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 diagramPlatonic solidfinite subgroup of $SO(3)$finite subgroup of $SU(2)$simple Lie group
$A_l$cyclic groupcyclic groupspecial unitary group
$D_l$dihedron/hosohedrondihedral groupbinary dihedral groupspecial orthogonal group
$E_6$tetrahedrontetrahedral groupbinary tetrahedral groupE6
$E_7$cube/octahedronoctahedral groupbinary octahedral groupE7
$E_8$dodecahedron/icosahedronicosahedral groupbinary icosahedral groupE8

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 $S$.

## Properties

### Group order

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 icosahedral group is $60 \times 2 = 120$.

### Exceptional isomorphisms

There is an exceptional isomorphism $I \simeq PSL_2(\mathbb{F}_5)$, with $2I \simeq SL_2(\mathbb{F}_5)$ covering this isomorphism.

($PSL_2$ the projective special linear group, $SL_2$ the special linear group, $\mathbb{F}_5$ the prime field for $p = 5$)

### 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

## References

Revised on September 7, 2016 03:19:10 by Urs Schreiber (82.113.121.135)