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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"> <p class="show_diff"> Showing changes from revision #12 to #13: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='group_theory'>Group Theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/group+theory'>group theory</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/group'>group</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-group'>∞-group</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/group+object'>group object</a>, <a class='existingWikiWord' href='/nlab/show/diff/groupoid+object+in+an+%28infinity%2C1%29-category'>group object in an (∞,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian group</a>, <a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/super+abelian+group'>super abelian group</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/action'>group action</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-action'>∞-action</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/representation'>representation</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-representation'>∞-representation</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/progroup'>progroup</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/homogeneous+space'>homogeneous space</a></li> </ul> <p><strong>Classical groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/general+linear+group'>general linear group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/unitary+group'>unitary group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/special+unitary+group'>special unitary group</a>. <a class='existingWikiWord' href='/nlab/show/diff/projective+unitary+group'>projective unitary group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/orthogonal+group'>orthogonal group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/special+orthogonal+group'>special orthogonal group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symplectic+group'>symplectic group</a></p> </li> </ul> <p><strong>Finite groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finite+group'>finite group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+group'>symmetric group</a>, <a class='existingWikiWord' href='/nlab/show/diff/cyclic+group'>cyclic group</a>, <a class='existingWikiWord' href='/nlab/show/diff/braid+group'>braid group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classification+of+finite+simple+groups'>classification of finite simple groups</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sporadic+finite+simple+group'>sporadic finite simple groups</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Monster+group'>Monster group</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mathieu+group'>Mathieu group</a></li> </ul> </li> </ul> <p><strong>Group schemes</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebraic+group'>algebraic group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/abelian+variety'>abelian variety</a></p> </li> </ul> <p><strong>Topological groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+topological+group'>compact topological group</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+group'>locally compact topological group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/maximal+compact+subgroup'>maximal compact subgroup</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/string+group'>string group</a></p> </li> </ul> <p><strong>Lie groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+group'>Lie group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+Lie+group'>compact Lie group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/supergroup'>super Lie group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/2-group'>2-group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/crossed+module'>crossed module</a>, <a class='existingWikiWord' href='/nlab/show/diff/strict+2-group'>strict 2-group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/n-group'>n-group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-group'>∞-group</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simplicial+group'>simplicial group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/crossed+complex'>crossed complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/k-tuply+groupal+n-groupoid'>k-tuply groupal n-groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/circle+n-group'>circle n-group</a>, <a class='existingWikiWord' href='/nlab/show/diff/string+2-group'>string 2-group</a>, <a class='existingWikiWord' href='/nlab/show/diff/fivebrane+6-group'>fivebrane Lie 6-group</a></p> </li> </ul> <p><strong>Cohomology and Extensions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/group+cohomology'>group cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/group+extension'>group extension</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-group+extension'>∞-group extension</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ext'>Ext-group</a></p> </li> </ul> <p><strong>Related concepts</strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/quantum+group'>quantum group</a></li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#definition'>Definition</a><ul><li><a href='#discrete_case'>Discrete case</a></li><li><a href='#category_of_orbits'>Category of orbits</a></li><li><a href='#topological_case'>Topological case</a></li></ul></li><ins class='diffins'><li><a href='#examples'>Examples</a></li></ins><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='definition'>Definition</h2> <h3 id='discrete_case'>Discrete case</h3> <p>Given an <a class='existingWikiWord' href='/nlab/show/diff/action'>action</a> <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>G\times X\to X</annotation></semantics></math> of a (<a class='existingWikiWord' href='/nlab/show/diff/discrete+group'>discrete</a>) <a class='existingWikiWord' href='/nlab/show/diff/group'>group</a> <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> on a set <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, any <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> of the form <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mi>x</mi><mo>=</mo><mo stretchy='false'>{</mo><mi>g</mi><mi>x</mi><mo stretchy='false'>|</mo><mi>g</mi><mo>∈</mo><mi>G</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>G x = \{g x|g\in G\}</annotation></semantics></math> for a fixed <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x\in X</annotation></semantics></math> is called an <strong>orbit</strong> of the action, or the <strong><math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-orbit through the point <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math></strong>. The set <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union'>disjoint union</a> of its orbits.