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value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="exceptional_structures">Exceptional structures</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/exceptional+structures">exceptional structures</a></strong>, <a class="existingWikiWord" href="/nlab/show/exceptional+isomorphisms">exceptional isomorphisms</a></p> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exceptional+finite+groups">exceptional finite groups</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monster+group">monster group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mathieu+group">Mathieu group</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Conway+group">Conway group</a></p> </li> </ul> </li> <li> <p>exceptional <a class="existingWikiWord" href="/nlab/show/finite+rotation+groups">finite rotation groups</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tetrahedral+group">tetrahedral group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/octahedral+group">octahedral group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/icosahedral+group">icosahedral group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exceptional+Lie+groups">exceptional Lie groups</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/G%E2%82%82">G₂</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/F%E2%82%84">F₄</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E%E2%82%86">E₆</a>, <a class="existingWikiWord" href="/nlab/show/E%E2%82%87">E₇</a>, <a class="existingWikiWord" href="/nlab/show/E%E2%82%88">E₈</a></p> </li> </ul> <p>and <a class="existingWikiWord" href="/nlab/show/Kac-Moody+groups">Kac-Moody groups</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/E%E2%82%89">E₉</a>, <a class="existingWikiWord" href="/nlab/show/E%E2%82%81%E2%82%80">E₁₀</a>, <a class="existingWikiWord" href="/nlab/show/E%E2%82%81%E2%82%81">E₁₁</a>, …</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dwyer-Wilkerson+H-space">Dwyer-Wilkerson H-space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exceptional+Lie+algebras">exceptional Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exceptional+Jordan+algebra">exceptional Jordan algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Albert+algebra">Albert algebra</a></li> </ul> </li> <li> <p>exceptional <a class="existingWikiWord" href="/nlab/show/Jordan+superalgebra">Jordan superalgebra</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mn>10</mn></msub></mrow><annotation encoding="application/x-tex">K_10</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E%E2%82%88+lattice">E₈ lattice</a>, <a class="existingWikiWord" href="/nlab/show/Leech+lattice">Leech lattice</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cayley+plane">Cayley plane</a></p> </li> </ul> <h2 id="interrelations">Interrelations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supersymmetry+and+division+algebras">supersymmetry and division algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+magic+square">Freudenthal magic square</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moonshine">moonshine</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mathieu+moonshine">Mathieu moonshine</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/umbral+moonshine">umbral moonshine</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/O%27Nan+moonshine">O'Nan moonshine</a></p> </li> </ul> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exceptional+geometry">exceptional geometry</a>, <a class="existingWikiWord" href="/nlab/show/exceptional+generalized+geometry">exceptional generalized geometry</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exceptional+field+theory">exceptional field theory</a></p> </li> </ul> <h2 id="philosophy">Philosophy</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/universal+exceptionalism">universal exceptionalism</a></li> </ul> </div></div> <h4 id="group_theory">Group Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/group+theory">group theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+object">group object</a>, <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object in an (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></li> <li><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+action">group action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></li> <li><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></li> <li><a class="existingWikiWord" href="/nlab/show/progroup">progroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/homogeneous+space">homogeneous space</a></li> </ul> <p><strong>Classical groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a>. <a class="existingWikiWord" href="/nlab/show/projective+unitary+group">projective unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+group">symplectic group</a></p> </li> </ul> <p><strong>Finite groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a>, <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classification+of+finite+simple+groups">classification of finite simple groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sporadic+finite+simple+groups">sporadic finite simple groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Monster+group">Monster group</a>, <a