+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Group Theory +-- {: .hide} [[!include group theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea By a _finite rotation group_ one means a [[finite]] [[subgroup]] of a [[group]] of [[rotations]], hence of a [[special orthogonal group]] $SO(n)$ or [[spin group]] $Spin(n)$ or similar. The finite subgroups of [[SO(3)]] and [[SU(2)]] follow an [[ADE classification]] (theorem \ref{ClassificationOfFiniteSubgroupsOfSO3} below). ## Classifications ### Finite subgroups of $O(3)$, $SO(3)$ and $Spin(3)$ +-- {: .num_theorem #ClassificationOfFiniteSubgroupsOfSO3} ###### Theorem **([[ADE classification]] of [[finite group|finite]] [[subgroups]] of [[SO(3)]] and [[spin group|Spin(3)]]$\simeq$ [[SU(2)]])** The [[finite group|finite]] [[subgroups]] of the [[special orthogonal group]] $SO(3)$ as well as the [[finite group|finite]] [[subgroups]] of the [[special unitary group]] [[SU(2)]] are, up to [[conjugation]], given by the following classification: [[!include ADE -- table]] Here under the [[double cover]] projection (using the [exceptional isomorphism](spin+group#ExceptionalIsomorphisms) $SU(2) \simeq Spin(3)$) $$ SU(2) \simeq Spin(3) \overset{\pi}{\longrightarrow} SO(3) $$ all the finite subgroups of $SU(2)$ except the [[odd number|odd]]-[[order of a group|order]] [[cyclic groups]] are the [[preimages]] of the corresponding finite subgroups of $SO(3)$, in that we have [[pullback]] [[diagrams]] $$ \array{ \left\langle \exp \left( \pi \mathrm{i} \tfrac{1}{n} \right) \right\rangle & = & \mathbb{Z}/(2n) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(2) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(3) \\ && \big\downarrow &{}^{(pb)}& \big\downarrow &{}^{(pb)}& \big\downarrow^{ \mathrlap{\pi} } \\ \left\langle Ad_{ \exp \left( \pi \mathrm{i} \tfrac{1}{n} \right) } \right\rangle & = & \mathbb{Z}/n &\overset{\phantom{AA}}{\hookrightarrow}& SO(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) } $$ exhibiting the [[even number|even]] [[order of a group|order]] [[cyclic groups]] as subgroups of [[Spin(2)]], including the minimal case of the [[group of order 2]] $$ \array{ \left\langle \exp \left( \pi \mathrm{i} \right) = -1 \right\rangle & = & \mathbb{Z}/2 &\overset{\phantom{AA}}{\hookrightarrow}& Spin(2) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(3) \\ && \big\downarrow &{}^{(pb)}& \big\downarrow &{}^{(pb)}& \big\downarrow^{ \mathrlap{\pi} } \\ \left\langle Ad_{ \exp \left( \pi \mathrm{i} \right) } = e \right\rangle & = & 1 &\overset{\phantom{AA}}{\hookrightarrow}& SO(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) } $$ as well as $$ \array{ \left\langle \exp\left( \pi \mathrm{i} \tfrac{1}{n} \right), \, \mathrm{j} \right\rangle &=& 2 D_{2n} &\overset{\phantom{AA}}{\hookrightarrow}& Pin_-(2) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(3) \\ && \big\downarrow &{}^{(pb)}& \big\downarrow &{}^{(pb)}& \big\downarrow^{ \mathrlap{\pi} } \\ \left\langle Ad_{\exp\left( \pi \mathrm{i} \tfrac{1}{n} \right) }, \, Ad_{\mathrm{j}} \right\rangle && D_{2n} &\overset{\phantom{AA}}{\hookrightarrow}& O(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) } $$ exhibiting the [[binary dihedral groups]] as sitting inside the [[Pin(2)]]-[[subgroup]] of [[Spin(3)]], but only [[commuting diagrams]] $$ \array{ \left\langle \exp \left( 2 \pi \mathrm{i} \tfrac{1}{{2n+1}} \right) \right\rangle & = & \mathbb{Z}/(2n+1) &&\overset{\phantom{AA}}{\hookrightarrow}&& Spin(3) \\ && \big\downarrow && && \big\downarrow^{ \mathrlap{\pi} } \\ \left\langle Ad_{ \exp \left( 2 \pi \mathrm{i} \tfrac{1}{2n+1} \right) } \right\rangle & = & \mathbb{Z}/(2n+1) &\overset{\phantom{AA}}{\hookrightarrow}& SO(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) } $$ for the [[odd number|odd]] [[order of a group|order]] [[cyclic group|cyclic]] [[subgroups]]. =-- This goes back to ([Klein 1884, chapter I](#Klein1884)). Full proof for $SO(3)$ is spelled out for instance in ([Rees 05, theorem 11](#Rees05), [De Visscher 11](#DeVisscher11)). The proof for the case of $SL(2,\mathbb{C})$ is spelled out in ([Miller-Blichfeldt-Dickson 16](#MillerBlichfeldtDickson16)) reviewed in ([Serrano 14, section 2](#Serrano14)). The proof of the case for $SU(2)$ given the result for $SO(3)$ is spelled out in [Keenan 03, theorem 4](#Keenan03). ### Finite subgroups of $O(4)$, [[SO(4)]] and [[Spin(4)]] {#FiniteSubgroupsOfO4} For classification of the finite subgroups of $O(4)$ see ([duVal 65](#DuVal65), [Conway-Smith 03](#ConwaySmith03)) For finite subgroups of [[Spin(4)]]: [MFF 12, appendix B](#MFF12). In this classification, the [[symmetry group]] of the [[120-cell]] and hence that of the [[600-cell]] is the [[quotient group]] $(2 I \times 2 I)/\mathbb{Z}_2$ by the [[cyclic group of order 2]] of the [[direct product group]] of two copies of the [[binary icosahedral group]] ([SadocMosseri 89, p. 172](#SadocMosseri89), see [MFF 12, table 16](#MFF12)). ### Finite subgroups of $O(5)$ For classification of the finite subgroups of $O(5)$ see [Mecchia-Zimmermann 10](#MecchiaZimmermann10) ## Properties ### Subgroup lattice {#SubgroupLattice} The [[subgroup]] [[lattice]] of [[SU(2)]] under the three exceptional [[finite group|finite]] subgroups [[2T]], [[2O]], [[2I]] (from Theorem \ref{ClassificationOfFiniteSubgroupsOfSO3}) looks as follows: <center> <img src="https://ncatlab.org/nlab/files/ADESubgroupLatticeUnderExceptional.jpg" width="490"/> </center> This is obtained from the subgroup lattice as shown on [GroupNames](https://people.maths.bris.ac.uk/~matyd/GroupNames/) for $2I \simeq$ <a href="https://people.maths.bris.ac.uk/~matyd/GroupNames/97/SL(2,5).html">SL(2,5)</a> and $2O \simeq $ <a href="https://people.maths.bris.ac.uk/~matyd/GroupNames/1/CSU(2,3).html">CSU(2,3)</a> See also [Goncalves-Guaschi 11, appendix](#GoncalvesGuaschi11). ### Group cohomology {#GroupCohomology} +-- {: .num_prop #GroupCohomologyOfFiniteSubgroupsOfSU2} ###### Proposition **([[group cohomology]] of [[finite subgroups of SU(2)]]) Let $G_{ADE} \xhookrightarrow{i} SU(2)$ be a [[finite subgroup of SU(2)]]. Then its [[group cohomology]] with [[integer]] [[coefficients]] is as follows: \[ \label{IntegralGroupCohomologyOfFiniteSubgroupsOfSU2} H^n_{grp}(G_{ADE}, \mathbb{Z}) \;\simeq\; \left\{ \array{ \mathbb{Z} &\vert& n = 0 \\ G_{ADE}^{ab} &\vert& n = 2 \, mod \, 4 \\ \mathbb{Z}/{\vert G_{ADE}\vert} &\vert& n \, \text{positive multiple of} \, 