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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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#character_table'>Character table</a></li> <li><a href='#matrix_representation'>Matrix representation</a></li> <li><a href='#SubgroupLattice'>Subgroup lattice</a></li> <li><a href='#group_cohomology'>Group cohomology</a></li> </ul> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>quaternion group</em> of <a class="existingWikiWord" href="/nlab/show/order+of+a+group">order</a> 8, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">Q_8</annotation></semantics></math>, is the <a class="existingWikiWord" href="/nlab/show/finite+subgroup+of+SU%282%29">finite subgroup of SU(2)</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub><mo>⊂</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>S</mi> <mn>3</mn></msup><mo>⊂</mo><mi>ℍ</mi></mrow><annotation encoding="application/x-tex">Q_8 \subset SU(2) \simeq S^3 \subset \mathbb{H}</annotation></semantics></math> of unit <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a> which consists of the canonical four <a class="existingWikiWord" href="/nlab/show/linear+basis">basis</a>-quaternions and their negatives:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">{</mo><mo>±</mo><mn>1</mn><mo>,</mo><mspace width="thinmathspace"></mspace><mo>±</mo><mi>i</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mo>±</mo><mi>j</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mo>±</mo><mi>k</mi><mo maxsize="1.2em" minsize="1.2em">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Q_8 \;=\; \big\{ \pm 1, \, \pm i, \, \pm j, \, \pm k \big\} \,. </annotation></semantics></math></div><div style="float:right;margin:0 10px 10px 0;"> <img src="https://upload.wikimedia.org/wikipedia/commons/5/59/Dynkin_diagram_D4.png" width="100" /> </div> <p>This is <a class="existingWikiWord" href="/nlab/show/isomorphic">isomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/binary+dihedral+group">binary dihedral group</a> of the same <a class="existingWikiWord" href="/nlab/show/order+of+a+group">order</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub><mo>≃</mo><mn>2</mn><msub><mi>D</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">Q_8 \simeq 2 D_4</annotation></semantics></math>. As such, the <a class="existingWikiWord" href="/nlab/show/Dynkin+diagram">Dynkin diagram</a> that corresponds to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">Q_8</annotation></semantics></math> under the <a class="existingWikiWord" href="/nlab/show/ADE-classification">ADE-classification</a> of <a class="existingWikiWord" href="/nlab/show/finite+subgroups+of+SU%282%29">finite subgroups of SU(2)</a> is <a class="existingWikiWord" href="/nlab/show/D4">D4</a>, the <a class="existingWikiWord" href="/nlab/show/triality">triality</a>-invariant one.</p> <blockquote> <p>graphics grabbed from Wikipedia <a href="https://upload.wikimedia.org/wikipedia/commons/5/59/Dynkin_diagram_D4.png">here</a></p> </blockquote> <p>This order-8 quaternion group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">Q_8</annotation></semantics></math> is the first in a row of generalized quaternion groups, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mrow><msup><mn>2</mn> <mi>n</mi></msup></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{2^n}</annotation></semantics></math>, which are also examples of dicyclic groups, which class forms part of an even larger family. We will treat both general dicyclic groups and the specific example of the quaternion group together.</p> <h2 id="definitions">Definitions</h2> <p>The <em>dicyclic</em> of order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn><mi>n</mi></mrow><annotation encoding="application/x-tex">4n</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n\geq 2</annotation></semantics></math>, is the <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><msub><mi>Dic</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">Dic_n</annotation></semantics></math> defined by the presentation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">|</mo><msup><mi>x</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>=</mo><msup><mi>x</mi> <mi>n</mi></msup><msup><mi>y</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow></msup><mo>=</mo><msup><mi>y</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>x</mi><mi>y</mi><mi>x</mi><mo>=</mo><mn>1</mn><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle x,y | x^{2n}= x^{n} y^{-2}=y^{-1}x y x=1\rangle</annotation></semantics></math>.</p> <p>The <em>quaternion group</em> (of order 8) is then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Dic</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">Dic_n</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n=2</annotation></semantics></math>.