The **icosahedron group** is the point group of the homogeneous icosahedron (and the homogeneous dodecahedron , which is dual to the icosahedron). It consists of the rotations and reflections that the icosahedron transforms into itself and has the order 120. It is closed isomorphic, where the alternating group is of order 5 (group of even permutations of 5 objects) andis the cyclic group of order 2 (consisting of the identity and the space reflection at the center of the icosahedron).

The too isomorphic subgroup, the icosahedron rotation group, consists of the orientation-maintaining symmetries of motion of the icosahedron (rotations). You can use this rotating group z. B. can be realized as a group of even permutations of the five cubes inscribed in a regular dodecahedron. is the smallest simple non-commutative group and has order 60.

The icosahedral group contains five-fold rotations and is therefore incompatible with long-range crystalline order ( *see * space group ). Quasicrystals , on the other hand, often have icosahedral symmetry.

The character table of the icosahedral group contains the golden ratio and related numbers, which is a direct consequence of the fivefold rotational symmetry.

Since the soccer ball is derived from a truncated icosahedron , it also has the icosahedral group as a symmetry group, as does the “soccer molecule” C _{60} (buckyball).

The icosahedron group has diverse applications in mathematics, which are presented in Felix Klein's classical work, *lectures on the icosahedron and the solution of the equations of the fifth degree* . ^{[1] }^{[2]} According to Galois theory, the general equation of the fifth degree has no solution in radicals, sinceis not resolvable (it is a finite simple group ).

In crystallography , the rotation group of the icosahedron is designated with the Schoenfliess symbol and the complete symmetry group with .

## Weblinks

- Icosahedral Group, Mathworld
- JH Eschenburg
*The icosahedron and the fifth degree equation according to Felix Klein*, pdf

## Individual evidence

- ^ New edition by Peter Slodowy , Birkhäuser 1993, originally Leipzig, Teubner 1884, online
- ↑ Slodowy
*The Icosahedron and the Fifth Degree Equation*, Mathematische Miniatures 3, Birkhäuser 1986