Icosahedron Binary Group

The binary icosahedron group 2I or <2,3,5> is a non-Abelian group of order 120. The group is an extension I or (2,3,5) of order 60 and the cyclic group of order 2 and is a prototype of the icosahedron group at 2: 1

\operatorname {Spin} (3)\to \operatorname {SO} (3)

{\ displaystyle \ operatorname {Spin} (3) \ to \ operatorname {SO} (3)}

{\ displaystyle \ operatorname {Spin} (3) \ to \ operatorname {SO} (3)}

special orthogonal group by a spinor group . It follows that the binary icosahedron group is a group Spin (3) of order 120.

This group should not be confused with the , having the same order of 120 but being a subgroup of the orthogonal group O (3).

The binary icosahedron group is best described as a discrete subgroup of unit quaternions , with isomorphism $\operatorname {Spin} (3)\cong \operatorname {Sp} (1)$ ${\ displaystyle \ operatorname {Spin} (3) \ cong \ operatorname {Sp} (1)}$ ${\ displaystyle \ operatorname {Spin} (3) \ cong \ operatorname {Sp} (1)}$ , where Sp (1) is the multiplicative group of unit quaternions ^[1] .

Content

Elements

The binary icosahedron group is explicitly defined by the union of 24 Hurwitz quaternions

{± 1, ± i , ± j , ± k , ½ (± 1 ± i ± j ± k )}

with all 96 quaternions obtained from

½ (0 ± i ± φ ⁻¹ j ± φ k )

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

In total, we get 120 elements (single icosians ). Their modulus is equal to unity, and therefore they lie in the group of units of the quaternions Sp (1). The convex hull of these 120 elements in 4-dimensional space forms a regular 4-dimensional polyhedron , known as a six-hundred-centimeter .

Properties

Central extension

The binary icosahedron group, denoted by 2 I , is the of the icosahedron group and therefore the is the perfect central extension of the simple group.

Specifically, the group fits into a short exact sequence

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

{\ displaystyle 1 \ to \ {\ pm 1 \} \ to 2I \ to I \ to 1.}

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

The center of the group 2 I is the subgroup {± 1}, so the group of inner automorphisms is isomorphic to I. The complete is isomorphic to S ₅ (a symmetric group of permutations of 5 letters), just like $I\cong A_{5}$ ${\ displaystyle I \ cong A_ {5}}$ - any automorphism 2 I fixes a nontrivial element of the center ( $-1$ ${\ displaystyle -1}$ ), and therefore reduces to an automorphism I, and vice versa, any automorphism I rises to an automorphism 2 I.

Super Excellence

The icosahedron binary group is a group, that is, it coincides with its commutant . In fact, 2 I is the only perfect group of order 120. It follows that 2 I is unsolvable .

Moreover, the icosahedron binary group is , which means that its first two zero - $H_{1}(2I;\mathbf {Z} )\cong H_{2}(2I;\mathbf {Z} )\cong 0.$ ${\ displaystyle H_ {1} (2I; \ mathbf {Z}) \ cong H_ {2} (2I; \ mathbf {Z}) \ cong 0.}$ This means that its abelization is trivial (the group does not have nontrivial abelian quotients) and that its is trivial (the group does not have nontrivial perfect central extensions). In fact, the icosahedron binary group is the smallest (non-trivial) super-perfect group.

The binary icosahedron group, however, is not , since H _n (2 I , Z ) is cyclic of order 120 for n = 4 k +3 and trivial for others n > 0 ^[2] .

Isomorphisms

The binary icosahedron group is a subgroup of Spin (3) and covers the icosahedron group, which is a subgroup of SO (3). The icosahedron group is isomorphic to the symmetry group of the 4-dimensional simplex , which is a subgroup of SO (4), and the binary icosahedron group is isomorphic to its double covering in Spin (4). Note that the symmetric group $S_{5}$ ${\ displaystyle S_ {5}}$ has a 4-dimensional representation (this is usually the smallest dimensionally irreducible representation of complete symmetries $(n-1)$ ${\ displaystyle (n-1)}$ -dimensional simplex), and therefore the complete set of symmetries of the 4-dimensional simplex is $S_{5},$ ${\ displaystyle S_ {5},}$ but this is not a complete icosahedron group (these are two different groups of the order of 120).

