Contents

group theory

# Contents

## Idea

The icosahedral group is the group of symmetries of an icosahedron.

As a symmetry group of one of the Platonic solids, the icosahedral group participates in the ADE pattern:

Dynkin diagram/
Dynkin quiver
dihedron,
Platonic solid
finite 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
$SU(n+1)$
A1cyclic group of order 2
$\mathbb{Z}_2$
cyclic group of order 2
$\mathbb{Z}_2$
SU(2)
A2cyclic group of order 3
$\mathbb{Z}_3$
cyclic group of order 3
$\mathbb{Z}_3$
SU(3)
A3
=
D3
cyclic group of order 4
$\mathbb{Z}_4$
cyclic group of order 4
$2 D_2 \simeq \mathbb{Z}_4$
SU(4)
$\simeq$
Spin(6)
D4dihedron on
bigon
Klein four-group
$D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$
quaternion group
$2 D_4 \simeq$ Q8
SO(8), Spin(8)
D5dihedron on
triangle
dihedral group of order 6
$D_6$
binary dihedral group of order 12
$2 D_6$
SO(10), Spin(10)
D6dihedron on
square
dihedral group of order 8
$D_8$
binary dihedral group of order 16
$2 D_{8}$
SO(12), Spin(12)
$D_{n \geq 4}$dihedron,
hosohedron
dihedral group
$D_{2(n-2)}$
binary dihedral group
$2 D_{2(n-2)}$
special orthogonal group, spin group
$SO(2n)$, $Spin(2n)$
$E_6$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

More in detail, there are variants of the icosahedral group corresponding to the stages of the Whitehead tower of O(3):

• the full icosahedral group is the subgroup of O(3)

$I_h \hookrightarrow O(3)$

which is the stabilizer of the standard embedding of the icosahedron into Cartesian space $\mathbb{R}^3$;

• the rotational icosahedral group $I \hookrightarrow SO(3)$ is the restriction to orientation-preserving symmetries, hence to SO(3); this is isomorphic to the alternating group $A_5$

• finally the binary icosahedral group is the double cover (see at covering of alternating group), hence the lift of $I$ to Spin(3)$\simeq$ SU(2);

• next there is a string 2-group lift $\mathcal{I} \hookrightarrow String_{SU(2)}$ of the icosahedral group (Epa 10, Epa-Ganter 16)

$\array{ String_{2I} &\hookrightarrow& String_{SU(2)} \\ \downarrow && \downarrow \\ 2 I &\hookrightarrow & Spin(3) \simeq SU(2) \\ \downarrow && \downarrow \\ I \simeq A_5 &\hookrightarrow& SO(3) \\ \downarrow && \downarrow \\ I_h \simeq A_5\times \mathbb{Z}/2 &\hookrightarrow & O(3) }$

## Definition

Regard the icosahedron, determined uniquely up to isometry on $\mathbb{R}^3$ as a regular convex polyhedron with $20$ faces, as a metric subspace $S$ of $\mathbb{R}^3$. Then the icosahedral group may be defined as the group of isometries of $S$.

(…)

The elements of the binary icosahedral group form the vertices of the 120-cell.

## Properties

### General properties

The subgroup of orientation-preserving symmetries of the icosahedron is the alternating group $A_5$ whose order is 60. The full icosahedral group is isomorphic to the Cartesian product $A_5 \times \mathbb{Z}_2$ (with the group of order 2).

Hence the order of the full icosahedral group is $60 \times 2 = 120$, as is that of the binary icosahedral group $2 I$.

###### Proposition

There is an exceptional isomorphism

$I \;\simeq\; PSL_2(\mathbb{F}_5)$

of the icosahedral group with the projective special linear group over the prime field $\mathbb{F}_5$.

and, covering this,

$2I \;\simeq\; SL_2(\mathbb{F}_5)$

of the binary icosahedral group with the special linear group over $\mathbb{F}_5$.

###### Proposition

The binary icosahedral group $2I$ is a perfect group: its abelianization is the trivial group.

In fact, up to isomorphism, the binary icosahedral group is the unique finite group of order 120 which is a perfect group.

###### Proposition

(quaternion group inside binary icosahedral group)

The binary icosahedral group contains the quaternion group of order 8, hence the binary dihedral group of order 8, as a subgroup (not normal):

$2 D_4 =Q_8 \subset 2 I \,.$

In fact the only finite subgroups of SU(2) which contain $2 D_4 =Q_8$ as a proper subgroup are the exceptional ones, hence the binary tetrahedral group, the binary octahedral group and the binary icosahedral group.

###### Proposition

(normal subgroups of binary icosahedral group

The only proper normal subgroup of the binary icosahedral group is its center $Z(2I) \simeq \mathbb{Z}/2$.

### Character table

linear representation theory of binary icosahedral group $2 I$

$\,$

group order: ${\vert 2I\vert} = 120$

conjugacy classes:12345A5B610A10B
their cardinality:1120301212201212

let $\phi \coloneqq \tfrac{1}{2}( 1 + \sqrt{5} )$ (the golden ratio)

character table over the complex numbers $\mathbb{C}$

irrep12345A5B610A10B
$\rho_1$111111111
$\rho_2$2-2-10$\phi - 1$$-\phi$1$\phi$$1 - \phi$
$\rho_3$2-2-10$-\phi$$\phi - 1$1$1-\phi$$\phi$
$\rho_4$330-1$1 - \phi$$\phi$0$\phi$$1-\phi$
$\rho_5$330-1$\phi$$1-\phi$0$1-\phi$$\phi$
$\rho_6$4410-1-11-1-1
$\rho_7$4-410-1-1-111
$\rho_8$55-1100-100
$\rho_9$6-600110-1-1

References

• Bockland, Character tables and McKay quivers (pdf)

### Quotient spaces

The coset space $SU(2)/2I$ is the Poincaré homology sphere.

### Group cohomology

For a little bit about the group cohomology (or at least the homology) of the binary icosahedral group $SL_2(\mathbb{F}_5)$, see Tomoda & Zvengrowski 08, Section 4.3, Epa & Ganter 16, p. 12 Groupprops

Origin:

• Felix Klein, chapter I.8 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