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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="category_theory">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></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> <h4 id="monoid_theory">Monoid theory</h4> <div class="hide"><div> <p><strong>monoid theory</strong> in <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>, <a class="existingWikiWord" href="/nlab/show/infinity-monoid">infinity-monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a>, <a class="existingWikiWord" href="/nlab/show/monoid+object+in+an+%28infinity%2C1%29-category">monoid object in an (infinity,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semiring">semiring</a>, <a class="existingWikiWord" href="/nlab/show/rig">rig</a>, <a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mon">Mon</a>, <a class="existingWikiWord" href="/nlab/show/CMon">CMon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+homomorphism">monoid homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/trivial+monoid">trivial monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/submonoid">submonoid</a>, <span class="newWikiWord">quotient monoid<a href="/nlab/new/quotient+monoid">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/divisor">divisor</a>, <span class="newWikiWord">multiple<a href="/nlab/new/multiple">?</a></span>, <span class="newWikiWord">quotient element<a href="/nlab/new/quotient+element">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverse+element">inverse element</a>, <a class="existingWikiWord" href="/nlab/show/unit">unit</a>, <a class="existingWikiWord" href="/nlab/show/irreducible+element">irreducible element</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal+in+a+monoid">ideal in a monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+ideal+in+a+monoid">principal ideal in a monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/tensor+product+of+commutative+monoids">tensor product of commutative monoids</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cancellative+monoid">cancellative monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GCD+monoid">GCD monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unique+factorization+monoid">unique factorization monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B%C3%A9zout+monoid">Bézout monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+ideal+monoid">principal ideal monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/absorption+monoid">absorption monoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/zero+divisor">zero divisor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+monoid">integral monoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/free+monoid">free monoid</a>, <a class="existingWikiWord" href="/nlab/show/free+commutative+monoid">free commutative monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graphic+monoid">graphic monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+action">monoid action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+over+a+monoid">module over a monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+a+monoid">localization of a monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+completion">group completion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+monoid">endomorphism monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+commutative+monoid">super commutative monoid</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/monoid+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#delooping'>Delooping</a></li> <li><a href='#generalizations'>Generalizations</a></li> <ul> <li><a href='#Internalization'>Internalization</a></li> <li><a href='#in_higher_categorical_and_homotopical_contexts'>In higher categorical and homotopical contexts</a></li> <li><a href='#weakened_axioms'>Weakened axioms</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#special_types_and_classes'>Special types and classes</a></li> <li><a href='#concrete_examples'>Concrete examples</a></li> <li><a href='#counterexamples'>Counterexamples</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#literature'>Literature</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>A <strong>group</strong> is an <a class="existingWikiWord" href="/nlab/show/algebraic+structure">algebraic structure</a> consisting of a <a class="existingWikiWord" href="/nlab/show/set">set</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> and a <a class="existingWikiWord" href="/nlab/show/binary+operation">binary operation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋆</mo></mrow><annotation encoding="application/x-tex">\star</annotation></semantics></math> that satisfies the <strong>group axioms</strong>, being:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/associativity">associativity</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><mi>G</mi><mo>:</mo><mo stretchy="false">(</mo><mi>a</mi><mo>⋆</mo><mi>b</mi><mo stretchy="false">)</mo><mo>⋆</mo><mi>c</mi><mo>=</mo><mi>a</mi><mo>⋆</mo><mo stretchy="false">(</mo><mi>b</mi><mo>⋆</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\forall a,b,c \in G: (a \star b) \star c = a \star (b \star c)</annotation></semantics></math></li> <li><a class="existingWikiWord" href="/nlab/show/identity">identity</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∃</mo><mi>e</mi><mo>∈</mo><mi>G</mi><mo>,</mo><mo>∀</mo><mi>a</mi><mo>∈</mo><mi>G</mi><mo>:</mo><mi>e</mi><mo>⋆</mo><mi>a</mi><mo>=</mo><mi>a</mi><mo>⋆</mo><mi>e</mi><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\exists e \in G, \forall a \in G: e \star a = a \star e = a</annotation></semantics></math></li> <li><a class="existingWikiWord" href="/nlab/show/inverse">inverse</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>a</mi><mo>∈</mo><mi>G</mi><mo>,</mo><mo>∃</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>∈</mo><mi>G</mi><mo>:</mo><mi>a</mi><mo>⋆</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>⋆</mo><mi>a</mi><mo>=</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">\forall a \in G, \exists a^{-1} \in G: a \star a^{-1} = a^{-1} \star a = e</annotation></semantics></math></li> </ul> <p>It follows that the <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">a^{-1}</annotation></semantics></math> is unique for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> and <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 non-empty.