4 \\ 0 &\vert& \text{otherwise} } \right. \] Here $G_{ADE}^{ab}$ denotes the [[abelianization]] of $G_{ADE}$ and $\vert G_{ADE}\vert$ its [[cardinality]], hence $\mathbb{Z}/{\vert G_{ADE}\vert}$ is the [[cyclic group]] whose [[order of a group|order]] is the cardinality of $G_{ADE}$. The [[group homology]] with [[integer]] [[coefficients]] is \[ \label{IntegralGroupHomologyOfFiniteSubgroupsOfSU2} H_n^{grp}(G_{ADE}, \mathbb{Z}) \;\simeq\; \left\{ \array{ \mathbb{Z} &\vert& n = 0 \\ G^{ab}_{ADE} &\vert& n = 1 \, mod \, 4 \\ \mathbb{Z}/{\vert G_{ADE}\vert} &\vert& n = 3 \,mod\, 4 \\ 0 &\vert& \text{otherwise} } \right. \] =-- This general statement is summarized in [Epa & Ganter 16, p. 12](#EpaGanter16). A detailed proof for the case of cyclic groups is spelled out in [BSST 07, Lem. 4.51](Pontrjagin+dual#BSST07). The three exceptional cases are worked out in explicit detail in [Tomoda & Zvengrowski 08, Section 4](#TomodaZvengrowski08). The vanishing of $H^3(G_{ADE},\mathbb{Z}) = H^2(G_{ADE}, \mathbb{C}^\times)$ is also [FHHP 01, Cor. 3.1](#FHHP01). +-- {: .num_remark #DiscreteTorsion} ###### Remark **([[C-field]] [[discrete torsion]])** In discussion of [[11-dimensional supergravity]] on [[spacetimes]] with [[ADE-singularities]], the special case $$ \underset{ \simeq H^3( \mathbb{C}^4 \sslash G_{ADE}, U(1)) }{ \underbrace{ H^4_{grp}(G_{ADE}, \mathbb{Z}) }} \;\simeq\; \mathbb{Z}/{\vert G_{ADE} \vert } $$ of Prop. \ref{GroupCohomologyOfFiniteSubgroupsOfSU2}, regarded as expressing [[orbifold cohomology]] of an [[ADE singularity]], as shown under the brace, witnesses the possible [[torsion subgroup|torsion]] [[supergravity C-field]] [[flux]] of [[M5-branes]] [[wrapped brane|wrapped]] on torsion homology 3-cycles ("[[discrete torsion]]", see [Aharony-Bergman-Jafferis 08, p. 8](discrete+torsion#AharonyBergmanJafferis08) and [BDHKMMS 01, section 4.6.2](discrete+torsion#BDHKMMS01)). See also at _[[Platonic 2-group]] -- [Relation to String 2-group](Platonic+2-group#RelationToTheString2Group)_. =-- \begin{proposition} \label{RestrictingTheSecondChernClassFromSU2ToFiniteSubgroup} For $G_{ADE} \xhookrightarrow{i} Spin(3)$ a [[finite subgroup of SU(2)]], the [[pullback in cohomology]] along the induced map $B G_{ADE} \xrightarrow{B i} B SU(2)$ of [[classifying spaces]] sends * the generator $c_2 \,\in\,H^4(B SU(2), \mathbb{Z}) \simeq \mathbb{Z}$ (the [[universal characteristic class|universal]] [[second Chern class]]) to * a generator of $H^4(B G_{ADE}, \mathbb{Z}) \simeq \mathbb{Z}/\left\vert G_{ADE}\right\vert$ (eq:IntegralGroupCohomologyOfFiniteSubgroupsOfSU2) in that the [[pullback in cohomology]] is identified with the [[quotient]] [[coprojection]] from the [[integers]] to the [[cyclic group]] of [[order of a group|order]] that of $G_{ADE}$: \[ \label{RestrictionOf4CohomologyOfBSU2ToBGADE} \array{ H^4_{grp}(SU(2), \mathbb{Z}) &=& H^4(B SU(2), \mathbb{Z}) &\simeq& \mathbb{Z} \\ {}^{\mathllap{i^\ast}}\big\downarrow && {}^{\mathllap{B i^\ast}}\big\downarrow && \big\downarrow {}^{\mathrlap{ {quotient} \atop {coprojection} }} \\ H^4_{grp}(G_{ADE}, \mathbb{Z}) &=& H^4(B G_{ADE}, \mathbb{Z}) &\simeq& \mathbb{Z}/\left\vert G_{ADE}\right\vert \mathrlap{\,.