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\section*{finite rotation group}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{classifcations}{Classifcations}\dotfill \pageref*{classifcations} \linebreak
\noindent\hyperlink{finite_subgroups_of___and_}{Finite subgroups of $O(3)$, $SO(3)$ and $Spin(3)$}\dotfill \pageref*{finite_subgroups_of___and_} \linebreak
\noindent\hyperlink{FiniteSubgroupsOfO4}{Finite subgroups of $O(4)$, [[SO(4)]] and [[Spin(4)]]}\dotfill \pageref*{FiniteSubgroupsOfO4} \linebreak
\noindent\hyperlink{finite_subgroups_of_}{Finite subgroups of $O(5)$}\dotfill \pageref*{finite_subgroups_of_} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{SubgroupLattice}{Subgroup lattice}\dotfill \pageref*{SubgroupLattice} \linebreak
\noindent\hyperlink{GroupCohomology}{Group cohomology}\dotfill \pageref*{GroupCohomology} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{finite_subgroups_of__and_}{Finite subgroups of $SO(3)$ and $Spin(3)$}\dotfill \pageref*{finite_subgroups_of__and_} \linebreak
\noindent\hyperlink{finite_subgroups_of__2}{Finite subgroups of $O(4)$}\dotfill \pageref*{finite_subgroups_of__2} \linebreak
\noindent\hyperlink{finite_subgroups_of__3}{Finite subgroups of $O(5)$}\dotfill \pageref*{finite_subgroups_of__3} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

By a \emph{finite rotation group} one means a [[finite]] [[subgroup]] of a rotation group, 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).

\hypertarget{classifcations}{}\subsection*{{Classifcations}}\label{classifcations}

\hypertarget{finite_subgroups_of___and_}{}\subsubsection*{{Finite subgroups of $O(3)$, $SO(3)$ and $Spin(3)$}}\label{finite_subgroups_of___and_}

\begin{theorem}
\label{ClassificationOfFiniteSubgroupsOfSO3}\hypertarget{ClassificationOfFiniteSubgroupsOfSO3}{}
\textbf{([[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 \href{spin+group#ExceptionalIsomorphisms}{exceptional isomorphism} $SU(2) \simeq Spin(3)$)

\begin{displaymath}
SU(2) \simeq Spin(3) \overset{\pi}{\longrightarrow} SO(3)
\end{displaymath}
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]]

\begin{displaymath}
\itexarray{
    \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)    
  }
\end{displaymath}
exhibiting the [[even number|even]] [[order of a group|order]] [[cyclic groups]] as subgroups of [[Spin(2)]], including the the minimal case of the [[group of order 2]]

\begin{displaymath}
\itexarray{
    \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)    
  }
\end{displaymath}
as well as

\begin{displaymath}
\itexarray{
    \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)    
  }
\end{displaymath}
exhibiting the [[binary dihedral groups]] as sitting inside the [[Pin(2)]]-[[subgroup]] of [[Spin(3)]],

but only [[commuting diagrams]]

\begin{displaymath}
\itexarray{
    \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)    
  }
\end{displaymath}
for the [[odd number|odd]] [[order of a group|order]] [[cyclic group|cyclic]] [[subgroups]].

\end{theorem}
This goes back to (\hyperlink{Klein1884}{Klein 1884, chapter I}). Full proof for $SO(3)$ is spelled out for instance in (\hyperlink{Rees05}{Rees 05, theorem 11}, \hyperlink{DeVisscher11}{De Visscher 11}). The proof for the case of $SL(2,\mathbb{C})$ is spelled out in (\hyperlink{MillerBlichfeldtDickson16}{Miller-Blichfeldt-Dickson 16}) reviewed in (\hyperlink{Serrano14}{Serrano 14, section 2}). The proof of the case for $SU(2)$ given the result for $SO(3)$ is spelled out in \hyperlink{Keenan03}{Keenan 03, theorem 4}.

\hypertarget{FiniteSubgroupsOfO4}{}\subsubsection*{{Finite subgroups of $O(4)$, [[SO(4)]] and [[Spin(4)]]}}\label{FiniteSubgroupsOfO4}

For classification of the finite subgroups of $O(4)$ see (\hyperlink{DuVal65}{duVal 65}, \hyperlink{ConwaySmith03}{Conway-Smith 03})

For finite subgroups of [[Spin(4)]]: \hyperlink{MFF12}{MFF 12, appendix B}.

