symmetry groups of the regular solids

The two paragraphs just explain the problem. You are supposed to prove that

**The group of rotational symmetries of a tetrahedron is **

**A****(4).**

**The full group of symmetries of a tetrahedron is **

**S****(4).**

**The group of rotational symmetries of a cube or of an octahedron**

**is ****S****(4).**

**The group of rotational symmetries of an icosahedron or of a dodecahedron ****is ****A(5).**

based on:

A regular solid is a 3-

dimensional polyhedron in which each face is a regular polygon. Any two

faces as well as any two vertices can be matched by an isometry (a rigid

motion) of the 3-dimensional space. It is convenient to use a symbol

{p, q}

for a regular solid whose faces are regular

p-gons with q of them situated

around each vertex. We have proved that there are FIVE regular solids:

•

TETRAHEDRON {3, 3},

•

CUBE {4, 3},

•

OCTAHEDRON {3, 4},

•

DODECAHEDRON {5, 3},

•

ICOSAHEDRON {3, 5}.

Any regular solid may be inscribed in a sphere, and then any symmetry of

any regular solid will leave the center of the sphere fixed and will transform

the surface of the sphere onto itself. We call a rotational symmetry of a

regular solid any rotation of the sphere (with respect to an axis passing

through its center) mapping the regular solid into inself. A symmetry of a

regular solid is either a rotational symmetry or a reflection with respect to

a plane (also passing through the center of the sphere), mapping the regular

solid onto itself.