The symmetry groups of the Platonic solids (centred at the origin in R3) are listed in the following table.
X                        Rot(X)  Symm(X)
tetrahedron               A4    S4
cube, octahedron          S4    S4xC2
dodecahedron,icosahedron  A5    A5xC2
Here An denotes the Alternating group, Sn denotes the Symmetric group and C2 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:R3->R3 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 C2 and Rot(X).

So to establish the second line of the table we have to show that Rot(X) is isomorphic to S4. 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)->S4. 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.

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