} } \] \end{proposition} This is equivalently the statement of [Epa & Ganter 16, Prop. 4.1](#EpaGanter16), whose proof is analogous to that of Lemma 3.1 there, where the analogue of (eq:RestrictionOf4CohomologyOfBSU2ToBGADE) is the top left square of the commuting diagram on [p. 11](https://arxiv.org/pdf/1605.09192.pdf#page=11). ## Related concepts * [[regular polytope]], [[regular polyhedron]] * [[ADE singularity]] * [[McKay correspondence]] * [[Platonic 2-group]] * [[wallpaper group]] ## References {#References} ### Finite subgroups of $SO(3)$ and $Spin(3)$ The classification in Theorem \ref{ClassificationOfFiniteSubgroupsOfSO3} goes back to * {#Klein1884} [[Felix Klein]], chapter I of _Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade_, 1884, translated as _Lectures on the Icosahedron and the Resolution of Equations of Degree Five_ by George Morrice 1888, [online version](https://archive.org/details/cu31924059413439) Expository review of the argument in [Klein 1884](#Klein1884): * {#Senechal90} Marjorie Senechal, *Finding the Finite Groups of Symmetries of the Sphere*, The American Mathematical Monthly **97** 4 (1990) 329-335 ([jstor:2324519](https://www.jstor.org/stable/2324519)) Textbook accounts: * {#MillerBlichfeldtDickson16} G. A. Miller, H. F. Blichfeldt, [[Leonard E. Dickson]], _Theory and applications of finite groups_, Dover, New York, 1916 ([ark:/13960/t0ht2kb4x](https://archive.org/details/theoryapplicatio00milluoft/)) * {#Lamotke86} [[Klaus Lamotke]], _Regular Solids and Isolated Singularities_, Vieweg 1986 ([doi:10.1007/978-3-322-91767-6](https://link.springer.com/book/10.1007/978-3-322-91767-6)) * [[Mark A. Armstrong]], chapter 19 of: *Groups and Symmetry*, Undergraduate Texts in Mathematics, Springer (1988) &lbrack;[doi:10.1007/978-1-4757-4034-9](https://doi.org/10.1007/978-1-4757-4034-9), [pdf](https://superoles.wordpress.com/wp-content/uploads/2014/10/lluvia.pdf)&rbrack; * {#Rees05} [[Elmer Rees]], _Notes on Geometry_, Springer 2005 see also * {#Serrano14} Javier Carrasco Serrano, _Finite subgroups of $SL(2,\mathbb{C})$ and $SL(3,\mathbb{C})$_, Warwick 2014 ([pdf](https://homepages.warwick.ac.uk/~masda/McKay/Carrasco_Project.pdf)) Complete proof of the classification of the finite subgroups of $SO(3)$ is also spelled out in * {#DeVisscher11} Maud De Visscher, _[Groups and Symmetry](http://www.staff.city.ac.uk/maud.devisscher.1/GS/)_ lecture notes handout: _Classification of finite rotation groups_ ([pdf](http://www.staff.city.ac.uk/maud.devisscher.1/GS/classif_rotation.pdf)) Based on the classification of the finite subgroups of $SO(3)$, full proof of that of the finite subgroups of $SU(2)$ is spelled out in * {#Keenan03} [[Adam Keenan]], _Which finite groups act freely on spheres?