</p> <p>In a broader sense, a group is a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> in which every element has a (necessarily unique) <a class="existingWikiWord" href="/nlab/show/inverse+element">inverse</a>. When written with a view toward <a class="existingWikiWord" href="/nlab/show/group+objects">group objects</a> (see <em><a href="#Internalization">Internalization</a></em> below), one should rather say that a group is a monoid together with an inversion operation.</p> <p>An <strong><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a></strong> is a group where the order in which two elements are multiplied is irrelevant. That is, it satisfies <em>commutativity</em>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>G</mi><mo>:</mo><mi>a</mi><mo>⋆</mo><mi>b</mi><mo>=</mo><mi>b</mi><mo>⋆</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\forall a,b \in G : a \star b = b \star a</annotation></semantics></math>.</p> <h2 id="delooping">Delooping</h2> <p>To some extent, a group “is” a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> with a single object, or more precisely a <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed</a> groupoid with a single object.</p> <p>The <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of a group <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 <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} G</annotation></semantics></math> with</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Obj</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mo>•</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">Obj(\mathbf{B}G) = \{\bullet\}</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mo>•</mo><mo>,</mo><mo>•</mo><mo stretchy="false">)</mo><mo>=</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">Hom_{\mathbf{B}G}(\bullet, \bullet) = G</annotation></semantics></math>.</p> </li> </ul> <p>Since for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>,</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G, H</annotation></semantics></math> two groups, <a class="existingWikiWord" href="/nlab/show/functors">functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \mathbf{B}H</annotation></semantics></math> are canonically in bijection with group homomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G \to H</annotation></semantics></math>, this gives rise to the following statement:</p> <p>Let <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a> be the 1-<a class="existingWikiWord" href="/nlab/show/category">category</a> whose objects are <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> and whose <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are <a class="existingWikiWord" href="/nlab/show/functors">functors</a> (discarding the <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a>). Let <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a> be the category of groups. Then the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo lspace="verythinmathspace">:</mo><mi>Grp</mi><mo>→</mo><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> \mathbf{B} \colon Grp \to Grpd </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a>. In terms of this functor we may regard groups as the full <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> of groupoids on groupoids with a single object.</p> <p>It is in this sense that a group really is a groupoid with a single object.</p> <p>But notice that it is unnatural to think of <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a> as a 1-category. It is really a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, namely the sub-2-category of <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> on groupoids.</p> <p>And the category of groups is <em>not</em> equivalent to the full sub-2-category of the 2-category of groupoids on one-object groupoids.</p> <p>The reason is that two functors:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>f</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mrow><annotation encoding="application/x-tex"> \mathbf{B}f_1, \mathbf{B}f_2 \colon \mathbf{B}G \to \mathbf{B}H </annotation></semantics></math></div> <p>coming from two group homomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>→</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">f_1, f_2 \colon G \to H</annotation></semantics></math> are related by a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>h</mi></msub><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>f</mi> <mn>1</mn></msub><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>f</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\eta_h \colon \mathbf{B}f_1 \to \mathbf{B}f_2</annotation></semantics></math> with single component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>h</mi></msub><mo lspace="verythinmathspace">:</mo><mo>•</mo><mo>↦</mo><mi>h</mi><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta_h \colon \bullet \mapsto h \in Mor(\mathbf{B} H)</annotation></semantics></math> for each element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∈</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">h \in H</annotation></semantics></math> such that the homomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">f_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f_2</annotation></semantics></math> differ by the <a class="existingWikiWord" href="/nlab/show/inner+automorphism">inner automorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ad</mi> <mi>h</mi></msub><mo lspace="verythinmathspace">:</mo><mi>H</mi><mo>→</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">Ad_h \colon H \to H</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>η</mi> <mi>h</mi></msub><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>f</mi> <mn>1</mn></msub><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⇔</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>Ad</mi> <mi>h</mi></msub><mo>∘</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\eta_h \colon \mathbf{B}f_1 \to \mathbf{B}f_2) \Leftrightarrow (f_2 = Ad_h \circ f_1) \,. </annotation></semantics></math></div> <p>To fix this, look at the category of <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed</a> groupoids with <span class="newWikiWord">pointed functors<a href="/nlab/new/pointed+functor">?</a></span> and pointed natural transformations. Between group homomorphisms as above, only identity transformations are pointed, so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Grp</mi></mrow><annotation encoding="application/x-tex">Grp</annotation></semantics></math> becomes a full sub-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Grpd</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">Grpd_*</annotation></semantics></math> (one that happens to be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/1-category">category</a>). (Details may be found in the appendix to <a class="existingWikiWord" href="/nlab/show/Lectures+on+n-Categories+and+Cohomology">Lectures on n-Categories and Cohomology</a> and should probably be added to <span class="newWikiWord">pointed functor<a href="/nlab/new/pointed+functor">?</a></span> and maybe also <a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a>.)</p> <h2 id="generalizations">Generalizations</h2> <h3 id="Internalization">Internalization</h3> <p>A <strong><a class="existingWikiWord" href="/nlab/show/group+object">group object</a></strong> <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> with finite <a class="existingWikiWord" href="/nlab/show/product">products</a> is an object <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> together with maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>mult</mi><mo>:</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">mult:G\times G\to G</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>id</mi><mo>:</mo><mn>1</mn><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">id:1\to G</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>inv</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">inv:G\to G</annotation></semantics></math> such that various diagrams expressing associativity, unitality, and inverses commute.</p> <p>Equivalently, it is a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Grp</mi></mrow><annotation encoding="application/x-tex">C^{op}\to Grp</annotation></semantics></math> whose underlying functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">C^{op} \to Set</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a>.</p> <p>For example, a group object in <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a> is a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>. A group object in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> is a <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a>. A group object in <a class="existingWikiWord" href="/nlab/show/Sch%2FS">Sch/S</a> (the category or <a class="existingWikiWord" href="/nlab/show/relative+schemes">relative schemes</a>) is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/group+scheme">group scheme</a>. And a group object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>CAlg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CAlg^{op}</annotation></semantics></math>, where <a class="existingWikiWord" href="/nlab/show/CAlg">CAlg</a> is the category of <a class="existingWikiWord" href="/nlab/show/commutative+algebras">commutative algebras</a>, is a (commutative) <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a>.</p> <p>A group object in <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a> is the same thing as an abelian group (see <a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+argument">Eckmann-Hilton argument</a>), and a group object in <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> is the same thing as an <a class="existingWikiWord" href="/nlab/show/internal+category">internal category</a> in <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a>, both being equivalent to the notion of <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a>.</p> <h3 id="in_higher_categorical_and_homotopical_contexts">In higher categorical and homotopical contexts</h3> <p>Internalizing the notion of <em>group</em> in <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher categorical</a> and <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopical</a> contexts yields various generalized notions. For instance</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a> is a group object in <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a></p> </li> <li> <p>an <a class="existingWikiWord" href="/nlab/show/n-group">n-group</a> is a group object internal to <a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a>s</p> </li> <li> <p>an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> is 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>.</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a> is a group object in <a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> </li> <li> <p>generally there is a notion of <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category">group object in an (infinity,1)-category</a>.