</p> <p>The <em>generalised quaternion group</em> of order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>2</mn> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2^{k+1}</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Dic</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">Dic_n</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><msup><mn>2</mn> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">n= 2^{k-1}</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <h3 id="general">General</h3> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Dic</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">Dic_n</annotation></semantics></math> is (isomorphic to) a finite subgroup of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℍ</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathbb{H}^\ast</annotation></semantics></math> as can be seen by taking generators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">x=j</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>=</mo><mi>cos</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">/</mo><mi>n</mi><mo stretchy="false">)</mo><mo>+</mo><mi>i</mi><mi>sin</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">/</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=\cos(\pi/n) + i\sin(\pi/n)</annotation></semantics></math>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n=2</annotation></semantics></math> this simply yields the subgroup generated by <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Dic</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">Dic_n</annotation></semantics></math> has another presentation as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>,</mo><mi>T</mi><mo stretchy="false">|</mo><msup><mi>R</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>T</mi> <mi>n</mi></msup><mo>=</mo><mi>R</mi><mi>S</mi><mi>T</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle R, S, T | R^2=S^2=T^n=R S T\rangle</annotation></semantics></math>. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>S</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">R S T</annotation></semantics></math> as a power of each of the generators is central and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Dic</mi> <mi>n</mi></msub><mo stretchy="false">/</mo><mo stretchy="false">⟨</mo><mi>R</mi><mi>S</mi><mi>T</mi><mo stretchy="false">⟩</mo><mo>=</mo><msub><mi>D</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Dic_n/\langle R S T\rangle= D_{2n}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>=</mo><mo stretchy="false">⟨</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>,</mo><mi>T</mi><mo stretchy="false">|</mo><msup><mi>R</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>T</mi> <mi>n</mi></msup><mo>=</mo><mi>R</mi><mi>S</mi><mi>T</mi><mo>=</mo><mn>1</mn><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">D_{2n}=\langle R, S, T | R^2=S^2=T^n=R S T=1\rangle</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/dihedral+group">dihedral group</a> of order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>n</mi></mrow><annotation encoding="application/x-tex">2n</annotation></semantics></math>.</p> </li> <li id="HamiltonianGroup"> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">Q_8</annotation></semantics></math> is a <strong>Hamiltonian group</strong> i.e. a <a class="existingWikiWord" href="/nlab/show/non-abelian+group">non-abelian group</a> such that every <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> is <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal</a>. Moreover, a general structure theorem for Hamiltonian groups by Baer (1933) says that every Hamiltonian group has a <a class="existingWikiWord" href="/nlab/show/direct+product+group">direct product group</a>-decomposition containing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">Q_8</annotation></semantics></math> as a factor hence, in particular, every Hamiltonian group contains <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">Q_8</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a>! (cf. <a href="#Scott87">Scott (1987, p.253</a>))</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">Q_8</annotation></semantics></math> is the multiplicative part of the quaternionic near-field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mn>9</mn></msub></mrow><annotation encoding="application/x-tex">J_9</annotation></semantics></math>. (cf. <a href="#Weibel07">Weibel (2007)</a>)</p> </li> </ul> <div class="num_prop" id="InclusionInLargerFininteSubgroupsOfSU2"> <h6 id="proposition">Proposition</h6> <p><strong>(inclusion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">Q_8</annotation></semantics></math> into <a class="existingWikiWord" href="/nlab/show/finite+subgroups+of+SU%282%29">finite subgroups of SU(2)</a>)</strong></p> <p>Among the <a class="existingWikiWord" href="/nlab/show/finite+subgroups+of+SU%282%29">finite subgroups of SU(2)</a> (hence among all “finite quaternion groups”) the quaternion group of <a class="existingWikiWord" href="/nlab/show/order+of+a+group">order</a> 8, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">Q_8</annotation></semantics></math> is a proper <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> precisely of the three <a class="existingWikiWord" href="/nlab/show/ADE-classification">exceptional cases</a>:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub><mo>⊂</mo><mn>2</mn><mi>T</mi></mrow><annotation encoding="application/x-tex">Q_8 \subset 2 T</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/binary+tetrahedral+group">binary tetrahedral group</a> (<a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal</a>),</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub><mo>⊂</mo><mn>2</mn><mi>O</mi></mrow><annotation encoding="application/x-tex">Q_8 \subset 2 O</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/binary+octahedral+group">binary octahedral group</a> (<a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal</a>)</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub><mo>⊂</mo><mn>2</mn><mi>I</mi></mrow><annotation encoding="application/x-tex">Q_8 \subset 2 I</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/binary+icosahedral+group">binary icosahedral group</a> (not <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal</a>)</p> </li> </ul> </div> <p>(e.g. <a