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group extension in nLab
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<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 extension </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/1833/#Item_13" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="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="cohomology">Cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>, <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a>, <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%3Fech+cohomology">?ech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#Definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#PropertiesGeneral'>General</a></li> <ul> <li><a href='#fibers_of_extensions_are_normal_subgroups'>Fibers of extensions are normal subgroups</a></li> <li><a href='#Torsors'>Extensions as torsors / principal bundles</a></li> <li><a href='#SplitExtensionsAndSemidirectProductGroups'>Split extensions and semidirect product groups</a></li> </ul> <li><a href='#PropertiesCentralGroupExtensions'>Central group extensions</a></li> <ul> <li><a href='#CentralExtensionClassificationByGroupCohomology'>Classification by group cohomology</a></li> <li><a href='#FormulationInHomotopyTheory'>Formulation in homotopy theory</a></li> </ul> <li><a href='#PropertiesAbelianGroupExtensions'>Abelian group extensions</a></li> <li><a href='#SchreierTheory'>Nonabelian group extensions (Schreier theory)</a></li> <ul> <li><a href='#SchreierTheoryTraditional'>Traditional description</a></li> <li><a href='#2Coboundaries'>Comparing different extensions; 2-coboundaries</a></li> <li><a href='#SchreierTheorynPOV'>Formulation in homotopy theory</a></li> </ul> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general_2'>General</a></li> <li><a href='#applications'>Applications</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>group extension</em> of a <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> by a group <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> is third group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat G</annotation></semantics></math> that sits in a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, that can usefully be thought of as a <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">A \to \hat G \to G</annotation></semantics></math>.</p> <h2 id="Definition">Definition</h2> <div class="num_defn" id="GroupExtension"> <h6 id="definition_2">Definition</h6> <p>Two consecutive <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> of <a class="existingWikiWord" href="/nlab/show/groups">groups</a></p> <div class="maruku-equation" id="eq:shortExtension"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>↪</mo><mi>i</mi></mover><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mi>p</mi></mover><mi>G</mi></mrow><annotation encoding="application/x-tex"> A \overset{i}\hookrightarrow \hat G\overset{p}\to G </annotation></semantics></math></div> <p>are a <strong><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a></strong> if</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a></p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/image">image</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is all of the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>im</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ker(p)\simeq im(i)</annotation></semantics></math>.</p> </li> </ol> <p>We say that such a short exact sequence exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat G</annotation></semantics></math> as an <strong>extension</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> by <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>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">A \hookrightarrow \hat G</annotation></semantics></math> factors through the <a class="existingWikiWord" href="/nlab/show/center">center</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat G</annotation></semantics></math> we say that this is a <strong><a class="existingWikiWord" href="/nlab/show/central+extension">central extension</a></strong>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Sometimes in the literature one sees <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat G</annotation></semantics></math> called an extension “of <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> by <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>”. This is however in conflict with terms such as <em><a class="existingWikiWord" href="/nlab/show/central+extension">central extension</a></em>, <em><a class="existingWikiWord" href="/nlab/show/extension+of+principal+bundles">extension of principal bundles</a></em>, etc, where the extension is always regarded <em>of the base, by the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a></em>. (On the other hand, our terminology conflicts with the usual meaning of “extension” in algebra. For example, in Galois theory if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is a field, then an extension of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> contains <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> as a subfield.)</p> </div> <p>Under the <a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a>-equivalence, this is equivalently reformulated as follows. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">G \in</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a> a <a class="existingWikiWord" href="/nlab/show/group">group</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a> for its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> groupoid.</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>A sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">A \to \hat G \to G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> of groups precisely if the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a></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><mi>A</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex"> \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a> in the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a>.</p> </div> <p>This says that group extensions are special cases of the general notion discussed at <em><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></em>. See there for more details.</p> <div class="num_defn" id="MorphismOfGroupExtensions"> <h6 id="definition_4">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of extensions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mover><mi>G</mi><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo>→</mo><msub><mover><mi>G</mi><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f : \hat G_1 \to \hat G_2</annotation></semantics></math> of a given <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> by a given <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> is a <a class="existingWikiWord" href="/nlab/show/group+homomorphism">group homomorphism</a> of this form which fits into a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mover><mi>G</mi><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mover><mi>G</mi><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \hat G_1 \\ & \nearrow && \searrow \\ A &&\downarrow^{\mathrlap{f}}&& G \\ & \searrow && \nearrow \\ && \hat G_2 } \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>A morphism of extensions as in def. <a class="maruku-ref" href="#MorphismOfGroupExtensions"></a> is necessarily an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> <div class="maruku-equation" id="eq:equivExt"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mn>1</mn><mo>→</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>i</mi></mover></mtd> <mtd><msub><mover><mi>G</mi><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mi>p</mi></mover></mtd> <mtd><mi>G</mi></mtd> <mtd><mo>→</mo><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo><mpadded width="0"><mo>=</mo></mpadded></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mpadded width="0"><mi>ϵ</mi></mpadded></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mpadded width="0"><mo>=</mo></mpadded></mtd> <mtd></mtd></mtr> <mtr><mtd><mn>1</mn><mo>→</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>i</mi><mo>′</mo></mrow></mover></mtd> <mtd><msub><mover><mi>G</mi><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><mi>p</mi><mo>′</mo></mrow></mover></mtd> <mtd><mi>G</mi></mtd> <mtd><mo>→</mo><mn>1</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ 1\to &A&\stackrel{i}\to &\hat G_1&\stackrel{p}\to &G&\to 1 \\ &\downarrow\mathrlap{=}&&\downarrow\mathrlap\epsilon&&\downarrow\mathrlap=& \\ 1\to &A&\stackrel{i'}\to &\hat G_2&\stackrel{p'}\to& G&\to 1 } \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>By the <a class="existingWikiWord" href="/nlab/show/short+five+lemma">short five lemma</a>.</p> </div> <h2 id="properties">Properties</h2> <p>We discuss properties of group extensions in stages,</p> <ul> <li> <p><a href="#PropertiesGeneral">General properties</a></p> </li> <li> <p><a href="#PropertiesCentralGroupExtensions">Central group extensions</a></p> </li> <li> <p><a href="#PropertiesAbelianGroupExtensions">Abelian group extensions</a></p> </li> <li> <p><a href="#SchreierTheory">Schreier theory for nonabelian extensions</a></p> </li> </ul> <h3 id="PropertiesGeneral">General</h3> <h4 id="fibers_of_extensions_are_normal_subgroups">Fibers of extensions are normal subgroups</h4> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">A \hookrightarrow \hat G \to G</annotation></semantics></math> a group extension, the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">A \hookrightarrow \hat G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a> inclusion.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>We need to check that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">a \in A \hookrightarrow G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g \in G</annotation></semantics></math> the result of the <a class="existingWikiWord" href="/nlab/show/adjoint+action">adjoint action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mi>a</mi><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">g a g^{-1}</annotation></semantics></math> formed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat G</annotation></semantics></math> is again in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>↪</mo><mi>i</mi></mover><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">A \stackrel{i}{\hookrightarrow} \hat G</annotation></semantics></math>.</p> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">p : \hat G \to G</annotation></semantics></math> is a group homomorphism we have that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mi>a</mi><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} p(g a g^{-1}) & = p(q) p(a) p(g^{-1}) \\ & = p(g) p(a) p(g)^{-1} \\ & = p(g) p(g)^{-1} \\ & = 1 \end{aligned} </annotation></semantics></math></div> <p>and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mi>a</mi><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">g a g^{-1}</annotation></semantics></math> is in the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>. By the defining exactness property therefore it is in the <a class="existingWikiWord" href="/nlab/show/image">image</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>.</p> </div> <h4 id="Torsors">Extensions as torsors / principal bundles</h4> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>i</mi></mover><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mi>p</mi></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">A \stackrel{i}{\to} \hat G \stackrel{p}{\to} G</annotation></semantics></math> a group extension, we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">p : \hat G \to G</annotation></semantics></math> is an <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>-<a class="existingWikiWord" href="/nlab/show/torsor">torsor</a> over <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> where the <a class="existingWikiWord" href="/nlab/show/action">action</a> of <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> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat G</annotation></semantics></math> is defined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>A</mi><mo>×</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>Id</mi><mo stretchy="false">)</mo></mrow></mover><mover><mi>G</mi><mo stretchy="false">^</mo></mover><msub><mo>×</mo> <mi>G</mi></msub><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mo>⋅</mo></mover><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \rho : A \times \hat G \stackrel{(i,Id)}{\to} \hat G \times_G \hat G \stackrel{\cdot}{\to} \hat G \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>That <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> is indeed an action <em>over</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> in that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><mo>×</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>ρ</mi></mover></mtd> <mtd></mtd> <mtd><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>p</mi><mo>∘</mo><msub><mi>p</mi> <mn>2</mn></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>p</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A \times \hat G &&\stackrel{\rho}{\to}&& \hat G \\ & {}_{\mathllap{ p \circ p_2}}\searrow && \swarrow_{\mathrlap{p}} \\ && G } </annotation></semantics></math></div> <p>follows from the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a group homomorphism and that <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> is in its <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>.</p> <p>That <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> is actually <em>equal</em> to the kernel gives the principality condition</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><mi>ρ</mi><mo>,</mo><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>:</mo><mi>A</mi><mo>×</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mo>≃</mo></mover><mover><mi>G</mi><mo stretchy="false">^</mo></mover><msub><mo>×</mo> <mi>G</mi></msub><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\rho, p_2) : A\times \hat G \stackrel{\simeq}{\to} \hat G \times_G \hat G \,. </annotation></semantics></math></div></div> <p>For <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> an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> we may understand the <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>-torsor/<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>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat G</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mstyle mathvariant="bold"><mi>BG</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{B} \hat G \to \mathbf{BG}</annotation></semantics></math> that is classified by (is the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of) the 2-<a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">c : \mathbf{B}G \to \mathbf{B}^2 A</annotation></semantics></math> that classifies the extension.</p> <p>All this is then summarized by the statement that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mover><mo>→</mo><mi>c</mi></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>A</mi></mrow><annotation encoding="application/x-tex"> A \to \hat G \to G \to \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G \stackrel{c}{\to} \mathbf{B}^2 A </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a> in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> (or in <span class="newWikiWord">?LieGrpd<a href="/nlab/new/%3FLieGrpd">?</a></span> if we have <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> extensions, etc).</p> <p>Here we may think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\mathbf{B}\hat G</annotation></semantics></math> as being the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a> over <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> classified by <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>. See the examples discussed at <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a>.</p> <h4 id="SplitExtensionsAndSemidirectProductGroups">Split extensions and semidirect product groups</h4> <div class="num_defn" id="SplitExtension"> <h6 id="definition_5">Definition</h6> <p>A group extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mi>p</mi></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">A \to \hat G \stackrel{p}{\to} G</annotation></semantics></math> is called <strong>split</strong> if there is a <a class="existingWikiWord" href="/nlab/show/section">section</a> <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\sigma \colon G \to \hat G</annotation></semantics></math>, hence a group homomorphism such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mi>σ</mi><mo>=</mo><msub><mi>id</mi> <mi>G</mi></msub></mrow><annotation encoding="application/x-tex">p \circ \sigma = id_G</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>It is important here that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> is itself required to be a group homomorphism, not just a <a class="existingWikiWord" href="/nlab/show/function">function</a> on the underlying sets. The latter always exists as soon as the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> holds, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> by definition.