The

symmetry groups of the

Platonic solids (centred at the

origin
in

**R**^{3}) are listed in the following
table.

X Rot(X) Symm(X)
-----------------------------------------------------------------
tetrahedron A_{4} S_{4}
cube, octahedron S_{4} S_{4}xC_{2}
dodecahedron,icosahedron A_{5} A_{5}xC_{2}

Here

*A*_{n} denotes the

Alternating group,

*S*_{n} denotes the

Symmetric group and

*C*_{2} is the

cyclic group of

order 2.

*Symm(X)* and

*Rot(X)* are defined in

symmetry group.

Note that dual Platonic solids (like the cube and tetrahedron)
have the same symmetry groups.

Let's see how to prove these statements in the case of the cube.
First of all notice a non-rotational symmetry of the cube.
The function *-I*:**R**^{3}->**R**^{3}
given by *-I(x)=-x* leaves the cube invariant. (This symmetry
has order 2 (*(-I)*^{2}=I) and its not hard to see that
it commutes with any *nxn* matrix. It is not too hard to prove from
this that *Symm(X)* is isomorphic to the direct product of
*C*_{2} and *Rot(X)*.

So to establish the second line of the table we have to show that
*Rot(X)* is isomorphic to *S*_{4}. A long diagonal of the
cube is the line segment that starts at one vertex of the cube, passes through
the origin at the centre of the cube, and then finishes at the opposite
vertex. The cube has 4 such long diagonals and every rotation of the
cube will take each long diagonal to another long diagonal.

This gives us a group homomorphism
*f*:*Rot(X)->S*_{4}.
To finish the proof we have to show that this map is surjective and injective.
For example, to show surjectivity it is enough to show that every
transposition in the symmetric group is in the image of *f*.
To see this consider the line *L* that passes through the midpoint of an edge
of the cube, the origin at the centre and then the midpoint of the
opposite edge. A rotation about this axis clearly leaves the cube
invariant. It is easy to see that this rotation has the following effect on
the 4 long diagonals. It swaps over two of them (the ones which begin
or end at a vertex of the cube on the same edge that *L* passes through)
but it leaves the other two long diagonals where they are. In other
words *f* maps this rotation to a transposition. By varying
the choice of *L* we can get all transpositions.