Binary icosahedral group

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Binary icosahedral group

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the binary icosahedral group is an extension

Group extension

In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence...
of the icosahedral group I of order 60 by a cyclic group

Cyclic group

In group theory, a cyclic group or monogenous group is a group that can be generating set of a group by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g ....
of order 2. It can be defined as the preimage of the icosahedral group under the 2:1 covering homomorphism where Sp(1) is the multiplicative group of unit quaternion

Quaternion

Quaternions, in mathematics, are a non-commutative number system that extends the complex numbers. The quaternions were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space....
s. (For a description of this homomorphism see the article on quaternions and spatial rotation

Quaternions and spatial rotation

quaternion provide a convenient mathematical notation for representing orientations and rotations of objects in three dimensions. Compared to Euler angles they are simpler to function composition and avoid the problem of gimbal lock....
s.) 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

Icosahedral symmetry

File:Soccer ball.svgA regular icosahedron has 60 rotational symmetries, and a total of 120 symmetries including transformations that combine a reflection and a rotation....
, which is a different group of order 120.

with all 96 quaternions obtained from

½(0 ± i ± φ⁻¹j ± φk)

by an even permutation of coordinates (all possible sign combinations).

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Encyclopedia

In mathematics

Mathematics

Group extension

Cyclic group

Quaternion

Quaternions and spatial rotation

Icosahedral symmetry

Elements

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

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

Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller....
.

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

Convex hull

In mathematics, the convex hull or convex envelope for a Set of points X in a real vector space V is the minimal convex set containing X....
of these 120 elements in 4-dimensional space form a convex regular 4-polytope

Convex regular 4-polytope

In mathematics, a convex regular 4-polytope is 4-dimensional polytope which is both regular polytope and convex set. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
called the 600-cell

600-cell

In geometry, the 600-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . Its boundary is composed of 600 tetrahedron cell with 20 meeting at each vertex....
.

Properties

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

Semidirect product

In mathematics, especially in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup....
of by I. In fact, there is no subgroup of 2I isomorphic to I.

The center of 2I is the subgroup , so that the inner automorphism group is isomorphic to I. The full automorphism group is isomorphic to S₅ (the symmetric group

Symmetric group

In mathematics, the symmetric group on a Set X, denoted by SX, or Sym, is the group whose underlying set is the set of all bijective function s from X to X, in which the group operation is that of Function composition, i.e., two such functions f and g can be composed to yield a new bijective function ,...
on 5 letters).

The binary icosahedral group is perfect

Perfect group

In mathematics, in the realm of group theory, a Group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian group quotient group....
, meaning that it is equal to its commutator subgroup

Commutator subgroup

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generating set of a group by all the commutators of the group....
. In fact, 2I is the unique perfect group of order 120. It follows that 2I is not solvable

Solvable group

In the history of mathematics, the origins of group theory lie in the search for a Mathematical_proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory....
.

Isomorphisms

One can show that the binary icosahedral group is isomorphic to the special linear group

Special linear group

In mathematics, the special linear group of degree n over a field F is the set of n×n Matrix with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
SL(2,5) — the group of all 2×2 matrices over the finite field

Finite field

In abstract algebra, a finite field or Galois field is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory....
F₅ with unit determinant; this covers the exceptional isomorphism

Alternating group

In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on the set is called the alternating group of degree n, or the alternating group on n letters and denoted by An or Alt....
of with the projective special linear group PSL(2,5).

Presentation

The group 2I has a presentation given by or equivalently, Generators with these relations are given by

Subgroups

The only proper normal subgroup

Normal subgroup

In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they can be used to construct quotient groups from a given group ....
of 2I is the center .

By the third isomorphism theorem, there is a Galois connection

Galois connection

In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets . Galois connections generalize the correspondence between subgroups and field investigated in Galois theory....
between subgroups of 2I and subgroups of I, where the closure operator

Closure operator

A closure operator on a set S is a function cl: P ? P from the power set of S to itself which satisfies the following conditions for all sets X,Y ? S....
on subgroups of 2I is multiplication by .

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 group

Cyclic group

In group theory, a cyclic group or monogenous group is a group that can be generating set of a group by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g ....
s 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
Quaternion group

In group theory, the quaternion group is a nonabelian group group of order 8. It is often denoted by Q or Q8 and written in multiplicative form, with the following 8 elementsHere 1 is the identity element, 2 = 1, and a = a = −a for all a in Q....
consisting of the 8 Lipschitz units forms a subgroup of index 15, which is also the dicyclic group
Dicyclic group

In group theory, a dicyclic group is a member of a class of group s Dicn , a non-abelian group of order 4n, which is an group extension of the cyclic group of order 2...
Dic₂; this covers the stabilizer of an edge.
The 24 Hurwitz units form an index 5 subgroup called the binary tetrahedral group
Binary tetrahedral group

In mathematics, the binary tetrahedral group is an group extension of the tetrahedral group T of order 12 by a cyclic group of order 2.It is the binary polyhedral group corresponding to the tetrahedral group, and as such can be defined as the preimage of the tetrahedral group under the 2:1 covering homomorphism...
; this covers a chiral tetrahedral group. This group is self-normalizing so its conjugacy class
Conjugacy class

In mathematics, especially group theory, the elements of any group may be partition of a set into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure....
has 5 members (this gives a map whose image is ).

Relation to 4-dimensional symmetry groups

The 4-dimensional analog of the icosahedral group is the symmetry group of the 600-cell

600-cell

In geometry, the 600-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . Its boundary is composed of 600 tetrahedron cell with 20 meeting at each vertex....
(also that of the 120-cell

120-cell

In geometry, the 120-cell is the convex regular 4-polytope with Schl?fli symbol .The boundary of the 120-cell is composed of 120 dodecahedral cell with 4 meeting at each vertex....
). This is the Coxeter group

Coxeter group

In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....
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)

In mathematics, SO is the four-dimensional rotation group; that is, the group of rotations about a fixed point in four-dimensional Euclidean space....
. SO(4) has a double cover called Spin(4)

Spin group

In mathematics the spin group Spin is the covering space of the special orthogonal group SO, such that there exists a short exact sequence of Lie groups...
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

Direct product

In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
2I×2I of order 14400. The rotational symmetry group of the 600-cell is then

[3,3,5]⁺ = 2I×2I/.

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

Applications

The coset space Sp(1)/2I is a spherical 3-manifold

Spherical 3-manifold

In mathematics, a spherical 3-manifold M is a 3-manifold of the formwhere Γ is a Finite group subgroup of Special orthogonal group Group action by rotations on the 3-sphere ....
called the Poincaré homology sphere. It is an example of a homology sphere

Homology sphere

In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n = 1. That is,...
, i.e. a 3-manifold whose homology groups are identical to those of a 3-sphere

3-sphere

In mathematics, a '3-sphere' is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space....
. The fundamental group

Fundamental group

In mathematics, more specifically algebraic topology, the fundamental group or Poincar? group is a group associated to any given pointed space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other....
of the Poincaré sphere is isomorphic to the binary icosahedral group.

Binary icosahedral group

Elements

Properties

Central extension

Isomorphisms

Presentation

Subgroups

Relation to 4-dimensional symmetry groups

Applications

See also