</p> </div> <div class="num_prop" id="SplitExtensionsAreSemidirectProducts"> <h6 id="proposition_4">Proposition</h6> <p>Split extensions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat G</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> by <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>, def. <a class="maruku-ref" href="#SplitExtension"></a>, are equivalently <a class="existingWikiWord" href="/nlab/show/semidirect+product+groups">semidirect product groups</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>≃</mo><mi>A</mi><msub><mo>⋊</mo> <mi>ρ</mi></msub><mi>G</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> A \hookrightarrow \hat G \simeq A \rtimes_\rho G \to G </annotation></semantics></math></div> <p>for some <a class="existingWikiWord" href="/nlab/show/action">action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>×</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\rho \colon G \times A \to A</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on <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>.</p> </div> <p>This means that the underlying set is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>A</mi><msub><mo>⋊</mo> <mi>ρ</mi></msub><mi>G</mi><mo stretchy="false">)</mo><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(A \rtimes_\rho G) = U(A) \times U(G)</annotation></semantics></math> and the group operation in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>⋊</mo> <mi>ρ</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">A \rtimes_\rho G</annotation></semantics></math> is</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>a</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>⋅</mo><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (a_1, g_1) \cdot (a_2, g_2) = (a_1 \cdot \rho(g_1)(a_2) , g_1 \cdot g_2) \,. </annotation></semantics></math></div> <p>The inclusion of <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> is by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> a \mapsto (a,e) </annotation></semantics></math></div> <p>and the projection to <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 by</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><mi>a</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>g</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (a,g) \mapsto g \,. </annotation></semantics></math></div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>Given a split extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>i</mi></mover><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mi>p</mi></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">A \stackrel{i}{\to} \hat G \stackrel{p}{\to} G</annotation></semantics></math> with splitting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\sigma \colon G \to \hat G</annotation></semantics></math>, define an <a class="existingWikiWord" href="/nlab/show/action">action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on <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> by the restriction of the <a class="existingWikiWord" href="/nlab/show/adjoint+action">adjoint action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>ad</mi></msub></mrow><annotation encoding="application/x-tex">\rho_{ad}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat G</annotation></semantics></math> on itself to <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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>×</mo><mi>G</mi><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow></mover><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>×</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mrow><msub><mi>ρ</mi> <mi>ad</mi></msub></mrow></mover><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex"> \rho \colon A \times G \stackrel{(i,\sigma)}{\to} \hat G \times \hat G \stackrel{\rho_{ad}}{\to} \hat G </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>g</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><mi>a</mi><mo>⋅</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (a,g) \mapsto \sigma(g)^{-1} \cdot a \cdot \sigma(g) \,. </annotation></semantics></math></div> <p>Then (…)</p> </div> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>A split extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">A \to \hat G \to G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/central+extension">central extension</a> precisely if the <a class="existingWikiWord" href="/nlab/show/action">action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> induced from it as in prop. <a class="maruku-ref" href="#SplitExtensionsAreSemidirectProducts"></a> is trivial.</p> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>For it to be a central extension the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>A</mi><msub><mo>⋊</mo> <mi>ρ</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">A \to A \rtimes_\rho G</annotation></semantics></math> has to land in the <a class="existingWikiWord" href="/nlab/show/center">center</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>⋊</mo> <mi>ρ</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">A \rtimes_\rho G</annotation></semantics></math>, hence all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \in A</annotation></semantics></math> have to commute as elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>A</mi><msub><mo>⋊</mo> <mi>ρ</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">(a,e) \in A \rtimes_\rho G</annotation></semantics></math> with all elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>⋊</mo> <mi>ρ</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">A \rtimes_\rho G</annotation></semantics></math>. But consider elements of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>A</mi><msub><mo>⋊</mo> <mi>ρ</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">(e,g) \in A \rtimes_\rho G</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g \in G</annotation></semantics></math>. Then</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><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (a,e) \cdot (e,g) = (a, g) </annotation></semantics></math></div> <p>but</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><mi>e</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (e,g) \cdot (a,e) = (\rho(g)(a), g) \,. </annotation></semantics></math></div> <p>For these to be equal for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \in A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho(g)</annotation></semantics></math> has to be the identity. Since this is to be true for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g \in G</annotation></semantics></math>, the action has to be trivial.</p> </div> <div class="num_prop"> <h6 id="remark_3">Remark</h6> <p>This means in particular that split central extensions are product groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">A \to G</annotation></semantics></math>. If all groups involved are <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>, then these are equivalently the <a class="existingWikiWord" href="/nlab/show/direct+sums">direct sums</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊕</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">A \oplus G</annotation></semantics></math> of abelian groups. In this way the notion of split group extension reduces to that of <a class="existingWikiWord" href="/nlab/show/split+short+exact+sequences">split short exact sequences</a> of abelian groups.</p> </div> <div class="num_prop"> <h6 id="remark_4">Remark</h6> <p>If we have a split extension the different splittings are given by <a class="existingWikiWord" href="/nlab/show/derivation+on+a+group">derivation</a>s, but with possibly non-abelian values. In fact if we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>A</mi><mo>⋊</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">s: G\to A\rtimes G</annotation></semantics></math> is a section then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s(g) = (a(g),g)</annotation></semantics></math>, and the multiplication in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⋊</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">A\rtimes G</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a: G\to A</annotation></semantics></math> is a derivation. These are considered as the (possibly non-abelian) 1-cocycles of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> with (twisted) coefficients in <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>, as considered in, for instance, Serre’s notes on <a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a>.</p> </div> <h3 id="PropertiesCentralGroupExtensions">Central group extensions</h3> <p>We discuss properties of <em>central</em> group extensions, those where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">A \hookrightarrow \hat G</annotation></semantics></math> factors through the <a class="existingWikiWord" href="/nlab/show/center">center</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat G</annotation></semantics></math>. This is a special case of the general discussion below in <em><a href="#SchreierTheory">Nonabelian group extensions (Schreier theory)</a></em> but is considerably less complex to write out in components.</p> <p>We first discuss the</p> <ul> <li><a href="#CentralExtensionClassificationByGroupCohomology">Classification by group cohomology</a></li> </ul> <p>of central extensions in components, and then show in</p> <ul> <li><a href="#FormulationInHomotopyTheory">Formulation in homotopy theory</a></li> </ul> <p>how this follows from a more systematic abstract theory.</p> <h4 id="CentralExtensionClassificationByGroupCohomology">Classification by group cohomology</h4> <p>We discuss the classification of <em>central</em> extensions by <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>. This is a special case of the more general (and more complicated) discussion below in <em><a href="#SchreierTheory">Nonabelian group extensions (Schreier theory)</a></em>.</p> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/group">group</a> and <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> an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>grp</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>Ab</mi></mrow><annotation encoding="application/x-tex"> H^2_{grp}(G,A) \;\; \in Ab </annotation></semantics></math></div> <p>for the degree-2 <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <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 write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ext</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>Ab</mi></mrow><annotation encoding="application/x-tex"> Ext(G,A) \;\;\in Ab </annotation></semantics></math></div> <p>for the group of central extensions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> by <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>.</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/natural+equivalence">natural equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ext</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>H</mi> <mi>Grp</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ext(G,A) \simeq H^2_{Grp}(G,A) \,. </annotation></semantics></math></div></div> <p>We prove this below as prop. <a class="maruku-ref" href="#ExtractionAndReconstructionConsituteEquivalence"></a>. Here we first introduce stepwise the ingredients that go into the proof.</p> <div class="num_defn" id="CentralExtensionAssociatedTo2Cocycle"> <h6 id="definition_6">Definition</h6> <p><strong>(central extension associated to group 2-cocycle)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>c</mi><mo stretchy="false">]</mo><mo>∈</mo><msubsup><mi>H</mi> <mi>Grp</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[c] \in H^2_{Grp}(G,A)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a> class represented by a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">c \colon G \times G \to A</annotation></semantics></math>, define a group</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><msub><mo>×</mo> <mi>c</mi></msub><mi>A</mi><mo>∈</mo><mi>Grp</mi></mrow><annotation encoding="application/x-tex"> G \times_c A \in Grp </annotation></semantics></math></div> <p>as follows. The underlying set is the <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(G) \times U(A)</annotation></semantics></math> of the underlying sets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and <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>. The group operation on this is given by</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>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>a</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>a</mi> <mn>2</mn></msub><mo>+</mo><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (g_1, a_1) \cdot (g_2, a_2) \coloneqq (g_1 \cdot g_2 ,\; a_1 + a_2 + c(g_1, g_2)) \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition_6">Proposition</h6> <p>This defines indeed a group: the <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> condition on <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> gives precisely the <a class="existingWikiWord" href="/nlab/show/associativity">associativity</a> of the product on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><msub><mo>×</mo> <mi>c</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">G \times_c A</annotation></semantics></math>. Moreover, the construction extends to a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of groups</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Rec</mi><mo>:</mo><msubsup><mi>H</mi> <mi>Grp</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Ext</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Rec : H^2_{Grp}(G,A) \to Ext(G,A) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>Forming the product of three elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><msub><mo>×</mo> <mi>c</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">G \times_c A</annotation></semantics></math> bracketed to the left is, according to def. <a class="maruku-ref" href="#CentralExtensionAssociatedTo2Cocycle"></a>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mo>(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>)</mo></mrow><mo>⋅</mo><mrow><mo>(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>2</mn></msub><mo>)</mo></mrow><mo>)</mo></mrow><mo>⋅</mo><mrow><mo>(</mo><msub><mi>g</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>3</mn></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><msub><mi>g</mi> <mn>3</mn></msub><mspace width="thickmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>a</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>a</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>a</mi> <mn>3</mn></msub><mo>+</mo><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>c</mi><mrow><mo>(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>3</mn></msub><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \left(g_1, a_1\right) \cdot \left(g_2, a_2\right) \right) \cdot \left( g_3, a_3 \right) = \left( g_1 g_2 g_3 \;,\; a_1 + a_2 + a_3 + c(g_1, g_2) + c\left( g_1 g_2, g_3 \right) \right) \,. </annotation></semantics></math></div> <p>Bracketing the same three elements to the right yields</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>)</mo></mrow><mo>⋅</mo><mrow><mo>(</mo><mrow><mo>(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>2</mn></msub><mo>)</mo></mrow><mo>⋅</mo><mrow><mo>(</mo><msub><mi>g</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>3</mn></msub><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><msub><mi>g</mi> <mn>3</mn></msub><mspace width="thickmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>a</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>a</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>a</mi> <mn>3</mn></msub><mo>+</mo><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>c</mi><mrow><mo>(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><msub><mi>g</mi> <mn>3</mn></msub><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left(g_1, a_1\right) \cdot \left( \left(g_2, a_2\right) \cdot \left( g_3, a_3 \right) \right) = \left( g_1 g_2 g_3 \;,\; a_1 + a_2 + a_3 + c(g_2, g_3) + c\left( g_1 , g_2 g_3 \right) \right) \,. </annotation></semantics></math></div> <p>The difference between the two expressions is read off to be precisely</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><mn>1</mn><mo>,</mo><mo stretchy="false">(</mo><mi>d</mi><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (1, (d c) (g_1, g_2, g_3)) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">d c</annotation></semantics></math> denotes the group cohomology differential of <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>. Hence this vanishes precisely if <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> is a group 2-cocycle, hence we have an associative product.</p> <p>To see that it has inverses, notice that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(g,a)</annotation></semantics></math> we have</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><mi>g</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>a</mi><mo>−</mo><mi>c</mi><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>a</mi><mo>−</mo><mi>a</mi><mo>−</mo><mi>c</mi><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi>c</mi><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (g,a) \cdot (g^{-1}, - a - c(g,g^{-1})) = (e, a - a - c(g,g^{-1})+ c(g,g^{-1}) ) </annotation></semantics></math></div> <p>and hence inverses are given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>a</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><mo stretchy="false">(</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>a</mi><mo>−</mo><mi>c</mi><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(g,a)^{-1} = (g^{-1}, -a - c(g,g^{-1}))</annotation></semantics></math>. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><msub><mo>×</mo> <mi>c</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">G \times_c A</annotation></semantics></math> is indeed a group.</p> <p>By the discussion at <a href="group+cohomology#Degree2">group cohomology – degree-2</a> we may assume without restriction that <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> is a normalized cocycle, hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c(e,-) = c(-,e) = 0</annotation></semantics></math>. Using this we find that the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>G</mi><msub><mo>×</mo> <mi>c</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex"> i \colon A \to G \times_c A </annotation></semantics></math></div> <p>given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a \mapsto (e,a)</annotation></semantics></math> is a group homomorphism. Moreover, the projection on the underlying sets evidently yields a group homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><msub><mo>×</mo> <mi>c</mi></msub><mi>A</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">p \colon G \times_c A \to G</annotation></semantics></math> given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">(g,a) \mapsto g</annotation></semantics></math>. The kernel of this is <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 hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>↪</mo><mi>i</mi></mover><mi>G</mi><msub><mo>×</mo> <mi>c</mi></msub><mi>A</mi><mover><mo>→</mo><mi>p</mi></mover><mi>G</mi></mrow><annotation encoding="application/x-tex"> A \stackrel{i}{\hookrightarrow} G \times_c A \stackrel{p}{\to} G </annotation></semantics></math></div> <p>is indeed a group extension. It is a <a class="existingWikiWord" href="/nlab/show/central+extension">central extension</a> again using the assumption that <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> is normalized <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c(g,e) = c(e,g) = 0</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><mi>g</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mover><mi>a</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>a</mi><mo>+</mo><mover><mi>a</mi><mo stretchy="false">˜</mo></mover><mo>+</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mover><mi>a</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (g,a) \cdot (e,\tilde a) = (g, a + \tilde a + 0) = (e,\tilde a) \cdot (g,a) \,. </annotation></semantics></math></div> <p>Finally to see that the construction is independent of the choice of coycle <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> representing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>c</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[c]</annotation></semantics></math>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>c</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde c</annotation></semantics></math> be another representative which differs by a <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">h \colon G \to A</annotation></semantics></math> with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>c</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>≔</mo><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde c (g_1,g_2) \coloneqq c(g_1,g_2) - h(g_1) - h(g_2) + h(g_1 g_2) \,. </annotation></semantics></math></div> <p>We claim that then we have a homomorphism of central extensions (hence an isomorphism) of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>G</mi><msub><mo>×</mo> <mi>c</mi></msub><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>G</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>G</mi></msub><mo>,</mo><msub><mi>p</mi> <mn>2</mn></msub><mo>−</mo><mi>h</mi><mo>∘</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>=</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>G</mi><msub><mo>×</mo> <mover><mi>c</mi><mo stretchy="false">˜</mo></mover></msub><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\to& G \times_c A &\to& G \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{(id_G, p_2 -h \circ p_1)}} && \downarrow^{\mathrlap{=}} \\ A &\to& G \times_{\tilde c} A &\to& G } \,. </annotation></semantics></math></div> <p>To see this we check for all elements that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>−</mo><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>2</mn></msub><mo>−</mo><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>a</mi> <mn>2</mn></msub><mo>−</mo><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>a</mi> <mn>2</mn></msub><mo>+</mo><mover><mi>c</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} (g_1, a_1 - h(g_1)) \cdot (g_2, a_2 - h(g_2)) & = (g_1 g_2, a_1 + a_2 - h(g_1) - h(g_2) + c(g_1, g_2)) \\ & = (g_1 g_2, a_1 + a_2 + \tilde c(g_1, g_2) - h(g_1 g_2) ) \end{aligned} \,. </annotation></semantics></math></div> <p>Hence the construction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><msub><mo>×</mo> <mi>c</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">G \times_c A</annotation></semantics></math> indeed defines a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>Grp</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>CentrExt</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^2_{Grp}(G,A) \to CentrExt(G,A)</annotation></semantics></math>.</p> </div> <p>Assume the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> in the ambient <a class="existingWikiWord" href="/nlab/show/foundations">foundations</a>.</p> <div class="num_defn" id="2CocycleExtractedFromCentralExtension"> <h6 id="definition_7">Definition</h6> <p><strong>(2-cocycle extracted from central extension)</strong></p> <p>Given a central extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">A \to \hat G \to G</annotation></semantics></math> define a group 2-cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>:</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">c : G \times G \to A</annotation></semantics></math> as follows.</p> <p>Choose a <a class="existingWikiWord" href="/nlab/show/section">section</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>:</mo><mi>U</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sigma : U(G) \to U(\hat G)</annotation></semantics></math> of the underlying <a class="existingWikiWord" href="/nlab/show/sets">sets</a> (which exists by the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> and the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">p : \hat G \to G</annotation></semantics></math> is by definition an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>). Then define <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> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>↦</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>∈</mo><mi>A</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> c \colon (g_1, g_2) \mapsto \sigma(g_1) \sigma(g_2) \sigma(g_1 g_2)^{-1} \in A \,, </annotation></semantics></math></div> <p>where on the right we are using that by the section-property of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> and the group homomorphism property of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex"> p(\sigma(g_1) \sigma(g_2) \sigma(g_1 g_2)^{-1}) = 1 </annotation></semantics></math></div> <p>and hence by the exactness of the extension the argument is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">A \hookrightarrow \hat G</annotation></semantics></math>.</p> </div> <p>Below in remark <a class="maruku-ref" href="#RelationBetweenComponentCocycleExtractionAndHomotopyTheory"></a> is a discussion of how this construction arises from a more systematic discussion in <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>.</p> <div class="num_prop"> <h6 id="proposition_7">Proposition</h6> <p>The construction of prop. <a class="maruku-ref" href="#2CocycleExtractedFromCentralExtension"></a> indeed yields a 2-cocycle in <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>. It extends to a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Extr</mi><mo lspace="verythinmathspace">:</mo><mi>Ext</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>H</mi> <mi>Grp</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Extr \colon Ext(G,A) \to H^2_{Grp}(G,A) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>The cocycle condition to be checked is that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>0</mn></msub><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex"> c(g_1, g_2) - c(g_0 g_1, g_2) + c(g_0, g_1 g_2) - c(g_0, g_1) = 1 </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g_0, g_1, g_2 \in G</annotation></semantics></math>. Writing this out with def. <a class="maruku-ref" href="#2CocycleExtractedFromCentralExtension"></a> yields</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><msup><mrow><mo>(</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>0</mn></msub><msub><mi>g</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>0</mn></msub><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>0</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>0</mn></msub><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><msup><mrow><mo>(</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>0</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>0</mn></msub><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \sigma(g_1)^{-1} \sigma(g_2)^{-1} \sigma(g_1 g_2) \left(\sigma(g_0 g_1)^{-1} \sigma(g_2)^{-1} \sigma(g_0 g_1 g_2)\right)^{-1} \sigma(g_0)^{-1} \sigma(g_1 g_2)^{-1} \sigma(g_0 g_1 g_2) \left( \sigma(g_0)^{-1} \sigma(g_1)^{-1} \sigma(g_0 g_1) \right)^{-1} \,. </annotation></semantics></math></div> <p>Here it is sufficient to observe that for every term also the inverse term appears.</p> <p>To see that this is a well-defined map to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>grp</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^2_{grp}(G,A)</annotation></semantics></math> we need to check that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>σ</mi><mo stretchy="false">˜</mo></mover><mo>:</mo><mi>G</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\tilde \sigma : G \to \hat G</annotation></semantics></math> a different choice of section, the corresponding cocycles differ by a group coboundary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>c</mi><mo stretchy="false">˜</mo></mover><mo>−</mo><mi>c</mi><mo>=</mo><mi>d</mi><mi>h</mi></mrow><annotation encoding="application/x-tex">\tilde c - c = d h</annotation></semantics></math>. Clearly this is obtained by setting</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo lspace="verythinmathspace">:</mo><mi>g</mi><mo>↦</mo><mover><mi>σ</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>g</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> h \colon g \mapsto \tilde \sigma(g)\sigma(g)^{-1} \,, </annotation></semantics></math></div> <p>where we use that the right hand side is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">A \hookrightarrow \hat G</annotation></semantics></math> since because both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>σ</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde \sigma</annotation></semantics></math> are sections of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, the image of the right hand under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is the neutral element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> </div> <div class="num_prop" id="ExtractionAndReconstructionConsituteEquivalence"> <h6 id="proposition_8">Proposition</h6> <p>The two morphisms of def. <a class="maruku-ref" href="#CentralExtensionAssociatedTo2Cocycle"></a> and def. <a class="maruku-ref" href="#2CocycleExtractedFromCentralExtension"></a> exhibit the <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>Grp</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mover><munder><mo>→</mo><mi>Rec</mi></munder><munderover><mo>←</mo><mo>≃</mo><mi>Extr</mi></munderover></mover><mi>CentrExt</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^2_{Grp}(G,A) \stackrel{\underoverset{\simeq}{Extr}{\leftarrow}}{\underset{Rec}{\to}} CentrExt(G,A) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>c</mi><mo stretchy="false">]</mo><mo>∈</mo><msubsup><mi>H</mi> <mi>Grp</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[c] \in H^2_{Grp}(G,A)</annotation></semantics></math>. Then by construction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>≔</mo><mi>G</mi><msub><mo>×</mo> <mi>c</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">\hat G \coloneqq G \times_c A</annotation></semantics></math> there is a canonical section of the underlying function of sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>G</mi><msub><mo>×</mo> <mi>c</mi></msub><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(G \times_c A) \to U(G)</annotation></semantics></math> given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mrow><mi>U</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mi>U</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(id_{U(G)}, 0) U(G) \to U(G) \times U(A)</annotation></semantics></math>. The cocycle induced by this section sends</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><mn>0</mn><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} (g_1, g_2) & \mapsto (g_1, 0) (g_2, 0) (g_1 g_2, 0)^{-1} \\ & = (g_1, 0) (g_1, 0) ((g_1 g_2)^{-1}, - c(g_1 g_2, (g_1 g_2)^{-1}) ) \\ & = (g_1 g_2, c(g_1, g_2) ) ((g_1 g_2)^{-1}, - c(g_1 g_2, (g_1 g_2)^{-1}) ) \\ & = (e, c(g_1, g_2) - c(g_1 g_2, (g_1 g_2)^{-1}) + c(g_1 g_2, (g_1 g_2)^{-1})) \\ & = (e, c(g_1, g_2)) \end{aligned} \,, </annotation></semantics></math></div> <p>which is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>A</mi><mo>↪</mo><mi>G</mi><msub><mo>×</mo> <mi>c</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">c(g_1, g_2) \in A \hookrightarrow G \times_c A</annotation></semantics></math>, and hence this recovers the 2-cocycle that we started with.</p> <p>This shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Extr</mi><mo>∘</mo><mi>Rec</mi><mo>=</mo><mi>id</mi></mrow><annotation encoding="application/x-tex">Extr \circ Rec = id</annotation></semantics></math> and in particular that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Rec</mi></mrow><annotation encoding="application/x-tex">Rec</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/surjection">surjection</a>. It is readily seen that the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Rec</mi></mrow><annotation encoding="application/x-tex">Rec</annotation></semantics></math> is trivial, and so it is an equivalence.</p> </div> <h4 id="FormulationInHomotopyTheory">Formulation in homotopy theory</h4> <p>We discuss the classification of central group extensions by degree-2 <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a> in the more abstract context of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> (via the translation discussed at <em><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></em>), complementing the <a href="CentralExtensionClassificationByGroupCohomology ">above</a> component-wise discussion.</p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> A \hookrightarrow \hat G \to G </annotation></semantics></math></div> <p>be a central group extension, def. <a class="maruku-ref" href="#GroupExtension"></a>, hence with <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> an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> included in the <a class="existingWikiWord" href="/nlab/show/center">center</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. Then <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> is in particular a <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a> and hence the homorphism</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><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (A \to \hat G) </annotation></semantics></math></div> <p>may be regarded as a <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a> of groups. This is equivalently a <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a> structure on the <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> whose objects are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat G</annotation></semantics></math> and whose morphisms are labeled in <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></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><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>g</mi><mover><mo>→</mo><mi>a</mi></mover><mi>a</mi><mo>⋅</mo><mi>g</mi><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (A \to \hat G) = \{ g \stackrel{a}{\to} a \cdot g \} \,. </annotation></semantics></math></div> <div class="num_defn"> <h6 id="definition_8">Definition</h6> <p>Write</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 stretchy="false">(</mo><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>∈</mo><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> \mathbf{B}(A \to \hat G) \in Grpd </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of this <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a> to a one-object <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-nerve">∞-nerve</a> (or <a class="existingWikiWord" href="/nlab/show/Duskin+nerve">Duskin nerve</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>∈</mo></mrow><annotation encoding="application/x-tex">N \mathbf{B}(A \to \hat G) \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> of this is a (3-<a class="existingWikiWord" href="/nlab/show/coskeleton">coskeletal</a>) <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> that realizes this as a <a class="existingWikiWord" href="/nlab/show/truncated+object+of+an+%28infinity%2C1%29-category">2-truncated</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>.