</p> </li> </ul> <p>And the notion of <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> and <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> makes sense (at least) in any <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a>.</p> <p>Notice that the relation between group objects and deloopable objects becomes more subtle as one generalizes this way. For instance not every <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category">group object in an (infinity,1)-category</a> is <a class="existingWikiWord" href="/nlab/show/delooping">deloopable</a>. But every group object in an <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos">(infinity,1)-topos</a> is.</p> <h3 id="weakened_axioms">Weakened axioms</h3> <p>Following the practice of <a class="existingWikiWord" href="/nlab/show/centipede+mathematics">centipede mathematics</a>, we can remove certain properties from the definition of group and see what we get:</p> <ul> <li>remove inverses to get <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a>, then remove the identity to get <a class="existingWikiWord" href="/nlab/show/semigroups">semigroups</a>;</li> <li>or remove associativity to get <a class="existingWikiWord" href="/nlab/show/loop+%28algebra%29">loops</a>, then remove the identity to get <a class="existingWikiWord" href="/nlab/show/quasigroups">quasigroups</a>;</li> <li>or remove all of the above to get <a class="existingWikiWord" href="/nlab/show/magma">magmas</a>;</li> <li>or instead allow (in a certain way) for the binary operation to be partial to get <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a>, then remove inverses to get <a class="existingWikiWord" href="/nlab/show/categories">categories</a>, and then remove identities to get <a class="existingWikiWord" href="/nlab/show/semicategory">semicategories</a></li> <li>etc.</li> </ul> <h2 id="examples">Examples</h2> <h3 id="special_types_and_classes">Special types and classes</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simple+group">simple group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a>, <a class="existingWikiWord" href="/nlab/show/progroup">progroup</a></p> <ul> <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> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/finite+abelian+group">finite abelian group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/divisible+group">divisible group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/acyclic+group">acyclic group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kac-Moody+group">Kac-Moody group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+of+Lie+type">group of Lie type</a></p> </li> </ul> <h3 id="concrete_examples">Concrete examples</h3> <p>Standard examples of <a class="existingWikiWord" href="/nlab/show/finite+groups">finite groups</a> include the</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group+of+order+2">group of order 2</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\;\mathbb{Z}/2\mathbb{Z}</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><msub><mi>Σ</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\;\Sigma_n</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Br</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">Br_n</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Monster+group">Monster group</a></p> </li> </ul> <p>Standard examples of non-finite groups include thr</p> <ul> <li> <p>group of <a class="existingWikiWord" href="/nlab/show/integers">integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> (under <a class="existingWikiWord" href="/nlab/show/addition">addition</a>);</p> </li> <li> <p>group of <a class="existingWikiWord" href="/nlab/show/real+number">real number</a>s without 0 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}\setminus \{0\}</annotation></semantics></math> under <a class="existingWikiWord" href="/nlab/show/multiplication">multiplication</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pr%C3%BCfer+group">Prüfer group</a></p> </li> </ul> <p>Standard examples of <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a> include the</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spin+group">Spin group</a>, <a class="existingWikiWord" href="/nlab/show/spin%5Ec+group">spin^c group</a></p> </li> </ul> <p>Standard examples of <a class="existingWikiWord" href="/nlab/show/topological+groups">topological groups</a> include</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+group">string group</a></li> </ul> <h3 id="counterexamples">Counterexamples</h3> <p>For more see <em><a class="existingWikiWord" href="/nlab/show/counterexamples+in+algebra">counterexamples in algebra</a></em>.</p> <ol> <li> <p>A non-<a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> <a class="existingWikiWord" href="/nlab/show/group">group</a>, all of whose <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a>s are <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>≔</mo><mo stretchy="false">⟨</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">|</mo><msup><mi>a</mi> <mn>4</mn></msup><mo>=</mo><mn>1</mn><mo>,</mo><msup><mi>a</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>b</mi> <mn>2</mn></msup><mo>,</mo><mi>a</mi><mi>b</mi><mo>=</mo><mi>b</mi><msup><mi>a</mi> <mn>3</mn></msup><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> Q \coloneqq \langle a, b | a^4 = 1, a^2 = b^2, a b = b a^3 \rangle </annotation></semantics></math></div></li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/finitely+presented+group">finitely presented</a>, infinite, <a class="existingWikiWord" href="/nlab/show/simple+group">simple group</a></p> <p><a class="existingWikiWord" href="/nlab/show/Thomson%27s+group">Thomson's group</a> T.