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data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <b>exceptional isomorphism</b>, also called an <b>accidental isomorphism</b>, is an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> between members <i>a</i><sub><i>i</i></sub> and <i>b</i><sub><i>j</i></sub> of two families, usually infinite, of mathematical objects, which is incidental, in that it is not an instance of a general pattern of such isomorphisms.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> These coincidences are at times considered a matter of trivia,<sup id="cite_ref-FOOTNOTEWilson2009[httpwwwmathsqmulacuk~rawfsgs_filesintrops_Chapter_1:_Introduction]_2-0" class="reference"><a href="#cite_note-FOOTNOTEWilson2009[httpwwwmathsqmulacuk~rawfsgs_filesintrops_Chapter_1:_Introduction]-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> but in other respects they can give rise to consequential phenomena, such as <a href="/wiki/Exceptional_object" title="Exceptional object">exceptional objects</a>.<sup id="cite_ref-FOOTNOTEWilson2009[httpwwwmathsqmulacuk~rawfsgs_filesintrops_Chapter_1:_Introduction]_2-1" class="reference"><a href="#cite_note-FOOTNOTEWilson2009[httpwwwmathsqmulacuk~rawfsgs_filesintrops_Chapter_1:_Introduction]-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In the following, coincidences are organized according to the structures where they occur. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Groups">Groups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exceptional_isomorphism&action=edit&section=1" title="Edit section: Groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Finite_simple_groups">Finite simple groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exceptional_isomorphism&action=edit&section=2" title="Edit section: Finite simple groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The exceptional isomorphisms between the series of <a href="/wiki/Finite_simple_group" class="mw-redirect" title="Finite simple group">finite simple groups</a> mostly involve <a href="/wiki/Projective_special_linear_group" class="mw-redirect" title="Projective special linear group">projective special linear groups</a> and <a href="/wiki/Alternating_group" title="Alternating group">alternating groups</a>, and are:<sup id="cite_ref-FOOTNOTEWilson2009[httpswebarchiveorgweb20070824011953httpswwwmathsqmulacuk~rawfsgs_filesintrops_Chapter_1:_Introduction]_3-0" class="reference"><a href="#cite_note-FOOTNOTEWilson2009[httpswebarchiveorgweb20070824011953httpswwwmathsqmulacuk~rawfsgs_filesintrops_Chapter_1:_Introduction]-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <ul><li>PSL<sub>2</sub>(4) ≅ PSL<sub>2</sub>(5) ≅ A<sub>5</sub>, the smallest non-abelian simple group (order 60);</li> <li>PSL<sub>2</sub>(7) ≅ PSL<sub>3</sub>(2), the second-smallest non-abelian simple group (order 168) – <a href="/wiki/PSL(2,7)" title="PSL(2,7)">PSL(2,7)</a>;</li> <li>PSL<sub>2</sub>(9) ≅ A<sub>6</sub>;</li> <li>PSL<sub>4</sub>(2) ≅ A<sub>8</sub>;</li> <li>PSU<sub>4</sub>(2) ≅ PSp<sub>4</sub>(3), between a <a href="/wiki/Projective_special_unitary_group" class="mw-redirect" title="Projective special unitary group">projective special unitary group</a> and a <a href="/wiki/Projective_symplectic_group" class="mw-redirect" title="Projective symplectic group">projective symplectic group</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Alternating_groups_and_symmetric_groups">Alternating groups and symmetric groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exceptional_isomorphism&action=edit&section=3" title="Edit section: Alternating groups and symmetric groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/The_exceptional_outer_automorphism_of_S6" class="mw-redirect" title="The exceptional outer automorphism of S6">The exceptional outer automorphism of S6</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Compound_of_five_tetrahedra.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Compound_of_five_tetrahedra.png/220px-Compound_of_five_tetrahedra.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Compound_of_five_tetrahedra.png/330px-Compound_of_five_tetrahedra.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Compound_of_five_tetrahedra.png/440px-Compound_of_five_tetrahedra.png 2x" data-file-width="1000" data-file-height="1000" /></a><figcaption>The <a href="/wiki/Compound_of_five_tetrahedra" title="Compound of five tetrahedra">compound of five tetrahedra</a> expresses the exceptional isomorphism between the chiral icosahedral group and the alternating group on five letters.