</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 stretchy="false">(</mo><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>∈</mo><mn>∞</mn><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}(A \to \hat G) \in \infty Grpd \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition_9">Proposition</h6> <p>The obvious strict <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></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 stretchy="false">(</mo><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mrow><annotation encoding="application/x-tex"> \mathbf{B}(A \to \hat G) \stackrel{\simeq}{\to} \mathbf{B}H </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">equivalence</a> of 2-groupoids.</p> </div> <div class="proof"> <h6 id="proof_9">Proof</h6> <p>One way to see this is to notice that this is a <a class="existingWikiWord" href="/nlab/show/k-surjective+functor">k-surjective functor</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">k \in \{0,1,2,3\}</annotation></semantics></math>, hence a weak equivalence in the <a class="existingWikiWord" href="/nlab/show/folk+model+structure">folk model structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-groupoids. Equivalently, under the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> the morphism of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>→</mo><mi>N</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mrow><annotation encoding="application/x-tex"> N\mathbf{B}(A \to \hat G) \to N \mathbf{B}H </annotation></semantics></math></div> <p>is an acyclic <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>, hence a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a> in the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a>.</p> </div> <div class="num_prop"> <h6 id="proposition_10">Proposition</h6> <p>The extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">A \to \hat G \to G</annotation></semantics></math> sits in a long <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a> in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mover><mo>→</mo><mstyle mathvariant="bold"><mi>c</mi></mstyle></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>A</mi></mrow><annotation encoding="application/x-tex"> A \to \hat G \to G \to \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^2 A </annotation></semantics></math></div> <p>which in <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a>/<a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> is <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presented</a> by the <a class="existingWikiWord" href="/nlab/show/zigzag">zigzag</a> of <a class="existingWikiWord" href="/nlab/show/n-functors">n-functors</a> between <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a> (sequence of <a class="existingWikiWord" href="/nlab/show/2-anafunctors">2-anafunctors</a>) of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0"><mo>≃</mo></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && && && && && \mathbf{B}(A \to \hat G) &\to & \mathbf{B}(A \to 1) = \mathbf{B}^2 A \\ && && && && && {}^{\mathrlap{\simeq}}\downarrow \\ && && (A \to \hat G) &\to& \mathbf{B}A &\to& \mathbf{B}\hat G &\to& \mathbf{B}G \\ && && \downarrow^{\mathrlap{\simeq}} \\ A &\to& \hat G &\to& G } \,. </annotation></semantics></math></div> <p>In particular, the induced <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></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>c</mi></mstyle><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>A</mi></mrow><annotation encoding="application/x-tex"> \mathbf{c} : \mathbf{B}G \to \mathbf{B}^2 A </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a> <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> that classifies the delooped extension as 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>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>.</p> </div> <div class="proof"> <h6 id="proof_10">Proof</h6> <p>One sees directly that the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\hat G \to \mathbf{B}G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}(A \to \hat G ) \to \mathbf{B}^2 A </annotation></semantics></math> as well as their loopings <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\hat G \to G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">(A \to \hat G) \to G</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/Kan+fibrations">Kan fibrations</a>. By the discussion at <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> this means that the set-theoretic <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> of these morphisms are models for their <a class="existingWikiWord" href="/nlab/show/homotopy+fibers">homotopy fibers</a>. But the ordinary <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}(A \to \hat G) \to \mathbf{B}(A \to 1) = \mathbf{B}^2 A</annotation></semantics></math> is manifestly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\mathbf{B} \hat G</annotation></semantics></math>, and so on.</p> </div> <div class="num_remark" id="RelationBetweenComponentCocycleExtractionAndHomotopyTheory"> <h6 id="remark_5">Remark</h6> <p>The construction in def. <a class="maruku-ref" href="#2CocycleExtractedFromCentralExtension"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Rec</mi><mo>:</mo><mi>CentrExt</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>H</mi> <mi>Grp</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Rec : CentrExt(G,A) \to H^2_{Grp}(G,A) </annotation></semantics></math></div> <p>is precisely the result of moving set-theoretically through the <a class="existingWikiWord" href="/nlab/show/zigzag">zigzag</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}(A \to \hat G) &\to& \mathbf{B}(A \to 1) = \mathbf{B}^2 A \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}G } </annotation></semantics></math></div> <p>from the bottom left to the top right, and that this is well-defined on <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> comes down to the statement that the vertical morphism is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>.</p> <p>This is a nonabelian analog of the discussion at <em><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></em> in the section <em><a href="mapping%20cone#HomologyExactSequencesAndFiberSequences">Homology exact sequences and fiber sequences</a></em>.</p> </div> <h3 id="PropertiesAbelianGroupExtensions">Abelian group extensions</h3> <div class="num_remark"> <h6 id="remark_6">Remark</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>G</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">A, G \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Ab">Ab</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">\hookrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a> even a <a class="existingWikiWord" href="/nlab/show/central+extension">central extension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat G</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> by <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> is not necessarily itself an abelian group.</p> </div> <p>But by prop. <a class="maruku-ref" href="#ExtractionAndReconstructionConsituteEquivalence"></a> above it is so if the group 2-cocycle that classifies the extension is symmetric:</p> <div class="num_defn"> <h6 id="definition_9">Definition</h6> <p>A 2-cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">c \colon G \times G \to A</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a> is <strong>symmetric</strong> if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∀</mo> <mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>∈</mo><mi>G</mi></mrow></msub><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \forall_{g_1, g_2 \in G} c(g_1, g_2) = c(g_2, g_1) \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_7">Remark</h6> <p>A group 2-cocycle cohomologous to a symmetric group 2-cocycle is itself symmetric. Hence we may speak of symmetric group cohomology classes in degree 2.</p> </div> <div class="num_defn"> <h6 id="definition_10">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>Grp</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>sym</mi></msub><mo>↪</mo><msubsup><mi>H</mi> <mi>Grp</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^2_{Grp}(G,A)_{sym} \hookrightarrow H^2_{Grp}(G,A) </annotation></semantics></math></div> <p>for the set (group) of classes of symmetric group 2-cocycles on <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> with coefficients in <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>.</p> </div> <div class="num_defn"> <h6 id="definition_11">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>,</mo><mi>A</mi><mo>∈</mo><mi>Ab</mi><mo>↪</mo><mi>Grp</mi></mrow><annotation encoding="application/x-tex">G,A \in Ab \hookrightarrow Grp</annotation></semantics></math>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ext(G,A)</annotation></semantics></math> for the subset of equivalence class of abelian group extensions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> by <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>.</p> </div> <p>The theory of <em>abelian group extensions</em> in <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> is naturally and classically treated with tools of <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, such as the theory of <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a>-functors.</p> <p>For the moment see at <em><a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></em> the section</p> <ul> <li><em><a href="projective+resolution#ProjectiveResolutionsForGroupCocycles">Projective resolutions adapted to group cocycles</a></em></li> </ul> <p>and</p> <ul> <li><em><a href="projective+resolution#DerivedHomFunctor">Derived Hom-functor / Ext-functor</a></em> .</li> </ul> <h3 id="SchreierTheory">Nonabelian group extensions (Schreier theory)</h3> <p>We discuss the classification theory for the general case of nonabelian group extensions, first in the form of</p> <ul> <li><em><a href="#SchreierTheoryTraditional">Traditional Schreier theory</a></em></li> </ul> <p>and then more abstractly in the language of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> in</p> <ul> <li><em><a href="#SchreierTheorynPOV">Homotopy theory perspective on Schreier theory</a></em></li> </ul> <h4 id="SchreierTheoryTraditional">Traditional description</h4> <p><a class="existingWikiWord" href="/nlab/show/Otto+Schreier">Otto Schreier</a> (1926) and <a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Eilenberg</a>-<a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Mac Lane</a> (late 1940-s) developed a theory of classification of nonabelian extensions of abstract groups leading to the low dimensional <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>. This is sometimes called <strong>Schreier’s theory</strong> of nonabelian group extensions.</p> <p>The traditional Schreier-Mac Lane way to obtain nonabelian group 2-cocycle from a group extension as above starts with choosing a set-theoretic section of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">p:G\to B</annotation></semantics></math>.</p> <p><strong>Note.</strong> The exposition which follows in this long “traditional” section of this entry is mainly from personal notes of Zoran Škoda from 1997.</p> <p>Each element <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> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> defines an inner automorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(g)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mi>k</mi><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\phi(g)(k) = g k g^{-1}</annotation></semantics></math>. The restriction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><msub><mo stretchy="false">|</mo> <mi>K</mi></msub></mrow><annotation encoding="application/x-tex">\phi|_K</annotation></semantics></math> takes (by definition) values in the subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Int</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Int(K)</annotation></semantics></math> of inner automorphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>. In fact <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>Inn</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi:G\to Inn(G)\subset Aut(K)</annotation></semantics></math> is a homomorphism of groups.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">g_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">g_2</annotation></semantics></math> are in the same left coset, that is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><mi>K</mi><mo>=</mo><msub><mi>g</mi> <mn>2</mn></msub><mi>K</mi></mrow><annotation encoding="application/x-tex">g_1K = g_2K</annotation></semantics></math>, then there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">k \in K</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>g</mi> <mn>2</mn></msub><mi>k</mi></mrow><annotation encoding="application/x-tex">g_1 = g_2k</annotation></semantics></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>k</mi><mo>′</mo><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">\forall k' \in K</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mi>k</mi><mo>′</mo><mo stretchy="false">)</mo><mo>=</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mi>k</mi><mi>k</mi><mo>′</mo><mo stretchy="false">)</mo><mo>=</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>k</mi><mi>k</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(g_1k') = \phi(g_2k k') = \phi(g_2)\phi(k k')</annotation></semantics></math> and therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mi>K</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(g_1K) \subset \phi(g_2)Int(K)</annotation></semantics></math>. Thus we obtain a well-defined map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mo>*</mo></msub><mo>:</mo><mi>G</mi><mo stretchy="false">/</mo><mi>K</mi><mo>→</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi_* : G/K \rightarrow Aut(K)/Int(K)</annotation></semantics></math>. Choose a set-theoretic section of the projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">p : G \rightarrow B</annotation></semantics></math> and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mover><mo>=</mo><mi>def</mi></mover><mi>ϕ</mi><mo>∘</mo><mi>σ</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">\psi \stackrel{def}{=} \phi \circ \sigma: B \rightarrow Aut(K).</annotation></semantics></math></div> <p><strong>Warning.</strong> Unlike <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>, the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math> is <em>not</em> a homomorphism of groups.</p> <p>We attempt to reconstruct <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> from the knowledge of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>. As a set, <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> can be naturally identified with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>×</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">B \times K</annotation></semantics></math>. Indeed, write each element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g \in G</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mi>k</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>B</mi><mo>,</mo><mi>k</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">\sigma(b)k, b \in B, k \in K</annotation></semantics></math> by setting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>,</mo><mi>k</mi><mo>=</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>g</mi></mrow><annotation encoding="application/x-tex">b = p(g), k = \sigma(p(g))^{-1}g</annotation></semantics></math>. Elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">k \in K</annotation></semantics></math> in that decomposition are unique, and we get a bijection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>×</mo><mi>K</mi><mo>∋</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mi>k</mi><mo>∈</mo><mi>G</mi><mo>,</mo></mrow><annotation encoding="application/x-tex"> B\times K\ni (b,k)\mapsto\sigma(b)k \in G, </annotation></semantics></math></div> <p>whose inverse is the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>,</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g \mapsto (p(g), \sigma(p(g))^{-1}g)</annotation></semantics></math>. By means of that bijection, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>×</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">B \times K</annotation></semantics></math> inherits the group structure from <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>. Let us figure out the multiplication rule on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>×</mo><mi>K</mi><mo>.