{"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":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":2.0000,"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\":49828,\"journal\":{\"name\":\"Memoirs of the American Mathematical Society\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":2.0000,\"publicationDate\":\"2022-03-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"9\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Memoirs of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/memo/1355\",\"RegionNum\":4,\"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":"Memoirs of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/memo/1355","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","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.
期刊介绍：
Memoirs of the American Mathematical Society is devoted to the publication of research in all areas of pure and applied mathematics. The Memoirs is designed particularly to publish long papers or groups of cognate papers in book form, and is under the supervision of the Editorial Committee of the AMS journal Transactions of the AMS. To be accepted by the editorial board, manuscripts must be correct, new, and significant. Further, they must be well written and of interest to a substantial number of mathematicians.
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