class="existingWikiWord" href="/nlab/show/Mathieu+group">Mathieu group</a></li> </ul> </li> </ul> <p><strong>Group schemes</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+variety">abelian variety</a></p> </li> </ul> <p><strong>Topological groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+topological+group">compact topological group</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+group">locally compact topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+group">string group</a></p> </li> </ul> <p><strong>Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kac-Moody+group">Kac-Moody group</a></p> </li> </ul> <p><strong>Super-Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a></p> </li> </ul> <p><strong>Higher groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a>, <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-group">n-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+groupal+n-groupoid">k-tuply groupal n-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a>, <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-group">fivebrane Lie 6-group</a></p> </li> </ul> <p><strong>Cohomology and Extensions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a>, <a class="existingWikiWord" href="/nlab/show/Ext-group">Ext-group</a></p> </li> </ul> <p><strong>Related concepts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The sporadic finite simple groups are the <a class="existingWikiWord" href="/nlab/show/exceptional+structures">exceptional structures</a> among <a class="existingWikiWord" href="/nlab/show/finite+groups">finite groups</a>:</p> <p>According to the <a class="existingWikiWord" href="/nlab/show/classification+of+finite+simple+groups">classification of finite simple groups</a>, there are 18 <a class="existingWikiWord" href="/nlab/show/countable+set">countably infinite families</a> and 26 <em>sporadic</em> simple groups. The latter groups do not fit into any systematic classification, but there are a number of links between them. For example, the <a class="existingWikiWord" href="/nlab/show/Monster+group">Monster group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, the largest of the sporadic groups, contains all but six (the ‘<a class="existingWikiWord" href="/nlab/show/pariah+groups">pariah groups</a>’, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>J</mi> <mn>3</mn></msub><mo>,</mo><mi>Ru</mi><mo>,</mo><mi>ON</mi><mo>,</mo><mi>Ly</mi><mo>,</mo><msub><mi>J</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">J_1, J_3,Ru,ON,Ly,J_4</annotation></semantics></math>) of the other sporadic groups as <a class="existingWikiWord" href="/nlab/show/subquotients">subquotients</a>. These 20 sporadic groups comprise what is termed the ‘<a class="existingWikiWord" href="/nlab/show/Happy+Family">Happy Family</a>’ by <a class="existingWikiWord" href="/nlab/show/Robert+Griess">Robert Griess</a>, (<a href="#Griess98">Griess 98</a>).</p> <p>The <em>Happy Family</em> is taken to be formed of three generations</p> <ul> <li>First: five subquotients of the <a class="existingWikiWord" href="/nlab/show/Mathieu+group">Mathieu group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mn>24</mn></msub></mrow><annotation encoding="application/x-tex">M_{24}</annotation></semantics></math> = symmetries of the <a class="existingWikiWord" href="/nlab/show/binary+Golay+code">binary Golay code</a></li> <li>Second: seven subquotients of the <a class="existingWikiWord" href="/nlab/show/Conway+group">Conway group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Co</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">Co_1</annotation></semantics></math> = symmetries of the <a class="existingWikiWord" href="/nlab/show/Leech+lattice">Leech lattice</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>mod</mi><mo stretchy="false">{</mo><mo>±</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">mod \{\pm 1\}</annotation></semantics></math>.</li> <li>Third: eight subquotients of the <a class="existingWikiWord" href="/nlab/show/Monster+group">Monster group</a> = conformal symmetries of the <a class="existingWikiWord" href="/nlab/show/monster+vertex+operator+algebra">monster vertex operator algebra</a></li> </ul> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Conway+group">Conway group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monster+group">monster group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mathieu+group">Mathieu group</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Moonshine">Moonshine</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphisms+of+super+vertex+operator+algebras">automorphisms of super vertex operator algebras</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Robert+A.+Wilson">Robert A. Wilson</a>, <em>The Finite Simple Groups</em>. Springer, Graduate Mathematics series <strong>251</strong> (2009).</p> </li> <li id="Griess98"> <p><a class="existingWikiWord" href="/nlab/show/Robert+Griess">Robert Griess</a>, <em>Twelve Sporadic Groups</em>, Springer, 1998 (<a href="https://link.springer.com/book/10.1007/978-3-662-03516-0">doi:10.1007/978-3-662-03516-0</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on April 20, 2023 at 10:27:24. 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