</mo></mrow><annotation encoding="application/x-tex">B \times K.</annotation></semantics></math> If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>k</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>g</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\sigma(b_1)k_1 = g_1</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><msub><mi>k</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>g</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\sigma(b_2)k_2 = g_2</annotation></semantics></math>, then,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><mo>=</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>k</mi> <mn>1</mn></msub><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><msub><mi>k</mi> <mn>2</mn></msub><mo>=</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>k</mi> <mn>1</mn></msub><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><msub><mi>k</mi> <mn>2</mn></msub><mo>.</mo></mrow><annotation encoding="application/x-tex"> g_1g_2 = \sigma(b_1)k_1\sigma(b_2)k_2 = \sigma(b_1)\sigma(b_2)\sigma(b_2)^{-1}k_1\sigma(b_2)k_2. </annotation></semantics></math></div> <p>Now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(\sigma(b_1)\sigma(b_2)) = p(b_1b_2)</annotation></semantics></math> so</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mover><mo>=</mo><mi>def</mi></mover><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>K</mi><mo>.</mo></mrow><annotation encoding="application/x-tex">\chi(b_1,b_2) \stackrel{def}{=} \sigma(b_1b_2)^{-1}\sigma(b_1)\sigma(b_2) \in K. </annotation></semantics></math></div> <p>This formula clearly defines a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo>:</mo><mi>B</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">\chi : B \times B \rightarrow K</annotation></semantics></math>. In this notation,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>k</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><msub><mi>k</mi> <mn>2</mn></msub><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ g_1g_2 & = & \sigma(b_1b_2)\chi(b_1,b_2)\phi(\sigma(b_2)^{-1})(k_1)k_2 \\ & = & \sigma(b_1b_2)\chi(b_1,b_2)[\psi(b_2)^{-1}(k_1)] k_2. } </annotation></semantics></math></div> <p>and using bijection of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>×</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">B\times K</annotation></semantics></math> this can be expressed in terms of elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>×</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">B\times K</annotation></semantics></math> so that</p> <div class="maruku-equation" id="eq:mrule"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>k</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>k</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><msub><mi>k</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> (b_1,k_1)(b_2,k_2) = (b_1b_2,\chi(b_1,b_2)[\psi(b_2)^{-1}(k_1)] k_2). </annotation></semantics></math></div> <p>According to this formula, <em>all the information about the multiplication is encoded in functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo>:</mo><mi>B</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi : B \times B \rightarrow Aut(K)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\psi : B \rightarrow Aut(K)</annotation></semantics></math>, and we may forget about <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></em> at this point. However, <em>not every pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>χ</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\chi,\psi)</annotation></semantics></math> will give some multiplication rule on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>×</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">B \times K</annotation></semantics></math></em>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">a,b,c \in B</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>=</mo><msub><mi>e</mi> <mi>K</mi></msub></mrow><annotation encoding="application/x-tex">e = e_K</annotation></semantics></math> be the unity element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>. Then</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><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mi>c</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> [(a,e)(b,e)](c,e) = (a b, \chi(a,b))(c,e) = (a b c, \chi(a b,c) \psi(c)^{-1}(\chi(a,b))). </annotation></semantics></math></div> <p>From the other side, this has to be the same, by associativity, to</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><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>bc</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mi>c</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (a,e)[(b,e)(c,e)] = (a,e)(bc,\chi(b,c)) = (a b c,\chi(a,b c)\chi(b,c)) </annotation></semantics></math></div> <p>where we took into account that expressions like <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>ψ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">[\psi^{-1}(b)(e)] = e</annotation></semantics></math>, because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\psi(b)</annotation></semantics></math> is an <em>automorphism</em> for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>.</p> <p>Comparing the expressions above we obtain</p> <div class="maruku-equation" id="eq:psi1"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>for</mi><mi>all</mi><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><mi>B</mi><mo>.</mo></mrow><annotation encoding="application/x-tex"> \chi(a b,c)\psi(c)^{-1}(\chi(a,b)) = \chi(a,b c)\chi(b,c), for all a,b,c \in B. </annotation></semantics></math></div> <p>If the pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>χ</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\chi,\psi)</annotation></semantics></math> is constructed as above, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mi>k</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>ϕ</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>k</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>σ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mi>k</mi><mi>σ</mi><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>σ</mi><mo stretchy="false">(</mo><mi>a</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>σ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mi>k</mi><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>σ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>k</mi><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \psi(a)\psi(b)k & = & \phi(\sigma(a))\phi(\sigma(b))k \\ & = & \sigma(a)\sigma(b)k\sigma(b)^{-1}\sigma(a)^{-1} \\ & = & \sigma(a b)\chi(a,b)k\chi(a,b)^{-1}\sigma(a b)^{-1} \\ & = & \psi(a b) Ad_K(\chi(a,b)) k, } </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ad</mi> <mi>K</mi></msub></mrow><annotation encoding="application/x-tex">Ad_K</annotation></semantics></math> is the canonical map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>→</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K \rightarrow Int(K)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>↦</mo><mi>k</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msup><mi>k</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">k\mapsto k(-)k^{-1}</annotation></semantics></math>.</p> <p>Thus we obtain the relation</p> <div class="maruku-equation" id="eq:psi2"><span class="maruku-eq-number">(5)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \psi(a)\psi(b) = \psi(a b) Ad_K(\chi(a,b)) </annotation></semantics></math></div> <div class="num_defn"> <h6 id="definition_12">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> be two groups. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo>:</mo><mi>B</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">\chi: B \times B \rightarrow K</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\psi : B \rightarrow Aut(K)</annotation></semantics></math> satisfy <a class="maruku-eqref" href="#eq:psi1">(4)</a> and <a class="maruku-eqref" href="#eq:psi2">(5)</a>. Then we call that the family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo>∈</mo><mi>B</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\chi(b_1,b_2)| b_1,b_2 \in B\}</annotation></semantics></math> is a factor system (This term is due Schreier(1924)) or a <strong>nonabelian group 2-cocycle with automorphisms</strong>, and the family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>b</mi><mo>∈</mo><mi>B</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\psi(b) | b \in B \}</annotation></semantics></math> – a system of automorphisms</p> </div> <p>A 2-cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math> is <strong>counital</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">\chi(b,e) = \chi(e,b) = e</annotation></semantics></math>, for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is commutative, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math> is always a homomorphism (cf. <a class="maruku-eqref" href="#eq:psi2">(5)</a>). Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is a right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-module through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\psi(-)^{-1}</annotation></semantics></math>. That justifies the sometimes used term “(right) cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-module” for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>ψ</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(K,\psi,\chi)</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math> is trivial (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Id</mi> <mi>K</mi></msub><mo>,</mo><mo>∀</mo><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\psi(b) = Id_K, \forall b \in B</annotation></semantics></math>) then the cocycle condition <a class="maruku-eqref" href="#eq:psi1">(4)</a> becomes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> \chi(a b,c)\chi(a,b) = \chi(a,b c)\chi(b,c). </annotation></semantics></math></div> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>If formulas <a class="maruku-eqref" href="#eq:psi1">(4)</a> and <a class="maruku-eqref" href="#eq:psi2">(5)</a> are both satisfied, then the formula <a class="maruku-eqref" href="#eq:mrule">(3)</a> for multiplication of pairs defines a group multiplication on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>×</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">B \times K</annotation></semantics></math>. That set, together with multiplication <a class="maruku-eqref" href="#eq:mrule">(3)</a> is called the <strong>cocycle cross product</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> with cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math> and action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math>. If the cocycle is trivial i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo>,</mo><mo>⋅</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>e</mi> <mi>K</mi></msub></mrow><annotation encoding="application/x-tex">\chi(\cdot,\cdot) = e_K</annotation></semantics></math>, we call it the <strong>(external) semidirect product</strong>.</p> </div> <div class="proof"> <h6 id="proof_11">Proof</h6> <p>We have checked above the <a class="existingWikiWord" href="/nlab/show/associativity">associativity</a> for pairs of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,e)</annotation></semantics></math> etc. This was useful to find the cocycle condition correctly. Now the general associativity should be a similar calculation with general elements. Using <a class="maruku-eqref" href="#eq:psi1">(4)</a> and <a class="maruku-eqref" href="#eq:psi2">(5)</a> it can be done.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mi>k</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>k</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mi>k</mi><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \psi(a)\psi(e)k & =& \psi(a)Ad_K(\chi(a,e))k \\ & = &\psi(a)\chi(a,e)k\chi(a,e)^{-1} } </annotation></semantics></math></div> <p>where we used <a class="maruku-eqref" href="#eq:psi2">(5)</a>.</p> <p>Thus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ad_K(\chi(a,e)) = \psi(e)</annotation></semantics></math> and therefore it does not depend on <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>.</p> <p>Then use <a class="maruku-eqref" href="#eq:psi1">(4)</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>=</mo><mi>c</mi><mo>=</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">b = c = e</annotation></semantics></math> to get <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>,</mo><mo>∀</mo><mi>a</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\psi(e)^{-1}(\chi(a,e)) = \chi(e,e), \forall a \in B</annotation></semantics></math>.</p> <p>Thus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi(a,e)^{-1}(\chi(a,e))\chi(a,e) = \chi(e,e)</annotation></semantics></math>, that is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi(a,e)</annotation></semantics></math> does not depend on <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>.</p> <p>Now we claim that the <em>unit</em> element is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(e, \chi(e,e)^{-1})</annotation></semantics></math>. To verify that it is also a right unit we compute</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>b</mi><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ (a,b)(e,\chi(e,e)^{-1}) & = & (a, \chi(a,e) \psi(e)^{-1}(b)\chi(e,e)^{-1}) \\ & = & (a, \chi(a,e)\chi(e,e)^{-1}b\chi(e,e)\chi(e,e)^{-1}) } </annotation></semantics></math></div> <p>what is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,b)</annotation></semantics></math> by just proved statement that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi(a,e)</annotation></semantics></math> does not depend on <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>.</p> <p>Now use <a class="maruku-eqref" href="#eq:psi1">(4)</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>=</mo><mi>b</mi><mo>=</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">a = b = e</annotation></semantics></math> to get</p> <div class="maruku-equation" id="eq:psiac"><span class="maruku-eq-number">(6)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mo>∀</mo><mi>c</mi><mo>∈</mo><mi>B</mi><mo>.</mo></mrow><annotation encoding="application/x-tex"> \psi(c)^{-1}(\chi(e,e)) = \chi(e,c), \forall c \in B. </annotation></semantics></math></div> <p>Thus we can verify that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(e, \chi(e,e)^{-1})</annotation></semantics></math> is a left unit too by a calculation as follows. Namely</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><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(e,\chi(e,e)^{-1})(a,b)= (a,\chi(e,a)\psi(a)^{-1}(\chi(e,e)^{-1})b)</annotation></semantics></math></div> <p>by the definition of the product. Then by <a class="maruku-eqref" href="#eq:psiac">(6)</a>, this equals to</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><mi>a</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,\psi(a)^{-1}(\chi(e,e))\psi(a)^{-1}(\chi(e,e)^{-1})b)</annotation></semantics></math></div> <p>and, because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\psi(a)^{-1}</annotation></semantics></math> is an antiautomorphism, this is finally equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,b)</annotation></semantics></math>.</p> <p>Now check that each element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,b)</annotation></semantics></math> can be factorized as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,e)(e,\chi(e,e)^{-1}b)</annotation></semantics></math>. In order to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,b)</annotation></semantics></math> has an inverse it is then enough to show that both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,e)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(e,\chi(e,e)^{-1}b)</annotation></semantics></math> have inverses.</p> <p>Claim: the inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,e)</annotation></semantics></math> is</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><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>a</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">(a^{-1},\chi(a,a)^{-1}\chi(e,e)^{-1}).</annotation></semantics></math></div> <p>To this aim, we calculate</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><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">(a,e)(a^{-1},\chi(a,a^{-1})^{-1}\chi(e,e)^{-1}) = (e,\chi(a,a^{-1})\psi(a^{-1})^{-1}(e)\chi(a,a^{-1})^{-1}\chi(e,e)^{-1}) =(e,\chi(e,e^{-1}),</annotation></semantics></math></div> <p>because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">\psi(a)^{-1}(e) = e</annotation></semantics></math>. Furthermore,</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><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">(a,e)(a^{-1},\chi(a,a^{-1})^{-1}\chi(e,e)^{-1}) = (e,\chi(a,a^{-1})\psi(a^{-1})^{-1}(e)\chi(a,a^{-1})^{-1}\chi(e,e)^{-1}) = (e,\chi(e,e^{-1}),</annotation></semantics></math></div> <p>because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">\psi(a)^{-1}(e) = e</annotation></semantics></math>. Next,</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><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a^{-1},\chi(a,a^{-1})^{-1}\chi(e,e)^{-1})(a,e) = (e,\chi(a^{-1},a)\psi(a)^{-1}(\chi(a,a^{-1}) \chi(e,e)^{-1})) </annotation></semantics></math></div> <p>what equals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(e,\chi(e,e)^{-1})</annotation></semantics></math>.