Contents

group theory

# Contents

## Properties

### Cohomology

###### Proposition

The integral cohomology ring of the classifying space $B SO(3)$ is

$H^\bullet\big( B SO(3), \mathbb{Z} \big) \;\simeq\; \mathbb{Z}\big[ p_1, W_3\big] / (2 W_3) \,,$

where

This is a special case of Brown 82, theorem 1.5, which is also reviewed as Rudolph-Schmidt 17, Thm. 4.2.23 with Remark 4.2.25.

### Finite subgroups

###### Theorem

(ADE classification of finite subgroups of SO(3) and Spin(3)$\simeq$ SU(2))

The finite subgroups of the special orthogonal group $SO(3)$ as well as the finite subgroups of the special unitary group SU(2) are, up to conjugation, given by the following classification:

Dynkin diagram/
Dynkin quiver
Platonic solidfinite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
$A_{n \geq 1}$cyclic group
$\mathbb{Z}_{n+1}$
cyclic group
$\mathbb{Z}_{n+1}$
special unitary group
$SU(n+1)$
D4Klein four-group
$D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$
quaternion group
$2 D_4 \simeq$ Q8
SO(8)
$D_{n \geq 4}$dihedron,
hosohedron
dihedral group
$D_{2(n-2)}$
binary dihedral group
$2 D_{2(n-2)}$
special orthogonal group
$SO(2n)$
$E_6$tetrahedrontetrahedral group
$T$
binary tetrahedral group
$2T$
E6
$E_7$cube,
octahedron
octahedral group
$O$
binary octahedral group
$2O$
E7
$E_8$dodecahedron,
icosahedron
icosahedral group
$I$
binary icosahedral group
$2I$
E8

Here under the double cover projection (using the exceptional isomorphism $SU(2) \simeq Spin(3)$)

$SU(2) \simeq Spin(3) \overset{\pi}{\longrightarrow} SO(3)$

all the finite subgroups of $SU(2)$ except the odd-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 order cyclic groups as subgroups of Spin(2), including the 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 order cyclic subgroups.

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

sp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
SO(3)Spin(3)
SO(4)Spin(4)
SO(5)Spin(5)Pin(5)
Spin(6)
Spin(7)
SO(8)Spin(8)SO(8)
SO(9)Spin(9)
$\vdots$$\vdots$
SO(16)Spin(16)SemiSpin(16)
SO(32)Spin(32)SemiSpin(32)

## References

• Jason Hanson, Rotations in three, four, and five dimensions (arXiv:1103.5263)

• Edgar H. Brown, The Cohomology of $B SO_n$ and $BO_n$ with Integer Coefficients, Proceedings of the American Mathematical Society, Vol. 85, No. 2 (Jun., 1982), pp. 283-288 (jstor:2044298)