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/group">group</a> that is not the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> of any <a class="existingWikiWord" href="/nlab/show/3-manifold">3-manifold</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℤ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex"> \mathbb{Z}^4 </annotation></semantics></math></div></li> <li> <p>Two <a class="existingWikiWord" href="/nlab/show/finite+group">finite</a> non-<a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> groups with the same <a class="existingWikiWord" href="/nlab/show/order+profile">order profile</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>4</mn></msub><mo>×</mo><msub><mi>C</mi> <mn>4</mn></msub><mo>,</mo><mspace width="2em"></mspace><msub><mi>C</mi> <mn>2</mn></msub><mo>×</mo><mo stretchy="false">⟨</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mo stretchy="false">|</mo><msup><mi>a</mi> <mn>4</mn></msup><mo>=</mo><mn>1</mn><mo>,</mo><msup><mi>a</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>b</mi> <mn>2</mn></msup><mo>,</mo><mi>a</mi><mi>b</mi><mo>=</mo><mi>b</mi><msup><mi>a</mi> <mn>3</mn></msup><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> C_4 \times C_4, \qquad C_2 \times \langle a, b, | a^4 = 1, a^2 = b^2, a b = b a^3 \rangle </annotation></semantics></math></div></li> <li> <p>A counterexample to the converse of <a class="existingWikiWord" href="/nlab/show/Lagrange%27s+theorem">Lagrange's theorem</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/alternating+group">alternating group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">A_4</annotation></semantics></math> has order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>12</mn></mrow><annotation encoding="application/x-tex">12</annotation></semantics></math> but no <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>6</mn></mrow><annotation encoding="application/x-tex">6</annotation></semantics></math>.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a> in which the product of two <a class="existingWikiWord" href="/nlab/show/commutator">commutator</a>s is not a commutator.</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>=</mo><mo stretchy="false">⟨</mo><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>e</mi><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mi>h</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>i</mi><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>j</mi><mi>l</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>m</mi><mi>o</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>n</mi><mi>p</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>e</mi><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mi>k</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>m</mi><mi>o</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>e</mi><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>g</mi><mi>h</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>m</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>o</mi><mi>p</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>i</mi><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>k</mi><mi>l</mi><mo stretchy="false">)</mo><mo stretchy="false">⟩</mo><mo>⊆</mo><msub><mi>S</mi> <mn>16</mn></msub></mrow><annotation encoding="application/x-tex"> G = \langle (a c)(b d), (e g)(f h), (i k)(j l), (m o)(n p), (a c)(e g)(i k), (a b)(c d)(m o), (e f)(g h)(m n)(o p), (i j)(k l)\rangle \subseteq S_{16} </annotation></semantics></math></div></li> </ol> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></p> </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> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>, <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a>,</p> </li> <li> <p><strong>group</strong>, <a class="existingWikiWord" href="/nlab/show/group+object">group object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/order+of+a+group">order of a group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/p-primary+group">p-primary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a>, <a class="existingWikiWord" href="/nlab/show/profinite+group">profinite group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finitely+generated+group">finitely generated group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/torsion+subgroup">torsion subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilizer">stabilizer</a>, <a class="existingWikiWord" href="/nlab/show/centralizer">centralizer</a>, <a class="existingWikiWord" href="/nlab/show/normalizer">normalizer</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isogeny">isogeny</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coset">coset</a>, <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/anabelian+group">anabelian group</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+completion">group completion</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+commutator">group commutator</a>, <a class="existingWikiWord" href="/nlab/show/commutator+subgroup">commutator subgroup</a>, <a class="existingWikiWord" href="/nlab/show/abelianization">abelianization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+character">group character</a></p> </li> <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/normed+group">normed group</a>, <a class="existingWikiWord" href="/nlab/show/bornological+group">bornological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+group">loop group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cogroup">cogroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multivalued+group">multivalued group</a></p> </li> <li> <p>is a commutative pregroup as mentioned in <a class="existingWikiWord" href="/nlab/show/pregroup+grammar">pregroup grammar</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/ring+object">ring object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphism+group">automorphism group</a>, <a class="existingWikiWord" href="/nlab/show/automorphism+2-group">automorphism 2-group</a>, <a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-group">automorphism ∞-group</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+bisections">group of bisections</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/center">center</a>, <a class="existingWikiWord" href="/nlab/show/center+of+an+%E2%88%9E-group">center of an ∞-group</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+automorphism+group">inner automorphism group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/outer+automorphism+group">outer automorphism group</a>, <a class="existingWikiWord" href="/nlab/show/outer+automorphism+%E2%88%9E-group">outer automorphism ∞-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+presentation">group presentation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupal+setoid">groupal setoid</a></p> </li> </ul> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/algebra">algebraic</a> <a class="existingWikiWord" href="/nlab/show/mathematical+structure">structure</a></th><th><a class="existingWikiWord" href="/nlab/show/oidification">oidification</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/magma">magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/magmoid">magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/pointed+set">pointed</a> <a class="existingWikiWord" href="/nlab/show/magma">magma</a> with an <a class="existingWikiWord" href="/nlab/show/endofunction">endofunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/setoid">setoid</a>/<a class="existingWikiWord" href="/nlab/show/Bishop+set">Bishop set</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unital+magma">unital magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unital+magmoid">unital magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasigroupoid">quasigroupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/loop+%28algebra%29">loop</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/loopoid">loopoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/semicategory">semicategory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/category">category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/anti-involution">anti-involutive</a> <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dagger+category">dagger category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/associative+quasigroup">associative quasigroup</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/associative+quasigroupoid">associative quasigroupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/group">group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+magma">flexible magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+magmoid">flexible magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+magma">alternative magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+magmoid">alternative magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/absorption+monoid">absorption monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/absorption+category">absorption category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cancellative+monoid">cancellative monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cancellative+category">cancellative category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rig">rig</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/CMon">CMon</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonunital+ring">nonunital ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>-<a class="existingWikiWord" href="/nlab/show/enriched+magmoid">enriched</a> <a class="existingWikiWord" href="/nlab/show/semicategory">semicategory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonassociative+ring">nonassociative ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>-<a class="existingWikiWord" href="/nlab/show/enriched+magmoid">enriched</a> <a class="existingWikiWord" href="/nlab/show/unital+magmoid">unital magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ring">ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ringoid">ringoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative unital algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unital+magmoid">unital</a> <a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonunital+algebra">nonunital algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear</a> <a class="existingWikiWord" href="/nlab/show/semicategory">semicategory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+category">linear category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C-star+category">C-star category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/differential+algebra">differential algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/differential+algebroid">differential algebroid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+algebra">flexible algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+magmoid">flexible</a> <a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+algebra">alternative algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+magmoid">alternative</a> <a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+poset">monoidal poset</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-poset">2-poset</a></td></tr> <tr><td style="text-align: left;"><span class="newWikiWord">strict monoidal groupoid<a href="/nlab/new/strict+monoidal+groupoid">?</a></span></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+%282%2C1%29-category">strict (2,1)-category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+2-groupoid">strict 2-groupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+monoidal+category">strict monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+groupoid">monoidal groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/bigroupoid">bigroupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>/<a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></td></tr> </tbody></table> </div> <h2 id="literature">Literature</h2> <p>For more see also the references at <em><a class="existingWikiWord" href="/nlab/show/group+theory">group theory</a></em>.