</p> <p>Indeed, <a class="maruku-eqref" href="#eq:psi1">(4)</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>=</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>=</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>c</mi><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">a = a, b = a^{-1}, c = a</annotation></semantics></math> reads <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi(e,a) \psi(a)^{-1}(\chi(a, a^{-1}))</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">=\chi(a,e)\chi(a^{-1},a)</annotation></semantics></math>.</p> <p>Then apply <a class="maruku-eqref" href="#eq:psiac">(6)</a> and take inverse of both sides to obtain</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>χ</mi><mo stretchy="false">(</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>a</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>.</mo></mrow><annotation encoding="application/x-tex">\psi(a)^{-1}(\chi(a,a^{-1})^{-1}\chi(e,e)^{-1})) = \chi(a^{-1},a)^{-1}\chi(a,e)^{-1}. </annotation></semantics></math></div> <p>Then recall that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi(a,e)</annotation></semantics></math> does not depend on <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 multiply by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi(a^{-1},a)</annotation></semantics></math> from the left.</p> <p>Claim: the inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(e,\chi(e,e)^{-1}k)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><msup><mi>k</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(e,\chi(e,e)k^{-1})</annotation></semantics></math>. Here the verification is symmetric (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> vs. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">k^{-1}</annotation></semantics></math>) for the left and for the right inverse and immediate.</p> </div> <p>Given groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> and any maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math> satisfying <a class="maruku-eqref" href="#eq:psi1">(4)</a> and <a class="maruku-eqref" href="#eq:psi2">(5)</a>, needed to define a cocycle cross product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mo>×</mo> <mi>χ</mi></msub><mi>K</mi></mrow><annotation encoding="application/x-tex">B\times_\chi K</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, one defines the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>K</mi><mo>→</mo><mi>B</mi><msub><mo>×</mo> <mi>χ</mi></msub><mi>K</mi></mrow><annotation encoding="application/x-tex">i : K \rightarrow B \times_\chi K</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k \mapsto (e,\chi(e,e)^{-1}k)</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is a monomorphism of groups, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i(K)</annotation></semantics></math> is a normal subroup of the cocycle cross product of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>, and there is a canonical isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>≅</mo><mi>G</mi><mo stretchy="false">/</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">B \cong G/K</annotation></semantics></math>. We define the set-theoretic maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>′</mo><mo>,</mo><mi>χ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\sigma',\chi'</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\psi'</annotation></semantics></math> as follows. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>′</mo><mo>:</mo><mi>B</mi><mo>→</mo><mi>B</mi><mo>×</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">\sigma' : B \rightarrow B \times K</annotation></semantics></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sigma'(b) = (b, e)</annotation></semantics></math> , for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo>′</mo><mo>:</mo><mi>B</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>i</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi' : B \times B \to i(K)</annotation></semantics></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>σ</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>σ</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>σ</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi'(b_1,b_2) = \sigma'(b_1b_2)^{-1}\sigma'(b_1)\sigma'(b_2)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>′</mo><mo>:</mo><mi>B</mi><mo>→</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\psi' : B \to Aut(i(K))</annotation></semantics></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mi>i</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mi>σ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mi>i</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mi>σ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\psi'(b)i(k) = \sigma'(b)i(k)\sigma'(b)^{-1}</annotation></semantics></math>. Using the natural identifications <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>K</mi><mo>≅</mo><mi>i</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i : K \cong i(K)</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>Aut</mi></msub><mo>:</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≅</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i_{Aut} : Aut(i(K)) \cong Aut(K)</annotation></semantics></math>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>′</mo><mo>=</mo><msub><mi>i</mi> <mi>Aut</mi></msub><mo>∘</mo><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi' = i_{Aut}\circ \psi</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo>′</mo><mo>=</mo><mi>i</mi><mo>∘</mo><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi' = i \circ \chi</annotation></semantics></math>. Now</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>χ</mi><mo>′</mo><mo>=</mo><mi>i</mi><mo>∘</mo><mi>χ</mi></mtd> <mtd><mo>⇔</mo></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>⇔</mo></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>⇔</mo></mtd> <mtd><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo>=</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>k</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \chi'=i\circ\chi &\Leftrightarrow&(b_1,e)(b_2,e)(e,\chi(e,e)^{-1}k) =(b_1 b_2,e)(e,\chi(e,e)^{-1}\chi(b_1,b_2)k)\\ &\Leftrightarrow& (b_1b_2,\chi(b_1,b_2))(e,\chi(e,e)^{-1}k) = (b_1 b_2,\chi(b_1 b_2,e)\chi(e,e)^{-1}\chi(b_1,b_2)k)\\ &\Leftrightarrow& \chi(b_1 b_2,e)\psi(e)^{-1}(\chi(b_1,b_2))\chi(e,e)^{-1}k = \chi(b_1 b_2,e)\chi(e,e)^{-1}\chi(b_1,b_2)k } </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b_1,b_2 \in B</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">k \in K</annotation></semantics></math> in all these lines. The last line is true by <a class="maruku-eqref" href="#eq:psi1">(4)</a>.</p> <p>Similarly, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>′</mo><mo>=</mo><msub><mi>i</mi> <mi>Aut</mi></msub><mo>∘</mo><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi' = i_{Aut} \circ \psi</annotation></semantics></math> iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(b,e)(e,\chi(e,e)^{-1}k) = (e,\chi(e,e)^{-1}\psi(b)k)(b,e)</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> <p>Here the LHS computes as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(b,k)</annotation></semantics></math> using <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi(b,e) = \chi(e,e)</annotation></semantics></math>, and the RHS is</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><mi>e</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(e,\psi(b)k)(b,e) = (b, \chi(e,b)\psi(b)^{-1}(\chi(e,e)^{-1}\psi(b)(k))) = (b, k) </annotation></semantics></math></div> <p>by <a class="maruku-eqref" href="#eq:psiac">(6)</a>.</p> <div class="num_prop"> <h6 id="proposition_11">Proposition</h6> <p>The following are equivalent</p> <ul> <li> <p>(i) extension <a class="maruku-eqref" href="#eq:shortExtension">(1)</a> is split</p> </li> <li> <p>(ii) for extension <a class="maruku-eqref" href="#eq:shortExtension">(1)</a> there is a subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mn>1</mn></msub><mo>⊂</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">B_1 \subset G</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mn>1</mn></msub><mo>∩</mo><mi>i</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">B_1 \cap i(K) = 1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mn>1</mn></msub><mi>i</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>=</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">B_1i(K) = G</annotation></semantics></math> (<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 an internal semidirect product of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">B_1</annotation></semantics></math>).</p> </li> <li> <p>(iii) extension <a class="maruku-eqref" href="#eq:shortExtension">(1)</a> is isomorphic to an external semidirect product of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>.</p> </li> </ul> </div> <div class="proof"> <h6 id="proof_12">Proof</h6> <p>(i) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math> (ii) If the extension is split then there is a <em>homomorphism</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\sigma : B \rightarrow G</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mi>σ</mi><mo>=</mo><msub><mi>id</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">p \circ \sigma = id_B</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mn>1</mn></msub><mo>=</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_1 = \sigma(B)</annotation></semantics></math>. By exactness of <a class="maruku-eqref" href="#eq:shortExtension">(1)</a>), all elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i(K)</annotation></semantics></math> map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> sends to 1, and by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mi>σ</mi><mo>=</mo><msub><mi>id</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">p \circ \sigma = id_B</annotation></semantics></math> map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>B</mi> <mn>1</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">p|_{B_1}</annotation></semantics></math> is injection, therefore the only element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i(K)</annotation></semantics></math> which belongs to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">B_1</annotation></semantics></math> is 1.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mn>1</mn></msub><mi>i</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>=</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">B_1i(K) = G</annotation></semantics></math> is also obvious: e.g. for given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi><mo>,</mo></mrow><annotation encoding="application/x-tex">g \in G,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mi>σ</mi><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(g) = p\sigma p(g)</annotation></semantics></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>σ</mi><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p((\sigma p (g))^{-1}g) = 1</annotation></semantics></math> what means <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>σ</mi><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>g</mi><mo>∈</mo><mi>Ker</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\sigma p (g))^{-1}g \in {Ker}(p)</annotation></semantics></math> so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>=</mo><mo stretchy="false">(</mo><mi>σ</mi><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>i</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g = (\sigma p (g))i(k)</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">k \in K</annotation></semantics></math> by exactness.</p> <p>(ii) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math> (iii) Our previous elaborate discussion of cocycle cross products makes it obvious: choosing a section <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> which is a homomorphism gives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\chi(a,b) = 1</annotation></semantics></math>, and we can construct equivalent external semidirect product as a cocycle cross product with trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math>.</p> <p>(iii) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math> (i) Equivalence of extensions preserves the property of the corresponding short exact sequence to be split. Every external semidirect product is as a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>×</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">K\times B</annotation></semantics></math> and the product is given by formula <a class="maruku-eqref" href="#eq:mrule">(3)</a> without a cocycle. The map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\sigma : B \rightarrow G</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>∋</mo><mi>b</mi><mo>↦</mo><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>K</mi></msub><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>K</mi><mo>×</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">B \ni b \mapsto (1_K,b) \in K \times B</annotation></semantics></math>, splits the sequence.</p> </div> <div class="num_defn"> <h6 id="definition_13">Definition</h6> <p>An extension <a class="maruku-eqref" href="#eq:shortExtension">(1)</a> is Abelian iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is Abelian. An Abelian extension <a class="maruku-eqref" href="#eq:shortExtension">(1)</a> is central iff it is isomorphic to a cocycle cross product extension with all the automorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>,</mo><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\psi(b), b \in B</annotation></semantics></math> trivial. We say that the extension <a class="maruku-eqref" href="#eq:shortExtension">(1)</a> is Abelian iff <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 Abelian.</p> </div> <p>Remarks. (i) Note that <a class="maruku-eqref" href="#eq:psi2">(5)</a> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math> is a homomorphism if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> in the case of Abelian extensions (for any choice of set-theoretic section <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>.</p> <p>(ii) If <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 Abelian then <a class="maruku-eqref" href="#eq:shortExtension">(1)</a> is central, but not every central extension is corresponding to an Abelian <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>. Abelian extensions in terms of the above definition trivially (strictly!) include both central extensions and extensions with <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> central. By abuse of language one sometimes says for <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> to be an extension of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> what leads to strange expression that not every Abelian extension (as extension – in terms of the definition above) is Abelian (as a group).</p> <h4 id="2Coboundaries">Comparing different extensions; 2-coboundaries</h4> <p>Let us now investigate when two extensions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">G_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>, given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>,</mo><mi>χ</mi></mrow><annotation encoding="application/x-tex">\psi,\chi</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>′</mo><mo>,</mo><mi>χ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\psi',\chi'</annotation></semantics></math> respectively, are equivalent, cf. diagram <a class="maruku-eqref" href="#eq:equivExt">(2)</a>.</p> <p>We know that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><msub><mo stretchy="false">|</mo> <mi>K</mi></msub><mo>:</mo><mi>i</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mover><mo>↦</mo><mi>ϵ</mi></mover><mi>i</mi><mo>′</mo><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\epsilon|_K : i(k) \stackrel{\epsilon}\mapsto i'(k)</annotation></semantics></math>, for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">k \in K</annotation></semantics></math>. The formula for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> in \luse{crossform} says that whenever we represent an extension as a cocycle extension we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">i(k) = (e,\chi(e,e)^{-1}k).</annotation></semantics></math> Thus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\epsilon(e,\chi(e,e)^{-1}k) = (e,\chi'(e,e)^{-1}k)</annotation></semantics></math>, for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>K</mi><mo>.</mo></mrow><annotation encoding="application/x-tex">k \in K.