</figcaption></figure> <p>There are coincidences between symmetric/alternating groups and small <a href="/wiki/Groups_of_Lie_type" class="mw-redirect" title="Groups of Lie type">groups of Lie type</a>/<a href="/wiki/Polyhedral_groups" class="mw-redirect" title="Polyhedral groups">polyhedral groups</a>:<sup id="cite_ref-FOOTNOTEWilson2009Chapter_3_4-0" class="reference"><a href="#cite_note-FOOTNOTEWilson2009Chapter_3-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <ul><li>S<sub>3</sub> ≅ PSL<sub>2</sub>(2) ≅ <a href="/wiki/Dihedral_group_of_order_6" title="Dihedral group of order 6">dihedral group of order 6</a>,</li> <li>A<sub>4</sub> ≅ PSL<sub>2</sub>(3),</li> <li>S<sub>4</sub> ≅ PGL<sub>2</sub>(3) ≅ PSL<sub>2</sub>(<b>Z</b> / 4),</li> <li>A<sub>5</sub> ≅ PSL<sub>2</sub>(4) ≅ PSL<sub>2</sub>(5),</li> <li>S<sub>5</sub> ≅ PΓL<sub>2</sub>(4) ≅ PGL<sub>2</sub>(5),</li> <li>A<sub>6</sub> ≅ PSL<sub>2</sub>(9) ≅ Sp<sub>4</sub>(2)′,</li> <li>S<sub>6</sub> ≅ Sp<sub>4</sub>(2),</li> <li>A<sub>8</sub> ≅ PSL<sub>4</sub>(2) ≅ O<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:0.9;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">+</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">6</sub></span></span>(2)′,</li> <li>S<sub>8</sub> ≅ O<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:0.9;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">+</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">6</sub></span></span>(2).</li></ul> <p>These can all be explained in a systematic way by using linear algebra (and the action of S<sub><i>n</i></sub> on affine <i>n</i>space) to define the isomorphism going from the right side to the left side. (The above isomorphisms for A<sub>8</sub> and S<sub>8</sub> are linked via the exceptional isomorphism <span class="nowrap">SL<sub>4</sub> / <i>μ</i><sub>2</sub> ≅ SO<sub>6</sub></span>.) </p><p>There are also some coincidences with symmetries of <a href="/wiki/Regular_polyhedra" class="mw-redirect" title="Regular polyhedra">regular polyhedra</a>: the alternating group A<sub>5</sub> agrees with the chiral <a href="/wiki/Icosahedral_group" class="mw-redirect" title="Icosahedral group">icosahedral group</a> (itself an exceptional object), and the <a href="/wiki/Covering_groups_of_the_alternating_and_symmetric_groups" title="Covering groups of the alternating and symmetric groups">double cover</a> of the alternating group A<sub>5</sub> is the <a href="/wiki/Binary_icosahedral_group" title="Binary icosahedral group">binary icosahedral group</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Trivial_group">Trivial group</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exceptional_isomorphism&action=edit&section=4" title="Edit section: Trivial group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Trivial_group" title="Trivial group">trivial group</a> arises in numerous ways. The trivial group is often omitted from the beginning of a classical family. For instance: </p> <ul><li>C<sub>1</sub>, the cyclic group of order 1;</li> <li>A<sub>0</sub> ≅ A<sub>1</sub> ≅ A<sub>2</sub>, the alternating group on 0, 1, or 2 letters;</li> <li>S<sub>0</sub> ≅ S<sub>1</sub>, the symmetric group on 0 or 1 letters;</li> <li>GL(0, <i>K</i>) ≅ SL(0, <i>K</i>) ≅ PGL(0, <i>K</i>) ≅ PSL(0, <i>K</i>), linear groups of a 0-dimensional vector space;</li> <li>SL(1, <i>K</i>) ≅ PGL(1, <i>K</i>) ≅ PSL(1, <i>K</i>), linear groups of a 1-dimensional vector space</li> <li>and many others.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Spheres">Spheres</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exceptional_isomorphism&action=edit&section=5" title="Edit section: Spheres"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The spheres <i>S</i><sup>0</sup>, <i>S</i><sup>1</sup>, and <i>S</i><sup>3</sup> admit group structures, which can be described in many ways: </p> <ul><li><i>S</i><sup>0</sup> ≅ Spin(1) ≅ O(1) ≅ (<b>Z</b> / 2<b>Z</b>)<sup>+</sup> ≅ <b>Z</b><sup>×</sup>, the last being the group of units of the integers;</li> <li><i>S</i><sup>1</sup> ≅ Spin(2) ≅ SO(2) ≅ U(1) ≅ <b>R</b> / <b>Z</b> ≅ <a href="/wiki/Circle_group" title="Circle group">circle group</a>;</li> <li><i>S</i><sup>3</sup> ≅ Spin(3) ≅ SU(2) ≅ Sp(1) ≅ <a href="/wiki/Versors" class="mw-redirect" title="Versors">unit quaternions</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Spin_groups">Spin groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exceptional_isomorphism&action=edit&section=6" title="Edit section: Spin groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In