The icosahedron binary group can be considered as a $A_{5},$ ${\ displaystyle A_ {5},}$ , $2\cdot A_{5}\cong 2I$ ${\ displaystyle 2 \ cdot A_ {5} \ cong 2I}$ . This isomorphism covers the isomorphism of the icosahedron group with the alternating group $A_{5}\cong I$ ${\ displaystyle A_ {5} \ cong I}$ and can be considered as subgroups of Spin (4) and SO (4) (as well as subgroups of the symmetric group $S_{5}$ ${\ displaystyle S_ {5}}$ and any of her double coatings $2\cdot S_{5}^{\pm }$ ${\ displaystyle 2 \ cdot S_ {5} ^ {\ pm}}$ , which, in turn, are subgroups of both the pin group and the orthogonal group $\operatorname {Pin} ^{\pm }(4)\to \operatorname {O} (4)$ ${\ displaystyle \ operatorname {Pin} ^ {\ pm} (4) \ to \ operatorname {O} (4)}$ )

Unlike the icosahedral group, which is in three-dimensional space, these tetrahedral and alternating groups (and their double coverings) exist in all dimensions.

It can be shown that the icosahedral group is isomorphic to the special linear group SL (2,5) - the group of all 2 × 2 matrices over a finite field F ₅ with the identity determinant.

Group Task

Group 2 I has an assignment

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

{\ displaystyle \ langle r, s, t \ mid r ^ {2} = s ^ {3} = t ^ {5} = rst \ rangle}

which is equivalent

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

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

The generators of this relation are given by the formula

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

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

Subgroups

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

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

Element $-1$ ${\ displaystyle -1}$ is the only element of order 2, and therefore it is contained in all subgroups of even order — any subgroup of group 2 I either has an odd order or is a pre-image of a subgroup of group I. In addition to cyclic groups formed by various elements (which may have an odd order), other subgroups of group 2 I (up to conjugation) can only be:

orders 12 and 20 (covering the dihedral groups D ₃ and D ₅ in I ).
The quaternion group , consisting of 8 Lipschitz units , forms a subgroup with an index of 15, which is also a Dic ₂ dicyclic group . This subgroup covers the stabilizer rib .
The 24 quaternions of Hurwitz form a subgroup with index 5, the binary group of the tetrahedron . This subgroup covers the chiral tetrahedral group . The latter is self-normalized , so its conjugacy class has 5 elements (this gives a mapping $2I\to S_{5}$ ${\ displaystyle 2I \ to S_ {5}}$ whose image is $A_{5}$ ${\ displaystyle A_ {5}}$ )

Connection with 4-dimensional symmetry groups

The 4-dimensional analogue I _h is the symmetric group of the six hundred cell (and also the dual hundred and twenty cell ). The first is a group of type H ₃ , and the second is a group of type H ₄ with the same notation [3,3,5]. Its subgroup of rotations, , is a group of order 7200 living in SO (4) . SO (4) has a ( Spin (4) ) in exactly the same way as Spin (3) is a covering group SO (3). Like the isomorphism Spin (3) = Sp (1), the group Spin (4) is isomorphic to Sp (1) × Sp (1).

The inverse image of [3,3,5] ⁺ in Spin (4) (the four-dimensional analogue of 2 I ) is exactly the direct product of 2 I × 2 I of order 14400. The group of rotations of the six-hundredth is

[3,3,5] ⁺ = (2 I × 2 I ) / {± 1}.

Various other four-dimensional symmetric groups can be formed from 2 I. See Conway and Smith Conway for details. ^[3]

Applications

The space of adjacent classes Spin (3) / 2 I = S ^3/2 I is a , called the Poincare sphere . This is an example of a homological sphere, that is, a 3-manifold whose homology groups are equal to the same 3-sphere groups. The fundamental group of the Poincare sphere is isomorphic to the binary group of the icosahedron, since the Poincare sphere is a factor group of the 3-sphere with respect to the binary group of the icosahedron.

Notes

↑ A description of this homomorphism can be found in the article “ Quaternions and rotation of space ”.
↑ Adem, Milgram, 1994 , p. 279.
↑ Conway, Smith, 2003 .

Links

Alejandro Adem, R. James Milgram. Cohomology of finite groups. - Berlin, New York: Springer-Verlag , 1994. - T. 309. - (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]). - ISBN 978-3-540-57025-7 .
HSM Coxeter, WOJ Moser. Generators and Relations for Discrete Groups, 4th edition. - New York: Springer-Verlag, 1980 .-- ISBN 0-387-09212-9 . 6.5 The binary polyhedral groups, p. 68
John H. Conway, Derek A. Smith. On Quaternions and Octonions. - Natick, Massachusetts: AK Peters, Ltd, 2003 .-- ISBN 1-56881-134-9 .

[1] A description of this homomorphism can be found in the article “ Quaternions and rotation of space ”.

[_2b7c0404386f8f58-2] Adem, Milgram, 1994 , p. 279.

[_90ca30271fc88620-3] Conway, Smith, 2003 .