</annotation></semantics></math> Also recall (or recalculate) that every element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,k)</annotation></semantics></math> in <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> can be factorized as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,e)(e,\chi(e,e)^{-1}k)</annotation></semantics></math>. By the definition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> is a homomorphism of groups, so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ϵ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mi>ϵ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\epsilon(a,k) = \epsilon(a,e)\epsilon(e,\chi(e,e)^{-1}k)</annotation></semantics></math>. Also the cosets are preserved, so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\epsilon(a,e) = (a,\lambda(a))</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">\lambda : B \rightarrow K</annotation></semantics></math> is some set-theoretic map. Thus</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ϵ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>χ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>χ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>k</mi><mo stretchy="false">)</mo><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \epsilon(a,k) & = & (a,\lambda(a))(e,\chi'(e,e)^{-1}k) \\ & = & (a, \chi'(a,e)\psi'(e)^{-1}(\lambda(a))\chi'(e,e)^{-1}k) \\ & = & (a, \lambda(a)k). } </annotation></semantics></math></div> <p>Now multiply more general elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ϵ</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>k</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>k</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><mi>λ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>k</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo>,</mo><mi>λ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><msub><mi>k</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>=</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub><mo>,</mo><mi>χ</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>k</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>λ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><msub><mi>k</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \epsilon((b_1,k_1)(b_2,k_2)) = (b_1,\lambda(b_1)k_1)(b_2,\lambda(b_2)k_2) \\ = (b_1b_2,\chi'(b_1,b_2)\psi'(b_2)^{-1}(\lambda(b_1)k_1)\lambda(b_2)k_2) } </annotation></semantics></math></div> <p>what should be the same as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>k</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub><mo>,</mo><mi>λ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>k</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \epsilon((b_1b_2,\chi(b_1,b_2)\psi(b_2)^{-1}(k_1)k_2)) = (b_1b_2,\lambda(b_1b_2)\chi(b_1b_2)\psi(b_2)^{-1}(k_1)k_2) </annotation></semantics></math></div> <p>In a special case, when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>k</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>e</mi> <mi>K</mi></msub></mrow><annotation encoding="application/x-tex">k_1 = e_K</annotation></semantics></math> we have therefore</p> <div class="maruku-equation" id="eq:equiv1"><span class="maruku-eq-number">(7)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>λ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>χ</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>λ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \chi(b_1,b_2) = \lambda(b_1b_2)^{-1}\chi'(b_1,b_2) \psi'(b_2)^{-1}(\lambda(b_1))\lambda(b_2) </annotation></semantics></math></div> <p>In order to obtain a relation between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\psi'(b)(k)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\psi(b)(k)</annotation></semantics></math> note that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>χ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> \epsilon ((e,\chi(e,e)^{-1}k)(b,e)) = (e, \chi'(e,e)^{-1}k)(b,\lambda(b)). </annotation></semantics></math></div> <p>That is equivalent to any in the following chain of formulas:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ϵ</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>χ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">)</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>χ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>χ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>e</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">)</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \epsilon (b,\chi(e,b)\psi(b)^{-1}(\chi(e,e)^{-1}k)) &=& (b, \psi'(b)^{-1}(\chi'(e,e)^{-1}k)\lambda(b)) \\ \Leftrightarrow \lambda(b)\chi(e,b)\psi(b)^{-1}(\chi(e,e)^{-1}k)) &=& \chi'(e,b)\psi'(b)^{-1}(\chi'(e,e)^{-1}k)\lambda(b) } </annotation></semantics></math></div> <p>Then by <a class="maruku-eqref" href="#eq:psiac">(6)</a>, it follows that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>λ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>χ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mi>χ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mo stretchy="false">(</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \lambda(b)\chi(e,b)\chi(e,b)^{-1}\psi(b)^{-1}(k) &=& \chi'(e,b)\chi'(e,b)^{-1}\psi'(b)^{-1}(k)\lambda(b) \\ \Leftrightarrow \lambda(b)\psi(b)^{-1}(k)) &=& (\psi'(b)^{-1}(k))\lambda(b) \\ \Leftrightarrow (Ad_K(\lambda(b)) \circ\psi(b)^{-1})(k) &=& \psi'(b)^{-1}(k) } </annotation></semantics></math></div> <p>Now invert the maps in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Aut(K)</annotation></semantics></math> to obtain</p> <div class="maruku-equation" id="eq:equiv2"><span class="maruku-eq-number">(8)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \psi'(b) = \psi(b)Ad_K(\lambda(b)^{-1}) </annotation></semantics></math></div> <p>Thus we obtain</p> <div class="num_theorem"> <h6 id="theorem_3">Theorem</h6> <p>Two extensions of a group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> by group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> with corresponding systems <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ψ</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\psi,\chi)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ψ</mi><mo>′</mo><mo>,</mo><mi>χ</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\psi',\chi')</annotation></semantics></math> are equivalent iff there is a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">\lambda: B \rightarrow K</annotation></semantics></math> such that the relations <a class="maruku-eqref" href="#eq:equiv1">(7)</a> and <a class="maruku-eqref" href="#eq:equiv2">(8)</a> are valid.</p> </div> <div class="proof"> <h6 id="proof_13">Proof</h6> <p>If function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> takes values in the center of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> then <a class="maruku-eqref" href="#eq:equiv2">(8)</a> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>′</mo><mo>=</mo><mi>ψ</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\psi' = \psi : B \rightarrow Aut(K)</annotation></semantics></math> and conversely.</p> <p>If instead of functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\psi'</annotation></semantics></math> we consider the respective maps into the group of external automorphisms (cosets of automorphisms with respect to the group of internal homomorphisms) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">]</mo><mo>:</mo><mo>~</mo><mi>B</mi><mo>→</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[\psi], [\psi']:~B \rightarrow Aut(K)/Int(K)</annotation></semantics></math>, then the equivalent extensions define the same maps. By <a class="maruku-eqref" href="#eq:psi2">(5)</a> these maps are actually homomorphisms (unlike e.g.<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math>). For a given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math> if there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math> so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ψ</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\psi,\chi)</annotation></semantics></math> does define an extension of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> we say that the extension is <em>associated</em> to (the homomorphism) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\psi]</annotation></semantics></math>. That does not mean that any given homomorphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>hom</mi> <mi>Group</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hom_{Group}(B,Aut(K)/Int(K))</annotation></semantics></math> is associated to any extension, nor it means that if a homomorphism is associated to some extension, that every its representative in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hom_{Set}(B,Aut(K))</annotation></semantics></math> is a part of a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ψ</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\psi,\chi)</annotation></semantics></math> defining an extension. To see that situation in more detail we start with a <em>given</em> automorphism, which we call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> , and <em>choose</em> an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>θ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\psi(a)\in\theta(a)</annotation></semantics></math>, the representative of a coset in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Aut(K)/Int(K)</annotation></semantics></math>; that choice should be specified for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">a \in B</annotation></semantics></math>. Note that for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>∈</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>,</mo><mi>a</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">\rho \in Aut(K), a \in K</annotation></semantics></math> we have, by direct inspection, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><msup><mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho Ad_K(a)\rho^{-1} = Ad_K(\rho(a))</annotation></semantics></math>. Thus there is a well-defined function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Ad</mi> <mi>K</mi></msub><mo>∘</mo><mi>h</mi><mo>:</mo><mi>B</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo>∘</mo><mi>h</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ad_K \circ h : B \times B \rightarrow Int(K), \,\,\, (Ad_K\circ h)(a,b) := \psi(a b)^{-1}\psi(a)\psi(b) </annotation></semantics></math></div> <ul> <li>indeed</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>r</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\psi(a)Ad_K(r_1)\psi(b)Ad_K(r_2) = \psi(a)\psi(b)Ad_K(\psi(b)^{-1}(r_1)r_2)</annotation></semantics></math></div> <p>so choosing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\psi(a b) \in [\psi(a b)]</annotation></semantics></math> is the same as choosing it in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\psi(a)][\psi(b)]</annotation></semantics></math> and guarantees that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\psi(a b)^{-1}\psi(a)\psi(b)</annotation></semantics></math> is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Int</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Int(K)</annotation></semantics></math>. Let us choose some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ad</mi> <mi>K</mi></msub><mo>∘</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">Ad_K \circ h</annotation></semantics></math> is interpretable as a genuine composition.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ (\psi(a)\psi(b))\psi(c) & = & \psi(a b)Ad_K(h(a,b))\psi(c) \\ & = & \psi(a b)\psi(c)\psi(c)^{-1}Ad_K(h(a,b))\psi(c) \\ & = & \psi(a b c)Ad_K(h(a b,c))Ad_K(\psi(c)^{-1}h(a,b)) \\ & = & \psi(a b c)Ad_K(h(a b,c)\psi(c)^{-1}h(a,b)) } </annotation></semantics></math></div> <p>what is by associativity the same as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \psi(a)(\psi(b)\psi(c)) & = & \psi(a)\psi(b c)Ad_K(h(b,c)) \\ & = & \psi(a b c)Ad_K(h(a,b c))Ad_K(h(b,c)) \\ & = & \psi(a b c)Ad_K(h(a,b c)h(b,c)). } </annotation></semantics></math></div> <p>Thus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">Ad_K(h(a b,c)\psi(c)^{-1}h(a,b)) = Ad_K(h(a,b c)h(b,c)).</annotation></semantics></math> Two elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> generate the same automorphism iff they differ by a central element. Thus</p> <div class="maruku-equation" id="eq:2semicoc"><span class="maruku-eq-number">(9)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> h(a b,c)\psi(c)^{-1}h(a,b) = h(a,b c)h(b,c)z(a,b,c) </annotation></semantics></math></div> <p>for a unique central element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Z</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">z(a,b,c) \in Z(K).</annotation></semantics></math> The correspondence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>:</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">z : (a,b,c) \mapsto z(a,b,c)</annotation></semantics></math> maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>×</mo><mi>B</mi><mo>×</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">B \times B \times B</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(K)</annotation></semantics></math>.</p> </div> <div class="num_prop"> <h6 id="proposition_12">Proposition</h6> <p><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">z</annotation></semantics></math> is an (Abelian) 3-cocycle with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo></mo><mo stretchy="false">/</mo></msub><mi>Z</mi><mo stretchy="false">(</mo><mi>K</mi><msub><mo stretchy="false">)</mo> <mrow><msup><mi>ψ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></msub></mrow><annotation encoding="application/x-tex">_/Z(K)_{\psi^{-1}}</annotation></semantics></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(K)</annotation></semantics></math> understood as trivial-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ψ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\psi^{-1}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-bimodule):</p> <div class="maruku-equation" id="eq:3coc"><span class="maruku-eq-number">(10)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>d</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mi>d</mi><mo stretchy="false">)</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> z(b,c,d)z(a,b c,d)\psi(d)^{-1}z(a,b,c) = z(a,b,c d)z(a b,c,d) </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_14">Proof</h6> <p>To see this we calcuate</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>d</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>d</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mi>d</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">(</mo><mi>b</mi><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>d</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mi>d</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mi>d</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mi>z</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>d</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ h(a b c,d)[\psi(d)^{-1}h(a b,c)\psi(c)^{-1}h(a,b)] & = h(a b c,d)[\psi(d)^{-1}h(a,b c)h(b,c)z(a,b,c)] \\ & = h(a,b c d)h(b c,d)z(a,b c,d)[\psi(d)^{-1}h(b,c)z(a,b,c)] \\ & = h(a,b c d)h(b,c d)h(c,d)z(b,c,d)z(a,b c,d)\psi(d)^{-1}z(a,b,c) } </annotation></semantics></math></div> <p>Compare</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>d</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>d</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>d</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>d</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>h</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><mi>d</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>h</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mi>d</mi><mo stretchy="false">)</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><mi>d</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>h</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mi>d</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mi>d</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mi>d</mi><mo stretchy="false">)</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ h(a b c,d)[\psi(d)^{-1}h(a b,c)\psi(c)^{-1}h(a,b)] & = h(a b c,d)[\psi(d)^{-1}h(a,b c)]\psi(d)^{-1}\psi(c)^{-1}h(a,b) \\ & = h(a b c,d)[\psi(d)^{-1}h(a b,c)]h(c,d)^{-1}[\psi(c d)^{-1}h(a,b)]h(c,d) \\ & = h(a b c,d)h(a b,c d)z(a b,c,d)[\psi(c d)^{-1}h(a,b)]h(c,d) \\ & = h(a,b c d)h(b,c d)h(c,d)z(a,b,c d)z(a b,c,d) } </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition_13">Proposition</h6> <p>(i) If we choose a different <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">Ad_K(h(a,b)) = \psi(a b)^{-1}\psi(a)\psi(b),</annotation></semantics></math></div> <p>then <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">z</annotation></semantics></math> will change only up to a 3-coboundary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi><mo>,</mo></mrow><annotation encoding="application/x-tex">d f,</annotation></semantics></math> i.e. there is a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>B</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>Z</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f : B \times B \rightarrow Z(K)</annotation></semantics></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>′</mo><mo>=</mo><mo stretchy="false">(</mo><mi>d</mi><mi>f</mi><mo stretchy="false">)</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">z' = (d f)z</annotation></semantics></math> where</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><mi>d</mi><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>for</mi><mi>all</mi><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><mi>B</mi><mo>.