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]] (\hyperlink{SadocMosseri89}{SadocMosseri 89, p. 172}, see \hyperlink{MFF12}{MFF 12, table 16}).

\hypertarget{finite_subgroups_of_}{}\subsubsection*{{Finite subgroups of $O(5)$}}\label{finite_subgroups_of_}

For classification of the finite subgroups of $O(5)$ see \hyperlink{MecchiaZimmermann10}{Mecchia-Zimmermann 10}

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{SubgroupLattice}{}\subsubsection*{{Subgroup lattice}}\label{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:

This is obtained from the subgroup lattice as shown on \href{https://people.maths.bris.ac.uk/~matyd/GroupNames/}{GroupNames} for $2I \simeq$  and $2O \simeq$ 

See also \hyperlink{GoncalvesGuaschi11}{Goncalves-Guaschi 11, appendix}.

\hypertarget{GroupCohomology}{}\subsubsection*{{Group cohomology}}\label{GroupCohomology}

\begin{prop}
\label{GroupCohomologyOfFiniteSubgroupsOfSU2}\hypertarget{GroupCohomologyOfFiniteSubgroupsOfSU2}{}
\textbf{([[group cohomology]] of [[finite subgroups of SU(2)]])}

Let $G_{ADE}$ be a [[finite subgroup of SU(2)]]. Then its [[group cohomology]] with [[integer]] [[coefficients]] is as follows:

\begin{displaymath}
H^n_{grp}(G_{ADE}, \mathbb{Z})
  \;\simeq\;
  \left\{
  \itexarray{
    \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.
\end{displaymath}
Here $G_{ADE}^{ab}$ denotes the [[abelianization]] of $G_{ADE}$ and $\vert G_{ADE}\vert$ its [[cardinality]], hence $\mathbb{Z}/{\vert G_{ADE}\vert}$ the [[cyclic group]] whose [[order of a group|order]] is the cardinality of $G_{ADE}$.

The [[group homology]] with [[integer]] [[coefficients]] is

\begin{displaymath}
H_n^{grp}(G_{ADE}, \mathbb{Z})
  \;\simeq\;
  \left\{
  \itexarray{
    \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.
\end{displaymath}
\end{prop}
For pointers to proofs see for instance \hyperlink{EpaGanter16}{Epa-Ganter 16, section 4}.

\begin{remark}
\label{}\hypertarget{}{}
In discussion of [[11-dimensional supergravity]] on [[spacetimes]] with [[ADE-singularities]], the special case

\begin{displaymath}
\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 }
\end{displaymath}
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 \href{discrete+torsion#AharonyBergmanJafferis08}{Aharony-Bergman-Jafferis 08, p. 8} and \href{discrete+torsion#BDHKMMS01}{BDHKMMS 01, section 4.6.2}).

\end{remark}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[regular polytope]], [[regular polyhedron]]


\item [[ADE singularity]]


\item [[McKay correspondence]]


\item [[Platonic 2-group]]


\item [[wallpaper group]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

\hypertarget{finite_subgroups_of__and_}{}\subsubsection*{{Finite subgroups of $SO(3)$ and $Spin(3)$}}\label{finite_subgroups_of__and_}

The classification in Theorem \ref{ClassificationOfFiniteSubgroupsOfSO3} goes back to

\begin{itemize}%
\item [[Felix Klein]], chapter I of \emph{Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade}, 1884, translated as \emph{Lectures on the Icosahedron and the Resolution of Equations of Degree Five} by George Morrice 1888, \href{https://archive.org/details/cu31924059413439}{online version}

\end{itemize}
Textbook accounts include

\begin{itemize}%
\item G. A. Miller, H. F. Blichfeldt, L. E. Dickson, \emph{Theory and applications of finite groups}, Dover, New York, 1916


\item [[Klaus Lamotke]], \emph{Regular Solids and Isolated Singularities}, Vieweg 1986


