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

• Groupnames, SL(2,5)

• 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 Groupprops