</mo></mrow><annotation encoding="application/x-tex"> (d f)(a,b,c) = f^{-1}(b,c)f^{-1}(a,b c)f(a b,c)\psi(c)^{-1}f(a,b),\,\,\,\,\, for all a,b,c \in B. </annotation></semantics></math></div> <p>(ii) Conversely, if <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">z</annotation></semantics></math> is a 3-cocycle obtained from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math> as above and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">d f</annotation></semantics></math> is a 3-coboundary, then there is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">h'</annotation></semantics></math> determining the same inner automoprhism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> such that the corresponding 3-cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">z'</annotation></semantics></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>df</mi><mo stretchy="false">)</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">(df)z</annotation></semantics></math>.</p> <p>(iii) Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>,</mo><mi>ψ</mi><mo>′</mo><mo>:</mo><mi>B</mi><mo>→</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\psi, \psi' : B \rightarrow Aut(K)</annotation></semantics></math> be two set-theoretic sections so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">]</mo><mo>=</mo><mi>θ</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[\psi] = [\psi'] = \theta : B \rightarrow Aut(K)/Int(K)</annotation></semantics></math>, then (for arbitrary choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">h'</annotation></semantics></math>) the cocycles <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">z</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">z'</annotation></semantics></math> obtained as above differ only up to a 3-coboundary. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo></mrow><annotation encoding="application/x-tex">\|</annotation></semantics></math></p> </div> <div class="proof"> <h6 id="proof_15">Proof</h6> <p>(i) Choose two different <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>′</mo><mo>,</mo><mi>h</mi><mo>:</mo><mi>B</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">h',h: B \times B \rightarrow K</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo>′</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ad_K(h') = Ad_K(h)</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h'(a,b) = h(a,b)f(a,b)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>B</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>Z</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f : B \times B \rightarrow Z(K)</annotation></semantics></math> is some function with values in center of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>. A direct comparison of <a class="maruku-eqref" href="#eq:2semicoc">(9)</a> written for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>,</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">h,z</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>′</mo><mo>,</mo><mi>z</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">h',z'</annotation></semantics></math> respectively proves the assertion.</p> <p>(ii) Trivial: Any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>B</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>Z</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f : B \times B \rightarrow Z(K)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>′</mo><mo>=</mo><mi>hf</mi></mrow><annotation encoding="application/x-tex">h' = hf</annotation></semantics></math> will not change the inner automorphism. Thus any central 3-coboundary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>df</mi></mrow><annotation encoding="application/x-tex">df</annotation></semantics></math> can be obtained by changing a choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math>.</p> <p>(iii) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\psi'] = [\psi]</annotation></semantics></math> implies that exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>K</mi><mo>,</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">k : B \rightarrow K, \psi'(a) = \psi(a)Ad_K(k(a)).</annotation></semantics></math> Then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>k</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>k</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>k</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \psi'(a b)Ad_K(h'(a,b)) & = & \psi'(a)\psi'(b) = \psi(a)Ad_K(k(a))\psi(b)Ad_K(k(b))\\ & = & \psi(a)\psi(b)Ad_K([\psi(b)^{-1}k(a)]k(b)) \\ & = & \psi(a b)Ad_K(h(a,b)[\psi(b)^{-1}k(a)]k(b)) \\ & = & \psi'(a b)Ad_K(k(a b)^{-1}h(a,b)[\psi(b)^{-1}k(a)]k(b)). } </annotation></semantics></math></div> <p>Thus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>k</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex"> h'(a,b) = k(a b)^{-1}h(a,b)[\psi(b)^{-1}k(a)]k(b),</annotation></semantics></math> for appropriate choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">h'</annotation></semantics></math> - what can change <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">z'</annotation></semantics></math> up to coboundary - using the freedom from (i). If we want formula involving <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\psi'</annotation></semantics></math> instead than we use <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mi>Ad</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\psi'(a) = \psi(a)Ad_K(k(a))</annotation></semantics></math> to obtain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mi>k</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k(a b)h'(a,b) = h(a,b)k(b)[\psi'(b)^{-1}k(a)]</annotation></semantics></math>. Using that and previous identities,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>k</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>k</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>k</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mi>k</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>k</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>k</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>k</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>k</mi><mo stretchy="false">(</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>k</mi><mo stretchy="false">(</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">[</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>k</mi><mo stretchy="false">(</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>k</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ k(a b c)h'(a b,c)\psi'(c)^{-1}h'(a,b) &=& h(a b,c)k(c)[\psi'(c)^{-1}k(a b)]\psi'(c)^{-1}h'(a,b) \\ &=& h(a b,c)k(c)\psi'(c)^{-1}k(a b)h'(a,b) \\ &=& h(a b,c)k(c)\psi'(c)^{-1}h(a,b)k(b)[\psi'(b)^{-1}k(a)] \\ &=& h(a b,c)[\psi(c)^{-1}h(a,b)]k(c)\psi'(c)^{-1}k(b)[\psi'(b)^{-1}k(a)] \\ &=& h(a,b c)h(b,c)z(a,b,c)k(c) [\psi'(c)^{-1}k(b)][\psi'(c)^{-1}\psi'(b)^{-1}k(a)] \\ &=& h(a,b c)h(b,c)k(c)[\psi'(c)^{-1}k(b)] [\psi'(c)^{-1}\psi'(b)^{-1}k(a)]z(a,b,c) \\ &=& h(a,b c)k(b c)h'(b,c)[\psi'(c)^{-1}\psi'(b)^{-1}k(a)]z(a,b,c) \\ &=& h(a,b c)k(b c)h'(b,c)h'(b,c)^{-1}[\psi'(b c)^{-1}k(a)]h'(b,c)z(a,b,c)\\ &=& h(a,b c)k(b c)[\psi'(b c)^{-1}k(a)]h'(b,c)z(a,b,c)\\ &=& k(a b c)h'(a,b c)h'(b,c)z(a,b,c) } </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">a,b,c \in B</annotation></semantics></math>. Thus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mi>h</mi><mo>′</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>z</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h'(a b,c)\psi'(c)^{-1}h'(a,b) = h'(a,b c)h'(b,c)z(a,b,c)</annotation></semantics></math> i.e. our choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">h'</annotation></semantics></math> insured no change in <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">z</annotation></semantics></math>. Of course that means that in arbitrary choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">h'</annotation></semantics></math> we do not miss more than a coboundary by (i).</p> </div> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>A given homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>:</mo><mi>B</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\theta : B \times B \rightarrow Aut(K)/Int(K)</annotation></semantics></math> is associated to some extension of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> iff <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">z</annotation></semantics></math> is a 3-coboundary.</p> </div> <div class="proof"> <h6 id="proof_16">Proof</h6> <p>Indeed, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> is associated to an extension, then we know that there is an isomorphism of the extension with a cross product given by some cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math> and some automorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">]</mo><mo>=</mo><mi>θ</mi></mrow><annotation encoding="application/x-tex">[\psi] = \theta</annotation></semantics></math>. But using the identification, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo>=</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">\chi = h</annotation></semantics></math> for that particular choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">z = 1</annotation></semantics></math>. By the proposition, every other <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">z</annotation></semantics></math> obtained from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> is in the same cohomology class, thus every such <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">z</annotation></semantics></math> is a coboundary. Conversely, if <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">z</annotation></semantics></math> is a coboundary, then by the proposition, we can change it to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">z = 1</annotation></semantics></math>, and then we have all the conditions for a cross product extension satisfied.</p> </div> <h4 id="SchreierTheorynPOV">Formulation in homotopy theory</h4> <p>One may regard the above from the <a class="existingWikiWord" href="/nlab/show/nPOV">nPOV</a> as a special case of the way cocycles in the general notion of <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> classify their <a class="existingWikiWord" href="/nlab/show/homotopy+fibers">homotopy fibers</a>. More on this is at</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/nonabelian+group+cohomology">nonabelian group cohomology</a>.</p> </li> </ul> <h2 id="examples">Examples</h2> <p>By the above classification theorems, all the examples at <em><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a></em> equivalently induce examples for group extensions. And indeed by definition every <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> defines an extension.</p> <p>But examples of fundamental importance include for instance</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> as an extension of the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>→</mo><mi>ℝ</mi><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{Z} \to \mathbb{R} \to U(1) \,. </annotation></semantics></math></div></li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> as an extension of the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>→</mo><mi>Spin</mi><mo>→</mo><mi>SO</mi></mrow><annotation encoding="application/x-tex"> \mathbb{Z}_2 \to Spin \to SO </annotation></semantics></math></div></li> <li> <p>etc.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+central+extension">universal central extension</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a></p> </li> </ul> </li> <li> <p><strong>group extension</strong>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> <p><a class="existingWikiWord" href="/nlab/show/Ext-group">Ext-group</a>, <a class="existingWikiWord" href="/nlab/show/central+extension">central extension</a>, <a class="existingWikiWord" href="/nlab/show/maximal+central+extension">maximal central extension</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer+sum">Baer sum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/solvable+group">solvable group</a>, <a class="existingWikiWord" href="/nlab/show/nilpotent+group">nilpotent group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+extension">ring extension</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid+cohomology">∞-Lie groupoid cohomology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/central+extension+of+groupoids">central extension of groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+extension">Lie algebra extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/central+charge">central charge</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general_2">General</h3> <p>Original articles:</p> <ul> <li id="EilenbergMacLane42"> <p><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, <em>Group Extensions and Homology</em>, Annals of Mathematics <strong>43</strong> 4 (1942) 757-831 [<a href="https://doi.org/10.2307/1968966">doi:10.2307/1968966</a>, <a href="https://www.jstor.org/stable/1968966">jstor:1968966</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, <em>Cohomology theory in abstract groups. II. Group extensions with a non-Abelian kernel</em>, Ann. of Math. (2) 48, (1947). 326–341 <a href="http://www.jstor.org/pss/1969174">jstor:1969174</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, <em>Cohomology theory in abstract groups</em>. III. Operator homomorphisms of kernels. Ann. of Math. (2) 50, (1949). 736–761.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lawrence+Breen">Lawrence Breen</a>, Théorie de Schreier supérieure, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 465–514 <a href="http://www.numdam.org/item?id=ASENS_1992_4_25_5_465_0">numdam</a>.</p> </li> </ul> <p>Textbooks accounts</p> <ul> <li> <p>A. G. Kurosh, Theory of groups</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kenneth+Brown">Kenneth Brown</a>, <em>Cohomology of Groups</em>, Graduate Texts in Mathematics, <strong>87</strong>, Springer 1982 (<a href="https://link.springer.com/book/10.1007/978-1-4684-9327-6">doi:10.1007/978-1-4684-9327-6</a>)</p> </li> </ul> <p>Lecture notes and similar include</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Brian+Conrad">Brian Conrad</a>, <em>Group cohomology and group extensions</em> (<a href="http://math.stanford.edu/~conrad/249BPage/handouts/gpext.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Terry+Tao">Terry Tao</a>, <em>Some notes on group extensions</em> (<a href="http://terrytao.wordpress.com/2010/01/23/some-notes-on-group-extensions/">blog</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Patrick+Morandi">Patrick Morandi</a>, <em>Group extensions and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">H^3</annotation></semantics></math></em> (<a href="http://sierra.nmsu.edu/morandi/notes/GroupExtensions.pdf">pdf</a>)</p> <p><em>Nobabelian cohomology</em> (<a href="http://sierra.nmsu.edu/morandi/notes/nonabeliancohomology.pdf">pdf</a>)</p> </li> <li> <p>Raphael Ho, <em>Classifications of group extensions and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">H^2</annotation></semantics></math></em> (<a href="http://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Ho.pdf">pdf</a>)</p> </li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/R.+Brown">R. Brown</a>, <a class="existingWikiWord" href="/nlab/show/T.+Porter">T. Porter</a>, <em>On the Schreier theory of non-abelian extensions: generalisations and computations</em>, Proc. Roy. Irish Acad. Sect. A, 96 (1996), 213 – 227.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Manuel+Bullejos">Manuel Bullejos</a>, <a class="existingWikiWord" href="/nlab/show/Antonio+M.+Cegarra">Antonio M. Cegarra</a>, A 3-dimensional non-abelian cohomology of groups with applications to homotopy classification of continuous maps Canad. J. Math., vol. 43, (2), 1991, 1-32.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Antonio+M.+Cegarra">Antonio M. Cegarra</a>, <a class="existingWikiWord" href="/nlab/show/Antonio+R.+Garz%C3%B3n">Antonio R. Garzón</a>, A long exact sequence in non-abelian cohomology, Proc. Int. Conf. Como 1990., Lec. Notes in Math. 1488, Springer 1991.</p> </li> </ul> <p>A theory for central 2-group extensions is here:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Antonio+R.+Garz%C3%B3n">Antonio R. Garzón</a> and E.M. Vitale, On the second cohomology categorical group and a Hochschild-Serre 2-exact sequence, Theory and Applications of Categories, Vol. 30 (2015), 933-984. (<a href="http://www.tac.mta.ca/tac/volumes/30/27/30-27.pdf">pdf</a>)</li> </ul> <p>See also references to Dedecker listed <a class="existingWikiWord" href="/zoranskoda/show/Paul+Dedecker">here</a>.</p> <h3 id="applications">Applications</h3> <p>A bit of discussion of some occurences of central extensions of groups in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> is in</p> <ul> <li>G. Tuynman and W. Wiegerinck, <em>Central extensions of physics</em> (<a href="http://math.univ-lille1.fr/~gmt/PaperFolder/CentralExtensions.pdf">pdf</a>)</li> </ul> <p>(In fact there are many more than mentioned in that introduction.)</p> <p>Extensions of <a class="existingWikiWord" href="/nlab/show/supergroups">supergroups</a> are discussed in</p> <ul> <li id="Drupieski14"><a class="existingWikiWord" href="/nlab/show/Christopher+Drupieski">Christopher Drupieski</a>, <em>Strict polynomial superfunctors and universal extension classes for algebraic supergroups</em> (<a href="http://arxiv.org/abs/1408.5764">arXiv:1408.5764</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 23, 2023 at 14:06:39. 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