\item [[Elmer Rees]], \emph{Notes on Geometry}, Springer 2005



\end{itemize}
see also

\begin{itemize}%
\item Javier Carrasco Serrano, \emph{Finite subgroups of $SL(2,\mathbb{C})$ and $SL(3,\mathbb{C})$}, Warwick 2014 (\href{https://homepages.warwick.ac.uk/~masda/McKay/Carrasco_Project.pdf}{pdf})

\end{itemize}
Complete proof of the classification of the finite subgroups of $SO(3)$ is also spelled out in

\begin{itemize}%
\item Maud De Visscher, \emph{\href{http://www.staff.city.ac.uk/maud.devisscher.1/GS/}{Groups and Symmetry}} lecture notes handout: \emph{Classification of finite rotation groups} (\href{http://www.staff.city.ac.uk/maud.devisscher.1/GS/classif_rotation.pdf}{pdf})

\end{itemize}
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

\begin{itemize}%
\item Adam Keenan, \emph{Which finite groups act freely on spheres?}, 2003 (\href{http://www.math.utah.edu/~keenan/actions.pdf}{pdf})

\end{itemize}
See also

\begin{itemize}%
\item GroupProps \emph{}

\end{itemize}
Discussion of the [[lattice]] of [[subgroups]] of the three exceptional subgroups is in

\begin{itemize}%
\item Daciberg Lima Gonçalves, John Guaschi, \emph{The Subgroups of the Binary Polyhedral Groups}, Appendix of \emph{The classification of the virtually cyclic subgroups of the sphere braid groups}, Springer (Ed.) (2013) 112 (\href{https://arxiv.org/abs/1110.6628}{arXiv:1110.6628}, \href{https://link.springer.com/content/pdf/bbm%3A978-3-319-00257-6%2F1.pdf}{pdf})

\end{itemize}
The [[universal higher central extension]] of finite subgroups of $SU(2)$ (``[[Platonic 2-groups]]'') are discussed in

\begin{itemize}%
\item [[Narthana Epa]], [[Nora Ganter]], \emph{Platonic and alternating 2-groups}, Higher Structures 1(1):122-146, 2017 (\href{http://arxiv.org/abs/1605.09192}{arXiv:1605.09192})

\end{itemize}
\hypertarget{finite_subgroups_of__2}{}\subsubsection*{{Finite subgroups of $O(4)$}}\label{finite_subgroups_of__2}

\begin{itemize}%
\item [[Patrick du Val]], \emph{Homographies, Quaternions and Rotations}, Oxford Mathematical Monographs, Clarendon Press (1964)

also(?): Journal of the London Mathematical Society, Volume s1-40, Issue 1 (1965) (\href{https://doi.org/10.1112/jlms/s1-40.1.569b}{doi:10.1112/jlms/s1-40.1.569b})


\item [[John Conway]], D. A. Smith, \emph{On quaternions and octonions: their geometry, arithmetic and symmetry} A K Peters Ltd., Natick, MA, 2003


\item Paul de Medeiros, [[José Figueroa-O'Farrill]], appendix B of \emph{Half-BPS M2-brane orbifolds}, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. (\href{http://arxiv.org/abs/1007.4761}{arXiv:1007.4761}, \href{https://projecteuclid.org/euclid.atmp/1408561553}{Euclid})


\item J. F. Sadoc, R. Mosseri, \emph{Icosahedral order, curved space and quasicrystals}, in Jaric, Gratias (eds.) \emph{Extended icosahedral structures}, 1989 (\href{https://books.google.co.uk/books?id=HWyIUglWeXsC&pg=PA172&lpg=PA172#v=onepage&q&f=false}{GoogleBooks})



\end{itemize}
\hypertarget{finite_subgroups_of__3}{}\subsubsection*{{Finite subgroups of $O(5)$}}\label{finite_subgroups_of__3}

\begin{itemize}%
\item Mattia Mecchia, Bruno Zimmermann, \emph{On finite groups acting on homology 4-spheres and finite subgroups of $SO(5)$}, Topology and its Applications 158.6 (2011): 741-747 (\href{https://arxiv.org/abs/1001.3976}{arXiv:1001.3976})

\end{itemize}
[[!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]]



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