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.