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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="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='#definition'>Definition</a></li> <li><a href='#special_cases'>Special cases</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#of_free_groups'>Of free groups</a></li> <li><a href='#of_lie_groups'>Of Lie groups</a></li> </ul> <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>A <em>subgroup</em> of a <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a “smaller” group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> sitting inside <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <h2 id="definition">Definition</h2> <p>A <em>subgroup</em> is a <a class="existingWikiWord" href="/nlab/show/subobject">subobject</a> in the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a> of <a class="existingWikiWord" href="/nlab/show/group">group</a>s: a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> of groups</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>↪</mo><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K \hookrightarrow G \,. </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is <em>a subgroup of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></em>.</p> <h2 id="special_cases">Special cases</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilizer+subgroup">stabilizer subgroup</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/torsion+subgroup">torsion subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel+subgroup">Borel subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/parabolic+subgroup">parabolic subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+Fitting+subgroup">generalized Fitting subgroup</a></p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="of_free_groups">Of free groups</h3> <p>Every subgroup of a <a class="existingWikiWord" href="/nlab/show/free+group">free group</a> is itself free. This is the statement of the <em><a class="existingWikiWord" href="/nlab/show/Nielsen-Schreier+theorem">Nielsen-Schreier theorem</a></em>.</p> <h3 id="of_lie_groups">Of Lie groups</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \hookrightarrow G</annotation></semantics></math> a sub-<a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> inclusion write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}H \to \mathbf{B}G</annotation></semantics></math> for the induced map on <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/Lie+groupoids">Lie groupoids</a>. The <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of this map (in <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>) is the <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math>: there is a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G/H \to \mathbf{B}H \to \mathbf{B}G \,. </annotation></semantics></math></div> <p>Now let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>↪</mo><mi>K</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \hookrightarrow K \hookrightarrow G</annotation></semantics></math> be a sequence of two subgroup inclusions. By the above this yields the <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>K</mi><mo stretchy="false">/</mo><mi>H</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>G</mi><mo stretchy="false">/</mo><mi>K</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>K</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>K</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ K/H &\to& G/H &\to& G/K \\ \downarrow && \downarrow && \downarrow \\ \mathbf{B}H &\to& \mathbf{B}H &\to& \mathbf{B}K \\ \downarrow && \downarrow && \downarrow \\ \mathbf{B}K &\to& \mathbf{B}G &\to& \mathbf{B}G } </annotation></semantics></math></div> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/centralizer+subgroup">centralizer subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normalizer+subgroup">normalizer subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilizer+subgroup">stabilizer subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conjugate+subgroup">conjugate subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/congruence+subgroup">congruence subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Young+subgroup">Young subgroup</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lattice+of+subgroups">lattice of subgroups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/maximal+subgroup">maximal subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/index+of+a+subgroup">index of a subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a>, <a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BN-pair">BN-pair</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subring">subring</a>, <a class="existingWikiWord" href="/nlab/show/submodule">submodule</a></p> </li> </ul> <h2 id="references">References</h2> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/univalent+foundations+of+mathematics">univalent foundations of mathematics</a> (<a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> with the <a class="existingWikiWord" href="/nlab/show/univalence+axiom">univalence axiom</a>, but for 1-groups):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mart%C3%ADn+Escard%C3%B3">Martín Escardó</a>, <em><a href="https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html#subgroups-sip">Subgroups</a></em>, §33.12 in: <em>Introduction to Univalent Foundations of Mathematics with Agda</em> [<a href="https://arxiv.org/abs/1911.00580">arXiv:1911.00580</a>, <a href="https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html">webpage</a>]</p> <blockquote> <p>(in <a class="existingWikiWord" href="/nlab/show/Agda">Agda</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Marc+Bezem">Marc Bezem</a>, <a class="existingWikiWord" href="/nlab/show/Ulrik+Buchholtz">Ulrik Buchholtz</a>, <a class="existingWikiWord" href="/nlab/show/Pierre+Cagne">Pierre Cagne</a>, <a class="existingWikiWord" href="/nlab/show/Bj%C3%B8rn+Ian+Dundas">Bjørn Ian Dundas</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+R.+Grayson">Daniel R. Grayson</a>: Chapter 5 of: <em><a class="existingWikiWord" href="/nlab/show/Symmetry">Symmetry</a></em> (2021) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://unimath.github.io/SymmetryBook/book.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <p>See also</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Subgroup">Subgroup</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 4, 2023 at 11:38:07. 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