</p> <p>The terminology “group” was introduced (for what today would more specifically be called <em><a class="existingWikiWord" href="/nlab/show/permutation+groups">permutation groups</a></em>) in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%C3%89variste+Galois">Évariste Galois</a>, <em><a class="existingWikiWord" href="/nlab/show/Galois%27+last+letter">letter to Auguste Chevallier</a></em>, (May 1832)</li> </ul> <p>The original article that gives a definition equivalent to the modern definition of a group:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Heinrich+Weber">Heinrich Weber</a>, <em>Beweis des Satzes, dass jede eigentlich primitive quadratische Form unendlich viele Primzahlen darzustellen fähig ist</em>, Mathematische Annalen 20:3 (1882), 301–329 (<a href="http://dx.doi.org/10.1007/bf01443599">doi:10.1007/bf01443599</a>)</li> </ul> <p>Introduction of group theory into (<a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum</a>) <a class="existingWikiWord" href="/nlab/show/physics">physics</a> (cf. <em><a class="existingWikiWord" href="/nlab/show/Gruppenpest">Gruppenpest</a></em>):</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hermann+Weyl">Hermann Weyl</a>, §III in: <em>Gruppentheorie und Quantenmechanik</em>, S. Hirzel, Leipzig (1931), translated by H. P. Robertson: <em>The Theory of Groups and Quantum Mechanics</em>, Dover (1950) [<a href="https://store.doverpublications.com/0486602699.html">ISBN:0486602699</a>, <a href="https://archive.org/details/ost-chemistry-quantumtheoryofa029235mbp/page/n15/mode/2up">ark:/13960/t1kh1w36w</a>]</li> </ul> <p>Textbook account in relation to applications in <a class="existingWikiWord" href="/nlab/show/physics">physics</a>:</p> <ul> <li id="Sternberg94"><a class="existingWikiWord" href="/nlab/show/Shlomo+Sternberg">Shlomo Sternberg</a>, <em>Group Theory and Physics</em>, Cambridge University Press 1994 (<a href="https://www.cambridge.org/gb/academic/subjects/mathematics/algebra/group-theory-and-physics?format=PB&isbn=9780521558853">ISBN:9780521558853</a>)</li> </ul> <p>See also:</p> <ul> <li> <p>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Group_(mathematics)">Group_(mathematics)</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bananaspace">bananaspace</a>, <em><a href="https://www.bananaspace.org/wiki/%E7%BE%A4">群</a></em> (Chinese)</p> </li> </ul> <p id="TTFormalizations"> Formalization of group structure in <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>:</p> <p>in <a class="existingWikiWord" href="/nlab/show/Coq">Coq</a>:</p> <ul> <li>Farida Kachapova, <em>Formalizing groups in type theory</em> [<a href="https://arxiv.org/abs/2102.09125">arXiv:2102.09125</a>]</li> </ul> <p>and with the <a class="existingWikiWord" href="/nlab/show/univalence+axiom">univalence axiom</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/unimath">unimath</a> -> <a href="https://unimath.github.io/doc/UniMath/d4de26f//UniMath.Algebra.Groups.html">UniMath.Algebra.Groups</a></li> </ul> <p>in <a class="existingWikiWord" href="/nlab/show/Agda">Agda</a>:</p> <ul> <li> <p><a href="https://unimath.github.io/agda-unimath/">agda-unimath</a> -> <a href="https://unimath.github.io/agda-unimath/group-theory.groups.html">group-theory.groups</a></p> </li> <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#groups-sip">Groups</a></em>, §3.33.10 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> </li> </ul> <p>in <a class="existingWikiWord" href="/nlab/show/cubical+Agda">cubical Agda</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/1lab">1lab</a>: <em><a href="https://1lab.dev/Algebra.Group.html">Algebra.Group</a></em></li> </ul> <p>in <a class="existingWikiWord" href="/nlab/show/Lean">Lean</a>:</p> <ul> <li><a href="https://leanprover-community.github.io/">Lean Community</a> –> <a href="https://leanprover-community.github.io/mathlib-overview.html">mathlib</a> –> <a href="https://leanprover-community.github.io/mathlib_docs/algebra/group/defs.html#top">algebra.group.defs</a> –> <a href="https://leanprover-community.github.io/mathlib_docs/algebra/group/defs.html#group">group</a></li> </ul> <p>Exposition in a context of <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Egbert+Rijke">Egbert Rijke</a>, Section 19 in: <em>Introduction to Homotopy Type Theory</em>, Cambridge Studies in Advanced Mathematics, Cambridge University Press [<a href="https://arxiv.org/abs/2212.11082">arXiv:2212.11082</a>]</li> </ul> <p>Alternative discussion (under <a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a>) of groups in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> as pointed connected <a class="existingWikiWord" href="/nlab/show/homotopy+1-types">homotopy 1-types</a>:</p> <ul> <li><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 4 of: <em><a class="existingWikiWord" href="/nlab/show/Symmetry">Symmetry</a></em> (2021) [<a href="https://unimath.github.io/SymmetryBook/book.pdf">pdf</a>]</li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/group+theory">group theory</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on July 10, 2024 at 18:54:21. See the <a href="/nlab/history/group" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/group" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/1488/#Item_18">Discuss</a><span class="backintime"><a href="/nlab/revision/group/83" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/group" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/group" accesskey="S" class="navlink" id="history" rel="nofollow">History (83 revisions)</a> <a href="/nlab/show/group/cite" style="color: black">Cite</a> <a href="/nlab/print/group" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/group" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>