n\\n<p>Together with a recent result of Burness and Testerman for <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> the Coxeter number plus one, this proves that all maximal subgroups with socle <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper P normal upper S normal upper L Subscript 2 Baseline left-parenthesis p Superscript a Baseline right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"normal\\\">P</mml:mi>\\n <mml:mi mathvariant=\\\"normal\\\">S</mml:mi>\\n <mml:mi mathvariant=\\\"normal\\\">L</mml:mi>\\n </mml:mrow>\\n <mml:mn>2</mml:mn>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:msup>\\n <mml:mi>p</mml:mi>\\n <mml:mi>a</mml:mi>\\n </mml:msup>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathrm {PSL}_2(p^a)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> inside these finite almost simple groups are known, with three possible exceptions (<inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p Superscript a Baseline equals 7 comma 8 comma 25\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msup>\\n <mml:mi>p</mml:mi>\\n <mml:mi>a</mml:mi>\\n </mml:msup>\\n <mml:mo>=</mml:mo>\\n <mml:mn>7</mml:mn>\\n <mml:mo>,</mml:mo>\\n <mml:mn>8</mml:mn>\\n <mml:mo>,</mml:mo>\\n <mml:mn>25</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">p^a=7,8,25</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 E 7\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mi>E</mml:mi>\\n <mml:mn>7</mml:mn>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">E_7</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>).</p>\\n\\n<p>In the three remaining cases we provide considerable information about a potential maximal subgroup.</p>\",\"PeriodicalId\":2,\"journal\":{\"name\":\"ACS Applied Bio Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.6000,\"publicationDate\":\"2022-03-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"9\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Bio Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/memo/1355\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, BIOMATERIALS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/memo/1355","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: 9
摘要
我们研究了在G = f4, e6, 2,e6时，P s2 (p2a) \ mathm {PSL}_2(P ^a)在例外群G(p2b) G(P ^b)中的嵌入。E 7 g = f_4, e_6，{}²\!E_6 E_7和p p a '和a b a b正整数。除了一些可能的例外，我们证明了任何具有集合pssl 2(P a) \ mathm {PSL}_2(P ^a)的几乎简单群，在Lie型的几乎简单例外群f4f_4, e6e_6, 2e6 {}^2\!E_6和E_7 E_7，是代数群内对应的a1a_1型最大闭子群的Frobenius映射下的不动点。结合Burness和Testerman关于p p (Coxeter数+ 1)的最新结果，证明了在这些有限几乎单群中，除三种可能的例外(p a = 7,8，25p ^a=7,8,25对于e7 E_7)。在剩下的三种情况中，我们提供了关于潜在最大子群的大量信息。
Maximal 𝑃𝑆𝐿₂ Subgroups of Exceptional Groups of Lie Type
We study embeddings of PSL2(pa)\mathrm {PSL}_2(p^a) into exceptional groups G(pb)G(p^b) for G=F4,E6,2E6,E7G=F_4,E_6,{}^2\!E_6,E_7, and pp a prime with a,ba,b positive integers. With a few possible exceptions, we prove that any almost simple group with socle PSL2(pa)\mathrm {PSL}_2(p^a), that is maximal inside an almost simple exceptional group of Lie type F4F_4, E6E_6, 2E6{}^2\!E_6 and E7E_7, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type A1A_1 inside the algebraic group.
Together with a recent result of Burness and Testerman for pp the Coxeter number plus one, this proves that all maximal subgroups with socle PSL2(pa)\mathrm {PSL}_2(p^a) inside these finite almost simple groups are known, with three possible exceptions (pa=7,8,25p^a=7,8,25 for E7E_7).
In the three remaining cases we provide considerable information about a potential maximal subgroup.
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