</p> <h3 id='category_of_orbits'>Category of orbits</h3> <p>The <em>category of orbits</em> of a <a class='existingWikiWord' href='/nlab/show/diff/group'>group</a> <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> is the full subcategory of the category of sets with an action of <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>.</p> <p>Since any orbit of <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> is isomorphic to the orbit <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo stretchy='false'>/</mo><mi>H</mi></mrow><annotation encoding='application/x-tex'>G/H</annotation></semantics></math> for some group <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math>, the category of <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-orbits admits the following alternative description: its objects are subgroups <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> and morphisms <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>H</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>H</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>H_1\to H_2</annotation></semantics></math> are elements <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>g</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>G</mi><mo stretchy='false'>/</mo><msub><mi>H</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>[g]\in G/H_2</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>H</mi> <mn>1</mn></msub><mo>⊂</mo><mi>g</mi><msub><mi>H</mi> <mn>2</mn></msub><msup><mi>g</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>H_1\subset g H_2g^{-1}</annotation></semantics></math>.</p> <p>In particular, the group of automorphisms of a <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-orbit <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo stretchy='false'>/</mo><mi>H</mi></mrow><annotation encoding='application/x-tex'>G/H</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>N</mi> <mi>G</mi></msub><mo stretchy='false'>(</mo><mi>H</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><mi>H</mi></mrow><annotation encoding='application/x-tex'>N_G(H)/H</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>N</mi> <mi>G</mi></msub><mo stretchy='false'>(</mo><mi>H</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>N_G(H)</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/normalizer'>normalizer</a> of <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>.</p> <h3 id='topological_case'>Topological case</h3> <p>If <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a>, <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> and the action <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous</a>, then one can distinguish <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed</a> orbits from those which are not. Even when one starts with <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo>,</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>G,X</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff</a>, the space of orbits is typically non-Hausdorff. (This problem is one of the motivations of the <a class='existingWikiWord' href='/nlab/show/diff/noncommutative+geometry'>noncommutative geometry</a> of Connes’ school.)</p> <p>If the original space is <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact</a> Hausdorff, then every orbit <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mi>x</mi></mrow><annotation encoding='application/x-tex'>G x</annotation></semantics></math> as a topological <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-space is isomorphic to <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo stretchy='false'>/</mo><mi>H</mi></mrow><annotation encoding='application/x-tex'>G/H</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/stabilizer+group'>stabilizer subgroup</a> of <math class='maruku-mathml' display='inline' id='mathml_6006d5b14e05f738364b75503dffa610c4bb9b79_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>.</p> <ins class='diffins'><h2 id='examples'>Examples</h2></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li>An orbit of a <a class='existingWikiWord' href='/nlab/show/diff/cyclic+group'>cyclic</a> <a class='existingWikiWord' href='/nlab/show/diff/subgroup'>subgroup</a> of a <a class='existingWikiWord' href='/nlab/show/diff/permutation+group'>permutation group</a> is called a <em><a class='existingWikiWord' href='/nlab/show/diff/cycle+of+a+permutation'>permutation cycle</a></em>.</li> </ul></ins><ins class='diffins'> </ins><h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/orbit+category'>orbit category</a>, <a class='existingWikiWord' href='/nlab/show/diff/orbit+type'>orbit type</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/coadjoint+orbit'>coadjoint orbit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/coinvariant'>coinvariant</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/coset'>coset space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/orbit+method'>orbit method</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/slice+theorem'>slice theorem</a></p> </li> </ul> <h2 id='references'>References</h2><ins class='diffins'> </ins><ins class='diffins'><p>Textbook accounts:</p></ins> <ul> <li id='Bredon72'><a class='existingWikiWord' href='/nlab/show/diff/Glen+Bredon'>Glen Bredon</a>, Sections I.3, I.4 of: <em><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+compact+transformation+groups'>Introduction to compact transformation groups</a></em>, Academic Press 1972 (<a href='https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1'>ISBN 9780080873596</a>, <a href='http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf'>pdf</a>)</li> </ul> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on April 18, 2021 at 16:22:18. 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