href="#KocaMocKoca16">Koca-Moc-Koca 16, p. 8</a>, pointing to <a href="#CoxeterMoser65">Coxeter-Moser 65</a> and <a href="#Coxeter73">Coxeter 73</a>)</p> <h3 id="character_table">Character table</h3> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/linear+representation+theory">linear representation theory</a> of <a class="existingWikiWord" href="/nlab/show/binary+dihedral+group">binary dihedral group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mi>D</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">2 D_4</annotation></semantics></math></strong></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/dicyclic+group">dicyclic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Dic</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">Dic_2</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/quaternion+group">quaternion group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">Q_8</annotation></semantics></math></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p><a class="existingWikiWord" href="/nlab/show/group+order">group order</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mn>2</mn><msub><mi>D</mi> <mn>4</mn></msub><mo stretchy="false">|</mo></mrow><mo>=</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">{\vert 2D_4\vert} = 8</annotation></semantics></math></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/conjugacy+classes">conjugacy classes</a>:</th><th>1</th><th>2</th><th>4A</th><th>4B</th><th>4C</th></tr></thead><tbody><tr><td style="text-align: left;">their <a class="existingWikiWord" href="/nlab/show/cardinality">cardinality</a>:</td><td style="text-align: left;">1</td><td style="text-align: left;">1</td><td style="text-align: left;">2</td><td style="text-align: left;">2</td><td style="text-align: left;">2</td></tr> </tbody></table> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <table><thead><tr><th></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/splitting+field">splitting field</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{Q}(\alpha, \beta)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>α</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>β</mi> <mn>2</mn></msup><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\alpha^2 + \beta^2 = -1</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/field">field</a> generated by <a class="existingWikiWord" href="/nlab/show/characters">characters</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math></td></tr> </tbody></table> <p><strong><a class="existingWikiWord" href="/nlab/show/character+table">character table</a> over <a class="existingWikiWord" href="/nlab/show/splitting+field">splitting field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{Q}(\alpha,\beta)</annotation></semantics></math>/<a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/irrep">irrep</a></th><th>1</th><th>2</th><th>4A</th><th>4B</th><th>4C</th><th><a class="existingWikiWord" href="/nlab/show/Schur+index">Schur index</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\rho_1</annotation></semantics></math></td><td style="text-align: left;">1</td><td style="text-align: left;">1</td><td style="text-align: left;">1</td><td style="text-align: left;">1</td><td style="text-align: left;">1</td><td style="text-align: left;">1</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\rho_2</annotation></semantics></math></td><td style="text-align: left;">1</td><td style="text-align: left;">1</td><td style="text-align: left;">-1</td><td style="text-align: left;">1</td><td style="text-align: left;">-1</td><td style="text-align: left;">1</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\rho_3</annotation></semantics></math></td><td style="text-align: left;">1</td><td style="text-align: left;">1</td><td style="text-align: left;">1</td><td style="text-align: left;">-1</td><td style="text-align: left;">-1</td><td style="text-align: left;">1</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\rho_4</annotation></semantics></math></td><td style="text-align: left;">1</td><td style="text-align: left;">1</td><td style="text-align: left;">-1</td><td style="text-align: left;">-1</td><td style="text-align: left;">1</td><td style="text-align: left;">1</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mn>5</mn></msub></mrow><annotation encoding="application/x-tex">\rho_5</annotation></semantics></math></td><td style="text-align: left;">2</td><td style="text-align: left;">-2</td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;">2</td></tr> </tbody></table> <p><strong><a class="existingWikiWord" href="/nlab/show/character+table">character table</a> over <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math>/<a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/irrep">irrep</a></th><th>1</th><th>2</th><th>4A</th><th>4B</th><th>4C</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\rho_1</annotation></semantics></math></td><td style="text-align: left;">1</td><td style="text-align: left;">1</td><td style="text-align: left;">1</td><td style="text-align: left;">1</td><td style="text-align: left;">1</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\rho_2</annotation></semantics></math></td><td style="text-align: left;">1</td><td style="text-align: left;">1</td><td style="text-align: left;">-1</td><td style="text-align: left;">1</td><td style="text-align: left;">-1</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\rho_3</annotation></semantics></math></td><td style="text-align: left;">1</td><td style="text-align: left;">1</td><td style="text-align: left;">1</td><td style="text-align: left;">-1</td><td style="text-align: left;">-1</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\rho_4</annotation></semantics></math></td><td style="text-align: left;">1</td><td style="text-align: left;">1</td><td style="text-align: left;">-1</td><td style="text-align: left;">-1</td><td style="text-align: left;">1</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mn>5</mn></msub><mo>⊕</mo><msub><mi>ρ</mi> <mn>5</mn></msub></mrow><annotation encoding="application/x-tex">\rho_5 \oplus \rho_5</annotation></semantics></math></td><td style="text-align: left;">4</td><td style="text-align: left;">-4</td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;">0</td></tr> </tbody></table> <p><strong>References</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/GroupNames">GroupNames</a>, <em><a href="https://people.maths.bris.ac.uk/~matyd/GroupNames/1/Q8.html">Q8</a></em>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Groupprops">Groupprops</a>, <em><a href="https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_dicyclic_groups">Linear representation theory of dicyclic groups</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/James+Montaldi">James Montaldi</a>, <em><a href="http://www.maths.manchester.ac.uk/~jm/wiki/Representations/BinaryCubic">Real representations – Binary cubic – Q8</a></em></p> </li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/character+tables">character tables</a></div></div> <h3 id="matrix_representation">Matrix representation</h3> <p>There are lots of different ways of defining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>:</mo><mo>=</mo><msub><mi>Q</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">Q:=Q_8</annotation></semantics></math>. One is that it is the subgroup of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gl</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Gl(2,\mathbb{C})</annotation></semantics></math> generated by the matrices</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ξ</mi><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>i</mi></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\xi = \left(\array{i&0\\0&-i}\right)</annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd></mtr> <mtr><mtd><mn>1</mn></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>.</mo></mrow><annotation encoding="application/x-tex">\eta =\left(\array{0&-1\\1&0}\right).</annotation></semantics></math></div> <p>In this form it is a nice exercise to derive a presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">Q_8</annotation></semantics></math>. Clearly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ξ</mi> <mn>4</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\xi^4=1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> is not in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>ξ</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> \langle \xi\rangle</annotation></semantics></math> as is easiy checked, so the order of this group must be at least 8.</p> <p>We note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>η</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>ξ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\eta^2 = \xi^2</annotation></semantics></math> and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mi>ξ</mi><msup><mi>η</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mi>ξ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\eta \xi \eta^{-1}= \xi^{-1}</annotation></semantics></math>, so a guess for a presentation would be</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>x</mi><mo>,</mo><mi>y</mi><mo>:</mo><msup><mi>x</mi> <mn>4</mn></msup><mo>=</mo><mn>1</mn><mo>,</mo><msup><mi>y</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>x</mi> <mn>2</mn></msup><mo>,</mo><mi>y</mi><mi>x</mi><msup><mi>y</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mi>x</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"> \langle x,y : x^4=1, y^2=x^2, y x y^{-1}=x^{-1}\rangle. </annotation></semantics></math></div> <p>Let us call <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> the group presented by this presentation, then there is an obvious epimorphism 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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> sending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ξ</mi></mrow><annotation encoding="application/x-tex">\xi</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math>. This is an isomorphism as will be clear if we show that the order 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> is less than of equal to 8. Now every element 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> can be written in the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>i</mi></msup><msup><mi>y</mi> <mi>j</mi></msup></mrow><annotation encoding="application/x-tex">x^i y^j</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">0\leq i\leq 3</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0\leq j\leq 1</annotation></semantics></math>, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mi>x</mi><mo>=</mo><msup><mi>x</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>y</mi></mrow><annotation encoding="application/x-tex">y x=x^{-1}y</annotation></semantics></math> so powers of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> can be shifted to the right in any expression and then if the resulting power of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> is greater than 2 we can use <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>y</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>x</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">y^2=x^2</annotation></semantics></math> to replace even powers of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> by powers of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>. We must therefore have that the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> must contain at most 8 elements so the above presentation is a presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">Q_8</annotation></semantics></math>.