nLab
icosahedral group

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:

ADE classification

Dynkin diagramPlatonic solidfinite subgroup of SO(3)SO(3)finite subgroup of SU(2)SU(2)simple Lie group
A lA_lcyclic groupcyclic groupspecial unitary group
D lD_ldihedron/hosohedrondihedral groupbinary dihedral groupspecial orthogonal group
E 6E_6tetrahedrontetrahedral groupbinary tetrahedral groupE6
E 7E_7cube/octahedronoctahedral groupbinary octahedral groupE7
E 8E_8dodecahedron/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):

String 2I String SU(2) 2I Spin(3)SU(2) IA 5 SO(3) I hA 5×/2 O(3) \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 3\mathbb{R}^3 as a regular convex polyhedron with 2020 faces, as a metric subspace SS of 3\mathbb{R}^3. Then the icosahedral group may be defined as the group of isometries of SS.

More to be added.

Properties

Group order

The subgroup of orientation-preserving symmetries of the icosahedron is the alternating group A 5A_5 whose order is 60. The full icosahedral group is isomorphic to the Cartesian product A 5× 2A_5 \times \mathbb{Z}_2 (with the group of order 2).

Hence the order of the icosahedral group is 60×2=120 60 \times 2 = 120 .

Exceptional isomorphisms

There is an exceptional isomorphism IPSL 2(𝔽 5)I \simeq PSL_2(\mathbb{F}_5), with 2ISL 2(𝔽 5)2I \simeq SL_2(\mathbb{F}_5) covering this isomorphism.

(PSL 2PSL_2 the projective special linear group, SL 2SL_2 the special linear group, 𝔽 5\mathbb{F}_5 the prime field for p=5p = 5)

Quotient spaces

The coset space SU(2)/2ISU(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(𝔽 5)SL_2(\mathbb{F}_5), see Groupprops

References

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