In mathematics, the binary icosahedral group 2I or <2,3,5> is a certain nonabelian group of order 120. It is an extension of the icosahedral group I or (2,3,5) of order 60 by a cyclic group of order 2, and is the preimage of the icosahedral group under the 2:1 covering homomorphism
Contents
- Elements
- Central extension
- Superperfect
- Isomorphisms
- Presentation
- Subgroups
- Relation to 4 dimensional symmetry groups
- Applications
- References
of the special orthogonal group by the spin group. It follows that the binary icosahedral group is a discrete subgroup of Spin(3) of order 120.
It should not be confused with the full icosahedral group, which is a different group of order 120, and is rather a subgroup of the orthogonal group O(3).
The binary icosahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism
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 all the four coordinates 0, 1, φ−1, φ, and with all possible sign combinations. Here φ = ½ (1 + √5) is the golden ratio.
In total there are 120 elements, namely the unit icosians. They all have unit magnitude and therefore lie in the unit quaternion group Sp(1). The convex hull of these 120 elements in 4-dimensional space form a regular 4-polytope, known as the 600-cell.
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
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), just as for
Superperfect
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.
Further, the binary icosahedral group is superperfect, meaning abstractly that its first two group homology groups vanish:
The binary icosahedral group is not acyclic, however, as Hn(2I,Z) is cyclic of order 120 for n = 4k+3, and trivial for n > 0 otherwise, (Adem & Milgram 1994, p. 279).
Isomorphisms
Concretely, the binary icosahedral group is a subgroup of Spin(3), and covers the icosahedral group, which is a subgroup of SO(3). Abstractly, the icosahedral group is isomorphic to the symmetries of the 4-simplex, which is a subgroup of SO(4), and the binary icosahedral group is isomorphic to the double cover of this in Spin(4). Note that the symmetric group
The binary icosahedral group can be considered as the double cover of the alternating group
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
Note also the exceptional isomorphism
Presentation
The group 2I has a presentation given by
or equivalently,
Generators with these relations are given by
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 }.
Relation to 4-dimensional symmetry groups
The 4-dimensional analog of the icosahedral symmetry group Ih is the symmetry group of the 600-cell (also that of its dual, the 120-cell). Just as the former is the Coxeter group of type H3, the latter is the Coxeter group of type H4, also denoted [3,3,5]. Its rotational 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 Spin(3) is the double cover of SO(3). Similar to the isomorphism Spin(3) = Sp(1), the group Spin(4) is isomorphic to Sp(1) × Sp(1).
The preimage of [3,3,5]+ in Spin(4) (a four-dimensional analogue of 2I) 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).
Applications
The coset space Spin(3) / 2I = S3 / 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, as the Poincaré sphere is the quotient of a 3-sphere by the binary icosahedral group.