addition to Spin(1), Spin(2) and Spin(3) above, there are isomorphisms for higher dimensional <a href="/wiki/Spin_group" title="Spin group">spin groups</a>: </p> <ul><li>Spin(4) ≅ Sp(1) × Sp(1) ≅ SU(2) × SU(2)</li> <li>Spin(5) ≅ Sp(2)</li> <li>Spin(6) ≅ SU(4)</li></ul> <p>Also, <a href="/wiki/Spin(8)" class="mw-redirect" title="Spin(8)">Spin(8)</a> has an exceptional order 3 <a href="/wiki/Triality" title="Triality">triality</a> automorphism. </p> <div class="mw-heading mw-heading2"><h2 id="Coxeter–Dynkin_diagrams"><span id="Coxeter.E2.80.93Dynkin_diagrams"></span>Coxeter–Dynkin diagrams</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exceptional_isomorphism&action=edit&section=7" title="Edit section: Coxeter–Dynkin diagrams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Klein_correspondence" class="mw-redirect" title="Klein correspondence">Klein correspondence</a></div> <p>There are some exceptional isomorphisms of <a href="/wiki/Dynkin_diagram#Isomorphisms" title="Dynkin diagram">Dynkin diagrams</a>, yielding isomorphisms of the corresponding <a href="/wiki/Coxeter_group" title="Coxeter group">Coxeter groups</a> and of polytopes realizing the symmetries, as well as isomorphisms of <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a> whose root systems are described by the same diagrams. These are: </p> <table class="wikitable"> <tbody><tr> <th>Diagram </th> <th>Dynkin classification </th> <th>Lie algebra </th> <th>Polytope </th></tr> <tr align="center"> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span></td> <td>A<sub>1</sub> = B<sub>1</sub> = C<sub>1</sub></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {sl}}_{2}\cong {\mathfrak {so}}_{3}\cong {\mathfrak {sp}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>≅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {sl}}_{2}\cong {\mathfrak {so}}_{3}\cong {\mathfrak {sp}}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/769bec46b8ed2caccf0adac966673283498cbe20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:15.441ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {sl}}_{2}\cong {\mathfrak {so}}_{3}\cong {\mathfrak {sp}}_{1}}"></span></td> <td>— </td></tr> <tr align="center"> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cong }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cong }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a725ebc5ab8de11d7b71a8aa5a3706c2ea467885" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.049ex; margin-bottom: -0.22ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \cong }"></span> <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/d/d6/CDel_3x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span></td> <td>A<sub>2</sub> = I<sub>2</sub>(2)</td> <td>—</td> <td><a href="/wiki/Simplex" title="Simplex">2-simplex</a> is <a href="/wiki/Regular_polygon" title="Regular polygon">regular 3-gon</a> (<a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a>) </td></tr> <tr align="center"> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span></td> <td>BC<sub>2</sub> = I<sub>2</sub>(4)</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}_{5}\cong {\mathfrak {sp}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>≅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}_{5}\cong {\mathfrak {sp}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9816987b043c730020bc6864950787098c069a63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:9.607ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {so}}_{5}\cong {\mathfrak {sp}}_{2}}"></span></td> <td><a href="/wiki/N-cube" class="mw-redirect" title="N-cube">2-cube</a> is <a href="/wiki/Cross-polytope" title="Cross-polytope">2-cross polytope</a> is <a href="/wiki/Regular_polygon" title="Regular polygon">regular 4-gon</a> (<a href="/wiki/Square" title="Square">square</a>) </td></tr> <tr align="center"> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cong }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cong }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a725ebc5ab8de11d7b71a8aa5a3706c2ea467885" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.049ex; margin-bottom: -0.22ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \cong }"></span> <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/1f/CDel_nodes.