120 реттің емес абелдік тобы

Nonabelian group of order 120
In mathematics, the binary icosahedral group 2I or 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 the cyclic group of order 2, and is the preimage of the icosahedral group under the 2:1 covering homomorphism

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). In the algebra of quaternions, the binary icosahedral group is concretely realized as a discrete subgroup of the versors, which are the quaternions of norm one. For more information see Quaternions and spatial rotations.

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: Concretely, this means that its abelianization is trivial (it has no non trivial abelian quotients) and that its Schur multiplier is trivial (it has no non trivial perfect central extensions). In fact, the binary icosahedral group is the smallest (non trivial) superperfect group. 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, .

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 does have a 4 dimensional representation (its usual lowest dimensional irreducible representation as the full symmetries of the simplex), and that the full symmetries of the 4 simplex are thus not the full icosahedral group (these are two different groups of order 120). The binary icosahedral group can be considered as the double cover of the alternating group denoted this isomorphism covers the isomorphism of the icosahedral group with the alternating group Just as is a discrete subgroup of , is a discrete subgroup of the double over of , namely The 2 1 homomorphism from to then restricts to the 2 1 homomorphism from to
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 with the projective special linear group PSL(2,5). Note also the exceptional isomorphism which is a different group of order 120, with the commutative square of SL, GL, PSL, PGL being isomorphic to a commutative square of which are isomorphic to subgroups of the commutative square of Spin(4), Pin(4), SO(4), O(4).

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).

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.