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<![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> group </h1> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='category_theory'>Category theory</h4> <div class='hide'> <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/end'>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%E2%80%93Ulmer+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> <h4 id='group_theory'>Group Theory</h4> <div class='hide'> <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/infinity-group'>∞-group</a></li> <li><a class='existingWikiWord' href='/nlab/show/group+object'>group object</a>, <a class='existingWikiWord' href='/nlab/show/groupoid+object+in+an+%28infinity%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/action'>group action</a>, <a class='existingWikiWord' href='/nlab/show/infinity-action'>∞-action</a></li> <li><a class='existingWikiWord' href='/nlab/show/representation'>representation</a>, <a class='existingWikiWord' href='/nlab/show/infinity-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+group'>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/supergroup'>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/infinity-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+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/infinity-group+extension'>∞-group extension</a>, <a class='existingWikiWord' href='/nlab/show/Ext'>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> <h4 id='monoid_theory'>Monoid theory</h4> <div class='hide'> <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/monoid+in+a+monoidal+%28infinity%2C1%29-category'>infinity-monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/monoid+in+a+monoidal+category'>monoid object</a>, <a class='existingWikiWord' href='/nlab/show/monoid+in+a+monoidal+%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/category+of+monoids'>Mon</a>, <a class='existingWikiWord' href='/nlab/show/category+of+monoids'>CMon</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/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'>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+of+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+category'>graphic monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/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'>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> <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><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><li><a href='#examples'>Examples</a><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><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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> and a <a class='existingWikiWord' href='/nlab/show/magma'>binary operation</a> <math class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_2' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_3' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_4' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_5' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_6' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_8' xmlns='http://www.w3.org/1998/Math/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'>inverse</a>. When written with a view toward <a class='existingWikiWord' href='/nlab/show/group+object'>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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_9' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_10' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_11' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_12' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_13' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_14' xmlns='http://www.w3.org/1998/Math/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/functor'>functors</a> <math class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_15' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_16' xmlns='http://www.w3.org/1998/Math/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/groupoid'>groupoids</a> and whose <a class='existingWikiWord' href='/nlab/show/morphism'>morphisms</a> are <a class='existingWikiWord' href='/nlab/show/functor'>functors</a> (discarding the <a class='existingWikiWord' href='/nlab/show/natural+transformation'>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 class='maruku-mathml' display='block' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_17' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_18' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_19' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_20' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_21' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_22' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>f_1</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_24' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_25' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_26' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Grp</mi></mrow><annotation encoding='application/x-tex'>Grp</annotation></semantics></math> becomes a full sub-<math class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>-category of <math class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_29' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_30' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> with finite <a class='existingWikiWord' href='/nlab/show/cartesian+product'>products</a> is an object <math class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> together with maps <math class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_33' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_34' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_35' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_36' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_37' xmlns='http://www.w3.org/1998/Math/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/relative+scheme'>Sch/S</a> (the category or <a class='existingWikiWord' href='/nlab/show/relative+scheme'>relative schemes</a>) is an <math class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_38' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_39' xmlns='http://www.w3.org/1998/Math/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/CommAlg'>CAlg</a> is the category of <a class='existingWikiWord' href='/nlab/show/commutative+algebra'>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/infinity-group'>∞-group</a> is a <a class='existingWikiWord' href='/nlab/show/groupoid+object+in+an+%28infinity%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/abstraction'>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/monoid'>monoids</a>, then remove the identity to get <a class='existingWikiWord' href='/nlab/show/semigroup'>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/quasigroup'>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/groupoid'>groupoids</a>, then remove inverses to get <a class='existingWikiWord' href='/nlab/show/category'>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+group'>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+group'>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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_40' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_41' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_42' xmlns='http://www.w3.org/1998/Math/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/integer'>integers</a> <math class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}</annotation></semantics></math> (under <a class='existingWikiWord' href='/nlab/show/sum'>addition</a>);</p> </li> <li> <p>group of <a class='existingWikiWord' href='/nlab/show/real+number'>real number</a>s without 0 <math class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_44' xmlns='http://www.w3.org/1998/Math/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+group'>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%E1%B6%9C'>spin^c group</a></p> </li> </ul> <p>Standard examples of <a class='existingWikiWord' href='/nlab/show/topological+group'>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 class='maruku-mathml' display='block' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_45' xmlns='http://www.w3.org/1998/Math/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+presentable+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 class='maruku-mathml' display='block' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_46' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_47' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_48' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_49' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_50' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_12b24c8359d45d972ca99bf8c601dafb6c92c181_51' xmlns='http://www.w3.org/1998/Math/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/infinity-group'>∞-group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/monoid'>monoid</a>, <a class='existingWikiWord' href='/nlab/show/monoid+in+a+monoidal+category'>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+group'>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'>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'>automorphism group</a>, <a class='existingWikiWord' href='/nlab/show/automorphism+2-group'>automorphism 2-group</a>, <a class='existingWikiWord' href='/nlab/show/automorphism+infinity-group'>automorphism ∞-group</a>,</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/bisection+of+a+Lie+groupoid'>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+infinity-group'>center of an ∞-group</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/inner+automorphism'>inner automorphism group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/outer+automorphism'>outer automorphism group</a>, <a class='existingWikiWord' href='/nlab/show/outer+automorphism+infinity-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> <table><thead><tr><th><a class='existingWikiWord' href='/nlab/show/algebra'>algebraic</a> <a class='existingWikiWord' href='/nlab/show/structure'>structure</a></th><th><a class='existingWikiWord' href='/nlab/show/horizontal+categorification'>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/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/alternative+algebra'>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+algebra'>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/category+of+monoids'>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/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/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/Ab-enriched+category'>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/algebroid'>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/alternative+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+preorder'>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> <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+group'>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+mechanics'>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) [[ISBN:0486602699](https://store.doverpublications.com/0486602699.html), <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> [[arXiv:2102.09125](https://arxiv.org/abs/2102.09125)]</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+project'>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> [[arXiv:1911.00580](https://arxiv.org/abs/1911.00580), <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/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 [[arXiv:2212.11082](https://arxiv.org/abs/2212.11082)]</li> </ul> <p>Alternative discussion (under <a class='existingWikiWord' href='/nlab/show/looping'>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-type'>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+Grayson'>Daniel R. Grayson</a>: Chapter 4 of: <em><a class='existingWikiWord' href='/nlab/show/Symmetry'>Symmetry</a></em> (2021) [[pdf](https://unimath.github.io/SymmetryBook/book.pdf)]</li> </ul> <p> </p> <p><div class='property'> category: <a class='category_link' href='/nlab/list/group+theory'>group theory</a></div></p> </div>  </div>  </body> </html>