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span></td> <td>A<sub>1</sub> × A<sub>1</sub> = D<sub>2</sub></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {sl}}_{2}\oplus {\mathfrak {sl}}_{2}\cong {\mathfrak {so}}_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {sl}}_{2}\oplus {\mathfrak {sl}}_{2}\cong {\mathfrak {so}}_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/883462c91a4801ef7ed21ac83c07cb1f91065803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.041ex; width:14.672ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {sl}}_{2}\oplus {\mathfrak {sl}}_{2}\cong {\mathfrak {so}}_{4}}"></span></td> <td>— </td></tr> <tr align="center"> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cong }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cong }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a725ebc5ab8de11d7b71a8aa5a3706c2ea467885" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.049ex; margin-bottom: -0.22ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \cong }"></span> <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/1f/CDel_nodes.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span></td> <td>A<sub>3</sub> = D<sub>3</sub></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {sl}}_{4}\cong {\mathfrak {so}}_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>≅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {sl}}_{4}\cong {\mathfrak {so}}_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b12c10ef41186d8f87a839fa9036473c86c3e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.041ex; width:9.096ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {sl}}_{4}\cong {\mathfrak {so}}_{6}}"></span></td> <td><a href="/wiki/Simplex" title="Simplex">3-simplex</a> is <a href="/wiki/Demihypercube" title="Demihypercube">3-demihypercube</a> (<a href="/wiki/Regular_tetrahedron" class="mw-redirect" title="Regular tetrahedron">regular tetrahedron</a>) </td></tr> </tbody></table> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exceptional_isomorphism&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Exceptional_object" title="Exceptional object">Exceptional object</a></li> <li><a href="/wiki/Mathematical_coincidence" title="Mathematical coincidence">Mathematical coincidence</a>, for numerical coincidences</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exceptional_isomorphism&action=edit&section=9" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Because these series of objects are presented differently, they are not identical objects (do not have identical descriptions), but turn out to describe the same object, hence one refers to this as an isomorphism, not an equality (identity).</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exceptional_isomorphism&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-FOOTNOTEWilson2009[httpwwwmathsqmulacuk~rawfsgs_filesintrops_Chapter_1:_Introduction]-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEWilson2009[httpwwwmathsqmulacuk~rawfsgs_filesintrops_Chapter_1:_Introduction]_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEWilson2009[httpwwwmathsqmulacuk~rawfsgs_filesintrops_Chapter_1:_Introduction]_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFWilson2009">Wilson 2009</a>, <a rel="nofollow" class="external text" href="http://www.maths.qmul.ac.uk/~raw/fsgs_files/intro.ps">Chapter 1: Introduction</a></span> </li> <li id="cite_note-FOOTNOTEWilson2009[httpswebarchiveorgweb20070824011953httpswwwmathsqmulacuk~rawfsgs_filesintrops_Chapter_1:_Introduction]-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWilson2009[httpswebarchiveorgweb20070824011953httpswwwmathsqmulacuk~rawfsgs_filesintrops_Chapter_1:_Introduction]_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWilson2009">Wilson 2009</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070824011953/https://www.maths.qmul.ac.uk/~raw/fsgs_files/intro.ps">Chapter 1: Introduction</a></span> </li> <li id="cite_note-FOOTNOTEWilson2009Chapter_3-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWilson2009Chapter_3_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWilson2009">Wilson 2009</a>, Chapter 3</span> </li> </ol></div></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWilson2009" class="citation cs2"><a href="/wiki/Robert_Arnott_Wilson" title="Robert Arnott Wilson">Wilson, Robert A.</a> (2009), <i>The finite simple groups</i>, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a> 251, vol. 251, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-84800-988-2">10.1007/978-1-84800-988-2</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84800-987-5" title="Special:BookSources/978-1-84800-987-5"><bdi>978-1-84800-987-5</bdi></a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1203.20012">1203.20012</a>[www.maths.qmul.ac.uk/~raw/fsgs.html 2007 preprint]</cite><span 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