</p> <h3 id="SubgroupLattice">Subgroup lattice</h3> <p>The following shows the <a class="existingWikiWord" href="/nlab/show/subgroup+lattices">subgroup lattices</a> of the first few <a class="existingWikiWord" href="/nlab/show/generalized+quaternion+groups">generalized quaternion groups</a>:</p> <center> <img src="https://ncatlab.org/nlab/files/SubgroupLatticesOfQuaternionGroups.jpg" width="700" /> </center> <h3 id="group_cohomology">Group cohomology</h3> <p>The <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a> of the generalized quaternion groups: see <a href="#TomodaZvengrowski08">Tomoda & Zvengrowski 2008</a></p> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quaternion">quaternion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dihedral+group">dihedral group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pauli+group">Pauli group</a>, <a class="existingWikiWord" href="/nlab/show/stabilizer+code">stabilizer code</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+formula+of+myth">canonical formula of myth</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kenneth+Brown">Kenneth S. Brown</a>, <em>Cohomology of Groups</em> , GTM <strong>87</strong> Springer Heidelberg 1982. (pp.98-101)</p> </li> <li> <p>H. S. M. Coxeter, <em>The binary polyhedral groups, and other generalizations of the quaternion group</em> , Duke Math. J. <strong>7</strong> no.1 (1940) pp.367–379.</p> </li> <li> <p>T. Y. Lam, <em>Hamilton’s Quaternions</em> , pp.429-454 in <em>Handbook of Algebra III</em> , Elsevier Amsterdam 2004. (<a href="http://math.berkeley.edu/~lam/quat.ps">preprint</a>)</p> </li> <li id="Scott87"> <p>W. R. Scott, <em>Group Theory</em> , Dover New York 1987. (pp.189-194, 252-254)</p> </li> <li id="Weibel07"> <p><a class="existingWikiWord" href="/nlab/show/Charles+Weibel">Charles Weibel</a>, <em>Survey of Non-Desarguesian Planes</em> , Notices of the AMS <strong>54</strong> no.10 (2007) pp.1294–1303. (<a href="http://www.ams.org/notices/200710/tx071001294p.pdf">pdf</a>)</p> </li> <li id="KocaMocKoca16"> <p>Mehmet Koca, Ramazan Koç, Nazife Ozdes Koca, <em>Two groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>2</mn> <mn>3</mn></msup><mo>.</mo><msub><mi>PSL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2^3.PSL_2(7)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>2</mn> <mn>3</mn></msup><mo>:</mo><msub><mi>PSL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2^3:PSL_2(7)</annotation></semantics></math> of order 1344</em> (<a href="https://arxiv.org/abs/1612.06107">arXiv:1612.06107</a>)</p> </li> <li id="CoxeterMoser65"> <p>H.S.M. Coxeter, W. O. J. Moser, <em>Generators and Relations for Discrete Groups</em>, (Springer Verlag, 1965);</p> </li> <li id="Coxeter73"> <p>H.S.M. Coxeter, <em>Regular Complex Polytopes</em> (Cambridge; Cambridge University Press, 1973).</p> </li> </ul> <p>See also</p> <ul> <li> <p>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Quaternion_group">Quaternion group</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Groupprops">Groupprops</a>, <em><a href="https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_quaternion_group">Linear representation theory of the quaternion group</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GroupNames">GroupNames</a>, <em><a href="https://people.maths.bris.ac.uk/~matyd/GroupNames/quaternion.html">Quaternion groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mrow><msup><mn>2</mn> <mi>n</mi></msup></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{2^n}</annotation></semantics></math></a></em></p> </li> </ul> <p>On the <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>:</p> <ul> <li id="TomodaZvengrowski08"><a class="existingWikiWord" href="/nlab/show/Satoshi+Tomoda">Satoshi Tomoda</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Zvengrowski">Peter Zvengrowski</a>, <em>Remarks on the cohomology of finite fundamental groups of 3-manifolds</em>, Geom. Topol. Monogr. 14 (2008) 519-556 (<a href="https://arxiv.org/abs/0904.1876">arXiv:0904.1876</a>)</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/equivariant+ordinary+cohomology">equivariant ordinary cohomology</a> (<a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a>) over the point but in arbitrary <a class="existingWikiWord" href="/nlab/show/RO%28G%29-degree">RO(G)-degree</a> for <a class="existingWikiWord" href="/nlab/show/equivariance+group">equivariance group</a> the <a class="existingWikiWord" href="/nlab/show/quaternion+group">quaternion group</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yunze+Lu">Yunze Lu</a>, <em>On the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RO</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">RO(G)</annotation></semantics></math>-graded coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">Q_8</annotation></semantics></math> equivariant cohomology</em>, Topology and its Applications, 2021 (<a href="https://doi.org/10.1016/j.topol.2021.107921">doi:10.1016/j.topol.2021.107921</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 22, 2021 at 16:03:53. 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