Binary icosahedral group

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In mathematics, the binary icosahedral group is an extension of the icosahedral group I of order 60 by a cyclic group of order 2. It can be defined as the preimage of the icosahedral group under the 2:1 covering homomorphism

$\mathrm{Sp}(1) \to \mathrm{SO}(3).\,$

where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.) It follows that the binary icosahedral group is discrete subgroup of Sp(1) of order 120.

It should not be confused with the full icosahedral group, which is a different group of order 120.

1 Elements
2 Properties
3 Relation to 4-dimensional symmetry groups
4 Applications
5 See also
6 References

[edit] Elements

Explicitly, the binary icosahedral group is given as the union of the 24 Hurwitz units

{±1, ±i, ±j, ±k, ½(±1 ± i ± j ± k)}

with all 96 quaternions obtained from

½(0 ± i ± φ⁻¹j ± φk)

by an even permutation of coordinates (all possible sign combinations). Here φ = ½(1+√5) is the golden ratio.

All told there are 120 elements. They all have absolute value 1 and therefore lie in the unit quaternion group Sp(1). The convex hull of these 120 elements in 4-dimensional space form a convex regular 4-polytope called the 600-cell.

[edit] Properties

[edit] Central extension

The binary icosahedral group, denoted by 2I, is the universal perfect central extension of the icosahedral group, and thus is quasisimple: it is a perfect central extension of a simple group.

Explicitly, it fits into the short exact sequence

$1\to\{\pm 1\}\to 2I\to I \to 1.\,$

This sequence does not split, meaning that 2I is not a semidirect product of {±1} by I. In fact, there is no subgroup of 2I isomorphic to I.

The center of 2I is the subgroup {±1}, so that the inner automorphism group is isomorphic to I. The full automorphism group is isomorphic to S₅ (the symmetric group on 5 letters).

The binary icosahedral group is perfect, meaning that it is equal to its commutator subgroup. In fact, 2I is the unique perfect group of order 120. It follows that 2I is not solvable.

[edit] Isomorphisms

One can show that the binary icosahedral group is isomorphic to the special linear group SL(2,5) — the group of all 2×2 matrices over the finite field F₅ with unit determinant; this covers the exceptional isomorphism of $I\cong A_5$ with the projective special linear group PSL(2,5).

[edit] Presentation

The group 2I has a presentation given by

$\langle r,s,t \mid r^2 = s^3 = t^5 = rst \rangle$

or equivalently,

$\langle s,t \mid (st)^2 = s^3 = t^5 \rangle.$

Generators with these relations are given by

$s = \tfrac{1}{2}(1+i+j+k) \qquad t = \tfrac{1}{2}(\varphi+\varphi^{-1}i+j).$

[edit] Subgroups

The only proper normal subgroup of 2I is the center {±1}.

By the third isomorphism theorem, there is a Galois connection between subgroups of 2I and subgroups of I, where the closure operator on subgroups of 2I is multiplication by {±1}.

$- 1$ is the only element of order 2, hence it is contained in all subgroups of even order: thus every subgroup of 2I is either of odd order or is the preimage of a subgroup of I. Besides the cyclic groups generated by the various elements (which can have odd order), the only other subgroups of 2I (up to conjugation) are:

binary dihedral groups of orders 12 and 20 (covering the dihedral groups D₃ and D₅ in I).
The quaternion group consisting of the 8 Lipschitz units forms a subgroup of index 15, which is also the dicyclic group Dic₂; this covers the stabilizer of an edge.
The 24 Hurwitz units form an index 5 subgroup called the binary tetrahedral group; this covers a chiral tetrahedral group. This group is self-normalizing so its conjugacy class has 5 members (this gives a map $2I \to S_5$ whose image is $A 5$ ).

[edit] Relation to 4-dimensional symmetry groups

The 4-dimensional analog of the icosahedral group is the symmetry group of the 600-cell (also that of the 120-cell). This is the Coxeter group of type H₄, also denoted [3,3,5]. The rotation subgroup, denoted [3,3,5]⁺ is a group of order 7200 living in SO(4). SO(4) has a double cover called Spin(4) in much the same way that Sp(1) is the double cover of SO(3). The group Spin(4) is isomorphic to Sp(1)×Sp(1).

The preimage of [3,3,5]⁺ in Spin(4) is precisely the product group 2I×2I of order 14400. The rotational symmetry group of the 600-cell is then

[3,3,5]⁺ = 2I×2I/{±1}.

Various other 4-dimensional symmetry groups can be constructed from 2I. For details, see (Conway and Smith, 2003).

[edit] Applications

The coset space Sp(1)/2I is a spherical 3-manifold called the Poincaré homology sphere. It is an example of a homology sphere, i.e. a 3-manifold whose homology groups are identical to those of a 3-sphere. The fundamental group of the Poincaré sphere is isomorphic to the binary icosahedral group.

[edit] See also

[edit] References

Conway, John H.; Smith, Derek A. (2003). On Quaternions and Octonions. Natick, Massachusetts: AK Peters, Ltd. ISBN 1-56881-134-9.

Categories: Finite groups | Quaternions | Rotational symmetry