_, 2003 ([pdf](http://www.math.utah.edu/~keenan/actions.pdf)) See also * [[Groupprops]], _<a href="https://groupprops.subwiki.org/wiki/Classification_of_finite_subgroups_of_SO(3,R)">Classification of finite subgroups of SO(3,R)</a> Discussion of the [[lattice]] of [[subgroups]] of the three exceptional subgroups: * {#GoncalvesGuaschi11} [[Daciberg Lima Gonçalves]], [[John Guaschi]]: _The Subgroups of the Binary Polyhedral Groups_, Appendix of: _The classification of the virtually cyclic subgroups of the sphere braid groups_, Briefs in Mathematics, Springer (2013) &lbrack;[arXiv:1110.6628](https://arxiv.org/abs/1110.6628), [doi:10.1007/978-3-319-00257-6](https://doi.org/10.1007/978-3-319-00257-6), [pdf](https://link.springer.com/content/pdf/bbm%3A978-3-319-00257-6%2F1.pdf)&rbrack; Discussion of the [[group cohomology]]: * {#TomodaZvengrowski08} [[Satoshi Tomoda]], [[Peter Zvengrowski]], *Remarks on the cohomology of finite fundamental groups of 3-manifolds*, Geom. Topol. Monogr. 14 (2008) 519-556 ([arXiv:0904.1876](https://arxiv.org/abs/0904.1876)) see also * {#FHHP01} [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], Nikolaos Prezas, Cor. 3.1 in: *Discrete Torsion, Non-Abelian Orbifolds and the Schur Multiplier*, JHEP 0101:033, 2001 ([arXiv:hep-th/0010023](https://arxiv.org/abs/hep-th/0010023)) The [[universal higher central extension]] of finite subgroups of $SU(2)$ ("[[Platonic 2-groups]]") are discussed in * {#EpaGanter16} [[Narthana Epa]], [[Nora Ganter]], _Platonic and alternating 2-groups_, Higher Structures 1(1):122-146, 2017 ([arXiv:1605.09192](http://arxiv.org/abs/1605.09192)) ### Finite subgroups of $O(4)$ * {#DuVal65} [[Patrick du Val]], _Homographies, Quaternions and Rotations_, Oxford Mathematical Monographs, Clarendon Press (1964) also(?): Journal of the London Mathematical Society, Volume s1-40, Issue 1 (1965) ([doi:10.1112/jlms/s1-40.1.569b](https://doi.org/10.1112/jlms/s1-40.1.569b)) * {#ConwaySmith03} [[John Conway]], D. A. Smith, *On quaternions and octonions: their geometry, arithmetic and symmetry*, A K Peters Ltd., Natick, MA, 2003 * {#MFF12} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], appendix B of _Half-BPS M2-brane orbifolds_, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. ([arXiv:1007.4761](http://arxiv.org/abs/1007.4761), [Euclid](https://projecteuclid.org/euclid.atmp/1408561553)) * {#SadocMosseri89} J. F. Sadoc, R. Mosseri, _Icosahedral order, curved space and quasicrystals_, in Jaric, Gratias (eds.) _Extended icosahedral structures_, 1989 ([GoogleBooks](https://books.google.co.uk/books?id=HWyIUglWeXsC&pg=PA172&lpg=PA172#v=onepage&q&f=false)) ### Finite subgroups of $O(5)$ * {#MecchiaZimmermann10} Mattia Mecchia, Bruno Zimmermann, _On finite groups acting on homology 4-spheres and finite subgroups of $SO(5)$_, Topology and its Applications 158.6 (2011): 741-747 ([arXiv:1001.3976](https://arxiv.org/abs/1001.3976)) [[!redirects finite rotation groups]] [[!redirects classification of the finite rotation groups]] [[!redirects finite subgroup of SO(3)]] [[!redirects finite subgroups of SO(3)]] [[!redirects finite subgroup of SU(2)]] [[!redirects finite subgroups of SU(2)]] [[!redirects finite subgroup of Spin(3)]] [[!redirects finite subgroups of Spin(3)]] [[!redirects finite subgroup of O(4)]] [[!redirects finite subgroups of O(4)]] [[!redirects finite subgroup of SO(4)]] [[!redirects finite subgroups of SO(4)]] [[!redirects finite subgroup of O(5)]] [[!redirects finite subgroups of O(5)]] [[!redirects finite subgroup of SO(5)]] [[!redirects finite subgroups of SO(5)]] [[!redirects classification of finite rotation groups]] [[!redirects classifications of finite rotation groups]]